Simultaneous search for one of two targets is slower and less accurate than search for a single target. Within the Signal Detection Theoretic (SDT) framework, this can be attributed to the division of resources during the comparison of visual input against independently cued targets. The current study used one or two cues to elicit single- and dual-target searches for orientation targets among similar and dissimilar distractors. In Experiment The online version of this article (10.3758/s13414-019-01854-w) contains supplementary material, which is available to authorized users. Finding a target-object in the scene requires observers to compare visual input at different locations with an internal representation of the target’s features. This ability is thought to rely on components of selective attention that integrate information across frontal, parietal and visual cortical regions Ptak, . Prior tMost studies of search have focussed on the accuracy and speed of detection for a single target. In this situation, selection elicits a topographic map that represents objects in terms of their similarity to the target’s features . Evidence to suggest independently cued features can also mediate perceptual categorisation was reported by Roper and Vecera . They usThe results above suggest the selection of visual input for independently cued targets generalises from electrophysiological to behavioural responses during search. Evidence that MITs can also guide oculomotor behaviour has been reported by Beck and colleagues . All reported normal or corrected-to-normal visual acuity. Recruitment, consent and all experimental procedures conformed to American Psychological Association (APA) ethics standards.We used a small-The experiment was run on an IBM PC with a 19-in. CRT View Sonic G90fB monitor . The display resolution was 1,240 × 768 pixels and the frame rate was 85 Hz. Stimulus presentation and data collection were controlled using custom-built software in MATLAB  with Psychophysics toolbox extensions  Landholt’s C-shapes and grey  annuli that subtended 3.0° × 3.0°. Stimuli were presented at 12 equally spaced locations on the circumference of a virtual circle with a radius of 7.0°. C-shapes and annuli were presented on a uniform black  background.Displays contained red (C-shapes). Experimental blocks contained four repetitions of this structure . On each block of trials, two target and four distractor orientation values were assigned. Target orientations were sampled from 90° and 270° ± 15° to 30°. Distractor orientations for each observer differed from Target-Left or Target-Right by an angle of rotation that produced 80% accuracy on a pre-test (see below). Target and distractor values were fixed within experimental blocks to produce symmetric target-distractor similarity distributions for two targets among numerically equivalent subsets of leftward and rightward Cs. ; trial type (target-present or -absent); target identity (Target-Left or -Right), and set size . At set size two, one leftward and one rightward distractor were randomly sampled from the four alternatives. At set size one, a single leftward or rightward distractor was sampled with equal probability. On target-present trials, displays contained one C-shape at the orientation assigned to Target-Left or Target-Right. Targets always replaced a similar distractor from the same (leftward or rightward) group and displays always contained one C-shape from a group that was cued. Search displays were presented for 94 ms and followed by blank screen, which remained visible until a response was recorded. Short-duration displays were used to equate processing time and prevent eye movements on single- and dual-target searches.Figure C-shape in the same displays as the experimental session but varied the orientations of the target and distractor across the range ± 5° to 30° using a method of constant stimulus. Individual responses were fitted with a cumulative Gaussian function to estimate the angle of rotation required by each observer to distinguish targets from distractors on 80% of trials.Observers completed ten blocks of trials in a single experimental session. In each block, single- and dual-target cues were equally likely for leftward and rightward targets. A cued target appeared in the search display on 50% of trials, and the order of presentation for each target, search type and trial was randomly assigned for each block. Prior to experimental sessions, observers completed two pre-test blocks of 80 trials. These presented a single i to the visual response to each object in the display j. The result of each comparison describes the perceived similarity between the cue and the object, which is represented by a real number sij. To classify the display, observers evaluate whether the maximum similarity value is less than or equal to a response criterion. The cumulative probability distribution function P(sij ≤R) ;of sij depends on three factors: (1) The type of comparison C, which can be a target(t) similar distractor(s) or dissimilar distractor(d), depending on the relationship between the cue and the object in the display. (2) The set size D, which can equal 1, 2 or 4. (3) The number T of templates used to guide search, which is determined by the number of cues. These dependencies are indicated by the notation P. As a simplifying assumption, all comparisons between cues and objects with opposite orientations are modelled by the same probability distribution. In particular, the target is considered a dissimilar distractor for the second cue on dual-target searches. In agreement with standard SDT  and variance 1, i.e., P = Φ ), where Φ denotes the standard normal cumulative probability distribution . The statistical modelling of this process is based on signal detection theory (SDT) and the assumption of independent comparisons . Because of independence, the probability that none of the comparisons exceed the response criterion equals the product of the probabilities for each individual comparison being less than λ:To distinguish between theoretical accounts of attentional control during search, we introduce additional assumptions to derive two types of SDT model. Multiple-item template (MIT) models simulate display classification based on the evaluation of similarity values for independently cued feature values during dual-target search. We assume that the observer reports target-absent if none of the comparisons exceed a response criterion D,T). Conditional probabilities for hits and false alarms are obtained as 1-P(target-absent), depending on whether the target is present or absent in the display.In Eq. λ and 18 μ parameters . However, the experimental design yields only 12 observations for each subject . To derive useful and testable models, we therefore impose restrictions on the model parameters: First, we note that model predictions do not change if the same constant value is added to all parameters appearing together in a product term  = 0 . Target discriminability is modelled as the difference between the means of the distributions for the target and those for similar and dissimilar distractors. For similar distractors, d'S = μ-μ. For dissimilar distractors, d'D = μ-μ. We then distinguish between noise- and capacity-limited search. In the former, we assume that d’ is independent of set size and search type, i.e., d’ = d’(C). As a shorthand, we set d’(C=s) = d’S and d’(C=d) = d’D. For capacity-limited models, we assume that d’ scales with the inverse root of the total number of comparisons in each search: with d’(C=t)=0, this can be written as d’ = d’S/√(D*T) with d’S =d’. This parameterisation produces negative d’ values for distractors that are differentiated from the target distribution on the basis of their dissimilarity from the cue, while scaling d’ by 1/√(D*T) conforms to the decline in discriminability predicted by the SDT sample-size model of search  = λ(D*T), as the number of comparisons is the product of set size and cued orientations. The various λ values are denoted λ1, λ2, λ4, λ8. Equation Applying Eq. as there is one cue-target comparison (CL-TL), three comparisons with similar distractors (CL-DL & CR-DR * 2) and four comparisons with dissimilar distractors .λ. To represent the qualitative distinction between cues during the comparison process, SIT derivations contain separate terms to compute similarity values for objects depending on which template is active during search. For example, the estimated hit rate for the noise-limited SIT model on a dual-target search with a set size of 4 is given byIn contrast to MIT models, single-item template (SIT) models simulate search guided by a single attentional template. On a dual-target search, target detection is based on pairwise comparisons between the active attentional template and the objects in the display. We assume that the observer reports target absent if these comparisons fall below a response criterion d’ is independent of set size, i.e., d’ = d’(C). For the capacity-limited model, d’ scales with the inverse root of the number of objects in the display, because the number of active attentional templates is fixed at 1. With d’(C=t)=0, this can be written as d’ = d’S/√D with d’S =d’. Similarly, we set λ = λ(D) to model decision-noise and changes in threshold as a function of the number of comparisons between one attentional template and objects in the display. The various λ values are denoted λ1, λ2, λ4.In this equation, the observer randomly selects one of the cues and performs a single-target search based on the selected (active) attentional template. If the selected template matches the target in the display, performance will be equivalent to a single-target search. If the selected template matches the target that does not appear, hits are highly unlikely, because the cue-target comparison will yield a low similarity value. Accuracy across dual-target searches is predicted to be the average of the two terms. As for MIT models, we also distinguish noise- and capacity-limited derivations of SIT search. For the former, we assume d’ and three or four λ values) producing 5 or 6 error degrees of freedom. Fitting was carried out using maximum likelihood estimation and goodness of fit was assessed using a parametric bootstrap technique for Pearson’s χ2 statistic. To do this, SDT models were fitted to each observer’s data to compute the observed χ2 statistic. Parameter estimates for the fitted model were then used to simulate observations over K repetitions of the experiment (we used K = 500). By re-fitting and computing the χ2 statistic for each repetition, the χ2 distribution that would derive under the fitted model was estimated. Comparing the observed χ2 statistic against this χ2 distribution yielded the probability  that the fitted model generates data that are at least as extreme as those observed. The larger this probability, the ‘more typical’ the observed data are under the fitted model, and the stronger the evidence for its plausibility. As an additional, more intuitive measure of fit, we also calculated the mean absolute difference between corresponding observed and estimated probabilities. For brevity, we report mean group estimates in the text. Model estimates for individuals are reported in the Electronic Supplementary Material  at a set size of one.t(10) = 1.31, p > 0.20, Cohen’s d = 0.34), and the remaining analyses collapse responses across both targets during search. Table F = 144.80, p < 0.001 , ηp2 = .94, set size, F = 138.80, p < 0.001, ηp2 = .93, and a significant Search Type by Set Size interaction, F = 4.97, p = 0.018, ηp2 = 0.32. Two-template SDT models predict a reciprocal relationship between search accuracy and the number of comparisons required to classify the display. Planned contrasts to compare single- and dual-target searches across equivalent set sizes revealed non-significant differences at two, t(10) = .14, p = 0.89, d = 0.04, and four comparisons, t(10) = .67, p = 0.52, Cohen’s d = 0.21, respectively. These data provide initial evidence that accuracy is related to the number of comparisons required to classify the display rather than qualitatively different strategies of attentional control on single- and dual-target searches.Accuracy for leftward and rightward leaning targets did not differ significantly  and response bias (λ) across equivalent set sizes, as well as statistical comparisons for the four models. All reveal a monotonic increase in λ as a function of set size. Estimates of d’D were smaller than those for d’S for the both SIT models and the MIT, noise-limited model, reversing the expected pattern of discriminability for targets among similar and dissimilar distractors. Allowing d’s to scale with set size produced less variable λ estimates, and the expected relationship between d’S, and d’ D for the MIT model only. AIC values were also smaller for the MIT than SIT models, with the smallest value obtained under the MIT, capacity-limited model – indicating a better fit between the observed and predicted hits and false alarms when accuracy was inversely scaled by the product of set size and the number of cued orientations. This group-level advantage was replicated at the individual level, where AIC values were smallest and goodness-of-fit indices largest for the MIT, capacity-limited model for 10/11 observers. The mean absolute difference between observed and estimated data was also smaller for the MIT, capacity- (0.03) than the MIT, noise-limited (0.05), and SIT, capacity- (0.07) and noise-limited (0.06) models ). This interpretation is consistent with capacity-limited decision processes based on the concurrent evaluation of similarity distributions for independently cued targets.SDT models of single-target search have been successfully used to characterise the relationships between decision-noise, discriminability and set size for different stimuli . All reported normal or normal-to-corrected visual acuity.Twelve observers were recruited to the study: Two were male and age ranged from 18 to 52 years . The display resolution was 1,280 × 1,024 pixels and the frame rate 100 Hz. Stimulus presentation and data collection were controlled using custom-built software in MATLAB  with Psychophysics toolbox extensions  Landholt’s Cs presented on a black background . Cs subtended 5.0° × 5.0° at equally spaced locations on the circumference of a virtual circle with a radius of 7.0°. In contrast to Experiment Displays contained four red  and stimulus type . On each block, Target-Left and Target-Right orientations were sampled from 90° and 270° ± 15° to 30°, respectively. Distractor orientations were assigned as Target-Left and -Right ± 25°. Target and distractor orientations were fixed and the order of presentation for each target and search type were randomly assigned within each block.C-shape at the orientation assigned to Target-Left or Target-Right and one annulus. On dual-target searches the cue contained two C-shapes at the orientations assigned to Target-Left and Target-Right. Single- and dual-target cues were presented for 500 ms and 1,000 ms, respectively, before being replaced by a fixation cross for 1,000 ms. Search displays followed and always contained four objects – a cued target and one similar and two dissimilar distractors – which were randomly assigned to the four possible locations. Observers were instructed to fixate the object that matched a cue, and responses were measured using a box criterion to record the first fixation falling within the 5.0° × 5.0° area surrounding each C-shape in the display. This method excludes saccades falling outside stimulus locations but produces results similar to the nearest endpoint criteria , observers were prompted to press a key to begin each block of trials.Observers completed a short practice  followed by four blocks of 60 trials in a single experimental session. Before each block, a 5-point calibration sequence was performed to ensure fixations to each i = 1 to 2) with those in the display (j = 1 to 4) and fixates the comparison that produces the maximum similarity sij to select the most likely target. The probability of the cue-object comparison  to yield maximum similarity is given byThe observer’s task on each trial is to indicate which of four stimuli match the cue by fixating the target. Our statistical description of this task uses the SDT framework derived for Experiment p,D,T) denotes the probability density associated with P,D,T). The equation can be interpreted as follows: The product term in the integrand describes the probability that all comparisons different from  have results less than y. Weighing this value with the probability density for obtaining response y in the  comparison and integrating over all possible values of y yields the total probability for selecting comparison .In Eq. D fixed to 4 and setting μ = 0 as in Experiment s) and dissimilar (d) distractors on single- (ST) and dual-target (DT) searches; d’(sST), d’(dST), d’(sDT) and d’(dDT). For each observer, there are four observed quantities, i.e., the probabilities for selecting the target and similar distractor on single- and dual-target searches, respectively (the probability for choosing the dissimilar distractor is fixed because of P(t)+P(s)+2P(d) = 1). Without any further restrictions on the d’ parameters, the model is fully identified and would reproduce the observations perfectly. However, in analogy to Experiment d’(sST) = d’(sDT) = d’S and d’(dST) = d’(dDT) = d’D. We also compute estimates for a capacity-limited MIT model in which d’ scales with the number of comparisons required to search for one or two cued-targets  = d’S, d’(sDT) = d’S/√2 and d’(dST) = d’(dDT) = d’D). As set size is fixed across search types in this experiment, scaling d’s by √2 models a capacity limit that is specific to the MIT model when VSTM resources are distributed across independent attentional templates during search. All three models have two free parameters and 2 error degrees of freedom at the subject level.With i,j) but only the corresponding display object j. Therefore, the statistical model has to provide expressions for P(j) rather than P. This does not cause any complications for single-target searches, as there is only one cue i=1 and P(j) = P. For dual-target searches in the MIT model, saccades to object j could be based on the comparison for either cue, i.e.,  or  so that the corresponding probabilities have to be added: P(j) = P+P. For the SIT model, the overall probability of fixating the target in a DT search is given byThe subject does not directly report the chosen comparison . For MIT models, the overall probability of selecting the target object in the noise-limited model is given byThis equation is interpreted as follows. On a dual-target search, the participant performs one target comparison, three similar-distractor comparisons and four dissimilar-distractor comparisons. The first integral describes the probability that the subject selects the comparison between the target cue and target object. The second pertains to choosing the comparison between the cued target that is not present and the target object, which is considered a dissimilar-distractor comparison . Models are again fitted using maximum-likelihood estimation. For the computation of integrals, we make use of the fact that φ is a normal distribution and apply Gauss-Hermite quadrature . Accuracy for Target-Left and Target-Right were not significantly different, t(11) = 0.07, p > 0.95, Cohen’s d = 0.019, and the following analyses collapse responses across both targets. Table t(11) = 16.61, p < 0.001, Cohen’s d = 4.80, and increase in fixations to dissimilar distractors, t(11) = 12.32, p < 0.001, Cohen’s d = 3.56, on dual- compared to single-target searches. The proportion of fixations to similar distractors for each type of search was not significantly different, t(11) = 0.52, p > 0.05, Cohen’s d = 0.15.Trials in which fixations to a display object were recorded in less than 100 ms or longer than 3 standard deviations from the mean of single- and dual-target searches, respectively, were excluded from further analyses  = 3.79, p = 0.038, ηp2 = .26, and a significant Search by Object type interaction, F = 8.44, p = 0.002, ηp2 = .43. The main effect of search did not reach statistical significance, F = 4.57, p > 0.05, ηp2 = .29, Post hoc tests revealed median saccadic latencies to dissimilar distractors were significantly faster on single- than dual-target searches , while differences to targets  and similar distractors  were not statistically significant.Our focus in this study was to investigate changes in the accuracy of saccadic targeting during single- and dual-target searches. In Experiment Figure The results of Experiment d’ is independent of the number of comparisons required to localise the target. Group and individual comparisons also reveal better fit indices for the MIT, capacity-limited than the SIT, noise-limited model, supporting saccadic targeting based on separate similarity distributions for each cue. As with Experiment Figure The current study was designed to investigate the nature of the dual-target cost on perceptual decisions and saccadic targeting during covert and overt search. To do this, we applied SIT and MIT SDT models to predict observers’ accuracy when comparisons between visual input and VSTM are limited to (1) a single-item attentional template or (2) multiple-item templates for independently cued targets during dual-target search. In addition, we compared observers’ accuracy to noise- and capacity-limited derivations for both models of attentional control. Each assumes search entails the comparison of noisy internal representations of visual input with one or two cued-orientations. Noise-limited derivations model the dual-target cost as an increase in stochastic noise when the number of comparisons doubles on dual- compared to single-target searches. Capacity-limited derivations apply an inverse square-root relationship to model an additional dependency between discriminability and the number of comparisons during search


Namely, every clause must contain at least two satisfied literals. Because of its robustness, super solutions are concerned in combinatorial optimization problems and decision problems. In this paper, we investigate the existence conditions of the -super solution of  Then the solution is an -super solution. For example, the solution In combinatorial optimization problems and decision problems, the robustness of solutions is a valuable property. A robust solution is not sensitive to small changes in dynamic and uncertain environments, and guarantees the existence of a small set of repairs when the future changes in a small way. R. Weigel and C. Bliek in  introduca,b)-super solution of a CNF formula with special structure is undoubtedly of realistic significance. Therefore, we focus on some special k,s)-SAT has a SAT-UNSAT transition phenomenon and investigate the characteristics of the transition phenomenon.Encoding into a CNF formula is a common way to solve a practical problem. These CNF formulas often have some regular structures. A -super solution is a generalization of Supermodels in SAT problems. Some algorithms of finding super solutions were presented in [An -SAT where each clause has exactly k distinct literals. Zhang P in [k-SAT is in P for Specially, a -super solution is a satisfying assignment such that if any one variable is flipped to the opposite value, the new assignment is still a satisfying assignment. In other words, a -super solution of a CNF formula must satisfy at least two liters of every clause. The decision problem whether a CNF formulas has a -super solution is denoted as -SAT. -ang P in  proved thou G in  obtainedk-SAT is NP-complete. It implies that for k-SAT can not be solved in polynomial time. What happens if the number of occurrences of each variable is limited? So we expand -k-SAT to -d-regular d-regular k distinct literals per clause and at most s occurrences of each variable. A regular -CNF formula is a k-CNF formula, in which each variable occurs exactly in s clauses. A d-regular d. The NP-completeness of the special SAT problems worth further study.-3-SAT problem is in P. This shows that there is a polynomial time algorithm to decide whether a 3-CNF formula has a -super solution. For Kratochvíl in  pointed (i) for (ii) for s such that all shown in . In [12,shown in ,14,15, ik-SAT to -In this study, we give a polynomial time reduction that transforms (i) every (ii) -That is to say, for x or a negated propositional variable x is called a positive literal, and C is a disjunction of literals, F is a conjunction of clauses, F, and F. x in F.A literal is a propositional variable F, if F but their variable sets are disjoint. We divide variables into forced variables or unforced variables. A forced variable is a variable forced a same value by all satisfying assignments of a formula.If the formulas F is denoted by d-Hamming neighbourhood F is a satisfying assignment such that for any A satisfying assignment is also called a solution. The set of solutions of a CNF formula Definition 1.A CNF formula is called a regular Definition 2.For each Definition 3.Ψ is a forced-A -CNF formula (i) there exist two variables Ψ has a -super solution and for any -super solution τ of Ψ, it holds that (ii) Definition 4.The projection Lemma 1.An assignment is a -super solution of F if and only if it satisfies .An assLemma 2.If .If k≥3Lemma 3.The critical function .The crLemma 4.An instance of SAT is satisfiable if:.An ins(i) all of its clauses contain more than one variable, and(ii) each variable appears exactly once complemented and once uncomplemented.Lemma 5.If the representation matrix of a formula F isthen the formula is satisfiable and every satisfying assignment forces all variables to a same value.ith variable occurs positively in the jth clause; if ith variable occurs negatively in the jth clause; if F expresses a cyclic of implicationHere, if an element A -super solution has certain robustness, but some CNF formula with special structures may not have a -super solution.Theorem 1.Every regular -CNF Formula without pure literals does not have a -super solution.Proof. F is a regular -CNF without pure literals, then each clause has exactly 3 distinct literals and each variable occurs exactly 2 times (one negative occurrence and another positive occurrence). Suppose, F has F. However, a -super solution must satisfy at least two literals of every clause. That is to say, a -super solution must satisfy at least F. Clearly, there is not a -super solution. Therefore, no regular -CNF Formula without pure literals has a -super solution. □If a CNF formula By bounding the difference between positive and negative occurrences of every variable, we find other CNF formulas without a -super solution.Corollary 1.For Proof. F is a 1-regular s times, and the absolute value of the difference between positive and negative occurrences of each variable is at most 1. Suppose F has F, but every -super solution must satisfy at least F. For Let F. Suppose F has F . Because every -super solution must satisfy at least F, only an assignment F. In other words, if F does not have a -super solution. That is, if the formula has a -super solution, the -super solution must be Now we consider an arbitrary 1-regular -CNF formula Moreover, we also find some Theorem 2.Every regular Proof. m clauses. By definition, each clause has exactly 4 distinct literals, and each variable occurs exactly in 2 clauses (one negative occurrence and another positive occurrence). We divide each clauses Suppose iable in . SupposeTheorem 2 can be considered as an extension of Hall’s Marriage Theorem. Suppose we have a finite set of single men and women. If each man is attracted by two women and each woman is attracted by four man, then there must be a way that each woman could be married with two men at the same time.Using Lemma 4, it is easy to obtain the following corollaries by using the proof method of Theorem 2.Corollary 2.For Corollary 3.A CNF formula must have a -super solution if:(i) each clause has at least 4 distinct variables and(ii) each variable occurs exactly in two clauses(one negative occurrence and another positive occurrence).For the formula in Corollary 3, all clauses may not have the same number of variables.Theorem 3.All -CNF formulas and -CNF formulas must have a -super solution.Proof. Suppose 8)≤29 in , all -CNF formulas must have a -super solution by 9)≤51 in . □In , it has In this section we study in what conditions -Theorem 4.For Proof. We will present a polynomial time reduction method from -super solution of the formula f.Step 2 Let f with the variables f being renamed as Step 3 We construct the formula s clauses. So Obviously, every clause of It is supposed that It is supposed that For The critical function Corollary 4.If Proof. The statement follows directly from Lemma 2 and Theorem 4. □By Hall’s Marriage Theorem, -CNF and -CNF are satisfiable. But --SAT and --SAT are NP-complete because -SAT and -SAT are NP-complete.Next, we make some modifications of the reduction method in the proof of Theorem 4. We introduce a sufficient number of new variables, add some new clauses to make up the gap of the occurrence number of every variable, and guarantee that every new clause have at least a positive occurrence of two new variables -super solution). Because Corollary 5.If Here, -regular -super solution have a -super solution. First we flip a unsatisfied literal of any one clause with only one satisfied literal. If the formula still does not have a -super solution, then we flip a unsatisfied literal of other clause with only one satisfied literal, until the formula has a -super solution. When the formula has a -super solution, the flipped literal is a key-literal.It is supposed that a  can make c and only two literals of the flipped clause. This also indicates that all -super solutions satisfy the literal y -super solution. There are two cases to consider.It is supposed that a -super solution. Lemma 6 entails that we can construct a forced-k-SAT to an instance of --(s)-SAT exists, then Z occurs in three clauses and every variable of X occurs in two clauses. Obviously, the formula Using Corollary 2, if n cyclic of implication. That is, if It is assumed that an assignment It is assumed that k-SAT is NP-complete, - every ((ii) --super satisfiability of (k. Besides, for larger values of k, the upper and lower bounds of Although we obtain some results about


This is a comment on “Manifestations of Sasang Typology according to Common Chronic Diseases in Koreans” recently published by Hong et al.  that exaAs for the SPQ measuring biopsychological characteristics of Yin-Yang personality traits, the SPQ and its subscale scores of DM, hypertension, FD, and MDD were correctly illustrated, but their implications were not discussed in the context of recently published reports , 5. The The SPQ-E score of the FD group was significantly higher than that of the hypertension and adenomyosis groups, and the MDD group was similar to the FD group . A persoAs for the SDFI measuring good digestive function and appetite, the article of Hong et al.  made crir = −0.585) and Functional Dyspepsia-Related Quality of Life (r = −0.433), while the SDFI-E  has a positive correlation with Dutch Eating Behavior Questionnaire (r = 0.481) [r = 0.299) and SDFI-D (r = 0.310) scores [The operational definition of SDFI-D is a measure of good and hyperactivated digestive function, and a person with high SDFI-D should have a high BMI , 6–8. A  r = −0.43, while In the article by Hong et al. , the BMIFrom this perspective, the high score of SDFI-D would be a typical clinical feature of metabolic disease showing DM, hypertension, and obesity (high BMI) distinguished from other chronic diseases. The FD patients with low SDFI-D and high SPQ-E scores might be recognized as bad or hypoactive digestive function along with psychopathological vulnerability in traditional Korean medicine.The SDFI-A score of MDD group was significantly lower than that of DM, hypertension, FD, and adenomyosis groups in Hong's study . ConsideAs a conclusion, with consideration of the latest clinical studies –7, the c


The generator matrices of polar codes and Reed–Muller codes are submatrices of the Kronecker product of a lower-triangular binary square matrix. For polar codes, the submatrix is generated by selecting rows according to their Bhattacharyya parameter, which is related to the error probability of sequential decoding. For Reed–Muller codes, the submatrix is generated by selecting rows according to their Hamming weight. In this work, we investigate the properties of the index sets selecting those rows, in the limit as the blocklength tends to infinity. We compute the Lebesgue measure and the Hausdorff dimension of these sets. We furthermore show that these sets are finely structured and self-similar in a well-defined sense, i.e., they have properties that are common to fractals. The fractal dimension of In his book on fractal geometry, Falconer characterizes a set operties   still eludes us. Nevertheless, our results may apply in areas beyond channel coding: Arıkan’s polarization technique was used to polarize Rényi information dimension  and to cSince we consider the case Lemma 1Let  W20y2|u1:=12∑W20y2|u1: channelW21=2I(W)  (ProposiLemma 2 (Bounds on the Bhattacharyya Parameter).with equality in Proof. W is a BEC [The equality and inequality in  follow f≤1, from   the cardinality of the output alphabet increases exponentially in If we stop the polarization procedure at a finite blocklength ly in 2n  (Chapterly in 2n  (p. 36).ly in 2n  .A channel  degraded w.r.t. the channel W  channels remains upgraded (degraded) during polarization:Lemma 3Suppose that ( (Lemma 4Lemma 4( (p. 9) &Proof. By choosingls, take  .Let Let W is nontrivial, i.e., that If Proposition 2 (Denseness).Proof. See f is not injective. It is not obvious, however, that the intersection exhausts the set on which f is non-injective. A consequence of this proposition is that there is no interval that contains only good channels. This has implications for code construction techniques. Indeed, the authors of [It is not really surprising that thors of ,19 suggeProposition 3 (Symmetry).There exists a function ϑ, defined for almost all values in Proof. See alignment of the sets W and Proposition 3 has two implications. The first implication concerns the negative . Indeed,arameter .The second implication is that, at least for BECs, the sets recurring, i.e., there is a length-k sequence z close to zero and z close to one, the operation x, It is possible to define Example 2.Let Now suppose that W is a BEC with Bhattacharyya parameter Proposition 4 (Lebesgue Measure & Hausdorff Dimension).Proof. See W. The fact that W. A positive Lebesgue measure and a Hausdorff dimension equal to one are not indicators of fractality.Loosely speaking, the Lebesgue measure of quasi self-similar. Along the same lines, the quasi self-similarity of The last fractal property we consider is self-similarity. As Falconer notes  (p. xxviProposition 5 (Self-Similarity).Let  quasi self-similar in the sense that, for all n and all k, If W is symmetric, Proof. See In other words, at least for a symmetric channel, self see . The selExample 3.We want to determine whether r, length-Hamming weight of i-th row of r, length-An order-dexed by (20)F={iTo analyze the effect of doubling the block length, note thati-th row be indexed by Assume that we indicate the rows of his with  yieldswDefining n tends to infinity. An important ingredient in our proofs is the concept of normal numbers.In Definition 3 .A number  simply normal to base 2.Let fractional Hamming weight larger than a given threshold.Loosely speaking, the set of heavy codewords corresponds to those rows of Example 5.Proposition 6 (Denseness).For all n to be even and set n, and one needs to depart from intuition based on these finite-blocklength considerations.Similarly as for polar codes, also Reed–Muller codes are such that no interval is contained in either Proof. See Proposition 7 (Lebesgue Measure & Hausdorff Dimension).The Hausdorff dimension satisfieswhere Proof. See W. In contrast, the set W. Rather, Proposition 7 suggests that the order parameter phase transition for the rate of Reed–Muller codes: If Loosely speaking, the Lebesgue measure of Let us briefly consider the case The sets cale cf. . We nextal codes  (ProposiProposition 8 (Self-Similarity).Let  quasi self-similar in the sense that, for all n and all k, Proof. See The set has a fine structure, i.e., there is detail on arbitrarily small scales;It does not admit a description in traditional geometrical language, neither locally nor globally; it is irregular in some sense;It has some form of self-similarity, at least approximate or statistical;The fractal dimension of the set exceeds its topological dimension.That Kronecker product-based codes possess fractal properties has long been suspected. The present manuscript contains several results that back this suspicion with solid mathematical analyses. Specifically, we assumed that the blocklength tends to infinity and investigated the properties of the set operties  (p. xxviIndeed, the sets R. In other words, while fractional order of the code, R and that, thus, its Hausdorff dimension equals one. An appropriate definition of R, length-One reviewer pointed out that our definition of q is prime. One can show that this matrix is polarizing as long as it is not upper-triangular [ℓ-ary expansion of real numbers in q-ary Reed–Muller codes, e.g., [Another obvious extension of our work are non-binary polar and Reed–Muller codes. For example, consider an iangular  (Theorems, e.g., ,24.


Scientific Reports 10.1038/s41598-019-41202-4, published online 20 March 2019Correction to: The original PDF version of this Article contained a typographical error in the publication date ‘20th March 2019’ which was incorrectly given as ‘18th March 2019’. This has now been corrected in the PDF version of the Article. The HTML version was correct at the time of publication.


Nature Communications 10.1038/s41467-020-17611-9, published online 30 July 2020.Correction to: The original version of this Article contained an error in the Abstract, in which the word ‘erosive’ was misspelled ‘eros0ive’. This has now been corrected in the PDF and HTML versions of the Article.


R ≈1.In a comment on our recent paper , the comR ≈ 1. We do not do that. Throughout the paper and its extensive supporting information, we emphasize that the derivation of the estimates for the critical connectivity R ≈ 1, or, equivalently, D ≈ Dc.We are surprised to see that the commentators in ref. R ≈ 1 or D ≈ Dc are not sharp, and, as we state explicitly in ref. R ≈ 1. Within the context of the entire paper (1) and its supporting information, this is completely clear. It seems that the commentators in ref. The critical values A in ref. A in ref. In addition, the analysis of close-to-critical spreading on Erdős-Rényi (ER) networks in ref. It is fortunate, however, that this comment gives us the opportunity to cite a work that noted the possibility of linear spreading of diseases on SW networks before (figure 5 in ref.


Here, the zero-temperature phase behavior of bosonic particles living on the nodes of a regular spherical mesh (“Platonic mesh”) and interacting through an extended Bose-Hubbard Hamiltonian has been studied. Only the hard-core version of the model for two instances of Platonic mesh is considered here. Using the mean-field decoupling approximation, it is shown that the system may exist in various ground states, which can be regarded as analogs of gas, solid, supersolid, and superfluid. For one mesh, by comparing the theoretical results with the outcome of numerical diagonalization, I manage to uncover the signatures of diagonal and off-diagonal spatial orders in a finite quantum system. Gases of ultracold bosonic atoms loaded in an optical lattice provide the unique opportunity to study quantum many-body effects under controlled conditions ,2. To a In a system at zero temperature . Experig., Ref.  or dysprg., Ref. ) and molt, in absolute terms) to the on-site interaction (U) is reduced. The overall number density of particles is controlled via a chemical-potential parameter t and U, the lattice becomes increasingly filled with particles, but this can only occur outside the Mott regions since the insulator phase is incompressible [V is introduced between nearest-neighbor atoms (“extended BH model”), new phases may arise, in primis a supersolid phase [The usual BH model predicts a ressible . The BH ressible ,10,11, pressible ,14,15,16ressible ,18, to nid phase ,25,26,27→∞ limit ,30,31,32I hereafter present the results of yet another investigation of the extended BH model, now choosing a finite graph as hosting space for bosons. Even though clearcut phase transitions  cannot occur in a few-particle system, a convenient choice of boundary conditions may alleviate the difference with an infinite system, making the study of a finite quantum system valuable anyway. A practical solution is to use spherical boundary conditions (SBCs), which have often been exploited in the past to discourage long-range triangular ordering at high density ,36,37,38The plan of this paper is the following. After introducing the models in i can only be 0 or 1 . In theThe rationale behind the choice of a cubic mesh is now clear: by introducing a repulsion between occupied NN sites, we promote the occurrence of a Platonic “crystal”, i.e., the regular tetrahedron , the total energy (E), and the order parameters for tetrahedral  are considered below.H describes a system of bosons on a spherical mesh of M sites, U term in  a density wave at low t, as well as a supersolid phase at the boundary between the insulator and superfluid phases. These features are also present in the hard-core limit, as reported, e.g., in Refs. [In the BH model on a standard lattice, at in Refs. ,32.MF theory is the method of choice when a new many-body problem is attacked; it has been frequently applied for continuous quantum systems as well, as an effective means to identify the ground states and quantum transitions between them . i, respectively. For hard-core bosons, we readily obtain i-th site. For a bipartite mesh, sites are either of type A or B, hence the unknown parameters are four, namely In the decoupling approximation, the two-site terms in the Hamiltonian  are lineHMF withHMF=−t∑iFA has three neighbors, all belonging to grid B, and vice versa. Hence,A (B) for operators relative to a single A (B) site, the MF Hamiltonian reads:B “crystal”), A “crystal”), and For the QCT model, the mesh is bipartite and formed by two tetrahedral sub-meshes with four points each. A point of grid an reads:HMF=E0−12H for A and B; hence, our search can be restricted to homogeneous (superfluid) solutions: E isSuperfluid and supersolid “phases” have non-zero, possibly distinct, complex values of rrive at(λ−b)2(λ−root of (E=b−(a−b)oot of (1E=b−(a−b) to giveρ=μ+3t3V+By comparing the grand potentials of all the “phases”, we arrive at the ground-state diagram in A, 8 sites) and a co-cube . We have:The decoupling approximation for the QDC model works similarly as for the QCT model. Again, the starting point is Equation  and the en reads:HMF=E0−24Moving to phases with o be see :(31)E=12Equation ).E satisfy the following linear system: Using he state  describia priori assuming them to be equal in pairs  and diagonalizing the ensuing matrix. An extensive mapping of a few quantum averages enables us to clarify the nature of the “phases” present.Let us again reconsider the QCT model at A or B, they belong to. Then, I compute the average occupancies of A and B sites . In A and a B site, with the only exception of A-B inversion are the same. I show in N operator, implying that the ground state (which is non-degenerate for N. Like in MF theory, the occupancy is zero below t, with no abrupt transition from “crystal” to superfluid values.As far as the occupancies are concerned, exact diagonalization shows that they are always equal for an t and t lines; each jump discontinuity of Another difference with MF theory concerns the ground-state averages of Equation . In the t values in the range 0 to 0.5. In the MF curves, cusp singularities are associated with the crossing of first-order transition lines. We see that MF theory systematically overestimates exact values, the more so the larger t is.Finally, in I have worked out the zero-temperature phase diagram of two systems of hard-core bosons defined on the nodes of a regular spherical mesh. The interaction is of Bose–Hubbard type, with a further repulsion between neighboring particles. Choosing a suitable mesh, bosons are pushed to form a Platonic “crystal” in a range of chemical potentials. In the QCT model, the mesh is cubic and a tetrahedral “crystal” is formed; in the QDC model, the mesh is dodecahedral and a cubic “crystal” is formed instead.Using a mean-field approximation, I have obtained fully analytic results for the thermodynamic properties of the two models. In addition to a number of insulating phases, both systems also exhibit a superfluid ground state. In the QDC model, triple coexistence between two solids and the superfluid is superseded by the occurrence of a more stable supersolid phase. Clearly, while the predictions of mean-field theory are generally accurate for an infinite Bose–Hubbard system, deviations will unavoidably be observed in a small system, where no true singularity can occur. However, discrepancies are probably less strong for bosons on a regular spherical mesh, which has equivalent sites and is devoid of natural boundaries.To check this expectation for the QCT model, I have diagonalized its Hamiltonian exactly, finding the ground state and a number of ground-state averages. Overall, the exact


Scientific Reports, 10.1038/s41598-020-71679-3, published online 09 September 2020Correction to: The original version of this Article contained an error in the title of the paper, where the word “Humid” was incorrectly given as “Human”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.


We read with interest the paper by Jun S.Y. et al.  dealing As survival rate is similarly low in patients with Crohn’s disease-associated SBA and those with sporadic SBA ,5,6, it


This paper studies index coding with two senders. In this setup, source messages are distributed among the senders possibly with common messages. In addition, there are multiple receivers, with each receiver having some messages a priori, known as side-information, and requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate normalized codelength, which is expressed as the optimal broadcast rate. In this work, firstly, for a given TSUIC problem, we form three independent sub-problems each consisting of the only subset of the messages, based on whether the messages are available only in one of the senders or in both senders. Then, we express the optimal broadcast rate of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we discover the structural characteristics of TSUIC. For the proofs of our results, we utilize confusion graphs and coding techniques used in single-sender index coding. To adapt the confusion graph technique in TSUIC, we introduce a new graph-coloring approach that is different from the normal graph coloring, which we call two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. We further determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates. Finally, we generalize the results of a class of TSUIC problems to multiple senders. In this scenario, if the sender is informed about the side-information available at all receivers, then it can leverage that information whilst encoding to reduce the required number of broadcast transmissions, in comparison with a naive approach of transmitting all requested messages uncoded and separately. Such an encoding process is called index coding, and the resulting sequence of coded messages is known as an index code. Moreover, each receiver upon receiving the index code will be able to decode its required message by utilizing its side-information. The main aim of index coding is to find the optimal (shortest) codelength and the corresponding coding scheme. Index coding was introduced by Birk and Kol [Consider a communication scenario over a noiseless channel where a sender is required to broadcast messages to multiple receivers, each caching some messages requested by other receivers a priori. The messages cached at each receiver is known as its  and Kol ,2, and f and Kol ,11,12,13Macro-cell networks with caching helpers —cellularcooperative data exchange —peer-to-distributed storage—storage networks where data are distributed over multiple storage devices/locations.Most existing works on index coding deal only with a single sender, capturing scenarios with centralized transmissions. However, many communication scenarios such as the following have messages distributed among multiple senders:In addition, each sender can be constrained to know only a subset of the total messages due to reasons such as limited storage, or error whilst receiving some messages over noisy channels, or server failure to deliver all messages. In this case, distributed transmissions are required, where multiple senders broadcast messages to the receivers. One metric to maximize the transmission efficiency in this scenario is to minimize the aggregate number of transmissions from all senders in such a way that all receivers’ demands can be fulfilled. As this problem is more general than an index-coding problem with a single sender and is of practical interest , it is a useful research avenue to study index-coding problems with multiple senders, known as multi-sender index-coding problems.broadcast-rate formulation of the problems. In their work, they devised lower and upper bounds on the optimal broadcast rate by implementing a graph-theoretic approach. The results were established using information-flow graphs, which represent receivers’ request, and message graphs, which represent senders’ message setting. Furthermore, they showed problem instances for which the upper and lower bounds coincide. A class of such instances is where no two senders have messages in common.The multi-sender index-coding problem was first studied by Ong et al. . They counicast message setting, meaning each message is requested by only one receiver, each receiver requests only one message, and each receiver knows a subset of messages requested by other receivers a priori. Based on graph-theoretic approaches, they established upper bounds on the optimal broadcast rate. In particular, they focused on the two-sender case, called two-sender unicast index coding (TSUIC). They extended existing single-sender index-coding schemes, namely the cycle-cover scheme [In another work, Thapa et al.  considerr scheme ,19, the r scheme  and the r scheme  to the cSadeghi et al.  considerdecentralized data shuffling problems in which the receivers/workers can communicate with one another via a shared link. The decentralized data shuffling phase with uncoded storage (which stores a subset of bits of the data set) is equivalent to a multi-sender index coding problem. For this problem, they proposed converse and achievable bounds that are to within a factor of 3/2 of one another. Moreover, the proposed schemes were shown to be optimal for some classes of the problem. Recently, Porter et al. [embedded index coding (EIC), in which each node acts as both sender and receiver. With the help of several results, they showed the relationship between single-sender index coding and EICs. Furthermore, they developed heuristics to solve EIC problems efficiently.In a recent work by Li et al. , a new rr et al.  introducconfusion graphs in index coding [Different approaches have been attempted to solve the multi-sender index-coding problems. However, the problems are more difficult and computationally complex than their single-sender counterparts, and we know very little about the characteristics of the problems. This paper studies the broadcast-rate formulation of TSUIC problems by implementing a graph-theoretic approach. More precisely, in the same spirit of studying structural properties of index-coding capacity in the single-sender case by Arbabjolfaei et al. , we examx coding  along wiProposing a new coloring concept for confusion graphs in TSUIC, called two-sender graph coloring . However, for TSUIC, as the two senders (encoders) contain some messages in common, the standard method of graph coloring of the confusion graph may not lead us to an index code. In this regard, we need a different kind of coloring function in TSUIC, and thus, in this paper, we propose a novel coloring technique to color the confusion graphs in TSUIC, and its optimization gives the optimal broadcast rate and optimal index code.Presenting a way of grouping the vertices of confusion graphs in TSUIC : By exploiting the symmetry of the confusion graph, we propose a way of grouping its vertices for analysis purposes mainly in its two-sender graph coloring. In particular, this grouping helps us to analyze the number of colors used in two-sender graph coloring of a confusion graph.Deriving the optimal broadcast rates of TSUIC problems as a function of the optimal broadcast rates of its sub-problems (Theorems 4–8): We divide a TSUIC problem into three independent sub-problems based on the requested messages by receivers, specifically whether the messages are present in only one of the senders or in both senders. Now in TSUIC, considering the interactions (defined by side-information available at the receivers) between these three independent sub-problems, we derive the optimal broadcast rate (in both asymptotic and non-asymptotic regimes in the message size) of the problem as a function of the optimal broadcast rates of its sub-problems. Moreover, we bound the optimal broadcast rate, and show that the bounds are tight for several classes of TSUIC instances (sometimes with conditions). Furthermore, we find a class of TSUIC instances where a TSUIC scheme can achieve the same optimal broadcast rate as the same instances when the two senders form a single sender having all messages.Characterizing a class of TSUIC instances where a certain type of side-information is not critical (Corollary 1): For a class of TSUIC instances, we prove that certain interactions between the three independent sub-problems can be removed without affecting the optimal broadcast rate (in the asymptotic regime). This means that those interactions are not critical.Generalizing the results of some classes of TSUIC problems to multiple senders : For some classes of TSUIC problems, we generalize the two-sender graph coloring of confusion graphs and the proposed grouping of their vertices. Then, we compute the optimal broadcast rates of those problems as a function of the optimal broadcast rates of their sub-problems.The contributions of this paper are summarized as follows:After posting the first draft of this paper  on ArxivN independent messages t binary bits. There are N receivers S, having all N messages In this paper, we consider unicast index coding. There are Definition 1 (Two-sender index code).A two-sender index code ((i) an encoding function for each sender (ii) a decoding function for every receiver r, r receives sub-codewords from both senders without any noise, and decodes This means that each sender Now, we define the aggregate normalized codelength, which measures the performance of a code Definition 2 .The broadcast rate of an index code (with a single sender or two senders) is the total number of transmitted bits  per received message bit. In TSUIC, it is denoted by  achievable for a UIC problem if there exists an index code of normalized length ℓ.For the rest of the paper, we refer to normalized codelength simply as codelength.Definition 3 .The optimal broadcast rate for a given index-coding problem with t-bit messages is  SSUIC and  TSUIC. The optimal broadcast rate over all t is defined as ’s lemma .Remark 1.With the  broadcast rate as a performance metric, we can treat SSUIC as a special case of TSUIC when An index-coding problem can be modeled by graphs, which are defined as follows:Definition 4 (Directed graphs and undirected graphs).A directed graph is an ordered pair From now on in this paper, we call directed graphs simply digraphs, and undirected graphs simply graphs.N receivers, and the arc set i to vertex j if and only if receiver i has message j) in its side-information. Thus, in a side-information digraph, i in D. In this paper, for convenience, a receiver i is also referred to as a vertex i, and vice versa. We also use the compact form of representation of an instance of UIC problems as used by Arbabjolfaei et al. [The receivers’ message setting of a UIC problem is represented by a side-information digraph i et al. , where aprivate messages and common messages defined as follows: Let D, without loss of generality, we define the following sub-digraphs induced by the following vertex subsets that partition D induced by vertices D such that In TSUIC, Definition 5 (Constraint due to the two senders).The constraint due to the two senders is the following: whilst encoding, any two private messages sender-constraint graph.In TSUIC, to reflect the senders’ message setting, we introduce an undirected graph, denoted by D and t, and over all t, respectively. As a TSUIC problem is described by D, three sub-problems based on the type of messages are In a TSUIC problem, if there is no common message, i.e., roblems .For simplicity, an interaction between the sub-digraphs H, we get a total of 64 possible cases of the orientation of arcs among its vertices. As the vertices 1 and 2 of H can be swapped because we can interchange H. Now, depending upon the type of orientation of arcs among the vertices of H, we classify all unique cases into two categories: (i) CASE I—Acyclic orientation , and (ii) CASE II—with some cyclic orientation (22 cases). CASE II is further classified into smaller sub-cases II-A, II-B, II-C, and II-D. Refer to H is labeled D defines arcs between them (not within the sub-digraph), and the cases of interactions (acyclic or cyclic) are defined with respect to the orientation of the arcs between the sub-digraphs. In this paper, a fully-participated interaction and a partially-participated interaction between D are called a cyclic-fully-participated interaction and a cyclic-partially-participated interaction between the sub-digraphs, respectively, if and only if Considering the digraph D. Moreover, structural properties can be used to determine the criticality/non-criticality of arcs in TSUIC as in its SSUIC counterpart [For SSUIC, Arbabjolfaei and Kim   for an index-coding problem is represented by a graph called a confusion graph, defined as follows:For an index-coding problem modeled by a side-information digraph Definition 6 (Confusion graph).The confusion graph, denoted (i) (ii) Before proposing a notion of coloring for TSUIC, we first recall the standard definition of the graph coloring in the following:Definition 7 (Graph coloring and Chromatic number).A proper graph coloring of a graph G is an onto function sets of independent vertices where all vertices belonging to one set are assigned with the same color in the graph coloring. Here, a set of independent vertices refers to a vertex set where any pair of vertices are not connected by an edge in independent vertex set. The tuples representing vertices within an independent vertex set are not confusable, and hence they can be coded into the same codeword. Assigning each independent vertex set  a unique codeword provides us a valid index code having D with t-bit messages can be obtained by using confusion graphs. This is stated in the following theorem.Consider coloring a confusion graph Theorem 1. is used for a general case, and the remaining two indices . In ourIn the form of lemmas, we discuss two-sender graph coloring of In TSUIC, the two senders encode separately, so, in the aforementioned definition, we need to assign an ordered pair of colors for each vertex, where the first color is associated with Lemma 1.For any two distinct vertices Proof. Since In a similar reasoning as in the above proof (of Lemma 1), one can prove the following lemma:Lemma 2.For any two distinct vertices, If Lemma 3.For any two distinct vertices, If Lemma 4.For any two vertices, t-bit messages, we have the following theorem:For a TSUIC problem with Theorem 2.The optimal broadcast rate for a TSUIC problem with t-bit messages isProof. t), we getFor lting inβt=, we getβt≤ p2′ in ≥lity in <We illustrate two-sender graph coloring of a confusion graph in TSUIC, and a mapping function that maps colors to codewords at each sender from the following example.Example 1.Consider a TSUIC problem  1. The confusion graph  1 . Since any index code for D, but the converse is not always true, so we have the following:For any Lemma 5 (A simple lower bound).For any D and t-bit messages in the following lemma.In TSUIC, each sender Lemma 6 (A lower bound).For any two-sender index-coding problem Proof. For any two-sender index-coding problem +βt(D2) .βt(D2) ≥□To compute the simple lower bound to the optimal broadcast rate of a given problem in TSUIC, we utilize the following SSUIC results by Arbabjolfaei and Kim  (ii) D whose For a digraph II-A see , we haveTheorem 4 (CASE I).For any D having any interaction  between its sub-digraphs Proof. f =β≥βIn TSUIC, if we consider the sub-digraphs 3). From , 12) an anD1, D2aphs, soβ≤βExample 2.Consider a TSUIC problem of the following: Proposition 1.For any D having a fully-participated interaction between its sub-digraphs Proof. Refer to Definition 9 (Vertex-transitive graphs).An undirected graph G is vertex-transitive if for every pair Remark 3.The proof of the Proposition 1 is based on the analysis of the confusion graph  TSUIC problem whose sub-problems interact with each other in some way, we can reduce the complexity arising during its analysis .For any D having any interaction  between its sub-digraphs Proof. For convenience, let ition 1:βtNow, taking a limit sides in , we getRemark 4.For any D having non-empty  SSUIC, we have  TSUIC, Corollary 1.For any D having any interaction  between its sub-digraphs  TSUIC in the asymptotic regime in message size (considering infinitely long messages).Proof. The proof follows from Theorems 4 and 5. □Theorem 6 (CASE II-B).For any D having a fully participated interaction between its sub-digraphs Proof. Refer to Corollary 2.For any D having a fully participated interaction between its sub-digraphs Proof. The result follows from Theorems 3 and 6. □Remark 5.In TSUIC, for any D having a fully participated interaction between its non-empty sub-digraphs  TSUIC scheme achieves Example 3.Consider a TSUIC problem of the following: Theorem 7 (CASE II-C).For any D having a fully participated interaction between its sub-digraphs (i) (ii) (iii) (iv) Proof. Refer to Case Proposition 2 (CASE II-C).For any D having a fully participated interaction between its sub-digraphs Proof. D whose D where D with It follows from Theorem 7 that a 5), soβ≥β)≤β(D3),β≤βa 5), soβ≥β, we getβCorollary 3.For any D having a fully participated interaction between its sub-digraphs Proof. D, by applying Theorem 3, one can get For the given digraph Corollary 4.For any D such that (i) if (ii) if (a) if (b) if Proof. If Remark 6.Let (i) (ii) Example 4.Consider a TSUIC problem D of the following: Theorem 8 (CASE II-D).For any D having a fully participated interaction between its sub-digraphs (i) (ii) (iii) (iv) Proof. Refer to Proposition 3.For any D having a fully participated interaction between its sub-digraphs Proof. It follows from Theorem 8 that 2), thenβ≤β1), thenβ≤β). From ≤β, we getβ≥βNow, for UIC problems with more than two senders, we study some classes of interactions between the sub-digraphs of a digraph (representing the UIC problem) in the following sub-section.In this section, we illustrate how the method proposed in this paper can be generalized to scenarios with more than two senders.D, we have D associated with vertices of those requesting the private messages, and Let Observe that the extensions of Definition 5  between its sub-digraphs Remark 7.Extending the other cases of TSUIC to MSUIC is a laborious task as it involves a construction of multi-dimensional blocks and sub-blocks in a confusion graph.D and its sub-digraphs u of D. Now, we make the following two observations in TSUIC:The role of side-information of the vertices in  TSUIC: It is proved in SSUIC that, if the interaction between D can be removed without affecting the optimal broadcast rate of D; in other words, those are non-critical arcs. In this paper, we have proved that this result is also true in TSUIC (by Theorem 5). Moreover, in TSUIC, we have proved that, for D, if the vertices in Non-critical arcs in SSUIC are not necessarily non-critical in TSUIC: We illustrate this with an example. Consider the TSUIC problem stated in Example 1 , there have been many important contributions made in the literature. In this paper, we solved TSUIC instances by expressing the optimal broadcast rates in terms of that of SSUIC. To this end, we introduced a two-sender graph coloring of confusion graphs in TSUIC, and propose a way of grouping the vertices of a confusion graph for analysis. Using these techniques, we derived optimal broadcast rates of TSUIC problems, both in the asymptotic and non-asymptotic regime, as a function of the optimal broadcast rates of their sub-problems. We have also presented a class of TSUIC instances where the interactions between the sub-problems of the problem are not critical. We illustrated that our proposed approach to TSUIC can be extended to some cases with multiple senders.Study of the critical edges in the TSUIC problems: It is observed that the non-critical arcs in SSUIC can be critical arcs in TSUIC. This requires further study.Study of a general distributed index coding: As our study is a step towards understanding multi-sender index coding, it is left as a future work to extend the approaches implemented and the results obtained in this paper to more general setups.Finding the optimal broadcast rates of TSUIC problems with cyclic-partially-participated interactions: The analysis of D with partially-participated interactions between its sub-digraphs Some open problems for future works are the following:


The family of all ω-open sets of X which is finer than τ and this topology is denoted by ω-open sets played a vital role in general topology research see, A in θ-closure and the A is θ-closed A is θ-open θ-closed  Since by f is weakly continuous. Since (b) Suppose to the contrary that Theorem 2.39Every faintly-continuous function is faintly continuous.ProofThe implication in Example 2.40f is faintly continuous but not faintly Consider the identity function We can summarize the results and examples above by means of the following diagram:S. Al Ghour, B. Irshidat: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.The authors declare no conflict of interest.No additional information is available for this paper.


This is a novel version of de Finetti’s problem of coherence checking for probabilistic assessments. We show how to reduce the coherence checking problem in relational Bayesian networks to a validity problem in first-order logic augmented with a transitive closure operator and how to combine this logic-based approach with faster, but incomplete algorithms.There are several formalisms that enhance Bayesian networks by including relations amongst individuals as modeling primitives. For instance, Probabilistic Relational Models (PRMs) use diagrams and relational databases to represent repetitive Bayesian networks, while Relational Bayesian Networks (RBNs) employ first-order probability formulas with the same purpose. We examine the  N independent biased coins, so that p for each one of them. Then, the probability of a particular configuration of all coins is exactly n is the number of coherent as they are satisfied by a probability distribution [Most statistical models are couched so as to guarantee that they specify a single probability measure. For instance, suppose we have xactly pn−pN−n, whribution ,3, as a ribution ,5 and geribution ,7,8.In this paper, we examine the coherence checking problem for probabilistic models that enhance Bayesian networks with relations and first-order formulas: more precisely, we introduce techniques that allow one to check whether a given relational Bayesian network, or a given probabilistic relational model is guaranteed to specify a probability distribution. Note that “standard” Bayesian networks are, given some intuitive assumptions, guaranteed to be coherent ,10,11. TX is a parent of node Y if there is an edge from X to Y. The set of parents of node X is denoted To recap, a Bayesian network consists of a directed acyclic graph, where each node is a random variable, and a joint probability distribution over those variables, such that the distribution and the graph satisfy a Markov condition: each random variable is independent of its non-descendants given its parents.  Consider that now we have three people, (Scenario 2) It is also straightforward to expand our Bayesian network to accommodate n individuals (Scenario 3) Consider a slightly different situation with three people, where: (Scenario 4) Suppose we want to extend Scenario 3 to a town with n people. Without knowing which pairs are neighbors, there is no way we can predict in advance the structure of the resulting Bayesian network. However, we can reason about the possible networks: for instance, we know that each set of n people produces a valid Bayesian network, without any cycles amongst random variables.X at a time, we deal with parameterized random variables [x in a domain; if we consider individuals a and b in a domain, we obtain random variables There are many other scenarios where probabilistic modeling must handle repetitive patterns such as the ones described in the previous examples, for instance in social network analysis or in processing data in the semantic web ,13,14. Tariables . We writPlates offer a popular scheme to manipulate parameterized random variables [ariables . A platePlates appeared with the BUGS package, to facilitate the specification of hierarchical models, and have been successful in applications . One resSeveral other combinations of parameterized random variables and graph-theoretical representations have been proposed, often grouped under the loose term “Probabilistic Relational Model (PRM)” ,19,20. USuppose that we want a PRM to represent Scenario 4 in Example 2. Now, the class xpressed .burglary(x) = 0.001;alarm(x) = 0.9 * burglary(x) + 0.01 * (1-burglary(x));calls(x) = NoisyOR { alarm(y) | y; neighbor };Instead of resorting to diagrams, one may instead focus just on textual languages to specify repetitive Bayesian networks. A very solid formalism that follows this strategy is Jaeger’s Relational Bayesian Networks (RBNs) ,23. In ROne problem that surfaces when we want to use an expressive formalism, such as RBNs or PRMs, is whether a particular model is guaranteed to always produce consistent Bayesian networks. Consider a simple example .Example 3.Suppose we are modeling genetic relationships using the parameterized random variable If y and z are such that It we try to specify a PRM for this setting, we face a difficulty in that some instances of r et al. . We woulThe possibility that RBNs and PRMs may lead to cyclic (thus inconsistent) Bayesian networks has been noticed before. Jaeger  suggestecoherent in the sense that it can be always satisfied by a probability distribution? Is it coherent in the sense that it always produces a unique probability distribution? Such are the questions we address, by exploring a cross-pollination of ideas described in the previous paragraph. In doing so, we bring logical rigor to the problem of coherence of PRMs and present a practical alternative to identifying coherent RBNs.Thus, we have arrived at the problems of interest in this paper: Suppose we have an RBN or a PRM. Is it After formally introducing relational Bayesian networks in In this section, we briefly introduce the formalism of Relational Bayesian Networks (RBNs). We use the version of RBNs presented in , as thatS and R be disjoint sets of relation symbols, called the predefined relations and probabilistic relations, respectively. We assume that S contains the equality symbol =, to be interpreted in the usual way. Each predicate symbol is associated with a positive integer k, which is its arity. Given a finite domain V is a set of relation symbols (as R or S), a V-structureV into sets of tuples in D. Formally, a V-structure k into a subset of V-structures over a given finite domain D. Given a domain D, a k and a tuple ground V-atom. A V-structure v is mapped to a relation containing t, we say that satisfied byLet V, together with variables, quantifiers and Boolean connectives. We call these V-formulas, and their meaning is given by the first-order logic semantics, as usual, through the V-structures. We denote by V-formula where V-formula and V-structure, Employing the syntax of function-free first-order logic, we can construct formulas using a vocabulary of relations random relational structure model for S and R is a partial function P that takes an S-structure D, and returns a probability distribution R-structures on the same domain. As an R-structure can be seen as total assignment over the ground R-atoms, S-structure of neighbors and returns a joint probability distribution over ground atoms for D, implies a specific Bayesian network whose variables are the ground atoms for A S-structure into a ground Bayesian network that encodes a probability distribution over R-structures. To begin, this ground Bayesian network has nodes representing k is the arity of r. Thus, given the domain D of the input S-structure, the nodes in the corresponding Bayesian network are already determined. To define the arcs and parameters of the Bayesian network associated with an arbitrary S-structure, relational Bayesian networks employ their central notion of probability formula.Relational Bayesian networks provide a way to compactly represent random relational structure models. This is achieved by mapping each S-structure. Once an R-structure and an S-structure are fixed, then for elements D, a probability formula Probability formulas are syntactical constructions intended to link the probability of a ground atom combination functions, which are functions from finite multi-sets over the interval oisyOR is a combination function such that, if oisyORThe definition of probability formulas makes use of Definition 1.S and R of relation symbols and a tuple x of k variables, -probability formula if:(constants) k;(indicator functions) (convex combinations) y is a tuple of variables and S-formula.(combination functions) Given disjoint sets x is a tuple of k variables, with k the arity of Relational Bayesian networks associate a probability formula Definition 2.S and R, the predefined and probabilistic relations, a relational Bayesian network is a set S,R)-probability formula.Given disjoint sets of relation symbols S-structure D. Then, an R-structure D entails a numeric value for each ground probability formulat is tuple of elements in D. This is done inductively, by initially defining t. The numeric value of c. As the set of ground atoms whose evaluation is needed to compute S-structure t and Here, To have an idea of how probability formulas work, consider a fixed S-structure For a given Definition 3.Relation ⪯, over ground R-atoms, is defined as follows:S-structure D, a probability distribution over the R-structures D via:When this relation is acyclic, a relational Bayesian network Example 4.Scenario 4 of Example 2: We can define a relational Bayesian network that returns the corresponding Bayesian network for each number and configuration of neighbors. Let  Φ:oisyOR; a combination function.Note that, if  NoisyOR above are in Φ determines a joint probability distribution over the ground R-atoms, via a Bayesian network. If we take an S-structure Given an S-structure S-structures yield a cyclic dependency relation ⪯. When the relation ⪯ is cyclic for an S-structure, no probability distribution is defined over the R-structures. In such a case, we say incoherent for that S-structure. This notion can be generalized to a class of S-structures coherent for S-structure in S-structures is precisely one of the problems we are interested in this work.It may happen for a relational Bayesian network S-structures and the coherence of a relational Bayesian network S-structures, note that they can be seen as first-order structures over which S-formulas are interpreted. That is, an S-formula defines the set of S-structures satisfying it. If S-formula (without free variables), we say that S-structures S-formula satisfied exactly by the S-structures in a given class In order to reason about the relation between a class of S-structure. Ideally, we would like to have a (first-order) S-formula, say S-structures yielding acyclic dependency relations ⪯. If that formula were available, a decision about the coherence of S-structure in the class S-structure in S-formulas, only S-structures matter, and we could ignore any relation not in S. To be precise, if a first-order structure S-structure To encode the coherence of Alas, to encode cycles in a graph, one needs to encode the notion of path, which is the transitive closure of a relation encoding arcs. It is a well-known fact that first-order logic cannot express transitivity. To circumvent that, we can add a (strict) transitive closure operator to the logic, arriving at the so-called transitive closure logics, as described for instance in .S-formula S and R, one could hope to obtain a formula S-structure n, with one node per ground atom, so a cycle can also contain more nodes than n. There seems to be no direct way of employing the transitive closure operator to devise a formula n nodes and that is to be satisfied by some structures n elements. In the next sections, we review a technique  to encoV-formula V containing S, that is satisfiable only by V-structures that encode both an S-structure V-structures should contain, besides an S-structure Our idea to construct a formula S and v. Consider a relational Bayesian network S,R)-probability formula. Let V-structure satisfying S-structure V a unary predicate S-structure V-structure Suppose we have two disjoint vocabularies V that includes the predefined relations in S and contains the unary predicate S). Furthermore, V must contain a binary relation r and to tuples S-structure V binary relations Firstly, we need to define a vocabulary x in the domain represents the ground atom To save notation, we use V-structure Now, we proceed to list, step-by-step, the set of conjuncts required in We have to ensure that the elements in Each element in No element may correspond to ground atoms for two different Each element corresponding to a ground atom should correspond to exactly one tuple. To achieve that, let Each element corresponding to a ground atom for a Only elements in Each ground atom must be represented by at least one element  is s is s8) iD representing ground atoms S-structure R, are determined by the size of S-formulas that hold for the S-structure S-structures to be specified by For all The binary relation The formula above forces that S-structure S-structure S-formula S-formula that is satisfied by For a known If If k is the arity of t. These base cases are in line with the recursive definition of If Above, In other words, the computation of S-formula involved. Recall that if S-formula that is satisfied by an S-structure The more elaborated case happens when If The inner union is analogous to the definition of Now, we can employ the formulas For each pair t and x and y. We point out that such construction depends only on probability formulas in the relational Bayesian network S-structure. To build each S-formulas In the formula above, The relation y and For each pair R, and let y and y denote the tuples Let Definition 4.Given disjoint sets of relations S and R and a relational Bayesian network mulas in –12)..Φ={Fri∣rV-structures S are interpreted in S-structure the relations in b between the domain R-atoms formed by the tuples in there is a bijection each the relation For some fixed relational Bayesian networks V-structure satisfying V-structures S-structure V-structures satisfying it.The original formula V-structure D iff there exists a x to itself. Consequently, detecting y is x, in a V-structure k, we can define reachability through kk) is unbounded a priori, as the domain D can be arbitrarily large. Therefore, there is no means in the first-order logic language to encode reachability, via arbitrarily large paths, with a finite number of formulas. In order to circumvent this situation, we can resort to a transitive closure logic.There is a cycle with y is x, with a non-empty path. Accordingly, a V-structure D, satisfies Transitive closure logics enhance first-order logics with a transitive closure operator e strict . If φ is encoded directly by Employing the transitive closure operator, the existence of a V-structures D satisfying S-structure in acyclic ground Bayesian network either it encodes an S-structure in or it is not the case that At this point, the S-structures ensures the acyclicity of the resulting ground Bayesian network S-formula S-structures without cycles, for instance.Back to the coherence-checking problem, we need to decide, for a fixed relational Bayesian network S-structures over D, while S-structure is given only over S-structures S-structure D. The formula S-structures, like To check the coherence of mulas in , so no xmulas in , contradmulas in  that reqidity of:θS→ S-structures Finally, we can define the formula that encodes the coherence of a relational Bayesian network Definition 6.S and R, a given relational Bayesian network S-structures defined by For disjoint sets of relations Putting all those arguments together, we obtain the translation of the coherence-checking problem to the validity of a formula from the transitive closure logic:Theorem 1.Φ and a class of S-structures Φ is coherent for For disjoint sets of relations S and R, a given relational Bayesian network (De Bona and Cozman ).For diR to be unary and assuming a decidable As first-order logic in general is already well-known to be undecidable, adding a transitive closure operator clearly does not make things easier. Nevertheless, decidability remains an open problem, even restricting the relations in perators ,26). SimJaeger introduced the coherence problem for RBNs as checking whether every input structure in a given class yields a probability distribution via an acyclic ground Bayesian network. Alternatively, we might define the coherence of an RBN as the existence of at least one input structure, out of a given class, resulting in an acyclic ground Bayesian network. This is closer to the satisfiability-like notion of coherence discussed by de Finetti and closer to work on probabilistic logic ,28.In this section, we show that, if one is interested in a logical encoding for this type of coherence for RBNs, the transitive closure operator can be dispensed with.S-structure in Suppose we have an RBN The idea to construct V is augmented with the binary relation In order to build By construction, one can verify the following result:Theorem 2.Φ and a class of S-structures Φ is coherent for some structure in For disjoint sets of relations S and R, a given relational Bayesian network The fact that In this section, we introduce the machinery of PRMs by following the terminology by Getoor et al. , focusindescriptive attributesreference slotsX (denoted range type) specified in the schema. The domain type of X. We can view this reference slot To define a PRM, illustrated in Example 5, we need a relational model, with classes associated with descriptive attributes and reference slots that behave like foreign keys. Intuitively, each object in a class is described by the values of its descriptive attributes, and reference slots link different objects. Formally, a relational schema is described by a set of classes inverse slotx from the class slot chain if i. The function corresponding to a slot chain x from For any reference slot An instance relational skeletoncompletion of a relational skeleton Given a relational schema, a PRM defines a probability distribution over its instances. In the simplest form, on which we focus, objects and the relations between them are given as input, and there is uncertainty only over the descriptive attributes values. A dependency graph; and the set of parameters define the conditional probability tables. The attributes in formal parents, as they will be instantiated for each object x in X according to the relational skeleton. There two types of formal parents: K is a slot chain.The format of a PRM resembles that of a Bayesian network: for each attribute x, aggregation is used. The attribute In general, for an object Definition 7.probabilistic Relational modelA ground Bayesian network induced by a relational skeleton, where the descriptive attributes of each object are the random variables.The semantics of a PRM is given by the Definition 8.There is a node representing each attribute For each Each Each The CPD for The CPD for A PRM simplified ground Bayesian network. The latter can be employed to factor the joint probability distribution over the descriptive attributes:The joint probability distribution over the descriptive attributes can be factored as usual to compute the probability of a specific instance Viewing Example 5.Recall again Scenario 4 in Example 2. We can define a PRM that returns the corresponding Bayesian network for each number and configuration of neighbors. In our relational schema, we have a class We assume that the relation For each descriptive attribute in our relational schema, we associate a set of formal parents and a conditional probability table, forming the following PRM Π to encode Scenario 4:Given a relational skeleton  Π determines a joint probability distribution over the the descriptive attributes, via a Bayesian network. Consider a skeleton all possible relational skeletons. To achieve that, possible cycles are detected in the class dependency graph.As with RBNs, for the model to be coherent, one needs to guarantee that the ground Bayesian network is acyclic. Getoor et al.  focused Definition 9.class dependency graphType I arcs: Type II arcs: Given a PRM guaranteed acyclic if we are guaranteed that, for any possible relational skeleton x and When the class dependency graph is acyclic, so is the ground Bayesian network for any relational skeleton. Nevertheless, it may be the case that, even for cyclic class dependency graphs, any relational skeleton occurring in practice leads to a coherent model. In other words, there might be classes of skeletons for which the PRM is coherent. To easily recognize some of these classes, Getoor et al.  put forwDefinition 10.colored class dependency graphYellow arcs: Green arcs: Red arcs: Given a PRM x is different from y, and there is no cycle. If there is a red arc in a cycle, however, one may have a skeleton that produces a cycle.Intuitively, yellow cycles in the colored class dependency graph correspond to attributes of the same object, yielding a cycle in the ground Bayesian network. If we add some green arcs to such a cycle, then it is guaranteed that, departing from a node stratified if every cycle contains at least one green arc and no red arc. Then:A colored class dependency graph is Theorem 3.Given a PRM Π and a set of guaranteed acyclic slot chains .Given a priori knowledge on the possible skeletons in practice. For instance, consider a slot chain In the result above and in the definition of guaranteed acyclic slot chains, “possible relational skeleton” refers to the class of skeletons that can occur in practice. The user must detect the guaranteed acyclic slot chains, taking advantage of his Identifying guaranteed acyclic slot chains is by no means trivial. In fact, Getoor et al.  also defThe simplest approach to the coherence of PRMs, via the non-colored class dependency graph, is intrinsically incomplete, in the sense that some skeletons might yield a coherent ground Bayesian network even for cyclic graphs. The approach via colored class dependency graph allows some cyclic graphs (the stratified ones) to guarantee consistency for the class of all possible skeletons. However, this method depends on a pre-specified set of guaranteed acyclic slot chains, and the colored class dependency graph being stratified for this set is only a sufficient, not a necessary condition for coherence. Therefore, the colored class dependency graph method is incomplete, as well. Even using different sets of g.a. slot chains (corresponding to shades of green) to eventually capture all of them, it is still possible that a cycle with red arcs cannot entail incoherence in practice. Besides being incomplete, the graph-based method is not easily applicable to an arbitrary class of skeletons. Given a class of skeletons as input, the user would have to detect somehow which slot chains are guaranteed acyclic for that specific class; this can be considerably more difficult than ensuring acyclicity in the general case.To address these issues, thus obtaining a general, complete method for checking the coherence of PRMs for a given class of skeletons, we can resort to the logic-based approach we introduced for the RBNs in previous sections. The goal of this section is to adapt the logic-basic techniques to PRMs.S-structures to probability distributions on R-structures. While the S-vocabulary is used to specify classes of objects and relations between them , the R-vocabulary expresses the descriptive attributes of the objects. Employing this logical encoding of PRMs, we can apply the approach from PRMs can be viewed as RBNs, as conditional probability tables of the former can be embedded into combination functions of the latter. This translation is out of our scope though, and it suffices for our purposes to represent PRMs as random relational structures, taking To follow this strategy, we first show how a PRM can be seen as a random relational structure described by a logical language.S-structure over a set of objects and a set of unary relations R denotes their attributes, the PRM becomes a random relational structure.Consider a PRM S-structure D of S to denote the class of each object. Accordingly, for each We need to represent a skeleton S-structure S, all objects of a given class, as well as the relations between them specified in the reference slots. In other words, there is a computable function S-structures The first-order R-atom R-structure R-structures R-structure The probabilistic vocabulary of the random relational structure corresponding to a PRM is formed by the descriptive attributes of every class in the relational schema. We assume that attributes in different classes have different names, as well, in order to define the vocabulary of unary relations D, let R-structures such that Now, we can define a random relational structure R-atoms. The structure of this network is isomorphic to the simplified ground Bayesian network yielded by The probability distribution S-structures We want to construct a formula that is satisfied only by those R-atoms K, then x via To encode the arcs, we employ once more a relation K. Thus, we define Note that it is possible that both S-structure Analogously to S-structures satisfying a first-order formula Consider now a class We have thus succeeded in turning coherence checking for PRMs into a logical inference, by adapting techniques we developed for RBNs. In the next section, we travel, in a sense, the reverse route: we show how to adapt the existing graph-based techniques for coherence checking of PRMs to coherence checking of RBNs.only if such a formula is valid. Nonetheless, completeness and flexibility regarding the input class come at a very high price, as deciding the validity of this first-order formula involving a transitive closure operator may be computationally hard, if at all decidable. Therefore, RBNs users can benefit from the ideas introduced by Getoor et al. [a priori knowledge, the RBN user perhaps can attest to the acyclicity of the resulting ground Bayesian network for all possible inputs.The logic-based approach to the coherence problem for RBNs can be applied to an arbitrary class of input structures, as long as the class can be described by a first-order formula, possibly with a transitive closure operator. Given any class of input structures r et al.  for the r et al.  proposesr et al.  are inteAny arc Definition 11.R-dependency graphType I arcs: Type II arcs: Given an RBN R-dependency graph of an RBN D and any tuple S-structure, t. Type II arcs capture dependencies that are contingent on the S-relations holding in the input structure. In other words, a Type II arc S-formula D) in the output ground Bayesian network iff the input S-formula might be fairly complicated. Nevertheless, the point here is simply noting that, given a Type II arc S-formula parametrized by Intuitively, a Type I arc R-dependency graph, when the latter is acyclic, so will be the former, for any input structure. As it happens with class dependency graphs and PRMs, though, a cycle in the R-dependency graph does not entail a cycle in the ground Bayesian network if a Type II arc is involved. It might well be the case that the input structures R-dependency graph.As each arc in the ground Bayesian network corresponds to an arc on the R-dependency graph of an RBN, these arcs encode the dependency between relations R-dependency graph whose sequence will prevent cycles in the ground Bayesian network, via a partial ordering over the tuples.In the same way that Type I arcs in the class dependency graph of a PRM relate to attributes of different objects, in the S of predefined relations, let D. We say a set D, there is a partial ordering ⪯ over For a vocabulary Definition 12.R-dependency graph of an RBN colored R-dependency graphR-dependency graph;Yellow arcs: Type I arcs in the R-dependency graph such that Green arcs: Type II arcs R-dependency graph.Red arcs: The remaining (Type II) arcs in the Given the R-dependency graph correspond to relations t, yielding a cycle in the ground Bayesian network. If green arcs are added to a cycle, then it is guaranteed that, departing from a node stratified a R-dependency graph whose every cycle contains at least one green arc and no red arc, we have:Again, yellow cycles in the colored Theorem 4.Φ and a set Given the R-dependency graph of an RBN R-dependency graphs of RBNs is even harder than, as a generalization of, detecting guaranteed acyclic slot chains in PRMs. In any case, if the involved relations Of course, detecting guaranteed acyclic Type II arcs in coherence checking, a central problem in the foundations of probability as conceived by de Finetti. The simplest formulation of coherence checking takes a set of events and their probabilities and asks whether there can be a probability measure over an appropriate sample space [In this paper, we examined a new version of le space . This sole space . Unsurprle space . Our focle space .S cannot be followed as originally stated; as we have argued, the possible number of tuples built from a domain typically outnumbers its size, so that there is no straightforward way to encode the ground Bayesian network, whose nodes are ground atoms, into the input S-structure. Therefore, it is hard to think of a method that translates the acyclicity of the ground Bayesian network into a formula S (satisfying S-structure and the corresponding Bayesian network. We have also extended those results to PRMs, in fact mixing the existing graph-based techniques for coherence checking with our logic-based approach. Our results seem to be the most complete ones in the literature.We have first examined the problem of checking the coherence of relational Bayesian networks for a given class of input structures. We used first-order logic to encode the output ground Bayesian network into a first-order structure, and we employed a transitive closure operator to express the acyclicity demanded by coherence, finally reducing the coherence checking problem to that of deciding the validity of a logical formula. We conjecture that Jaeger’s original proposal concerning the format of the formula encoding the consistency of a relational Bayesian network Future work includes searching for decidable instances of the formula encoding the consistency of a relational Bayesian network for a class of input structures and exploring new applications for the logic techniques herein developed.


Nature Communications 10.1038/s41467-020-17945-4, published online 20 August 2020.Correction to: The original version of this Article contained an error in Fig. 4a, in which the value for the ‘partition coefficient’ for phenyltriethylammonium was incorrectly given as 0. The correct value for the ‘partition coefficient’ is 2.9.This has been corrected in both the PDF and HTML versions of the Article.