| { | |
| "paper_id": "C92-1021", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T12:34:29.924501Z" | |
| }, | |
| "title": "Hopfield Models as Nondeterministic Finite-State Machines", | |
| "authors": [ | |
| { | |
| "first": "Marc", | |
| "middle": [ | |
| "F J" | |
| ], | |
| "last": "Drossacrs", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Twente", | |
| "location": { | |
| "addrLine": "The Netherlands", | |
| "postBox": "P.O Box 217", | |
| "postCode": "7500 AE", | |
| "settlement": "Enschede" | |
| } | |
| }, | |
| "email": "[email protected]." | |
| } | |
| ], | |
| "year": "", | |
| "venue": null, | |
| "identifiers": {}, | |
| "abstract": "Tbe use of neural networks for integrated linguistic analysis may be profitable. This paper presents the first results of our research on that subject: a Hopfield model for syntactical analysis. We construct a neural network as an implementation of a bounded push-down automaton, which can accept context-free languages with limited center-embedding. The network's behavior can be predicted a priori, so the presented theory can be tested. The operation of the network as an implementation of the acceptor is provably correct. Furthermore we found a solution to the problem of spurious states in Hopfield models: we use them as dynamically constructed representations of sets of states of the implemented acceptor. The so-called neural-network aceeptor we propose, is fast but large.", | |
| "pdf_parse": { | |
| "paper_id": "C92-1021", | |
| "_pdf_hash": "", | |
| "abstract": [ | |
| { | |
| "text": "Tbe use of neural networks for integrated linguistic analysis may be profitable. This paper presents the first results of our research on that subject: a Hopfield model for syntactical analysis. We construct a neural network as an implementation of a bounded push-down automaton, which can accept context-free languages with limited center-embedding. The network's behavior can be predicted a priori, so the presented theory can be tested. The operation of the network as an implementation of the acceptor is provably correct. Furthermore we found a solution to the problem of spurious states in Hopfield models: we use them as dynamically constructed representations of sets of states of the implemented acceptor. The so-called neural-network aceeptor we propose, is fast but large.", | |
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| "section": "Abstract", | |
| "sec_num": null | |
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| ], | |
| "body_text": [ | |
| { | |
| "text": "Neural networks may be well suited for integrated linguistic analysis, as Waltz and Pollack [10] indicate. An integrated linguistic analysis is a parallel composition of several analyses, such as syntactical, semantical, and pragmatic analysis. When integrated, these analyses constrain each other interactively, and may thus suppress a combinatoric explosion of sentence structure and meaning representations.", | |
| "cite_spans": [ | |
| { | |
| "start": 74, | |
| "end": 96, | |
| "text": "Waltz and Pollack [10]", | |
| "ref_id": null | |
| } | |
| ], | |
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| "eq_spans": [], | |
| "section": "Introduction", | |
| "sec_num": "1" | |
| }, | |
| { | |
| "text": "This paper presents the first results of our research into the use of neural networks for integrated linguistic analysis: a Hopfield model for syntactical analysis. Syntactical analysis in the context of integration with other analyses boils down to the decision whether a sentence is an element of a language. A parse tree is superfluous here as an intermediary representation, since it will not be finished before the complete integrated analysis is. This fact allows us to deal with the problem of a restricted length of the sentences a neural par .... n handle, see e.g. [51, [7] , a problem that could not be elegantly solved, see e.g. [6] , [3] .", | |
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| "text": "[7]", | |
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| "text": "[3]", | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
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| "text": "In this paper we propose a formal model that recognizes syntactically correct sentences (section 2), a tlopfield model onto which we want to map this formar model (section 3), the parameters that makes the network operate as intended (section 4), and a way to map the formal model onto the Hopfield model, including a correctness result for the latter (section 5). The theoretically predicted behavior of the so~ obtained network has been verified, and a simple example provides the taste of it (section 6). We alas consider complexity aspects of the model (section 7). Section 8 consists of concluding remarks.", | |
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| "section": "Introduction", | |
| "sec_num": "1" | |
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| "text": "Although it is not an t.~tablished fact, it is asstoned here that natural languages are context-free, and consequently that sentences in a natural language can be recognized, by a push-down atttomaton (PDA). ilowever, we are not interested in modeling the competence of natural language users, but in modeling their performance. The human performance in natural language use is also characterized by a very limited degree of center-embedding. In terms of PDAs this means that there is n bound on the,number of items on the stack of a PI)A for a natural language. A bounded push-down automaton M = (Q,Y~,I',6, qs, Zo, F) is a PDA that has an upper limit k E ~ on the number of items on its stack, i.e. H ~< k for every instantaneous description (ID) (q, w, a) of M. The set of stack states of this PDA is delined to be: QST =-{a I (qo,w, Zs) P*~t (q,e,\u00a2~)}.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "A Bounded Push-Down Automaton", | |
| "sec_num": "2" | |
| }, | |
| { | |
| "text": "QsT is finite: IQsT[ <_ (IFI) ~, therefore we may define a nondeterministic finite~state aeceptor (NDA) M ~ that has QST ms its set of states.", | |
| "cite_spans": [], | |
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| "section": "A Bounded Push-Down Automaton", | |
| "sec_num": "2" | |
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| "text": "The class of PDAs of which we would like to map bounded versions onto NDAs is constrained, among others to the class of v-free PDAs. By this constraint we anticipate the situation that grammars are stored in a neural network by self-organization. In that sitnation a neural network will store e-productions only if examples of applications of e-productions are repeatedly presented to it. This requires e to have a representation in the machine, in which case it fails to accommodate its definition.", | |
| "cite_spans": [], | |
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| "section": "A Bounded Push-Down Automaton", | |
| "sec_num": "2" | |
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| "text": "Another restriction we would like to introduce is to grammars in 2-standard form with a minimal number of quadratic productions: productions of the form ACrEs DE COL~G-92, NANTES, 23-28 AOOT 1992 I I 3 PROC. OF COLING-92, NA~rEs, AIJG. 23-28, 1992 A ~ bCD where b is a terminal and C and D are variables. Such a grammar can be seen as a minimal extension of a right-linear grammar. Within such grammars, quadratic productions provide for the center-embedding. Since such grammars have a minimal number of quadratic productions, acceptance by a PDA defined for such grammars requires a minimal use of (stack) memory, and titus generates a minireal Qs'r. To maintain this minimal use of memory a restriction to one-state PDAs that accept by empty stack is also required: when a PDA is mapped onto an NDA, the information concerning its states is h)st, unless it was stored on the stack. An e-free, one-state PDA that simulates a contextfree grammar in 2-standard form with a minimal number of quadratic productions (and that accepts by empty stack) satisfies all our criteria. For every such PDA we can define an NDA, for which we can prove [4] that it accepts the same language ms the PDA does. In as far as a natural language is context-free, we claim that there is an instance of our aeeeptor that recognizes it.", | |
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| "end": 1144, | |
| "text": "[4]", | |
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| "start": 153, | |
| "end": 249, | |
| "text": "ACrEs DE COL~G-92, NANTES, 23-28 AOOT 1992 I I 3 PROC. OF COLING-92, NA~rEs, AIJG. 23-28, 1992", | |
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| "section": "A Bounded Push-Down Automaton", | |
| "sec_num": "2" | |
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| "text": "In this section a noiseless Hopfield model is proposed that is tailored to implement NDAs on. The model is based on the associative memory described by Buhmann et al. [2] and the theory of delayed synap~s from [8] . We chose the Itopfield model because of its analytical transparency, and its capability of sequence traversing, which agrees well with the sequential nature of language use at a pbenomenologica[ level. The Hopfield model proposed is a memory for temporal transitions extended with externa|-input synapses. Figure 1 shows the architecture involved. Ill this network only those neurons are active upon which a combined local field operates that transcends the threshold. The activity generated by such a lo~ cal field is the active overlap of the temporal image of past activity provided by so-called temporal synapses, and the image of input external activity provided by so-called input synapses. By the temporal synapses, this activity will later generate another (subthreshold) temporal image, so network activity may be considered a transition mechanism that brings the network from one temporal image to another. Active overlaps are unique with high probability if the activity patterns are chosen at random and represent low nrean network activity. This uniqueness makes tile selectivity of the network very plausible: if an external activity pattern is presented that does not match the current temporal image, then there will not he activity of any significance; tile input is not recognized.", | |
| "cite_spans": [ | |
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| "start": 167, | |
| "end": 170, | |
| "text": "[2]", | |
| "ref_id": "BIBREF1" | |
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| "text": "[8]", | |
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| "end": 530, | |
| "text": "Figure 1", | |
| "ref_id": "FIGREF0" | |
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| ], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| "text": "When an NDA is mapped onto this network, pairs of NDA-state q and input-symbol x, such that 6(q, x) y\u00a3 {~, are mapped onto activity patterns. Temporal relations in the network then serve to implement NDA transitions. Note that single NDA transitions arc mapped onto single network transitions. This results in complex representations of the NDA states and the input symbols. An NDA state is rap resented by all activity patterns that represent a pair containing that state, and input patterns are represented by a component-wise OR over all activity patterns containing that input symbol. A consequence is that mixed temporal images, the subthreshold analogue of mixture states, are a very natural phenomenon in this network, because tile temporal image of an active overlap comprises at least all activity patterns representing a successor state. But this is not all. Also the network will act as if it implements the deterministic equivalent of the NDA, i.e. it will trace all paths through state space the input allows for, concurrently. The representations of the states of this deterministic finite-state automaton (FSA) are dynamically constructed along the way; they are mixed temporal images. The concept of a \"dynamically constructed representation\" is borrowed from Touretzky [9] , who, by the way, argued that they could not exist in the current generation of neural networks, such ms ltopfield models.", | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| "text": "A time cycle of the network can be described as follows:", | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| "text": "1~ The network is allowed to evolve into a stable activity pattern that is the active overlap of a temporal image of past activity, and the input image of external input for a pe~'iod tr (= relaxation time), when an external activity pattern is presented to the network;", | |
| "cite_spans": [], | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| "text": "2. After some time the network reaches a state of stable activity and starts to construct a new temporal image. It is allowed to do this for a period t, (= active time)', 3 . Then the input is removed, and the network evolves towards inactivity. This takes again about a period t~;", | |
| "cite_spans": [ | |
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| "start": 171, | |
| "end": 172, | |
| "text": "3", | |
| "ref_id": "BIBREF2" | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| { | |
| "text": "Acids DE COLING-92, NANTES, 23-28 AOt~T 1992 1 1 4 PRO(:. OF COL]NG-92, NANTI~S, AUG. 23-28. 1992", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| { | |
| "text": "4. Not before a period ta (= delay time) has passed, a new input arrives. The new temporal image is forwarded by tile slow synapses during a period ta +iv, starting when td ends. The slow synapses have forgotten the old temporal image while the network was in its td.", | |
| "cite_spans": [], | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "The synapses modeled in the network collect the incoming activity over a period ~ + tr, and emit the time average over again a period ta + tr after having waited a period ta + t~. In the network this is modeled by computing a time average over prior neuronal activity, and then multiply it by the synoptic efficacy. Tile time average ranges over a period More formally the network can be described as follows: The choice for a transition time tr depends on tile probability with which one requires the network to get in the next stable activity state. This subject will be dealt with in section 7.", | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| "text": "5'i (5 {0,1}, wherei= 1 .... ,N, l 1 ifh~(t)>U .')'i(t+l) : 0 if hi(t)< U, h~(t) : h~(t)+hl(t),", | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| "text": "In the estimates of A t and the storage capacity below, an expression of the temporal transition term in terms of the overlap parameter m o is used, which will be introduced here first. The overlap parameter rnP(t) measures the overlap of a network state Assuming that N --* co while p remains fixed this is, after expansion of the Jq and ignoring infinitesimal terms, approximated by:", | |
| "cite_spans": [], | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| { | |
| "text": "At p N h:(t) - a(1-a)N Z(~+l-a) Z(~f-a);j(t) t~=l j=l t I\u00b0 A' ~,-.tf\u00a2+ , = (1 -a) ,=z'-~l'\" -a)t~l\"(t), where r -O ~o\u00b0\u00b0 m\"(t) ~ ~ m\"(t-t')w(t')dt'.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| { | |
| "text": "If the temporal image is {~\u00a2~} then h~ is about (N co):", | |
| "cite_spans": [], | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| { | |
| "text": "h~(t) = ~,(~+l _a)(r_ O) w(t)dt. 0", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| "text": "If a number of patterns in a mixture state have the same successor, that pattern may be activated. To prevent this A ~ will be chosen such that the slow synapses do not induce activity in the network autonomously, not even if all the neurons in the network are active. On average, the activity in the network is given by the parameter a. The total activity in a network is a quantity x such that z = 1/a, so what we require is that xh~ < U, i.e. that:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
| }, | |
| { | |
| "text": "a~(~+~ -.)(r -0) w(t)dt < U. aO", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| "text": "The interesting case is (i \"+1 = 1. Since the integral is at most O/(r -O) which is the strongest condition on the left side, the left expression can be written as )d(1 -a)/a. It was earlier demanded that only a combined local field can transcend the threshold, which implies that external input .~e(1 -a) < U, so we can take A ~ < A~a safely. This is small because a is small. Next a value for the threshold that optimizes storage capacity is estimated by signal-to-noise ratio analysis, following [1] and [2] , for N,p --* oo. Temporal effects are neglected because they effect signal and noise equally. It is also assumed that external input is present, so that the critical noise effects can be studied. In this model the external input synapses do not add noise, they do not contain any information apart from the magnitude of the incoming signal. Now suppose the system is in state {S} = {~}. The signal is that part of the input that excites the current pattern:", | |
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| "start": 499, | |
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| "text": "[1]", | |
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| { | |
| "text": "s = A'(~, ~-a).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
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| "text": "The noise is the part of the input that excites other patterns. It can be seen as a random variable with zero mean and it is estimated by its variance: ~t v'~-a where a = p/N. We want that given the right input h i > U, if both the temporal and the external input excite Si, and that hi < U if the temporal input does not excite Si. This gives signal-to-noise ratios: Pt = (,V+At)(1-a)-U, and", | |
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| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| { | |
| "text": "U + ,Va -,~'(1 -a) PO = ,v", | |
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| "eq_spans": [], | |
| "section": "An Input-Driven Sequencing Hopfield Model", | |
| "sec_num": "3" | |
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| "text": "Recall is optimal in case Po = Pl which is true for a threshold:", | |
| "cite_spans": [], | |
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| "eq_spans": [], | |
| "section": "vCg~", | |
| "sec_num": null | |
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| { | |
| "text": "Uor,, = At(1-a)+,V(\u00bd-a).", | |
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| "section": "vCg~", | |
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| "text": "Substituted in either P0 or Pt it results in Pore = l This result is the same as obtained by Buhmann et al. [2] , and they found a storage capacity ct c .~ -(ainu) -1 where ac = prna,:/N. The storage capacity is large for small a, so a will be chosen a << 0.5. A last remark concerns an initial input for the network. In case there has been no input for the network for some time, it does not contain a temporal image anymore, and consequently has to be restarted. This can be done by preceding a sequence by an extra strong first input, a kind of warning signal of magnitude e.g. A\" + At. This input creates a memory of previous activity and serves as a starting point for the temporal sequences stored in the network.", | |
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| "start": 108, | |
| "end": 111, | |
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| { | |
| "text": "Neural-Network Acceptors", | |
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| "section": "5", | |
| "sec_num": null | |
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| "text": "In this section it is shown how NDAs from definition 2.1 can be mapped onto networks as described in sections 3 and 4. Such networks can only be used for cyclic recognition runs. Where \"cyclic\" indicates that both the initial and the accepting state of an NDA are mapped onto the same network state. If this were not done, the accepting state is not assigned an activity vector, since no transition departs from it, see definition 5.2 below. Cyclic recognition in its turn can only be correctly done for grammars that generate end-of-sentence markers. Any grammar can be extended to do this. An NDA is related to a network by a parameter list. (a, A ~, A t , N, p, prnaz, ta, td, tr, U) , where: 9. U=M(1-a)+M(\u00bd-a).", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
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| "start": 644, | |
| "end": 686, | |
| "text": "(a, A ~, A t , N, p, prnaz, ta, td, tr, U)", | |
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| } | |
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| "section": "5", | |
| "sec_num": null | |
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| { | |
| "text": "Note that there arc an infinite number of parameter lists for each NDA. The mapping of an NDA onto a network starts by mapping basic entities of the NDA onto activity patterns. Such a pattern is called a code.", | |
| "cite_spans": [], | |
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| "section": "5", | |
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| "text": "Let M = (Q,E,ti, Qo, F) he an NDA, let t\" : (a,A e,M, N,p, pma,,ta,ta,tr,U) The set Pq of activity patteT~s for q E Q is:", | |
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| { | |
| "start": 44, | |
| "end": 75, | |
| "text": "(a,A e,M, N,p, pma,,ta,ta,tr,U)", | |
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| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "G = {c(%z) l x e ~}.", | |
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| "eq_spans": [], | |
| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The set P:: of activity patterns for\" x E E is:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "P, = {c(q,x) I q E Q}.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Then a network transition is defined ms a matrix operator specified by the network's storage prescription, and related to NDA transitions using the previously defined partition of the set of codes. Now that we have a neural-uetwork acceptor, we triay also Wahl to use it to judge the legality of strings against a given grammar with it. Let M = (Q,E,6, Qo, F) be an NDA, let P --(a, M, A t, N, p, Pm~x, is, Gt, G, U) be a parameter list defined according to definition 5.1, let H = (T, f, U, S) be an NNA defined according to definition 5.5 that takes its parameters from P, and let ai E E, q E Q0, and q' E F. H is said to accept a string w = al \" \" \" an if and only if H evolves through a series of temporal images that ends at Pq, if started at Pq, if OR(P~),..., OR(Pa,) appears as external input for the network.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Next the correctness of an NNA is to be proven. Since an NNA is essentially a stochastic machine, this raises some extra problems. What we propose is to let various network parameters approach values for which the NNA has only exact properties, and prove that the uetwork is correct in the limit. Those \"various network parameters\" are defined below. The following lemma states that for neural-network acceptors that take their parameters from a list of large parameters both the probability that the network reaches the next stable state within relaxation time, and the probability that only the patterns that are temporally related to the previous activity pattern will become active, tend to unity. Essentially it means that such networks operate exactly as prescribed by the synapses. Such networks are intrinsically correct.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Definition 5.2", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Let M = (Q,~2,8, Qo, F) be an NDA, let P = (a, A', ~t, N, p, pm,::, ta, ta, tr, U) be a parameter list defined according to definitions 5.1 and 5.9, and let H = (T, f, U, S) be an NNA defined according to definition 5.5 that takes its parameters from P, then H is such that: Then the correctness of an NNA follows.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Lemma 5.10 intrinsical correctness", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "Theorem 5.11 (correctness of the NNA) Let M = (Q,Z,6, Q0, F) be an NDA, let P = (a, A', A t, N, p, p,,o~, t,, , ta, G, U) be a parameter list defined according to definitions 5.1 and 5.9, let H = (71, f, U, S) be an NNA defined according to definition 5.5 that takes its parameters from P, and let w E E +, then the probability that M accepts a string w if and only if H accepts w, tends to unity.", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 80, | |
| "end": 109, | |
| "text": "(a, A', A t, N, p, p,,o~, t,,", | |
| "ref_id": null | |
| } | |
| ], | |
| "eq_spans": [], | |
| "section": "Lemma 5.10 intrinsical correctness", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The proof of the theorem is given in [4] .", | |
| "cite_spans": [ | |
| { | |
| "start": 37, | |
| "end": 40, | |
| "text": "[4]", | |
| "ref_id": "BIBREF3" | |
| } | |
| ], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Lemma 5.10 intrinsical correctness", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "As an'example we constructed au NNA that accepts the language generated by a grammar with productions: INPUTS Plot of the system dynamics.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Simulation Results", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "S' ~ theBE", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Simulation Results", | |
| "sec_num": "6" | |
| }, | |
| { | |
| "text": "If a neural-network acceptor h~.s to process a sequence of n input patterns, it (worst ease) first has to construct its initial temporal image, when awakened by an initial input (that is not considered a part of the sequence), and then has to build n further temporal images length u as a function of the length of the input sequence is thus (T --O)(n + 1). The constant r al~ depends oti t,. which is chosen to let the network satisfy a certain probability that it reaches the next state in relaxation. This probability is given by (1 -.~/)o where B = tr/N. The time complexity of the neuralnetwork acceptor is O(n).", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Complexity Aspects", | |
| "sec_num": "7" | |
| }, | |
| { | |
| "text": "The upper limit on the number p of stored temporal relations between single activity patterns is [ Q 1:~ x I ~3 [2. The number of neurons in a network is then cx { Q ]e \u00d7 ] E 17, where e depends on the storage capacity and the chosen (low) probability that selection errors occur. The randomly chosen activity patterns overlap, so if a large number of patterns is active they may constitute, by overlap, other unselected activity patterns that will create tlreir own causal consequences. This is called a selection er= rot. The probability that this can happen can be estinrated by l'~,.,,,~(n) -~. 1 -1'(,'\u00a2, = 0), where the latter is:", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Complexity Aspects", | |
| "sec_num": "7" | |
| }, | |
| { | |
| "text": "(-l -2np ' ", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Complexity Aspects", | |
| "sec_num": "7" | |
| }, | |
| { | |
| "text": "the nmnber of activity patterns stored in the network, and m is the number of patterns that were supposed to be present in the mixture state. The probability q = 1 -p, and ,1 :_~ (aN) is the number of patterns that can bc constructed fi'om the aetiw~ neurons in the mix. S,, is the mnnber of wrongly selected activity patterns for a given n. l),rror(n) decreases with increa~ing N if the other parameters remain tixed.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "-_-(~ ), ,: is", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "The space complexity of the network, exprc&sed as the nnmber of neurons, and as a function of the number of NI)A states is O([ Q 17). This is large because Q = Qs'r <1 F I ~ for some PDA M. tlowever things conld have been worse. Not using mixed temporal images to represent FSA states would necessitate the use of a mnnber of temporal images of order 2 IQ'I~, So compared to a more conventional use of lloptield models, this approach yields a redaction of the space complexity of the network.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "-_-(~ ), ,: is", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "We proposed an receptor for all context-free languages with limited center-embedding, and a suitable variant of the Ilopfield mode]. The formal model was implemented on the lloptield model, and a correct= uess theorem for the latter was given. Simulation restilts provided initial corroboration ofonr theory. The obtamed neural-network receptor is fast but large.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusions", | |
| "sec_num": "8" | |
| }, | |
| { | |
| "text": "Continuation of this research in the near fnture consists of the design of an adaptive variant of this mode[, one that learns a grammar from examples in an unsupervised fashion.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Conclusions", | |
| "sec_num": "8" | |
| } | |
| ], | |
| "back_matter": [ | |
| { | |
| "text": "1 am very grateful for the indispensable support of Mannes Poel and Anton Nijholt in making this paper. Special thanks go out to Bert llelthuis who wrote tile very beautiful simulation program with which the theory was corroborated.", | |
| "cite_spans": [], | |
| "ref_spans": [], | |
| "eq_spans": [], | |
| "section": "Acknowledgements", | |
| "sec_num": null | |
| } | |
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| "ref_entries": { | |
| "FIGREF0": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "Let M = ({*},Z,I',6,*,Zo,{~) be an e-free bounded PDA with hound k. An NDA defined for M is an NDA M' = (Q', Y2', 6', Q'o, f\") such that: Q' = QST, as defined above; E' = E; 6' : QsT x E ~ 2 osT is defined by: ~'(~l, ~)= {~ I (*,~,~)~M (,, w,~)} Q~ : {z0}; and F' = {el (empty stack)." | |
| }, | |
| "FIGREF1": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "correctness of the NDA) Let M be an e.free one-state PDA with bound k, if M' is an NDA a.~ defined in definition 2.1, then M accepts a string by empty stack tf and only if M' accepts it by accepting state." | |
| }, | |
| "FIGREF2": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "2t~ + la + 31~)/N -(l, + I~)/N. The first argument is the total time span in the network, covering two active periods and an intervening delay time, including their transition times. The second argument is tile current period of network is activity, activity that cannot directly interfere with the network's dyna[lliCS." | |
| }, | |
| "FIGREF3": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "a)N jz~=l\"\"a)(~\u00a2~' -a), with J. = 0, ~(t) ~ r O-0 L ~ Sj(t-t')w(t')dt', where { 7~-ifO<t<r w(t) = 0 otherwise , h~(t) = x~(s,'(t) -a), the external input term, The Si are neuronal variables (5\",'. is a neuron in another network), hi is the total input on Si, U is a threshold value which is equal for all Si, Jij is the synaptic efficacy of the synapse connecting S i to Si, and A is the relative magnitude of the synapses. The average at time ~ is expressed by ~/(~), where r =-(2t. + ta + 3tr)/N and ~ -~ (ta + t~)/N. The function w(t) determines over which period activity is averaged. The input synapses are nonzero only in case i = j. These synapses carry a negative ground signal -A'a, which is equivalent to an extra threshold generated by the input synapses. The activity patterns {~'} ({~\"} ~ (~],~ ..... ~N) ) are statist,tally independent, and satisfy the same probability distribution as the patterns in the model of Buhmann et al. [2]: ...... -(1-a)b(~), where l if x :~ 0 ~(x) ~ 0 otherwise. If a \u00a2 \u00bd the pattern is biased. For N -~ co, I/N ~N=I ~' --* a. The updating p .... is a Monte Carlo random walk.D II~ IFigure 1: 7'he model for N = 3. Usually HopJield models consist of very many neurons. The arced arrows denote temporal synapses. The straiyht alT'ows denote input synapses. system parameters need to be related in order to make the model work correctly.Timing; is fairly important ill this network. Tile time the network is active (to) should not exceed tile delay time t~. If it does then ta+lr > ta+tr, and since no average is computed over a period ta + tr back in time, not the fldl time average of the previous activity need to be computed, consequently we choose ta < ta." | |
| }, | |
| "FIGREF4": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "{S} ~_ ($1,$2,...,SN) r at time t with stored pattern {~P}, and is delined by: N 1 ~(~' -.)s',(t). mo(t) = ~-N~=I m\" E i-a, I -a]. The expression for the temporal transition term is: N h~(t) = ~slj~j(1). j=l Acaqis OE COLING-92, NANTES, 23-28 ao~r 1992 1 1 5 PROC. Of COLING-92, N^NTgs, AUG. 23-28, 1992" | |
| }, | |
| "FIGREF5": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "be a parameter list defined according to definition 5.1. The coding [unction c is given by: c : QxE~{0,1} N, such that for q E Q, and x E :E: 1. if 6(q, x) :/: 0: c(q,x) = {~}, where ~i is chosen at random from {0, 1} with probability distribution r'(~) = a6(~ -I) + (t -a)~(~,), a.d 2. undefined otherwise. The set of codes is then partitioned: into sets of activity patterns corresponding to NDA states, and into sets of patterns corresponding to input symbols. Let M = (Q,E,6, Qs, I\") Ire an NDA, let P = (a, M, A t, N, p, PmaJ:, In, td, tr, U) be a parameter list defined according to definition 5.1." | |
| }, | |
| "FIGREF6": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "Let M = (Q,~,6, Qo, F) bc an NDA, let P = (a, M, A t , N, p, Pma*, ta, t d, G, U) be a parameter list defned according to definition 5.1. and let a be an N-dimensional vector, with each component a. The set Tr of network transitions is: At tTr = {d(\u00a5,q,x) [ d(q',q,x) = a(1 -a)N E (c(q',y) -a)(c(q,x) --a) T A c(q',y)eP u, qt E ~(q, x)},where each jt is an N x N matrix. This suffices to define a neuralonetwork acceptor.Definition 5.5 Let M = (Q,E,6, Qo, I\") be an NDA, let P = (a, A*, A*, N, p, Pmax, to, td, tr, U) be a parameter list defined according to definition 5.1. A neural-network acceptor (NNA) defined for an NDA M that takes its parameters from P is a quadruple H = (T, f, U, S), where: 1. tim topology T, a list of neurons per layer, is: (N); 2. the activation fimction f is given by: Si = 1 if E~J/j.~j+A'(S~-a)>U (} if EjJej~J+A'('d~-a)<-U' 3. the update procedure is a Monte Carlo random walk. 4. the synaptic coefficients are given by: 'It =: E jl (q,.q.~) C Tr, and je .: A~I where I is the identity matrix. In order to construct activity patterns that can serve xs external input x for the network a component-wise OR operation is performed over the set P~ as defined in definition 5.3. Definition 5.6 7he OR operation over a set t\u00bd of activity patterns is specified by: 1. oa({~v}, {U}) = {OR(~/',~)} ; ~. oit({\u00a2},..., W'}) :~ oft.(K'}, ()lt({~q ..... {~\" })) ; ~ud a OR(P~) = OR({~ ~} ..... {C'}) if \u2022 e~ = {W} ..... {~\"}}, At last a formal definition can be given of a temporal image a.s the set of all activity patterns for which there is a network input that makes such an activity pattern the network's next qua.st-stable state. Definition 5.7 Let M = (Q,E,6, Qc, F) I)e an NDA, let P = (a, M, M, N, p, p,.,,, t,, ta, it, U) be a parameter list defined according to definition 5.1, and let H = (T, f, U,S) be an NNA defined according to definition 5.5 that takes its parameters from P. A temporal image is a set: {c(q, x)[ input OIt(P~) for H implies m \u00a2(q'~) = 1 --a}, a set Pq, is a temporal image of a quasi-stable stale {S} = c(q, x) of H if and only if J~q,,q,,:) is a transition of H." | |
| }, | |
| "FIGREF7": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "Let M : (Q,E,6, Qo, F) he an NDA, let P = (a, M, M, N, p, P,na,,, ta, t,t~ t,., U) be a parameter list defined according to definition 5.1. A list of large parameters is a parameter list P such that: 1. a =_ cl/N where cl << N is a constant; 2. ~ ~ c~N/cl, where c2 is a small constant; 3. 0 < M < A~a,i.e. 0< M < c2; 4. Pm,~\" ---{alna)-l N; 5. N ~oo; 6. t~lN ~oo." | |
| }, | |
| "FIGREF8": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "for all neurons Si in H, P(SI is selected) ~ 1 during network evolution; and 2. /or all actwity patterns {c} ~ Upu, y ~ QuZ, if l~ 7 \u00a3 \", then P(~,~ = ~ = 1) ~ O, where i=l,...,N." | |
| }, | |
| "FIGREF10": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": ". The time required to process a string of AC'rE.S DE COLING-92, NaNTEs, 23-28 Aor~r 1992 1 I 8 PgOC. OF COLING-92, NAWrES. AUG. 23-28, 1992" | |
| }, | |
| "FIGREF11": { | |
| "type_str": "figure", | |
| "num": null, | |
| "uris": null, | |
| "text": "1.-2,,i)~ In this expressi .... P ----( ..... ,)/v wh ...." | |
| }, | |
| "TABREF0": { | |
| "type_str": "table", | |
| "content": "<table><tr><td>Definition 5.1</td></tr><tr><td>Let M = (Q,\u2022,6,</td></tr></table>", | |
| "num": null, | |
| "text": "Qo, F) be an NDA. A parameter list defined .for an NDA M is a list of the form", | |
| "html": null | |
| } | |
| } | |
| } | |
| } |