## Logic Programs and Connectionist Networks (2004)

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Venue: | Journal of Applied Logic |

Citations: | 44 - 15 self |

### BibTeX

@ARTICLE{Hitzler04logicprograms,

author = {Pascal Hitzler and Steffen Hölldobler and Anthony Karel Seda},

title = {Logic Programs and Connectionist Networks},

journal = {Journal of Applied Logic},

year = {2004},

volume = {2},

pages = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for first-order normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a wellknown theorem of Funahashi, and necessitate the study of when Funahasi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.

### Citations

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Citation Context ... technical details here and refer to the abovementioned literature. Such a move renders the kernels accessible to the backpropagation algorithm, a standard technique for training feedforward networks =-=[RHW86]-=-. Rule Extraction After training a feedforward network with sigmoidal units in the hidden layer, the knowledge encoded in the network is mostly inaccessible to a human without postprocessing. Numerous... |

1855 |
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Citation Context ...rograms, but their occurrence is awkward from the point of view of denotational semantics, especially if they occur in negated body literals since this leads to the so-called floundering problem, see =-=[Llo88]-=-. It is easy to see that, in the context of Herbrand-interpretations, and if function symbols are present, then the absence of local variables is equivalent to a program being of finite type. 4.6 Prop... |

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Citation Context ... the details. Similar considerations apply to the operator Ψ on Belnap’s four-valued logic [Fit02] and to the operators from [HS99]. We mention in passing the non-monotonic Gelfond-Lifschitz operat=-=or [GL88]-=- in classical two-valued logic, whose fixed points yield the stable models of the program in question. It turns out that this operator is not a consequence operator in the sense discussed in this pape... |

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Citation Context ...input-output functions of 3-layer feedforward networks. � The restriction to programs with continuous consequence operator is not entirely satisfactory. There is another approximation theorem, due t=-=o [HSW89], w-=-hich requires only measurability of the functions in question. 4.16 Theorem Suppose that φ is a monotone increasing function from R onto (0, 1) . Let f : R r → R be a Borel-measurable function and ... |

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Citation Context ...re its meaning. It is an important fact that the models just mentioned are fixed points of various operators determined by programs. In particular, the supported models, or Clark completion semantics =-=[Cla78]-=-, of a normal logic program P coincide with the fixed points of the immediate consequence operator TP . Furthermore, the fixed points themselves are frequently found by iterating the corresponding ope... |

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Citation Context ...nother natural question concerns carrying over the programme given here for the supported model semantics of a normal logic program to the stable model semantics [GL88] and the well-founded semantics =-=[vGRS91]-=-, and one possible means of doing this is provided by the results of [Wen02]. From the connectionist point of view, the main open question is how to build a connectionist network given a first-order l... |

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Citation Context ...gies such as Q comes from our desire to be able to control the iterative behaviour of semantic operators. Topologies which are closely related to order structures, as common in denotational semantics =-=[AJ94]-=-, are of limited applicability since non-monotonic operators frequently arise naturally in the logic programming context. See also [Hit01] for a study of these issues. We proceed next with studying a ... |

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Citation Context ...h the semantic operators themselves and also their fixed 2spoints, at least if the feedforward networks are turned into recurrent ones. Our methods here are based on a well-known theorem of Funahashi =-=[Fun89] w-=-hich shows that every continuous function on the reals can be uniformly approximated by a 3-layer feedforward neural network. However, application of Funahashi’s theorem depends on TP itself being c... |

311 |
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Citation Context ...ple, consider the definite program P4 = {A1} ∪ {Ai+1 ← Ai | 1 ≤ i < n}. The least fixed point of TP is the interpretation which evaluates each Ai , 1 ≤ i ≤ n , to t . Using the technique des=-=cribed in [DG84]-=- and [Scu90], it can be computed in O(n) steps. 5 Obviously, the parallel computational model needs as many steps. More generally, let P be a definite program containing n clauses. The time needed by ... |

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Citation Context ... in the following subsection is whether or not even simpler networks, viz. recurrent networks with a 2-layer feedforward kernel of binary threshold units will do. Such networks are called perceptrons =-=[Ros62]-=-. It is well-known that their computing capabilities are limited to computing solutions for linearly separable problems [MP72]. 3.1 Hidden Layers are Needed Usually, the need for a hidden layer is sho... |

234 |
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Citation Context ...e knowledge encoded in the network is mostly inaccessible to a human without postprocessing. Numerous techniques have been proposed to extract rules from trained feedforward networks (see for example =-=[ADT95]-=- and [dGBG01]). We can now envision a cycle in which a given (preliminary) logic program is translated into a feedforward network, this network is trained by examples using backpropagation, and a new ... |

195 | Physical Symbol Systems - Newell - 1980 |

150 | Knowledge-based artificial neural networks
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Citation Context ... logic program is translated into a feedforward network, this network is trained by examples using backpropagation, and a new (refined) logic program is extracted from the network after training (see =-=[TS94]-=-). The reference [dGBG02] contains several examples of such cyclic knowledge processing. Propositional Modal Logics The approach discussed so far has been extended to (propositional) modal programs, w... |

126 | Reasoning about termination of pure prolog programs
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Citation Context ...acyclic programs next since we will need this notion in subsequent discussions. To do this, we need first to recall the notion of level mapping, familiar in the context of studies of termination, see =-=[AP93] for ex-=-ample. A level mapping for a program P is a mapping l : BP → α for some ordinal α . As usual, we always assume that l has been extended to all literals by setting l(¬A) = l(A) for each A ∈ BP .... |

108 | Holographic reduced representations
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Citation Context ...ber of first-order clauses, but cannot copy clauses on demand and, thus, the entailment relation is decidable. Connectionist mechanisms for representing terms like holographic reduced representations =-=[Pla91]-=- or recursive auto-associative memories [Pol88] and variations thereof can handle some recursive structures, but as soon as the depth of the represented terms increases, the performance of these metho... |

105 | Fixpoint semantics for logic programming — A survey
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(Show Context)
Citation Context ...ise naturally in the logic programming context. See also [Hit01] for a study of these issues. We proceed next with studying a rather general notion of semantic operator, akin to Fitting’s approach i=-=n [Fit02], -=-which generalizes standard notions occurring in the literature. 18s4.3 Definition An operator T on IP is called a consequence operator for P if for every I ∈ IP the following condition holds: for ev... |

82 |
General Topology
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Citation Context ...al, we have f(x0) = x0 . One of the main results concerning contraction mappings defined on complete metric spaces is the following well-known theorem. 2.1 Theorem (Banach Contraction Mapping Theorem =-=[Wil70]) Le-=-t f be a contraction mapping defined on a complete metric space (X, d) . Then f has a unique fixed point x0 ∈ X . Furthermore, the sequence x, f(x), f(f(x)), . . . converges to x0 for any x ∈ X . ... |

60 | Metric methods: Three examples and a theorem
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(Show Context)
Citation Context ...h that there exists a metric which renders TP a contraction, then Theorem 2.1 shows that P has a unique supported model. Semantic analysis of logic programs along these general lines was initiated in =-=[Fit94]-=-, and has subsequently been studied and generalized by a number of authors. The recent publication [HS03b] contains both a state-of-the-art treatment using this approach and a comprehensive list of re... |

60 |
Recursive auto-associative memory: Devising compositional distributed representations
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Citation Context ...uses on demand and, thus, the entailment relation is decidable. Connectionist mechanisms for representing terms like holographic reduced representations [Pla91] or recursive auto-associative memories =-=[Pol88]-=- and variations thereof can handle some recursive structures, but as soon as the depth of the represented terms increases, the performance of these methods degrades quickly [McI00]. Furthermore, whils... |

55 | Approximating the semantics of logic programs by recurrent neural networks
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- 1999
(Show Context)
Citation Context ...l units are real numbers, and we will use real numbers to represent interpretations. In Figure 6, the recurrent nets considered in this section are sketched. This section extends results published in =-=[HKS99]-=- and therefore we review the previous work in the following subsection. 16s4.1 Previous Work The reference [HKS99] was concerned with the following problem. Suppose we are given a first-order logic pr... |

47 | Symbolic knowledge extraction from trained neural networks: A sound approach
- Garcez, Broda, et al.
(Show Context)
Citation Context ...encoded in the network is mostly inaccessible to a human without postprocessing. Numerous techniques have been proposed to extract rules from trained feedforward networks (see for example [ADT95] and =-=[dGBG01]-=-). We can now envision a cycle in which a given (preliminary) logic program is translated into a feedforward network, this network is trained by examples using backpropagation, and a new (refined) log... |

46 | Characterizing termination of logic programs with level mapping - Bezem - 1992 |

42 |
Towards a massively parallel computational model for logic programming
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(Show Context)
Citation Context ...rks ∗ To appear in the Journal of Applied Logic, Special Edition on Neural-Symbolic Systems. This is a revised and extended treatment of results which to date have appeared only in the workshop pape=-=r [HK94]-=- and the conference papers [HS00, HS03a]. 1sare quite complementary. For example, there is a widespread belief that the ability to represent and reason about structured objects and structure-sensitive... |

41 | Quasi-metrics and the semantics of logic programs
- Seda
- 1997
(Show Context)
Citation Context ...e results of Section 3 to the first-order case by means of approximation. This involves a fairly detailed study of the (topological) continuity of semantic operators, extending results to be found in =-=[Sed95], -=-before we can ultimately take up the question of applying results such as Funahashi’s theorem and discussing measures of approximation appropriate to the study of neural networks. Finally, in Sectio... |

34 |
The Elements of Integration
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Citation Context ...given conditions that BA is finite for all A ∈ BP , which implies that T is locally finite. � We next take a short detour from our discussion of continuity to study the weaker notion of measurabil=-=ity [Bar66] for -=-consequence operators. For a collection M of subsets of a set X , we denote by σ(M) the smallest σ -algebra containing M , called the σ -algebra generated by M . Recall that a function f : X → X ... |

31 |
and Vijaya Ramachandran, “Parallel algorithms for shared-memory machines
- Karp
- 1990
(Show Context)
Citation Context ...ing in the program. This comes as no surprise as it follows from [JL77] that satisfiability of propositional Horn formulae is P -complete and, thus, is unlikely to be in the class NC (see for example =-=[KR90]). On the ot-=-her hand, consider the program P5 = {Ai | 1 ≤ i ≤ n and i even} ∪ {Ai+1 ← Ai | 1 ≤ i ≤ n and i even}. 3 A parallel computational model requiring p(n) processors and t(n) time to solve a pr... |

31 |
A Kripke-Kleene-semantics for general logic programs
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(Show Context)
Citation Context ...s in the body of C . In the case of Kleene’s strong three-valued logic, with set of truth values T = {t, u, f} and logical connectives as in Table 1, the associated Fitting operator was introduced in =-=[Fit85]-=- and is denoted by ΦP , for a given program P . As in the case of classical two-valued logic, we obtain the following lemma. 4.14 Lemma If ΦP (I)(A) = t , then ΦP is locally finite for A and I . Furth... |

27 | Generalized metrics and uniquely determined logic programs
- Hitzler, Seda
(Show Context)
Citation Context ...pported model. Semantic analysis of logic programs along these general lines was initiated in [Fit94], and has subsequently been studied and generalized by a number of authors. The recent publication =-=[HS03b]-=- contains both a state-of-the-art treatment using this approach and a comprehensive list of references on this topic. The following definition will be very convenient for our purposes. 2.2 Definition ... |

23 |
Generalized Metrics and Topology in Logic Programming Semantics
- Hitzler
- 2001
(Show Context)
Citation Context ...lated to order structures, as common in denotational semantics [AJ94], are of limited applicability since non-monotonic operators frequently arise naturally in the logic programming context. See also =-=[Hit01] f-=-or a study of these issues. We proceed next with studying a rather general notion of semantic operator, akin to Fitting’s approach in [Fit02], which generalizes standard notions occurring in the lit... |

22 | Characterizations of classes of programs by three-valued operators
- Hitzler, Seda
- 1999
(Show Context)
Citation Context ... analogous to Theorem 4.13 is now straightforward, but tedious, and we omit the details. Similar considerations apply to the operator Ψ on Belnap’s four-valued logic [Fit02] and to the operators fr=-=om [HS99]-=-. We mention in passing the non-monotonic Gelfond-Lifschitz operator [GL88] in classical two-valued logic, whose fixed points yield the stable models of the program in question. It turns out that this... |

21 |
A Kripke/Kleene semantics for general logic programs
- Fitting
- 1985
(Show Context)
Citation Context ... three-valued logic. In the case of Kleene’s strong three-valued logic, with set of truth values T = {t, u, f} and logical connectives as in Table 1, the associated Fitting operator was introduced i=-=n [Fit85] an-=-d is denoted by ΦP , for a given program P . As in the case of classical two-valued logic, we obtain the following lemma. 4.14 Lemma If ΦP (I)(A) = t , then ΦP is locally finite for A and I . Furth... |

19 | Automated inferencing and connectionist models
- Hölldobler
- 1993
(Show Context)
Citation Context ...f symbolic computation. Systems like SHRUTI [SA93] or the BUR-calculus [HKW00] allow n -place predicate symbols and a finite set of constants and, thus, are propositional in nature. Systems like CHCL =-=[Höl93]-=- allow a fixed number of first-order clauses, but cannot copy clauses on demand and, thus, the entailment relation is decidable. Connectionist mechanisms for representing terms like holographic reduce... |

18 |
A note on Dowling and Gallier’s top-down algorithm for propositional Horn Satisfiability
- Scutella
- 1990
(Show Context)
Citation Context ...er the definite program P4 = {A1} ∪ {Ai+1 ← Ai | 1 ≤ i < n}. The least fixed point of TP is the interpretation which evaluates each Ai , 1 ≤ i ≤ n , to t . Using the technique described in [=-=DG84] and [Scu90]-=-, it can be computed in O(n) steps. 5 Obviously, the parallel computational model needs as many steps. More generally, let P be a definite program containing n clauses. The time needed by the network ... |

17 | Acyclic programs and the completeness of SLDNF-resolution - Cavedon - 1991 |

15 | Logic programs, iterated function systems, and recurrent radial basis function networks
- Bader, Hitzler
(Show Context)
Citation Context ...fly mentioning a few of them, as follows. First, there is the question of giving explicit constructions of networks for approximating TP in case that TP is continuous, and this point is considered in =-=[BH03]-=-. A question which is also related to the results given in [BH03] is that of providing good bounds on Lipschitz constants for fP , and this issue appears to be central to actually giving constructions... |

12 | Miroslaw Truszczynski. Approximating operators, stable operators, well-founded fixpoints and applications in non-monotonic reasoning - Denecker, Marek - 2000 |

11 | A connectionist inductive learning system for modal logic programming
- Garcez, S, et al.
- 2002
(Show Context)
Citation Context ... programs, where literals occurring in a clause may be prefixed by the modalities � and ♦ , clauses are labelled by the world in which they hold, and a finite set of relations between worlds is gi=-=ven [dGLG02]-=-. It was shown that Theorem 3.2 can be extended to such modal programs in that for each such program there exists a 3-layer connectionist network computing the modal fixed point operator of the given ... |

10 |
Topological model set deformations in logic programming
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(Show Context)
Citation Context ...endowed with the discrete topology. We note that this topology can be defined analogously for the non-Herbrand case. For n = 2 , the generalized atomic topology Q specializes to the query topology of =-=[BS89]-=- (in the Herbrand case) and to the atomic topology Q of [Sed95] (in the non-Herbrand case). The following results follow immediately since Q is a product of the discrete topology on a finite set, and ... |

10 | A note on relationships between logic programs and neural networks - Hitzler, Seda - 2000 |

10 |
Epistemological challenges for connectionism
- McCarthy
- 1988
(Show Context)
Citation Context ...bstacles to be overcome. For example, from the computational point of view, most connectionist systems developed so far are propositional in nature. John McCarthy called this a propositional fixation =-=[McC88]-=- in 1988, and not much has changed since then. Although it is known that connectionist systems are Turing-equivalent, we are unaware of any connectionist reasoning system which fully incorporates the ... |

8 | d’Avila Garcez and Gerson Zaverucha. The connectionist inductive lerarning and logic programming system - Artur - 1999 |

8 | Unfolding the well-founded semantics
- Wendt
- 2002
(Show Context)
Citation Context ...s out that this operator is not a consequence operator in the sense discussed in this paper, and attempts to characterize continuity of it will involve different methods (by means of the results from =-=[Wen02]-=-, for example). 4.3 Approximation by Artificial Neural Networks We have now finished our general preparations and continue next with our main task, namely, the study of the representability of logic p... |

7 |
de Carvalho. Logical inference and inductive learning in artificial neural networks
- Garcez, Zaverucha, et al.
- 1997
(Show Context)
Citation Context ... logic programs and constructed by the translation algorithm presented in the proof of Theorem 3.2 cannot be trained by the usual learning methods applied to connectionist systems. It was observed in =-=[dGZdC97]-=- (see also [dGZ99, dGBG02]) that results similar to Theorem 3.2 and Corollary 3.3 can be achieved if the binary threshold units occurring in the hidden layer of the feedforward kernels are replaced by... |

6 | A recursive neural network for reflexive reasoning
- Holldobler, Kalinke, et al.
- 1999
(Show Context)
Citation Context ...ctionist systems are Turing-equivalent, we are unaware of any connectionist reasoning system which fully incorporates the power of symbolic computation. Systems like SHRUTI [SA93] or the BUR-calculus =-=[HKW00] -=-allow n -place predicate symbols and a finite set of constants and, thus, are propositional in nature. Systems like CHCL [Höl93] allow a fixed number of first-order clauses, but cannot copy clauses o... |

4 | Continuity of semantic operators in logic programming and their approximation by artificial neural networks - Hitzler, Seda - 2003 |

4 |
Venkat Ajjanagadde. From associations to systematic reasoning: A connectionist representation of rules, variables and dynamic bindings using temporal synchrony
- Shastri
- 1993
(Show Context)
Citation Context ...ough it is known that connectionist systems are Turing-equivalent, we are unaware of any connectionist reasoning system which fully incorporates the power of symbolic computation. Systems like SHRUTI =-=[SA93] -=-or the BUR-calculus [HKW00] allow n -place predicate symbols and a finite set of constants and, thus, are propositional in nature. Systems like CHCL [Höl93] allow a fixed number of first-order clause... |

2 |
Complete problems for deterministic sequential time
- Jones, Laaser
- 1977
(Show Context)
Citation Context ...le down into the unique stable state is 3n in the worst case and, thus, the time is linear with respect to the number of clauses occurring in the program. This comes as no surprise as it follows from =-=[JL77] tha-=-t satisfiability of propositional Horn formulae is P -complete and, thus, is unlikely to be in the class NC (see for example [KR90]). On the other hand, consider the program P5 = {Ai | 1 ≤ i ≤ n a... |