Krysia Broda

Symbolic knowledge extraction from trained neural networks: a sound approach

Although neural networks have shown very good performance in many application domains, one of their main drawbacks lies in the incapacity to provide an explanation for the underlying reasoning mechanisms. The “explanation capability” of neural networks can be achieved by the extraction of symbolic knowledge. In this paper, we present a new method of extraction that captures nonmonotonic rules encoded in the network, and prove that such a method is sound. We start by discussing some of the main problems of knowledge extraction methods. We then discuss how these problems may be ameliorated. To this end, a partial ordering on the set of input vectors of a network is defined, as well as a number of pruning and simplification rules. The pruning rules are then used to reduce the search space of the extraction algorithm during a pedagogical extraction, whereas the simplification rules are used to reduce the size of the extracted set of rules. We show that, in the case of regular networks, the extraction algorithm is sound and complete. We proceed to extend the extraction algorithm to the class of non-regular networks, the general case. We show that non-regular networks always contain regularities in their subnetworks. As a result, the underlying extraction method for regular networks can be applied, but now in a decompositional fashion. In order to combine the sets of rules extracted from each subnetwork into the final set of rules, we use a method whereby we are able to keep the soundness of the extraction algorithm. Finally, we present the results of an empirical analysis of the extraction system, using traditional examples and real-world application problems. The results have shown that a very high fidelity between the extracted set of rules and the network can be achieved.

Hybrid Abductive Inductive Learning: A Generalisation of Progol

The learning system Progol5 and the underlying inference method of Bottom Generalisation are firmly established within Inductive Logic Programming (ILP). But despite their success, it is known that Bottom Generalisation, and therefore Progol5, are restricted to finding hypotheses that lie within the semantics of Plotkin’s relative subsumption. This paper exposes a previously unknown incompleteness of Progol5 with respect to Bottom Generalisation, and proposes a new approach, called Hybrid Abductive Inductive Learning, that integrates the ILP principles of Progol5 with Abductive Logic Programming (ALP). A proof procedure is proposed, called HAIL, that not only overcomes this newly discovered incompleteness, but further generalises Progol5 by computing multiple clauses in response to a single seed example and deriving hypotheses outside Plotkin’s relative subsumption. A semantics is presented, called Kernel Generalisation, which extends that of Bottom Generalisation and includes the hypotheses constructed by HAIL.

APPLYING CONNECTIONIST MODAL LOGICS TO DISTRIBUTED KNOWLEDGE REPRESENTATION PROBLEMS

Neural-Symbolic Systems concern the integration of the symbolic and connectionist paradigms of Artificial Intelligence. Distributed knowledge representation is traditionally seen under a symbolic perspective. In this paper, we show how neural networks can represent distributed symbolic knowledge, acting as multi-agent systems with learning capability (a key feature of neural networks). We apply the framework of Connectionist Modal Logics to well-known testbeds for distributed knowledge representation formalisms, namely the muddy children and the wise men puzzles. Finally, we sketch a full solution to these problems by extending our approach to deal with knowledge evolution over time.

Tableau Methods for Substructural Logics

Over the last few decades a good deal of research in logic has been prompted by the realization that logical systems can be successfully employed to formalize and solve a variety of computational problems. Traditionally, the theoretical framework for most applications was assumed to be classical logic. However, this assumption often turned out to clash with researchers’ intuitions even in well-established areas of application. Let us consider, for example, what is probably the most representative of these application areas: logic programming The idea that the execution of a Prolog program is to be understood as a derivation in classical logic has played a key role in the development of the area. This interpretation is the leitmotiv of Kowalski’s well-known [1979], whose aim is described as an attempt ‘to apply the traditional methods of [classical] logic to contemporary theories of problem solving and computer programming’. However, here are some quotations which are clearly in conflict with the received view (and with each other) as to the correct interpretation of logic programs: Relevance Logic not only shows promise as a standard for modular reasoning systems, but it has, in a sense, been already adopted by artificial intelligence researchers. The resolution method for Horn clauses appears to be based on classical logic, but procedural derivation (see [Kowalski, 1979]), the method actually used for logic programming, is not complete for classical logic, and is in fact equivalent to Relevance Logic.[...] the systems of modules which are actually used in computer science have the feature that relevance, not classical logic, provides a theory of their behaviour. [Carson, 1989, p.214] According to the standard view, a logic program is a definite set of Horn clauses. Thus logic programs are regarded as syntactically restricted first-order theories within the framework of classical logic. Correspondingly, the proof-theory of logic programs is considered as a specialized version of classical resolution, known as SLD-resolution. This view, however, neglects the fact that a program clause a 0 ← a l, a 2, ..., a n , is an expression of a fragment of positive logic [a subsystem of Intuitionistic Logic] rather than an implication formula of classical logic. The logical behaviour of such clauses is in no way related to any negation or complement operation. So (positive) logic programs are ‘sub-classical’. The classical interpretation seems to be a semantical overkill’ [Wagner, 1991, p.835].

A Hybrid Abductive Inductive Proof Procedure

This paper introduces a proof procedure that integrates Abductive Logic Programming (ALP) and Inductive Logic Programming (ILP) to automate the learning of first order Horn clause theories from examples and background knowledge. The work builds upon a recent approach called Hybrid Abductive Inductive Learning (HAIL) by showing how language bias can be practically and usefully incorporated into the learning process. A proof procedure for HAIL is proposed that utilises a set of user specified mode declarations to learn hypotheses that satisfy a given language bias. A semantics is presented that accurately characterises the intended hypothesis space and includes the hypotheses derivable by the proof procedure. An implementation is described that combines an extension of the Kakas-Mancarella ALP procedure within an ILP procedure that generalises the Progol system of Muggleton. The explicit integration of abduction and induction is shown to allow the derivation of multiple clause hypotheses in response to a single seed example and to enable the inference of missing type information in a way not previously possible.

Transformation Methods in LDS

The methodology of Labelled Deductive Systems — or simply LDS1 — is a unifying framework for the study of logics and their interactions. It was proposed by Dov Gabbay a few years ago in response to conceptual pressure arising from application areas, and has now become a large and influential research programme providing logicians, both pure and applied, with a common language and a common set of basic principles in which to express and to solve their problems.

Generalised Kernel Sets for Inverse Entailment

The task of inverting logical entailment is of central importance to the disciplines of Abductive and Inductive Logic Programming (ALP & ILP). Bottom Generalisation (BG) is a widely applied approach for Inverse Entailment (IE), but is limited to deriving single clauses from a hypothesis space restricted by Plotkin’s notion of C-derivation. Moreover, known practical applications of BG are confined to Horn clause logic. Recently, a hybrid ALP-ILP proof procedure, called HAIL, was shown to generalise existing BG techniques by deriving multiple clauses in response to a single example, and constructing hypotheses outside the semantics of BG. The HAIL proof procedure is based on a new semantics, called Kernel Set Subsumption (KSS), which was shown to be a sound generalisation of BG. But so far KSS is defined only for Horn clauses. This paper extends the semantics of KSS from Horn clause logic to general clausal logic, where it is shown to remain a sound extension of BG. A generalisation of the C-derivation, called a K*-derivation, is introduced and shown to provide a sound and complete characterisation of KSS. Finally, the K*-derivation is used to provide a systematic comparison of existing proof procedures based on IE.

Policy conflict analysis using tableaux for on demand VPN framework

The medical field has a requirement for ubiquitous computing with secure and reliable access control to permit patient information to be logged as they go about their normal activities or to permit medics to access patient information remotely from various mobile devices. Healthcare involves many different people from multiple organizations - general practitioner, hospital doctor or nurse, social workers - who all need different information. Defining the required authorization policies can be very complex, resulting in conflicts, which could result in information leaks, with privacy implications, or prevent access to information needed. We propose an approach for detecting conflicts defined in an authorization policy by using free variable tableaux. Our method enables us not only to detect a conflicting policy statically, but also to obtain information that would be helpful to correct the policy by using abductive inference.

Abstract policy evaluation for reactive agents

This paper describes a method for constructing and evaluating teleo-reactive policies for one or more agents, based upon discounted-reward evaluation of policy-restricted subgraphs of complete situation-graphs. The combinatorial burden that would potentially ensue from state-perception associations can be ameliorated by suitable use of abstractions and empirical simulation results indicate that the method affords a good degree of scalability and predictive power. The paper formally analyses the predictive quality of two different abstractions, one for applications involving several agents and one for applications with large numbers of perceptions. Sufficient conditions for reasonable predictive quality are given.

A Decidable CLDS for Some Propositional Resource Logics

The compilation approach for Labelled Deductive Systems (CLDS) is a general logical framework. Previously, it has been applied to various resource logics within natural deduction, tableaux and clausal systems, and in the latter case to yield a decidable (first order) CLDS for propositional Intuitionistic Logic (IL). In this paper the same clausal approach is used to obtain a decidable theorem prover for the implication fragments of propositional substructural Linear Logic (LL) and Relevance Logic (RL). The CLDS refutation method is based around a semantic approach using a translation technique utilising first-order logic together with a simple theorem prover for the translated theory using techniques drawn from Model Generation procedures. The resulting system is shown to correspond to a standard LL(RL) presentation as given by appropriate Hilbert axiom systems and to be decidable.