Merging Connectionism and Logicism in Knowledge Representation

2018

The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the ”global” aspects of categories of logics in the vein of the categories Ss,Ls,As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the ”identity problem” for logics ([Bez]): for instance, the presentations of ”classical logics” (e.g., in the signature {¬,∨} and {¬,→}) are not Ls-isomorphic. In this work, we sketch a possible way to overcome this ”defect” (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) ”Morita equivalence” of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent a...

Graph Transformations, 2004

We show how edge-labelled graphs can be used to represent first-order logic formulae. This gives rise to recursively nested structures, in which each level of nesting corresponds to the negation of a set of existentials. The model is a direct generalisation of the negative application conditions used in graph rewriting, which count a single level of nesting and are thereby shown to correspond to the fragment ∃¬∃ of first-order logic. Vice versa, this generalisation may be used to strengthen the notion of application conditions. We then proceed to show how these nested models may be flattened to (sets of) plain graphs, by allowing some structure on the labels. The resulting formulae-as-graphs may form the basis of a unification of the theories of graph transformation and predicate transformation.

A method to represent first-order predicate logic (Horn clause logic) by a data-flow network is presented. Like a data-flow computer for a von Neumann program, the proposed network explicitly represents the logical structure of a declarative program by unlabeled edges and operation nodes. In the deduction, the network first propagates symbolic tokens to create an expanded AND/OR network by the backward deduction, and then executes unification by a newly developed method to solve simultaneous equations buried in the network. The paper argues the soundness and completeness of the network in a conventional way, then explains how a kind of ambiguous solution is obtained by the newly developed method. Numerical experiments are also conducted with some data-flow networks, and the method's convergence ability and scaling property to larger problems are investigated.

Foundations of Artificial Intelligence, 2008

Mathematical logicians had developed the art of formalizing declarative knowledge long before the advent of the computer age. But they were interested primarily in formalizing mathematics. Because of the important role of nonmathematical knowledge in AI, their emphasis was too narrow from the perspective of knowledge representation, their formal languages were not sufficiently expressive. On the other hand, most logicians were not concerned about the possibility of automated reasoning; from the perspective of knowledge representation, they were often too generous in the choice of syntactic constructs. In spite of these differences, classical mathematical logic has exerted significant influence on knowledge representation research, and it is appropriate to begin this handbook with a discussion of the relationship between these fields.

2008

It is well known that interleaving presentations is at the heart of fibring, as shown by the mechanism of fibring languages and deduction systems. This idea is abstractly introduced herein at the level of the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness results are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics and subsume all logics endowed with an algebraic semantics.

When we represent logical, connective implications by directed edges, the resulting set of directed edges can be regarded as a complex network. In this article, we compose a network model that represents a deductive-logic-like structure composed solely of implications. The proposed network model grows like the BA model reported by Barabási and Albert [Science 286, 509 (1999)]. Though the BA model references the whole of the existing network when a node is added, our model references only part of the existing network. In this view, our model is more realistic than the BA model. However, it also exhibits power law characteristics.

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an m-graph where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is an m-graph where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics, and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided.

2000

2000

A graph-theoretic account of bring of logics is developed, capital- izing on the interleaving characteristics of bring. Signatures, interpre- tation structures and deductive systems are dened as enriched graphs. This graph-theoretic view is general enough to accommodate very dier- ent propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial seman- tics, substructural logics,

This work is an attempt to lay foundations for a theory of interactive computation and bring logic and theory of computing closer together. It semantically introduces a logic of computability and sets a program for studying various aspects of that logic. The intuitive notion of (interactive) computational problems is formalized as a certain new, procedural-rule-free sort of games (called static games) between the machine and the environment, and computability is understood as existence of an interactive Turing machine that wins the game against any possible environment. The formalism used as a speciÿcation language for computational problems, called the universal language, is a non-disjoint union of the formalisms of classical, intuitionistic and linear logics, with logical operators interpreted as certain-most basic and natural-operations on problems. Validity of a formula is understood as being "always computable", and the set of all valid formulas is called the universal logic. The name "universal" is related to the potential of this logic to integrate, on the basis of one semantics, classical, intuitionistic and linear logics, with their seemingly unrelated or even antagonistic philosophies. In particular, the classical notion of truth turns out to be nothing but computability restricted to the formulas of the classical fragment of the universal language, which makes classical logic a natural syntactic fragment of the universal logic. The same appears to be the case for intuitionistic and linear logics (understood in a broad sense and not necessarily identiÿed with the particular known axiomatic systems). Unlike classical logic, these two do not have a good concept of truth, and the notion of computability restricted to the corresponding two fragments of the universal language, based on the intuitions that it formalizes, can well qualify as "intuitionistic truth" and "linear-logic truth". The paper also provides illustrations of potential applications of the universal logic in knowledgebase, resourcebase and planning systems, as well as constructive applied theories. The author has tried to make this article easy to read. It is fully self-contained and can