Vivid logic. Knowledge-based reasoning with two kinds of negation
The book aims at defining a cognitively adequate and computationally feasible model for commonsense reasoning. It proposes a concept of `vivid' knowledge representation and reasoning system (VKRS), extending computational systems like databases and logic programs. VKRSs are characterized by the presence of two kinds of negation, that respectively model default-implicit falsity and constructive-explicit falsity. Rather than developing yet another KRS, the author identifies some minimal requirements any VKRS must satisfy. The idea of vividness as a design principle for KRSs has originally been proposed by Levesque. Levesque defines vivid knowledge as complete knowledge about entities and relationships of interest in the world, that can be decomposed into atomic knowledge. He identifies existential, universal, disjunctive, and negated sentences as sources of incompleteness, and, consequently, he allows in a vivid knowledge base (VKB) only ground atoms. He requires VKBs to be consistent and complete, satisfying the domain closure axiom, the closed-world assumption and the usual equality axioms. In the book, such a notion of VKB is refined by (i) allowing the universe of discourse to dynamically change (no domain closure axiom), (ii) adding a notion of strong negation \(\sim\) to codify explicit negative information, thus making positive and negative statements perfectly symmetric, (iii) restricting the closed-world assumption to predicates which are totally represented in the knowledge base, and (iv) tolerating inconsistency. Accordingly, a VKRS has to satisfy the properties of restricted reflexivity, constructivity, and non-explosiveness. The first property restricts reflexivity to consistent formulas. The second property constrains the inference relation to satisfy the conditions of constructible truth (a disjunction can only by true if one of its disjuncts is) and constructible falsity (a conjunction can only be false if one of its conjuncts is). The first condition prevents one to conclude, from an arbitrary sentence \(F\), that, for any sentence \(G\), \(G \vee \sim G\) holds; the second one excludes, for instance, the principle of contradiction \(\sim (G \wedge \sim G)\). The third property invalidates the principle `ex contradictione sequitur quodlibet' requiring the commonality of contents between premises and conclusions of inferences. It prevents one to conclude, from a pair of contradictory sentences \(F \wedge \sim F\), that any sentence \(G\) holds. The author also provides a logical account of VKRSs. He identifies two main sources of intractability in the entailment problem: the formula structure and the reasoning structure. The first problem can be reduced by normalization; the second one by using sufficiently weak logics. In this respect, the logic of VKRSs (vivid logic) can be viewed as a generalization of Belnap's 4-valued logic and Nelson's paraconsistent constructive logic. Belnap's 4-valued logic deals with partial and possibly incomplete information. Nelson's paraconsistent logic conservatively extends it by adding a notion of constructive implication. Vivid logic further extends Nelson's paraconsistent constructive logic by adding weak negation. In the second part of the book, the author considers in detail logic programs and knowledge-based systems. He first shows how positive logic programs can be extended by adding a notion of strong negation. Such a notion allows one to distinguish between exact predicates, satisfying the classical principle `tertium non datur', and inexact predicates, which have truth value gaps, and are useful to reason about empirical domains. He also investigates the problem of providing logic programs with a Lindenbaum-algebraic semantics. He shows that the algebraic structure underlying positive logic programs is a distributive lattice with a greatest element. In case of positive logic programs extended with strong negation, he only gives a negative result showing that a distributive lattice endowed with a unary operation satisfying the law of double negation and the DeMorgan rules (DeMorgan algebra) is, in general, not appropriate. With regard to knowledge-based systems, facts and rules of VKRSs are not restricted to simple facts and definite clauses anymore. In the general case, the knowledge base includes both positive and negative facts; moreover, the premises of inference rules may involve both kinds of negation, and their conclusions can be either positive atoms or strongly negated atoms. Finally, in order to deal with looping problems, the author defines the concepts of wellfounded and weakly wellfounded knowledge base, corresponding to the notions of acyclic and locally stratified logic programs, and defines a loop-tolerant recursive proof- theory for such knowledge bases.
`vivid' knowledge representation and reasoning system
scientific article; zbMATH DE number 514413