We introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers # new and # new which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predic...

Problems in logic are well-known to be hard to solve in the worst case. Two different strategies for dealing with this aspect are known from the literature: language restriction and theory approximation. In this paper we are concerned with the second strategy. Our main goal is to define a semantically well-founded logic for approximate reasoning, which is justifiable from the intuitive point of view, and to provide fast algorithms for dealing with it even when using expressive languages. We also want our logic to be useful to perform approximate reasoning in different contexts. We define a method for the approximation of decision reasoning problems based on multivalued logics. Our work expands and generalizes in several directions ideas presented by other researchers. The major features of our technique are: 1) approximate answers give semantically clear information about the problem at hand; 2) approximate answers are easier to compute than answers to the original problem; 3) approxim...

One way to think about STRIPS is as a mapping from databases to databases, in the following sense: Suppose we want to know what the world would be like if an action, represented by the STRIPS operator ff, were done in some world, represented by the STRIPS database D 0 . To find out, simply perform the operator ff on D 0 (by applying ff's elementary add and delete revision operators to D 0 ). We describe this process as progressing the database D 0 in response to the action ff. In this paper, we consider the general problem of progressing an initial database in response to a given sequence of actions. We appeal to the situation calculus and an axiomatization of actions which addresses the frame problem (Reiter [21]). This setting is considerably more general than STRIPS. Our results concerning progression are mixed. The (surprising) bad news is that, in general, to characterize a progressed database we must appeal to second order logic. The good news is that there are many useful spec...

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The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cut-free systems are anomalous from the proof-theoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the "computational scandal" that such systems cannot polynomially simulate the truth-tables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being "close" to Smullyan's tableau method, is not cut-free but includes a class...

An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms. 2 G. J. Chaitin Key Words and Phrases: complexity of sets, computational complexity, difficulty of theoremproving, entropy of sets, formal systems, Godel's incompleteness theorem, halting problem, information content of sets, information content of axioms, information theory, information time trade-offs, metamathematics, random strings, recursive functions, recursively enumerable sets, size of proofs, universal computers CR Categories: 5.21, 5.25, 5.27, 5.6 1. Introduct...

Recently, there has been much interest in the derivation of logic programming languages based on linear logic, a logic of resource-consumption. Such languages provide a notion of resource-oriented programming, often leading to programs that are more elegant and concise than their equivalents in languages, such as Prolog, based on classical logics. We discuss, with examples, the design, implementation and applications of Lygon, a linear logic programming language. Lygon is based on a proof-theoretic analysis of which occurrences of the linear connectives provide an adequate basis for programming. In common with other linear logic programming languages, Lygon allows clauses to be used exactly once in a computation, thereby avoiding the need for the explicit resource-counting often necessary in Prolog-like languages. Indeed, it appears that resource-sensitivity leads to significant differences between the natural programming methodologies in Lygon and Prolog. Just as linear logic...

One way to think about STRIPS is as a mapping from databases to databases, in the following sense: Suppose we want to know what the world would be like if an action, represented by the STRIPS operator ff, were done in some world, represented by the STRIPS database D 0 . To find out, simply perform the operator ff on D 0 (by applying ff's elementary add and delete revision operators to D 0 ). We describe this process as progressing the database D 0 in response to the action ff. In this paper, we consider the general problem of progressing an initial database in response to a given sequence of actions. We appeal to the situation calculus and an axiomatization of actions which addresses the frame problem (Reiter [13], Lin and Reiter [8]). This setting is considerably more general than STRIPS. Our results concerning progression are mixed. The (surprising) bad news is that, in general, to characterize a progressed database we must appeal to second order logic. The good news is that there...

The relation of inclusion between types has been suggested by the practice of programming, as it enriches the polymorphism of functional languages. We propose a simple (and linear) calculus of sequents for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful, sequents. In particular, transitivity and anti-symmetry will be derived from elementary logical principles, which stresses the power of sequents and Gentzen-style proof methods. Proof techniques based on cut-elimination will be at the core of our results. 1 Introduction 1.1 Motivations, Theories and Models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance...

The relation of inclusion between types has been suggested by the practice of programming as it enriches the polymorphism of functional languages. We propose a simple (and linear) sequent calculus for subtyping as logical entailment. This allows us to derive a complete and coherent approach to subtyping from a few, logically meaningful sequents. In particular, transitivity and anti-symmetry will be derived from elementary logical principles. 1 Introduction 1.1 Motivations, theories and models In recent years, several extensions of core functional languages have been proposed to deal with the notion of subtyping; see, for example, [CW85, Mit88, BL90, BCGS91, CMMS91, CG92, PS94, Tiu96, TU96]. These extensions were suggested by the practice of programming in computer science. In particular, they were inspired by the notion of inheritance as used in object-oriented programming languages, or by other concrete implementations of the following form of polymorphism: data living in a type oe, ...