In order to reason about specifications of computations that are given via the proof search or logic programming paradigm one needs to have at least some forms of induction and some principle for reasoning about the ways in which terms are built and the ways in which computations can progress. The literature contains many approaches to formally adding these reasoning principles with logic specifications. We choose an approach based on the sequent calculus and design an intuitionistic logic F Oλ ∆IN that includes natural number induction and a notion of definition. We have detailed elsewhere that this logic has a number of applications. In this paper we prove the cut-elimination theorem for F Oλ ∆IN, adapting a technique due to Tait and Martin-Löf. This cut-elimination proof is technically interesting and significantly extends previous results of this kind. 1

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Intuitionistic and linear logics can be used to specify the operational semantics of transition systems in various ways. We consider here two encodings: one uses linear logic and maps states of the transition system into formulas, and the other uses intuitionistic logic and maps states into terms. In both cases, it is possible to relate transition paths to proofs in sequent calculus. In neither encoding, however, does it seem possible to capture properties, such as simulation and bisimulation, that need to consider all possible transitions or all possible computation paths. We consider augmenting both intuitionistic and linear logics with a proof theoretical treatment of definitions. In both cases, this addition allows proving various judgments concerning simulation and bisimulation (especially for noetherian transition systems). We also explore the use of infinite proofs to reason about infinite sequences of transitions. Finally, combining definitions and induction into sequent calculus proofs makes it possible to reason more richly about properties of transition systems completely within the formal setting of sequent calculus.

What is the semantics of Negation-as-Failure in logic programming? We try to answer this question by proof-theoretic methods. A rule based sequent calculus is used in which a sequent is provable if, and only if, it is true in all three-valued models of the completion of a logic program. The main theorem is that proofs in the sequent calculus can be transformed into SLDNF-computations if, and only if, a program has the cut-property. A fragment of the sequent calculus leads to a sound and complete semantics for SLDNFresolution with substitutions. It turns out that this version of SLDNF-resolution is sound and complete with respect to three-valued possible world models of the completion for arbitrary logic programs and arbitrary goals. Since we are dealing with possibly nonterminating computations and constructive proofs, three-valued possible world models seem to be an appropriate semantics.

Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that the subject-reduction theorem holds. One challenge in developing a framework for such reasoning is that higherorder abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution,is difficult to treat in proofs involving induction. In this paper, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed-terms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of a definition is a proof-theoretic device that allows certain theories to be treated as “closed ” or as defining fixed points. The resulting meta-logic can specify various logical frameworks and a large range of judgments regarding programming languages and inference systems. We illustrate this point through examples, including the admissibility of cut for a simple logic and subject reduction, determinacy of evaluation, and the equivalence of SOS and natural semantics presentations of evaluation for a simple functional programming language. 1.

We define a translation between the language Golog and a fragment of linear logic augmented with definitions; we prove bisimulation for this translation and finally suggest some extensions to Golog motivated by the translation. Contents 0 Preliminaries 2 0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2 Girard's Fixpoint Theorem . . . . . . . . . . . . . . . . . . . . . 2 1 The Basic Representation 4 1.1 Situations and Fluents . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 A First Attempt . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.2 The Situation-Forming Operators . . . . . . . . . . . . . . 6 1.1.3 Permutabilities . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Representing Situations . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Basic Actions . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Representing Fluents . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 The Fluents . . . . . . . . . . . . . . ...