We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science --- LICS'96 (E. Clarke editor), pp. 264--275, New Brunswick, NJ, July 27--30 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cut-elimination. 1 Introduction A logical framework is a formal system desig...

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The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [12], provide data types, higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends the languages λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. As a meta-language, Forum greatly extends the expressiveness of these other logic programming languages. To illustrate its expressive strength, we specify in Forum a sequent calculus proof system and the operational semantics of a functional programming language that incorporates such nonfunctional features as counters and references. 1

The theory of cut-free sequent proofs has been used to motivate and justify the design of a number of logic programming languages. Two such languages, λProlog and its linear logic refinement, Lolli [15], provide for various forms of abstraction (modules, abstract data types, and higher-order programming) but lack primitives for concurrency. The logic programming language, LO (Linear Objects) [2] provides some primitives for concurrency but lacks abstraction mechanisms. In this paper we present Forum, a logic programming presentation of all of linear logic that modularly extends λProlog, Lolli, and LO. Forum, therefore, allows specifications to incorporate both abstractions and concurrency. To illustrate the new expressive strengths of Forum, we specify in it a sequent calculus proof system and the operational semantics of a programming language that incorporates references and concurrency. We also show that the meta theory of linear logic can be used to prove properties of the objectlanguages specified in Forum.

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

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.

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We show that using deductive systems to specify an offline partial evaluator allows its correctness to be mechanically verified. For a -mix-style partial evaluator, we specify binding-time constraints using a natural-deduction logic, and the associated program specializer using natural (aka "deductive") semantics. These deductive systems can be directly encoded in the Elf programming language --- a logic programming language based on the LF logical framework. The specifications are then executable as logic programs. This provides a prototype implementation of the partial evaluator. Moreover, since deductive system proofs are accessible as objects in Elf, many aspects of the partial evaluation correctness proofs (e.g., the correctness of binding-time analysis) can be coded in Elf and mechanically verified. This work illustrates the utility of declarative programming and of using deductive systems for defining program specialization systems: by exploiting the logical character of definit...

This document formally defines the syntax and semantics of the Extended ML language. It is based directly on the published semantics of Standard ML in an attempt to ensure compatibility between the two languages. LFCS, Department of Computer Science, University of Edinburgh, Edinburgh, Scotland. y Institute of Informatics, Warsaw University, and Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland. ii CONTENTS Contents 1 Introduction 1 1.1 Behavioural equivalence : : : : : : : : : : : : : : : : : : : : : : : : 3 1.2 Metalanguage : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3 2 Syntax of the Core 8 2.1 Reserved Words : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.2 Special constants : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 2.3 Comments : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.4 Identifiers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Lexical analysis : : : :...

. We address here the problem of automatically translating the Natural Semantics of programming languages to Coq, in order to prove formally general properties of languages. Natural Semantics [18] is a formalism for specifying semantics of programming languages inspired by Plotkin's Structural Operational Semantics [22]. The Coq proof development system [12], based on the Calculus of Constructions extended with inductive types (CCind), provides mechanized support including tactics for building goal-directed proofs. Our representation of a language in Coq is inAEuenced by the encoding of logics used by Church [6] and in the Edinburgh Logical Framework (ELF) [15, 3]. 1 Introduction The motivation for our work is the need for an environment to help develop proofs in Natural Semantics. The interactive programming environment generator Centaur [17] allows us to compile a Natural Semantics speciøcation of a given language into executable code (type-checkers, evaluators, compilers, program t...