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...

linear logic to reason about sequent

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.

The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives# of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives -#, & and #.

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

this paper, we do this by adding the #-quantifier: its role will be to declare variables to be new and of local scope. The syntax of the formula # x.B is like that for the universal and existential quantifiers. Following Church's Simple Theory of Types [Church 1940], formulas are given the type o, and for all types # not containing o, # is a constant of type (# o) o. The expression # #x.B is ACM Transactions on Computational Logic, Vol. V, No. N, October 2003. 4 usually abbreviated as simply # x.B or as if the type information is either simple to infer or not important

A powerful and declarative means of specifying computations containing abstractions involves meta-level, universally quantified generic judgments. We present a proof theory for such judgments in which signatures are associated to each sequent (used to account for eigenvariables of the sequent) and to each formula in the sequent (used to account for generic variables locally scoped over the formula). A new quantifier, ∇, is introduced to explicitly manipulate the local signature. Intuitionistic logic extended with ∇ satisfies cut-elimination even when the logic is additionally strengthened with a proof theoretic notion of definitions. The resulting logic can be used to encode naturally a number of examples involving name abstractions, and we illustrate using the π-calculus and the encoding of objectlevel provability.

We present a new logic, Linc, which is designed to be used as a framework for specifying and reasoning about operational semantics. Linc is an extension of firstorder intuitionistic logic with a proof theoretic notion of definitions, induction and coinduction, and a new quantifier #.

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.

We present the propositional fragment CLF0 of the Concurrent Logical Framework (CLF). CLF extends the Linear Logical Framework to allow the natural representation of concurrent computations in an object language. The underlying type theory uses monadic types to segregate values from computations. This separation leads to a tractable notion of definitional equality that identifies computations di#ering only in the order of execution of independent steps. From a logical point of view our type theory can be seen as a novel combination of lax logic and dual intuitionistic linear logic. An encoding of a small Petri net exemplifies the representation methodology, which can be summarized as "concurrent computations as monadic expressions ".