this paper we reconsider the foundations of modal logic, following MartinL of's methodology of distinguishing judgments from propositions [ML85]. We give constructive meaning explanations for necessity (2) and possibility (3). This exercise yields a simple and uniform system of natural deduction for intuitionistic modal logic which does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic [FM97] and find that it is already contained in modal logic, using the decomposition of the lax modality fl A as

Abstract not found

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 #.

Abstract not found

We prove a folklore theorem, that two derivations in a cut-free sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are inter-permutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are inter-permutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...

This paper gives an introduction to type theory, focusing on its recent use as a logical framework for proofs and programs. The first two sections give a background to type theory intended for the reader who is new to the subject. The following presents Martin-Lof's monomorphic type theory and an implementation, ALF, of this theory. Finally, a few small tutorial examples in ALF are given.

specication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 6.1.1 Description of events : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 6.1.2 The specication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 37 6.2 Implementation outline : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 39 6.2.1 Hardware specication : : : : : : : : : : : : : : : : : : : : : : : : : : 41 6.2.2 Implementation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 1 1 Introduction ABEL is a language together with a formal logic for use in program development. The overall goal has been to support specication and program development through semimechanical aids for reasoning and verication. It has been of major concern that the language and the associated reasoning formalism are such that consistency requirements and other proof obligations are as simple and manageable as possible. In particular we have sought to: ffl ooeer language constructs well suited fo...

This paper studies the problem of simplifying typings and the size-complexity of most general typings in typed programming languages with atomic subtyping. We define a notion of minimal typings relating all typings which are equivalent with respect to instantiation. The notion of instance is that of Fuh and Mishra [13], which supports many interesting simplifications. We prove that every typable term has a unique minimal typing, which is the logically most succinct among all equivalent typings. We study completeness properties, with respect to our notion of minimality, of well-known simplification techniques. Drawing upon these results, we prove a tight exponential lower bound for the worst case dag-size of constraint sets as well as of types in most general typings. To the best of our knowledge, the best previously proven lower bound was linear. 1 Introduction Subtyping is a fundamental idea in type systems for programming languages, which can in principle be integrated into standar...

In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a non-trivial translation procedure to extract human-oriented deductions from machine-oriented proofs. Previously known translation procedures, though complete, tend to produce unintuitive deductions. One of the major flaws in such procedures is that too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof. We present an algorithm, symmetric simplification, for discovering useful lemmas in deductions of theorems in first- and higher-order logic. This algorithm, which has been implemented in the TPS system, has the feature that resulting deductions may no longer have the weak subformula property. It is currently limited, however, in that it only generates lemmas of the form C ∨ ¬C ′ , where C and C ′ have the same negation normal form. 1

We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is well-suited for practical applications. We show that, with respect to provability, it is equivalent to other formulations in the literature, sketch a simple type checking algorithm, and prove subject reduction and the existence of canonical forms for well-typed terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and binding-time analysis for functional languages [6]. 1 Introduction Modal operators familiar from traditional logic have received renewed attention in computer science through their importance in linear logic. Typically, they are described axiomatically in the style of Hilbert or via sequent calculi. However, the Curry-Howard isomorphism between proofs and -terms is most poignant for natural deduction, so natural deduction formulations of modal and...