The primary focus of this thesis is the semantic gap between a fine-grained structural operational semantics and a set of rely/guarantee-style development rules. The semantic gap is bridged by considering the development rules to be a part of the same logical framework as the operational semantics, and a set of soundness proofs show that the development rules, though making development easier for a developer, do not add any extra power to the logical framework as a whole. The soundness proofs given are constructed to take advantage of the structural nature of the language and its semantics; this allows for the addition of new development rules in a modular fashion. The particular language semantics allows for very fine-grained concurrency. The language itself includes a construct for nested parallel execution of statements, and the semantics is written so that statements can interfere with each other between individual variable reads. The language also includes an atomic block construct for which the semantics is an embodiment of a form of software transactional memory. The inclusion of the atomic construct helps illustrate the inherent expressive weakness present in the rely/guarantee rules with respect to termination properties. As such, two development rules are proposed for the atomic construct, one of which has serious restrictions in its application, and another for which the termination property does not hold.

Abstract. Cut elimination is a central result of the proof theory. This paper proposes a new approach for proving the theorem for Gentzen’s intuitionistic sequent calculus LJ, that relies on completeness of the cutfree calculus with respect to Kripke Models. The proof defines a general framework to extend the cut elimination result to other intuitionistic deduction systems, in particular to deduction modulo provided the rewrite system verifies some properties. We also give an example of rewrite system for which cut elimination holds but that doesn’t enjoys proof normalization.

Abstract. Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cut-free ” sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose. 1

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A procedure for learning a lexical assignment together with a system of syntactic and semantic categories given a fixed type-logical grammar is briefly described. The logic underlying the grammar can be any cut-free decidable modally enriched extension of the Lambek calculus, but the correspondence between syntactic and semantic categories must be constrained so that no infinite set of categories is ultimately used to generate the language. It is shown that under these conditions various linguistically valuable subsets of the range of the algorithm are classes identifiable in the limit from data consisting of sentences labeled by simply typed lambda calculus meaning terms in normal form. The entire range of the algorithm is shown to be not a learnable class, contrary to a mistaken result reported in a preliminary version of this paper. It is informally argued that, given the right type logic, the learnable classes of grammars include members which generate natural languages, and thus that natural languages are learnable in this way.

Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent “cut-free ” sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose. 1

Linear logic, introduced by Girard in 1987, has been called a resource conscious logic. In order to express a dynamic change in process environment, it is useful to consider a concept of resource such as data consumption. The expressive power of linear logic is evidenced by some very natural encodings of computational models such as Petri nets, counter machines, Turing machines, and others. For example, in Petri nets, tokens are considered as resources that are consumed and transitions are considered as reusable resources. It is well known that the reachability problem for ordinary Petri nets is equivalent to the provability for the corresponding sequent of linear logic. Also, as a formal logical system, linear logic satisfies some basic theorems. In it the cut elimination theorem and the soundness and completeness theorems for phase semantics which is a standard semantics of linear logic hold true. In particular, the cut elimination theorem can be applied to logic programming, uniform proof and proof search, and so on. We think that linear logic has been given various applications in computer science through its resource consciousness and usefulness as a formal system. However, since linear logic does not include a concept of time directly, it is not enough to treat a dynamic change in environments with the passage of time such as execution time and waiting time. A typical example is the encoding of timed Petri nets. Although ordinary Petri nets can be encoded into linear logic naturally as stated above, the encoding of timed Petri nets into the corresponding sequent is too complex for linear logic since the reachability problem for timed Petri nets includes a time concept. Thus, it can be considered to extend linear logic with respect to the time concept. The aim of t...

. In this paper, we investigate translations from a classical cut-free sequent calculus LK into an intuitionistic cut-free sequent calculus LJ. Translations known from the literature rest on permutations of inferences in classical proofs. We show that, for some classes of firstorder formulae, all minimal LJ-proofs are non-elementary but there exist short LK-proofs of the same formula. Similar results are obtained even if some fragments of intuitionistic first-order logic are considered. The size of the corresponding minimal search spaces for LK and LJ are also nonelementarily related. We show that we can overcome these difficulties by extending LJ with an analytic cut rule. 1 Introduction Characterizing classes of formulae for which classical derivability implies intuitionistic derivability was one topic in the Leningrad group around Maslov in the sixties. Such classes are called (complete) Glivenko classes which were extensively characterized by Orevkov [7]. More recently, people ar...

Szabo's derivation systems on sequent calculi with exchange and product are not Church-Rosser. Thus his coherence results for categories having a symmetric product (either monoidal or cartesian) are false. 1 Introduction Gentzen's sequent calculi [9] have been applied extensively in category theory, e.g [2, 3, 4, 6, 7, 8]. Sequents correspond to morphisms of a category, and the rules of the calculus correspond to categorical structures (e.g. having an associative tensor product). Cut-elimination was then used to put bounds on the complexity of these structures, e.g. to produce exhaustive lists (perhaps with duplications) of the canonical natural transformations between given functors. For symmetric, monoidal closed categories it was shown in [12] how to decide in principle whether two such transformations are equal, while an effective, linear-time decision procedure was given in [1]. Derivation systems (reduction rules) can be used to eliminate some duplicates in the list of cut-free ...