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Categorical Proof Theory of Classical Propositional Calculus
, 2005
"... We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. ..."
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We investigate semantics for classical proof based on the sequent calculus. We show that the propositional connectives are not quite wellbehaved from a traditional categorical perspective, and give a more refined, but necessarily complex, analysis of how connectives may be characterised abstractly. Finally we explain the consequences of insisting on more familiar categorical behaviour.
A Classical Linear λcalculus
, 1997
"... This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natu ..."
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This paper proposes and studies a typed λcalculus for classical linear logic. I shall give an explanation of a multipleconclusion formulation for classical logic due to Parigot and compare it to more traditional treatments by Prawitz and others. I shall use Parigot's method to devise a natural deduction formulation of classical linear logic. This formulation is compared in detail to the sequent calculus formulation. In an appendix I shall also demonstrate a somewhat hidden connexion with the paradigm of control operators for functional languages which gives a new computational interpretation of Parigot's techniques.
On a Modal \lambdaCalculus for S4*
 Proceedings of the Eleventh Conference on Mathematical Foundations of Programming Sematics
, 1995
"... We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited 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 pr ..."
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We present !2 , a concise formulation of a proof term calculus for the intuitionistic modal logic S4 that is wellsuited 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 welltyped terms. Applications include a new formulation of natural deduction for intuitionistic linear logic, modal logical frameworks, and a logical analysis of staged computation and bindingtime 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 CurryHoward isomorphism between proofs and terms is most poignant for natural deduction, so natural deduction formulations of modal and...
Sharing Continuations: Proofnets for Languages With Explicit Control
"... sumes it. Yet evaluating expressions is very familiar, while evaluating continuations is considered esoteric, even though both are made ofthe same stuff. The incorporation ofcontinuations as firstclass citizens in programming languages was not welcomed like the Emancipation Proclamation, but instea ..."
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sumes it. Yet evaluating expressions is very familiar, while evaluating continuations is considered esoteric, even though both are made ofthe same stuff. The incorporation ofcontinuations as firstclass citizens in programming languages was not welcomed like the Emancipation Proclamation, but instead regarded warily as a kind ofwitchcraft, with implementation pragmatics that are illdefined and unclear. Ifexpressions and continuations are indeed dual, then so should be the technology oftheir implementation, and the flexibility with which we reason about them. Efficient evaluation ofone should reveal dual strategies for evaluating the other. In short, everything we know about expressions we ought to know about continuations. We take a significant step towards this equality by formulating a general version ofgraph reduction that implements the sharing and optimal incremental evaluation ofboth expressions and continuations, each evaluated using the same primitive operations. By founding our technology on generic tools from logic and programming language theory, specifically the CPS transform and its relation
Correctness Criteria based on a Homology of Proof Structures in Multiplicative Linear Logic
, 1996
"... Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : ..."
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Contents 1 Introduction 1 2 Multiplicative linear logic 5 2.1 The calculus : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 Cut elimination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.3 Strong normalization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 3 Graphs 15 3.1 Paired directed multigraphs : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 3.2 Constructions on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 17 3.3 Proof nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 4 Homology 27 4.1 General theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 4.2 Application on pdm's : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31
Petri Nets as Multiset Rewriting Systems in a Linear Framework
, 1994
"... Keywords: Petri Nets, Multiset rewriting systems, Concurrency, Process Theory, Linear Logic. This research was completed during an extended visit of the author at the Department of Computer Science of Carnegie Mellon University, Pittsburgh, PA 15213. His email address there is iliano@cs.cmu.edu. ..."
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Keywords: Petri Nets, Multiset rewriting systems, Concurrency, Process Theory, Linear Logic. This research was completed during an extended visit of the author at the Department of Computer Science of Carnegie Mellon University, Pittsburgh, PA 15213. His email address there is iliano@cs.cmu.edu. Contents 1 Introduction 1 2 Multisets and multiset rewriting systems 2 2.1 Multisets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.2 Multiset rewriting systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Beyond multiset rewriting systems : : : : : : : : : : : : : : : : : : : : : : : : : 7 3 Petri nets 13 3.1 Place/Transition nets with infinite place capacities : : : : : : : : : : : : : : : : 14 3.2 P/T ! systems with infinite markings : : : : : : : : : : : : : : : : : : : : : : : 18 3.3 Place/Transition nets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 3.4 Other net models : : : : : : : : : : : : : : : : :...
A Linear Approach to Modal Proof Theory
"... We show how to extend the proof theoretical analysis of sequent derivations in classical logic by means of linear logic, as exposed in Danos et al.(1994), to sequent derivations in S4, by constructing embeddings of S4 into classical linear logic that are correct with respect to proofs (i.e. define l ..."
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We show how to extend the proof theoretical analysis of sequent derivations in classical logic by means of linear logic, as exposed in Danos et al.(1994), to sequent derivations in S4, by constructing embeddings of S4 into classical linear logic that are correct with respect to proofs (i.e. define linear decorations of S4derivations). Among the immediate corollaries are cut elimination for S4, and reduction strategies that are strongly normalizing. "dedicated to dj" 1 Aims Linear logic 1 bans the structural rules of weakening and contraction from the formulation of classical logic as a sequent calculus, and reintroduces them in modalized form: structural manipulation to the left of the entailmentsign is allowed only for formulas prefixed by "!", to the right only for those prefixed by "?". (`Linear' logicians refer to !, ? as the exponentials.) The resulting calculus is a proof theoretical jewel, which combines the deep symmetries of classical, with the computational properties ...
Multiplicative Linear Logic
, 1996
"... this paper we concentrate on the socalled multiplicative fragment, i.e. the connectives\Omega ..."
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this paper we concentrate on the socalled multiplicative fragment, i.e. the connectives\Omega
Theorem Proving with the Inverse Method for Linear Logic
, 2004
"... Linear logic presents a unified framework for describing and reasoning about stateful systems. Because of its view of hypotheses as resources, it supports such phenomena as concurrency, external and internal choice, and state transitions that are common in such domains as protocol verification, conc ..."
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Linear logic presents a unified framework for describing and reasoning about stateful systems. Because of its view of hypotheses as resources, it supports such phenomena as concurrency, external and internal choice, and state transitions that are common in such domains as protocol verification, concurrent computation, process calculi and games. It accomplishes this unifying view by providing logical connectives whose behaviour is closely tied to the collection of resources, which is free of structural phenomena such as weakening (allowing more resources than necessary) or contraction (using a resource more than once). The usual (nonlinear) logic is embedded in this substructural framework by means of an exponential modal operator. The interaction of the rules for multiplicative, additive and exponential connectives gives rise to a wide and expressive array of behaviours. Various approaches have been taken to produce automated reasoning systems for fragments of linear logic, usually in the form of logic programming engines; but, due to the lack of the full complement of linear connectives, uses of such systems have an idiomatic commitment, for example as serializations or in continuationpassingstyle. This thesis addresses the need for automated reasoning for the complete set of operators for first order intuitionistic linear logic (i.e., ⊗, 1, ❜, &, ⊤, ⊕, 0,!, ∀, ∃), which removes the need for such idiomatic constructions and allows direct logical expression. The particular theorem proving technique used is the inverse method, which performs forward