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A Computational Interpretation of Modal Proofs
 Proof Theory of Modal Logics
, 1994
"... The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, exten ..."
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The usual (e.g. Prawitz's) treatment of natural deduction for modal logics involves a complicated rule for the introduction of the necessity, since the naive one does not allow normalization. We propose natural deduction systems for the positive fragments of the modal logics K, K4, KT, and S4, extending previous work by Masini on a twodimensional generalization of Gentzen's sequents (2sequents). The modal rules closely match the standard rules for an universal quantifier and different logics are obtained with simple conditions on the elimination rule for 2. We give an explicit term calculus corresponding to proofs in these systems and, after defining a notion of reduction on terms, we prove its confluence and strong normalization. 1. Introduction Proof theory of modal logics, though largely studied since the fifties, has always been a delicate subject, the main reason being the apparent impossibility to obtain elegant, natural systems for intensional operators (with the excellent ex...
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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Cited by 7 (0 self)
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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...
unknown title
"... The inference rules so far only model intuitionistic logic, and some classically true propositions such as A ∨¬A (for an arbitrary A) are not derivable, as we will see in Section??. There are three commonly used ways one can construct a system of classical natural deduction by adding one additional ..."
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The inference rules so far only model intuitionistic logic, and some classically true propositions such as A ∨¬A (for an arbitrary A) are not derivable, as we will see in Section??. There are three commonly used ways one can construct a system of classical natural deduction by adding one additional rule of inference. ⊥C is called Proof by Contradiction or Rule of Indirect Proof, ¬¬C is the Double Negation Rule, and XM is referred to as Excluded Middle.
.2 Classical Logic
"... Since hypotheses and their restrictions are critical for linear logic, we give here a formulation of natural deduction for intuitionistic logic with localized hypotheses, but not parameters. For this we need a notation for hypotheses which we call a context. Contexts \Gamma ::= \Delta j \Gamma; u:A ..."
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Since hypotheses and their restrictions are critical for linear logic, we give here a formulation of natural deduction for intuitionistic logic with localized hypotheses, but not parameters. For this we need a notation for hypotheses which we call a context. Contexts \Gamma ::= \Delta j \Gamma; u:A Here, "\Delta" represents the empty context, and \Gamma; u:A adds hypothesis ` A labelled u to \Gamma. We assume that each label u occurs at most once in a context in order to avoid ambiguities. The main judgment can then be written as \Gamma ` A, where \Delta; u 1 :A 1 ; : : : ; un :An ` A stands for u 1 ` A 1 : : :<F43.12
A Linear Logical Framework
"... syntax Concrete syntax Kinds type type \Pix:A: K x:AK A ? K K ! A P M P M ? !T? Types ANB A & B A(B A o B B o A \Pix:A: B x:AB A ? B B ! A h i () hM; Ni M,N fst M !fst? M Objects snd M !snd? M x:A: M [xA]M M N M N x:A: M [x:A]M M N M N The next table gives the relative p ..."
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syntax Concrete syntax Kinds type type \Pix:A: K x:AK A ? K K ! A P M P M ? !T? Types ANB A & B A(B A o B B o A \Pix:A: B x:AB A ? B B ! A h i () hM; Ni M,N fst M !fst? M Objects snd M !snd? M x:A: M [xA]M M N M N x:A: M [x:A]M M N M N The next table gives the relative precedence and associativity of these operators. Parentheses are available to override these behaviors. Note that o, ?, o, and ! all have the same precedence. Precedence Operator Position highest !fst? !snd? left prefix left associative & right associative o ? right associative o ! left associative , right associative : left associative lowest  : [ : ] [ ] left prefix As in Elf, a signature declaration c : A is represented by the program clause: c : A. Type family constants are declared similarly. For practical purposes, it is convenient to provide a means of declaring linear assumptions. Indeed, whenever the object formalism we want to represent requires numerous linear ...
Lambda! Considered Both as a Paradigmatic Language and as a MetaLanguage
"... Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an u ..."
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Intuitionistic Linear Logic (ILL) is a resourceconscious logic. The CurryHoward Isomorphism (CHI) applied to ILL, generates typed functionallike languages that have primitive constants by means of which the amount of resources (terms), used during the computation, is explicit. \Gamma ! is an untyped functionallike language inspired from a typed language joined at ILL by CHI. We want to use the resourceaware language \Gamma ! both as a paradigmatic programming language and as a metalanguage for implementing a fragment of the untyped calculus fi . For using \Gamma ! in the first way we give an algorithm for automatically assigning formulas of ILL as types to terms of \Gamma ! . Concerning the second kind of use, we introduce a onestep translation Tr from the fragment C of fi that can be typed a la Curry to the typable fragment of \Gamma ! in ILL. Tr preserves the linearbehaved terms of C and is both correct and complete, in a reasonable sense, w.r.t. the...
Termination
"... proof. Let us examine why. ; #M 1 :A 2#A 1 2 :A 2 #E. M 2 :A 1 We can make the following inferences. V 1 = #x:A 2 .M # 1 By type preservation and inversion At this point we cannot proceed: we need a derivation of [V 2 /x]M # 1 ## V for some V to complete the derivation of M 1 ..."
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proof. Let us examine why. ; #M 1 :A 2#A 1 2 :A 2 #E. M 2 :A 1 We can make the following inferences. V 1 = #x:A 2 .M # 1 By type preservation and inversion At this point we cannot proceed: we need a derivation of [V 2 /x]M # 1 ## V for some V to complete the derivation of M 1 M 2 ## V . Unfortunately, the induction hypothesis does not tell us anything about [V 2 /x]M # 1 . Basically, we need to extend it so it makes a statement about the result of evaluation ( #x:A 2 .M # 1 ,inthis case). Sticking to the case of linear application for the moment, we call a term M "good" if it evaluates to a "good" value V .AvalueVis "good" if it is a function #x:A 2 .M # 1 and if substituting a "good" value V 2 for x in M # 1 results in a "good" term. Note that this is not a proper definition, since to see if V is "good" we may need to substitute any "good" value V 2 into it, possibly including V itself. We can make this definition inductive if we observe that the value
Linear Type Checking
"... but ag it to indicate that it may not be exact, but that some of these linear hypotheses may be absorbed if necessary. In other words, in the judgment any of the remaining hypotheses in O need not be consumed in the other branches of the typing derivation. On the other hand, the judgment ; I n O ` ..."
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but ag it to indicate that it may not be exact, but that some of these linear hypotheses may be absorbed if necessary. In other words, in the judgment any of the remaining hypotheses in O need not be consumed in the other branches of the typing derivation. On the other hand, the judgment ; I n O ` 0 M : A indicates the M uses exactly the variables in I O . When we think of the judgment ; I n O ` i M : A as describing an algorithm, we think of , I and M as given, and O and the slack indicator i as part of the result of the computation. The type A may or may not be givenin one case it is synthesized, in the other case checked. This re nes our view as computation being described as the bottomup construction of a derivation to include parts of the judgment in dierent roles (as input, output, or bidirectional components). In logic programming, which is based on the notion of computationasproofsearch, these roles of the syntactic constituents of a judgment are called