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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, ..."
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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...
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...
Chapter 3 Sequent Calculus
, 2001
"... In the previous chapter we developed linear logic in the form of natural deduction, which is appropriate for many applications of linear logic. It is also highly economical, in that we only needed one basic judgment (A true) and two judgment forms (linear and unrestricted hypothetical judgments) to ..."
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In the previous chapter we developed linear logic in the form of natural deduction, which is appropriate for many applications of linear logic. It is also highly economical, in that we only needed one basic judgment (A true) and two judgment forms (linear and unrestricted hypothetical judgments) to explain the meaning of all connectives we have encountered so far. However, it is not immediately wellsuited for proof search, because it involves mixing forward and backward reasoning even if we restrict ourselves to searching for normal deductions. In this chapter we develop a sequent calculus as a calculus of proof search for normal natural deductions. We then extend it with a rule of cut that allows us to model arbitrary natural deductions. The central theorem of this chapter is cut elimination which shows that the cut rule is admissible. We obtain the normalization theorem for natural deduction as a direct consequence of this theorem. It was this latter application which led to the original discovery of the sequent calculus by Gentzen [Gen35]. There are many useful immediate corollaries of the cut elimination theorem, such as consistency of the logic, or the disjunction property. 3.1 CutFree Sequent Calculus In this section we transcribe the process of searching for normal natural deductions into an inference system. In the context of sequent calculus, proof search is seen entirely as the bottomup construction of a derivation. This means that elimination rules must be turned “upsidedown ” so they can also be applied bottomup rather than topdown. In terms of judgments we develop the sequent calculus via a splitting of the judgment “A is true ” intotwojudgments: “Ais a resource ” (Ares) and“A is a goal ” (Agoal). Ignoring unrestricted hypothesis for the moment, the main judgment w1:A1 res,...,wn:Anres = ⇒ C goal
weak
, 1996
"... A natural deduction system NDIL described here admits normalization and has subformula property. It has standard axioms A ⊢ A, ⊢ 1, standard introduction and elimination rules for &, − ◦ (linear implication), ⊕ and quantifiers. The rules for ⊗ are now standard too. Structural rules are (implic ..."
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A natural deduction system NDIL described here admits normalization and has subformula property. It has standard axioms A ⊢ A, ⊢ 1, standard introduction and elimination rules for &, − ◦ (linear implication), ⊕ and quantifiers. The rules for ⊗ are now standard too. Structural rules are (implicit) permutation plus contraction and weakening for mformulas. The rules for! use an idea of D. Prawitz. By a mformula we mean 1, any formula beginning with!, and any expression < Γ>!A, whereΓisa list of formulas and mformulas, and A is a formula. Derivable objects are sequents Γ ⊢ A where Γ is a multiset of formulas and mformulas, and A is a formula. The rules for!, weakening and contraction are as follows: Γ ⊢!A
Summary
, 1993
"... In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties ..."
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In this thesis we carry out a detailed study of the (propositional) intuitionistic fragment of Girard’s linear logic (ILL). Firstly we give sequent calculus, natural deduction and axiomatic formulations of ILL. In particular our natural deduction is different from others and has important properties, such as closure under substitution, which others lack. We also study the process of reduction in all three logical formulations, including a detailed proof of cut elimination. Finally, we consider translations between Intuitionistic Logic (IL) and ILL. We then consider the linear term calculus, which arises from applying the CurryHoward correspondence to the natural deduction formulation. We show how the various proof theoretic formulations suggest reductions at the level of terms. The properties of strong normalization and confluence are proved for these reduction rules. We also consider mappings between the extended λcalculus and the linear term calculus. Next we consider a categorical model for ILL. We show how by considering the linear term calculus as an equational logic, we can derive a model: a Linear category. We consider two alternative models: firstly, one due to Seely and then one due to Lafont. Surprisingly, we find that Seely’s model is not sound, in that equal terms are not modelled with equal morphisms. We show how after adapting Seely’s model (by viewing it in a more abstract setting) it becomes a particular instance
If ; ; =) A + then ; =) A.
"... r each one separately. In a realistic application, such as general theorem proving, logic programming, model checking, many of these highlevel optimizations may need to be applied simultaneously and their interaction considered. 4.3 Uni cation We begin with a discussion of uni cation, a techniq ..."
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r each one separately. In a realistic application, such as general theorem proving, logic programming, model checking, many of these highlevel optimizations may need to be applied simultaneously and their interaction considered. 4.3 Uni cation We begin with a discussion of uni cation, a technique for eliminating existential nondeterminism. When proving a proposition of the form 9x: A by its right rule in the sequent calculus, we must supply a term t and then prove [t=x]A. The domain of quanti cation may include in nitely many terms (such as the natural numbers), so this choice cannot be resolved simply by trying all possible terms t. Similarly, when we use a hypothesis of the form 8x: A we must supply a term t to substitute for x. Fortunately, there is a better technique which is sound and complete for syntactic equality between terms. The basic idea is quite simple: we postpone the choice of t and instead substitute a new existential variable (often called metavariable or lo
Chapter 4 Proof Search
"... an inference rule with which it might be inferred. We also may need to determine exactly how the conclusion of the rule matches the judgment. For example, in the R rule we need to decide how to split the linear hypotheses between the two premises. After these choices have been made, we reduce the ..."
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an inference rule with which it might be inferred. We also may need to determine exactly how the conclusion of the rule matches the judgment. For example, in the R rule we need to decide how to split the linear hypotheses between the two premises. After these choices have been made, we reduce the goal of deriving the judgment to a number of subgoals, one for each premise of the selected rule. If there are no premises, the subgoal is solved. If there are no subgoals left, we have derived the original judgment. One important observation about bottomup proof search is that some rules are invertible, that is, the premises are derivable whenever the conclusion is derivable. The usual direction states that the conclusion is evident whenver the premises are. Invertible rules can safely be applied whenever possible without losing completeness, although some care must be taken to retain a terminating procedure in the presence of unrestricted hypotheses. We also separate weakly invertible r