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A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4

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A Linear-Logical Reconstruction of Intuitionistic Modal Logic S4 A Linear-logical Reconstruction of Intuitionistic Modal Logic S4 Yosuke Fukuda Graduate School of Informatics, Kyoto University, Japan http://www.fos.kuis.kyoto-u.ac.jp/~yfukuda [email protected] Akira Yoshimizu French Institute for Research in Computer Science and Automation (INRIA), France http://www.cs.unibo.it/~akira.yoshimizu [email protected] Abstract We propose a modal linear logic to reformulate intuitionistic modal logic S4 (IS4) in terms of linear logic, establishing an S4-version of Girard translation from IS4 to it. While the Girard translation from intuitionistic logic to linear logic is well-known, its extension to modal logic is non-trivial since a naive combination of the S4 modality and the exponential modality causes an undesirable interaction between the two modalities. To solve the problem, we introduce an extension of intuitionistic multiplicative exponential linear logic with a modality combining the S4 modality and the exponential modality, and show that it admits a sound translation from IS4. Through the Curry–Howard correspondence we further obtain a Geometry of Interaction Machine semantics of the modal λ-calculus by Pfenning and Davies for staged computation. 2012 ACM Subject Classification Theory of computation → Linear logic; Modal and temporal logics; Proof theory; Type theory Keywords and phrases Linear logic, Modal logic, Girard translation, Curry–Howard correspondence, Geometry of Interaction, Staged computation Acknowledgements The authors thank Kazushige Terui for his helpful comments on our work and pointing out related work on multicolored linear logic, and Atsushi Igarashi for his helpful comments on earlier drafts. Special thanks are also due to anonymous reviewers of FSCD 2019 for their fruitful comments. This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. 1 Introduction Linear logic discovered by Girard [7] is, as he wrote, not an alternative logic but should be regarded as an “extension” of usual logics. Whereas usual logics such as classical logic and intuitionistic logic admit the structural rules of weakening and contraction, linear logic does not allow to use the rules freely, but it reintroduces them in a controlled manner by using arXiv:1904.10605v1 [cs.LO] 24 Apr 2019 the exponential modality ‘!’ (and its dual ‘?’). Usual logics are then reconstructed in terms of linear logic with the power of the exponential modalities, via the Girard translation. In this paper, we aim to extend the framework of linear-logical reconstruction to the (, ⊃)-fragment of intuitionistic modal logic S4 (IS4) by establishing what we call “modal linear logic” and an S4-version of Girard translation from IS4 into it. However, the crux to give a faithful translation is that a naive combination of the -modality and the !-modality causes an undesirable interaction between the inference rules of the two modalities. To solve the problem, we define the modal linear logic as an extension of intuitionistic multiplicative exponential linear logic with a modality ‘! ’ (pronounced by “bangbox”) that integrates ‘’ and ‘!’, and show that it admits a faithful translation from IS4. 2 A Linear-logical Reconstruction of Intuitionistic Modal Logic S4 Syntactic category Inference rule 0 Γ ` A Γ , A ` B !Γ ` A Formulae A, B, C ::= p | A B | !A Ax Cut !R ( A ` A Γ, Γ0 ` B !Γ ` !A Γ, A ` B Γ ` A Γ0, B ` C Γ, A ` B Γ ` B Γ, !A, !A ` B !W (R 0 (L !L Γ, !A ` B !C Γ ` A ( B Γ, Γ , A ( B ` C Γ, !A ` B Γ, !A ` B Figure 1 Definition of IMELL. dAe def def dΓe def dpe = p, dA ⊃ Be = (!dAe) ( dBe dΓe = {dAe | A ∈ Γ} Figure 2 Definition of the Girard translation from intuitionistic logic. As an application, we consider a computational interpretation of the modal linear logic. A typed λ-calculus that we will define corresponds to a natural deduction for the modal linear logic through the Curry–Howard correspondence, and it can be seen as a reconstruction of the modal λ-calculus by Pfenning and Davies [18, 5] for the so-called staged computation. Thanks to our linear-logical reconstruction, we can further obtain a Geometry of Interaction Machine (GoIM) for the modal λ-calculus. The remainder of this paper is organized as follows. In Section 2 we review some formalizations of linear logic and IS4. In Section 3 we explain a linear-logical reconstruction of IS4. First, we discuss how a naive combination of linear logic and modal logic fails to obtain a faithful translation. Then, we propose a modal linear logic with the ! -modality that admits a faithful translation from IS4. In Section 4 we give a computational interpretation of modal linear logic through a typed λ-calculus. In Section 5 we provide an axiomatization of modal linear logic by a Hilbert-style deductive system. In Section 6 we obtain a GoIM of our typed λ-calculus as an application of our linear-logical reconstruction. In Sections 7 and 8 we discuss related work and conclude our work, respectively. 2 Preliminaries We recall several systems of linear logic and modal logic. In this paper, we consider the minimal setting to give an S4-version of Girard translation and its computational interpretation. Thus, every system we will use only contain an implication and a modality as operators. 2.1 Intuitionistic MELL and its Girard translation Figure 1 shows the standard definition of the (!, ()-fragment of intuitionistic multiplicative exponential linear logic, which we refer to as IMELL. A formula is either a propositional variable, a linear implication, or an exponential modality. We let p range over the set of propositional variables, and A, B, C range over formulae. A context Γ is defined to be a multiset of formulae, and hence the exchange rule is assumed as a meta-level operation. A judgment consists of a context and a formula, written as Γ ` A. As a convention, we often write Γ ` A to mean that the judgment is derivable (and we assume similar conventions throughout this paper). The notation !Γ in the rule !R denotes the multiset {!A | A ∈ Γ}. Figure 2 defines the Girard translation1 from the ⊃-fragment of intuitionistic propositional 1 This is known to be the call-by-name Girard translation (cf. [12]) and we only follow this version in later discussions. However, we conjectured that our work can apply to other versions. Y. Fukuda and A. Yoshimizu 3 Syntactic category Inference rule Γ ` A Γ0, A ` B Ax Γ ` A Formulae A, B, C ::= p | A ⊃ B | A 0 Cut R A ` A Γ, Γ ` B Γ ` A Γ, A ` B Γ ` A Γ0, B ` C Γ, A ` B Γ ` B Γ, A, A ` B W ⊃R 0 ⊃L L C Γ ` A ⊃ B Γ, Γ , A ⊃ B ` C Γ, A ` B Γ, A ` B Γ, A ` B Figure 3 Definition of LJ. Syntactic category Reduction rule Types A, B, C ::= p | A ⊃ B | A (β ⊃)(λx : A.M) N M[x := N ] Terms M, N, L ::= x | λx : A.M | MN (β) let x = N in M M[x := N ] | M | let x = M in N Typing rule Ax Ax ∆; Γ, x : A ` x : A ∆, x : A;Γ ` x : A ∆; Γ, x : A ` M : B ∆; Γ ` M : A ⊃ B ∆; Γ ` N : A ⊃I ⊃E ∆; Γ ` (λx : A.M): A ⊃ B ∆; Γ ` MN : B ∆; ∅ ` M : A ∆; Γ ` M : A ∆, x : A;Γ ` N : B I E ∆; Γ ` M : A ∆; Γ ` let x = M in N : B Figure 4 Definition of λ. logic IL. For an IL-formula A, dAe will be an IMELL-formula; and dΓe a multiset of IMELL- formulae. Then, we can show that the Girard translation from IL to IMELL is sound. I Theorem 1 (Soudness of the translation). If Γ ` A in IL, then !dΓe ` dAe in IMELL. 2.2 Intuitionistic S4 We review a formalization of the (, ⊃)-fragment of intuitionistic propositional modal logic S4 (IS4). In what follows, we use a sequent calculus for the logic, called LJ. The calculus LJ used here is defined in a standard manner in the literature (e.g. it can be seen as the IS4-fragment of G1s for classical modal logic S4 by Troelstra and Schwichtenberg [23]). Figure 3 shows the definition of LJ. A formula is either a propositional variable, an intuitionistic implication, or a box modality. A context and a judgment are defined similarly in IMELL. The notation Γ in the rule R denotes the multiset {A | A ∈ Γ}. I Remark 2. It is worth noting that the !-exponential in IMELL and the -modality in LJ have similar structures. To see this, let us imagine the rules R and L replacing the symbol ‘’ with ‘!’. The results will be exactly the same as !R and !L. In fact, the !-exponential satisfies the S4 axiomata in IMELL, which is the reason we also call it as a modality. 2.3 Typed λ-calculus of the intuitionistic S4 We review the modal λ-calculus developed by Pfenning and Davies [18, 5], which we call → λ . The system λ is essentially the same calculus as λe in [5], although some syntax are changed to fit our notation in this paper. λ is known to correspond to a natural deduction system for IS4, as is shown in [18]. Figure 4 shows the definition of λ. The set of types corresponds to that of formulae of IS4. We let x range over the set of term variables, and M, N, L range over the set of terms. The first three terms are as in the simply-typed λ-calculus. The terms M and let x = M in N 4 A Linear-logical Reconstruction of Intuitionistic Modal Logic S4 is used to represent a constructor and a destructor for types A, respectively.
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