Martini, S. and Masini, A. (1994). A computational interpretation of modal proofs. In Wansing, H., editor, Proof Theory of Modal Logics. Kluwer. Workshop proceedings.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....study of modal logic via Kripke structures from the point of view of logical frameworks. In certain cases this can be simplified to obtain a formulation of natural deduction employing a stack of contexts, representing a path through the Kripke structure. Variations of this idea can be found in [MM94, PW95, DP99], including a very fine grained study of reduction in [GL96, GL97] These are natural for some applications of necessity, but it does not appear that similarly compact and elegant versions exist for possibility. 25 One particularly fruitful interpretation of 2A is as the intensional type for ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics. Kluwer, 1994. Workshop proceedings.

....of this paper is then as follows. In the following section, we start with L , an axiomatic formulation due to Stirling [13] for a small classical linear time temporal logic including fl. We then formulate a natural deduction system in a similar style to the modal systems of Martini and Masini [9], and prove that it has the same theorems as the axiomatic formulation. This allows us to directly apply the Curry Howard isomorphism to the natural deduction system, yielding the typed calculus with the fl operator in the types. In the second half of the paper we consider , which is ....

....of fl. Our natural deduction formulation uses a judgement annotated with a natural number n, representing the time of the conclusion and with each assumption A in Gamma also annotated by a time n. These are just like the levels in the modal natural deduction systems of Martini and Masini [9], and in fact our system is exactly the same as their rules for modal K, except that because of linearity we do not need any restriction on the introduction rule for fl. Our rules for the non temporal fragment are relatively standard for natural deduction for pure classical logic, which will later ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics. Kluwer, 1995. To appear.

....K. After that we extend our work on natural deduction and term assignment systems, with the further correspondence between typed calculus and category theory, that is usually referred as the extended Curry Howard isomorphism . Other approaches (e.g. Basin et al.[BMV98] Martini and Masini[MM96]) that tie in the semantics of modal logics (in terms of possible worlds) with their sytanctic presentation have been devised. We do not say much about this line of work here. 2 The Logical System 2.1 Sequent Calculus and Axiomatic System We take the sequent calculus described below as de ning ....

A. Massini and S. Martini. A Computational Interpretation of Modal Proofs. In Proof theory of Modal Logic, H. Wansing (editor), 213-241, 1996.

....familiar with Hilbert style systems, might be interested in proof editors supporting this style. Some claims (or disclaimers) on our work are in order. Our objective is not that of extending to modal logics the proposition as types , generalized # terms as proofs paradigm, as is the case in [21, 26]. We explore, rather, the possibility of extending to modal logics the judgements as dependent types , # terms as ND proofs paradigm of [15] To this end we do not try to invent radically new deductive systems or new proof figures as in [21, 26] possibly using special extensions of the ....

.... # terms as proofs paradigm, as is the case in [21, 26] We explore, rather, the possibility of extending to modal logics the judgements as dependent types , # terms as ND proofs paradigm of [15] To this end we do not try to invent radically new deductive systems or new proof figures as in [21, 26], possibly using special extensions of the # calculus. These systems, albeit very interesting for the new insights that they can provide in the conceptual understanding of modality, and the conceptual meaning of the corresponding normalization procedure, are beyond the scope of this paper. These ....

[Article contains additional citation context not shown here]

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994.

....with abstraction. Since modal logic S4 and all standard term constructors can be represented by proof polynomials, the Logic of Proofs can also emulate modal calculi. As it was shown in [8] 11] the intuitionistic version of LP naturally realizes the modal calculus for IS4 ( 23] [68], 84] cf. also [27] and thus supplies modal terms with standard provability semantics. This EXPLICIT PROVABILITY 31 result may be considered as a more general abstract version of the CurryHoward isomorphism which relates terms types with proofs formulas. x11. First order case. Theories based ....

S. Martini and A. Masini, A computational interpretation of modal proofs, Proof theory of modal logics. workshop proceedings (H. Wansing, editor), Kluwer, 1994.

....p( y) such that ILPG y : Gamma ) p( y) B. Since both modal logic S4 and all standard term constructors can be emulated by proof polynomials, the Logic of Proofs can also emulate modal calculi. As it was shown in [6] 7] ILPG naturally realizes the modal calculus for IS4 ( 10] [45], 60] cf. also [15] and thus supplies modal terms with standard provability semantics. This result may be considered as a more general abstract version of the well known Curry Howard isomorphism which relates terms types with proofs formulas. 10 Discussion Roughly speaking, LP is an advanced ....

S. Martini and A. Masini,"A computational interpretation of modal proofs", in Wansing, ed., Proof Theory of Modal Logics, (Workshop proceedings), Kluwer, 1994.

....in some fields of proof Department of Mathematics, Cornell University, Ithaca NY, 14853 email:artemov math.cornell.edu; Steklov Mathematical Institute, Russian Academy of Sciences, 42 Vavilova str. Moscow 117966, Russia 1 theory, automated deductions and the logical theory of computation. cf. [13] and [15] for an impressive current list of applications of proof motivated calculi) In the current paper we show how to represent the typed calculi in LP directly. Under this embedding, the formation rules for the terms become admissible rules of LP . In fact, the calculus for Int can be ....

....Horn formulas only, without proof checker , choice operations, or nesting of proof terms. The Intuitionistic Logic of Proofs ILP has a natural provability semantics with respect to Heyting Arithmetic HA; ILP is also a natural dynamic counterpart of the intuitionistic modal logic IS42 (cf. 4] [13], 15] All these give a provability semantics for the typed calculus and for the modal calculus ( 13] 15] 2 Logic of Proofs The language of LP contains sentence variables p 0 ; p n ; boolean constants ; proof variables x 0 ; x n ; boolean connectives ....

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S. Martini and A. Masini,"A computational interpretation of modal proofs", in Wansing, ed., Proof Theory of Modal Logics, (Workshop proceedings), Kluwer, 1994.

....for Int) can be realized in a small fragment of ILPN consisting of pure derivations only. We already have enough ingredients to demonstrate that the Logic of Proofs can emulate modal calculi. 52 Under IS42 we mean the intuitionistic modal logic on the basis of S4, introduced in [8] cf. also [30], 42] An inspection of the proof of theorem 6.2 (realization of modal logic) shows that this theorem holds also for ILP instead of LP and IS42 instead of S4. In other words, the intuitionistic logic of proofs is an explicit version of IS42 in the same sense that LP is an explicit version of ....

....shows that this theorem holds also for ILP instead of LP and IS42 instead of S4. In other words, the intuitionistic logic of proofs is an explicit version of IS42 in the same sense that LP is an explicit version of S4. We will show how ILPG naturally emulates the modal calculus for IS42 ( 8] [30], 42] cf. also [11] and thus supplies modal terms with standard provability semantics. 9.9 Theorem. Realization of modal calculus) There is an effective step by step realization r of any derivation x : Gamma ) t( x) A in the term calculus for IS42 as a derivation of x : Gamma r ....

S. Martini and A. Masini,"A computational interpretation of modal proofs", in Wansing, ed., Proof Theory of Modal Logics, (Workshop proceedings), Kluwer, 1994.

....for can also be realized as admissible rules in LPGi (cf. 2] 3] Since both modal logic IS4 and all standard term constructors can be emulated by proof polynomials, LPi can also emulate modal calculi. As it was shown in [2] 3] LPGi naturally realizes the modal calculus for IS4 ( 4] [6], 7] cf. also [5] 6 Deep realization of modalities by combinatory ( terms Realization algorithm from Section 4 recovers combinatory terms for every occurrence of modalities in any IS4 derivation. Natural fragments of S4 may be be now regarded as implicit description of the corresponding ....

....Here is a typical example. Consider the sequent 2F ) 22F derivable in IS4. There is no IS4 derivation of this sequent that ends with the necessitation rule F ) 2F 2F ) 22F ; since F ) 2F is not derivable in IS4. Hence there is no modal calculus realization of 2F ) 22F in the sense of [4] [6], 7] i.e. there is no modal term t(x) such that the modal calculus derives x : F ) t(x) 2F . On the other hand, the formula 2F ) 22F admits a relization in LPi, namely x : F ) x : x : F where x is the proof checker polynomial. 7 Standard provability interpretation of LPi Within this ....

S. Martini and A. Masini,\A computational interpretation of modal proofs", in Wansing, ed., Proof Theory of Modal Logics, (Workshop proceedings), Kluwer, 1994.

.... A number of authors have considered the question of providing natural deduction formulations of (intuitionistic) modal logics (some in response to our earlier work [6] These include Benevides and Maibaum [2] Bull and Segerberg [9, pages 29 30] Davies and Pfenning [13] Martini and Masini [26], Mints [27, Pages 221 294] and Simpson [37] However they all use extensions of one form or another to the nature of natural deduction (for example, by indexing formulae with possible worlds information) Again we reiterate the conceptual simplicity of our proposal we use no new features of ....

S. Martini and A. Masini. A computational interpretation of modal proofs. Technical Report TR--27/93, Dipartimento di informatica, Universita di Pisa, November 1993.

....providing an additional construct for the execution of code fragments, and cross stage persistence. Cross stage persistence is the ability to bind a variable at stage n and use it at stage n 1. Both features are important for pragmatic reasons. 2 and fl have clean, logical foundations [5, 4, 7, 6]: there is a Curry Howard isomorphism between fl and linear time temporal logic, and between 2 and modal logic S4. MetaML emphasizes the pragmatic importance of being able to combine cross stage persistence, evaluation under lambda (or symbolic computation ) and being able to execute ....

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logic. Kluwer, 1996.

....logic via Kripke structures from the point of view of logical frameworks. In certain cases this can be simplified to obtain a formulation of natural deduction employing a stack of contexts, representing a path through the Kripke structure. Variations of this idea can be found in several papers (Martini and Masini, 1994; Pfenning and Wong, 1995; Davies and Pfenning, 2000) including a very fine grained study of reduction (Goubault Larrecq, 1996; 1997) These are natural for some applications of necessity, but it does not appear that similarly compact and elegant versions exist for possibility. One particularly ....

Martini, S. and Masini, A. (1994). A computational interpretation of modal proofs. In Wansing, H., editor, Proof Theory of Modal Logics. Kluwer. Workshop proceedings.

....persistence is the ability to use at one level a variable declared at a lower level. Both features are important for pragmatic reasons. ffl AIM [11] revising and extending MetaML with a closed code type for expressivity and modularity. 2 and fl already have clean, logical foundations (see [4, 5, 7, 6]) there is a Curry Howard isomorphism between fl and linear time temporal logic, and between 2 and modal logic S4. MetaML had no such foundations, nor the formal hygiene they often promote. Indeed, MetaML had a complex type system and a number of ad hoc restrictions (see [12] which ....

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logic. Kluwer, 1996.

....study of modal logic via Kripke structures from the point of view of logical frameworks. In certain cases this can be simplified to obtain a formulation of natural deduction employing a stack of contexts, representing a path through the Kripke structure. Variations of this idea can be found in [MM94, PW95, DP99], including a very fine grained study of reduction in [GL96, GL97] These are natural for some applications of necessity, but it does not appear that similarly compact and elegant versions exist for possibility. 25 One particularly fruitful interpretation of 2A is as the intensional type for ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics. Kluwer, 1994. Workshop proceedings.

....an additional construct for the execution of code fragments, and cross stage persistence. Cross stage persistence is the ability to bind a variable at stage n and use it at stage n 1. Both features are important for pragmatic reasons. 2 and fl have clean, logical foundations [DP96, Dav96, Mas93, MM96]: there is a Curry Howard isomorphism between fl and linear time temporal logic, and between 2 and modal logic S4. MetaML emphasizes the pragmatic importance of being able to combine cross stage persistence, evaluation under lambda (or symbolic computation ) and being able to execute ....

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logic. Kluwer, 1996.

....has been proven correct in [PW95] As an example of this translation, it maps the above definition of power to the previous explicit one. It is important to note that the operational semantics induced by the translation is very different from the natural one defined directly on Mini ML 2 . In [MM94] a simple reduction semantics for a system similar to our implicit system is introduced which does not reflect binding time separation in any way. It is instead used to prove a Church Rosser theorem and strong normalization for a pure modal calculus. 4 A Two level Language In this section we ....

....that the translation of a two level term can always be type checked only using the tpi unbox and tpi pop rules when tpi unbox immediately follows tpi pop. This corresponds to a weaker modal logic, K, in which we drop the assumption in S4 that the accessibility relation is reflexive and transitive [MM94]. In fact, we can define a language Mini ML 2 K by replacing the unbox and pop constructors with one equivalent to unbox 1 as in [MM94] Then, Mini ML 2 K closely models Mini ML 2 , but permits an arbitrary number of phases, each of which can only execute the code generated by the immediately ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. Workshop proceedings, To appear.

....of LP formulas. A straightforward induction on a derivation in LP demonstrates that if LP F , then S4 F o . As it was shown in [3] 5] the converse also holds. Namely, LP suffices to realize any S4 theorem. Under IS4 we mean the intuitionistic modal logic on the basis of S4 (cf. 7] [16], 21] where IS4 was studied under the name IS42 ) Basically the same algorithm (below) provides a realization of IS4 in ILP. 3.1 Example. IS4 (2A 2B) 2(AB) In ILP the corresponding derivation is 1. A; B AB, by propositional logic 2. x : A; y : B t(x; y) AB) by Lifting 3. x : Ay ....

....ILPN (as well as calculus for Int) can be realized in a small fragment of ILPN consisting of pure derivations only. We already have enough ingredients to demonstrate that the Logic of Proofs can emulate modal calculi. We will show how ILPG naturally emulates the modal calculus for IS4 ( 7] [16], 21] cf. also [10] and thus supplies modal terms with standard provability semantics. 6.9 Theorem. Realization of modal calculus) There is an effective step by step realization r of any derivation x : Gamma ) t( x) A in the term calculus for IS4 as a derivation of x : Gamma r ) ....

S. Martini and A. Masini,"A computational interpretation of modal proofs", in Wansing, ed., Proof Theory of Modal Logics, (Workshop proceedings), Kluwer, 1994.

....has been proven correct in [PW95] As an example of this translation, it maps the above definition of power to the previous explicit one. It is important to note that the operational semantics induced by the translation is very different from the natural one defined directly on Mini ML 2 . In [MM94] a simple reduction semantics for a system similar to our implicit system is introduced which does not reflect binding time separation in any way. It is instead used to prove a Church Rosser theorem and strong normalization for a pure modal calculus. 4 A Two level Language In this section we ....

....that the translation of a two level term can always be type checked only using the tpi unbox and tpi pop rules when tpi unbox immediately follows tpi pop. This corresponds to a weaker modal logic, K, in which we drop the assumption in S4 that the accessibility relation is reflexive and transitive [MM94]. In fact, we can define a language Mini ML 2 K by replacing the unbox and pop constructors with one equivalent to unbox 1 as in [MM94] Then, Mini ML 2 K closely models Mini ML 2 , but permits an arbitrary number of phases, each of which can only execute the code generated by the ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics. Kluwer, 1994. Workshop proceedings, To appear.

....staging. This simplifies the study of staging properties of Mini ML 2 e , but it also makes it difficult to directly relate it to previous work on staged languages, such as twolevel languages [NN92] We thus consider a more implicit formulation of S4 motivated by its Kripke semantics following [MM94, PW95] and then augment it as before to form Mini ML 2 . With some syntactic sugar, Mini ML 2 is intended to serve as the basis for a conservative extension of ML with practical means to express and check staging of computation. The operational semantics of Mini ML 2 is given by a type preserving ....

....maps the definition of power from Section 5.3 to the one in Section 3.4. Note that the restructuring achieved by the compiler is similar to a staging transformation [JS86] The operational semantics induced by the translation is different from some obvious ones defined directly on Mini ML 2 . In [MM94], for example, a simple reduction semantics is introduced for a system similar to the pure fragment of our implicit system. It does not reflect staging, and is instead used to prove a Church Rosser theorem and strong normalization for a pure modal calculus. Similarly, in [PW95] an algorithm for ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. Workshop proceedings.

....evaluation semantics is listed in Figure 4. 4 System 2 : modal calculus for closed code The system 2 was initially developed by Pfenning and Wong [24] to be a term calculus for the modal logic S4, which is more suitable for type checking than a previous formulation by Martini and Masini [19]. Through CurryHoward isomorphism between proofs and programs, 2 gives a logicalexplanation of program staging, and is therefore well suited as a staging programming language. We first go over some basic ideas of modal logic S4 and how it is related to program staging (Section 4.1) and then we ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logic, Kluwer, 1994. Workshop proceedings.

....we expand this solution and present new alternatives, using judgements on proofs or exploiting the underlying calculus structure of the metalanguage. Our objective is not that of extending to modal logics the proposition as types , generalized terms as proofs paradigm, as is the case in [14, 17]. We explore, rather, the possibility of extending to modal logics the judgements as dependent types , terms asND proofs paradigm of [7] To this end we do not try to invent radically new deductive systems or new proof figures as in [14, 17] possibly using special extensions of the ....

.... terms as proofs paradigm, as is the case in [14, 17] We explore, rather, the possibility of extending to modal logics the judgements as dependent types , terms asND proofs paradigm of [7] To this end we do not try to invent radically new deductive systems or new proof figures as in [14, 17], possibly using special extensions of the calculus. These systems, albeit very interesting for the new insights that they can provide in the conceptual understanding of modality, are beyond the scope of this paper, because they cannot be used as the basis of an encoding of modal logics in ....

[Article contains additional citation context not shown here]

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994.

....for example, in the rule for the universal quantifier in the module above. As another example illustrating the generality of this approach, we sketch a presentation in rewriting logic of the 2 sequent calculus defined by A. Masini and S. Martini in order to develop a proof theory for modal logics [64, 61]. In their approach, a 2 sequent is an expression of the form Gamma Delta, where Gamma and Delta are not lists of formulas as usual, but they are lists of lists of formulas, so that sequents are endowed with a vertical structure. For example, A; B C D E; F G is a 2 sequent, which will ....

S. Martini and A. Masini, A computational interpretation of modal proofs, Technical report TR-27/93, Dipartimento di Informatica, Universit`a di Pisa, November 1993.

....Typically, they are described axiomatically in the style of Hilbert or via sequent calculi. However, the Curry Howard isomorphism between proofs and terms is most poignant for natural deduction, so natural deduction formulations of modal and linear logics have also been the subject of research [20,1,5,3,4,22,13]. This work is supported by NSF Grant CCR 9303383 and the Advanced Research Projects Agency under ARPA Order No. 8313. c fl 1995 Elsevier Publishers B. V. and the author. F. Pfenning and H. C. Wong Although most of the researchers in this area agree on the (potential) applications of modal ....

....checking algorithm. We then prove subject reduction and the existence of a canonical form for every well typed term. We omit a simpler and not so relevant strong normalization result. A similar, but more verbose system motivated by proof theoretic considerations has been given by Martini Masini [13]. They investigate normalization and the Church Rosser property which is complementary to our own meta theoretic study. A similar approach can be used beyond S4 to give an analogous formulation of intuitionistic linear logic which is beyond the scope of this paper. A related system without ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. To appear.

....more familiar with Hilbert style systems, might be interested in proof editors supporting this style. Some claims (or disclaimers) on our work are in order. Our objective is not that of extending to modal logics the proposition as types , generalized terms as proofs paradigm, as is the case in [21, 26]. We explore, rather, the possibility of extending to modal logics the judgements as dependent types , terms as ND proofs paradigm of [15] To this end we do not try to invent radically new deductive systems or new proof figures as in [21, 26] possibly using special extensions of the ....

.... terms as proofs paradigm, as is the case in [21, 26] We explore, rather, the possibility of extending to modal logics the judgements as dependent types , terms as ND proofs paradigm of [15] To this end we do not try to invent radically new deductive systems or new proof figures as in [21, 26], possibly using special extensions of the calculus. These systems, albeit very interesting for the new insights that they can provide in the conceptual understanding of modality, and the conceptual meaning of the corresponding normalization procedure, are beyond the scope of this paper. These ....

[Article contains additional citation context not shown here]

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994.

....all proof terms for intuitionistic S4 set theory. Finally, we suggest that geometric like realizations functors provide a recipe to build other models of intuitionistic propositional S4 proof terms. 1 Introduction There are now several different proof term languages for intuitionistic S4 [BdP92, BdP96, PW95, MM96, GL96a, GL96b], with applications in partial evaluation [DP96, WLP98] in higher order abstract syntax [Lel97] etc. These calculi are related, in that we can translate from one to any other. Some of these calculi are even intertranslatable in an untyped setting. Our goal in this paper is to develop a few ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logic, pages 213--241. Kluwer, 1996.

....binding time analysis and temporal logic. The structure of this paper is then as follows. In the following section, we start with a natural deduction formulation of intuitionistic linear time temporal logic with and fl. Our system has a similar style to the modal systems of Martini and Masini [13]. We then verify that our fl operator is the same as that considered elsewhere by showing that adding negation and classical reasoning leads to a system equivalent to L fl , an axiomatic formulation due to Stirling [17] for a small classical linear time temporal logic including fl. We then ....

....logic containing only fl and . Our formulation uses a judgement annotated with a natural number n, representing the time of the conclusion, and with each assumption A in G also annotated by a time n. These are just like the levels in the modal natural deduction systems of Martini and Masini [13], and in fact our system is exactly the same as their rules for modal K, except that because of linearity we do not need any restriction on the assumptions used in the introduction rule for fl. Our rules for the non temporal fragment are completely standard. A n in G V G n A G; A n 1 ....

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logics. Kluwer, 1996. To appear.

.... of modal logic and labels ; i.e. instead of A, we consider w:A, where w is a world , and A iff 8w 2 W( w:A) This provides a language in which we can formulate 2 Different approaches to proof under assumption in modal logics, based on modified sequent systems, have been proposed, e.g. [3, 7, 9, 17, 32, 33, 57]. We do not consider such systems here, but see [56] for detailed comparison and discussion. 3 We use the vocabulary of [2] which should be consulted for a technical discussion of consequence relations, and which notes (x5.5) that every ordinary, pure single conclusioned ND system can, e.g. ....

S. Martini and A. Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof Theory of Modal Logic, pages 213--241. Kluwer Academic Publishers, Dordrecht, 1996.

....operational semantics along the lines of Mini ML [CDDK86] Mini ML 2 e is somewhat awkward because it requires the broad syntactic structuring of the program to directly reflect staging. We thus consider a more implicit formulation of S4 directly motivated by its Kripke semantics following [MM94, PW95] and then augment it as before to form Mini ML 2 . With some syntactic sugar, Mini ML 2 is intended to serve as the basis for a conservative extension of ML with a practical means to express and check staging of computation. The operational semantics of Mini ML 2 is given by a ....

....the definition of power from Section 3.3 to the one in Section 2.4. Note that the restructuring achieved by the compiler is similar to a staging transformation [JS86] The operational semantics induced by the translation is very different from the obvious ones defined directly on Mini ML 2 . In [MM94] a simple reduction semantics is introduced for a system similar to the pure fragment of our implicit system. It does not reflect staging, and is instead used to prove a Church Rosser theorem and strong normalization for a pure modal calculus. Similarly, in [PW95] an algorithm for converting pure ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. Workshop proceedings.

....Linear Logic and Light Linear Logic. 1 Introduction There is not a single reason for proposing an extension of Gentzen s format for sequents; often, it is rather a blend of distinct issues to inspire the design of a particular system [Kri63, Dos85, GdQ92a, Wan94] The 2 sequent approach [Mas92, Mas93, MM95a, MM95b] is not an exception to that rule. The original goal was notational: providing symmetric and local (i.e. context free) rules for the minimal deontic logic KD. But soon, we discovered that 2 sequents could be used as a uniform tool for several logical systems. In particular, starting with a common ....

....could be used as a uniform tool for several logical systems. In particular, starting with a common core the logical rules we could shift from one system to another (e.g. from KD to S4) just changing the way in which syntactical objects are manipulated say, the structural rules. In [MM95b], we applied this scalar approach to the modal logics in the K S4 range. Here, we shall apply the same methodology to Girard s Linear Logic (LL) This study started in [MM95a] where we gave natural deduction style presentations for certain LL fragments. As a result, we discovered some unsuspected ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics, volume 2 of Applied Logic Series, pages 213--241, Kluwer, 1996.

....during the cut elimination process. A box is a global, explicitly given notion: each occurrence of an of course connective in the proof net comes together with a certain subgraph, its box. In [MM95] applying to linear logic ideas and techniques previously developed for modal logic, see [MM96] we discovered that a different, straightforward approach was possible, labeling with natural number indexes the formulas of the proof net. The approach of [MM95] moreover, allowed a clear recognition, at any time, of the boundary of the box. This suggested our new, simple absorption rule. The ....

....during an optimal reduction (see Section 3.3) We stress the presence of the absorption rule (B abs ) corresponding to the case when the mux reaches the border of a box (through one of its secondary doors) and has therefore exhausted its job. It is motivated by the proof theoretical work in [MM95,MM96] (see also Section 6) and it is a special case of a safe reduction [Asp95] Remark 7 Any rule of opt , but B abs , is admissible with respect to the re10 A k A k A k cut ax B ide A k O AOB k A k B k ffl A ffl B k A k B k cut B mul A k B k A k B k ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics, pages 213--241. Kluwer, 1996.

....during the cut elimination process. A box is a global, explicitly given notion: each occurrence of an of course connective in the proof net comes together with a certain subgraph, its box. In [MM95] applying to linear logic ideas and tech niques previously developed for modal logic, see [MM96] we discovered that a different, straightforward approach was possible, labelling with natural number indexes the formulas of the proof net. The approach of [MM95] moreover, allowed a clear recognition, at any time, of the boundary of the box. This suggested our new, simple absorption rule. The ....

....during an optimal reduction (see Section 2.2) We stress the presence of the absorption rule (B abs ) corresponding to the case when the mux reaches the border of a box (through one of its secondary doors) and has therefore exhausted its job. It is motivated by the proof theoretical work in [MM95, MM96] (see also Section 5) and it is a special case of a safe reduction [Asp95] A k cut A k ax A k B ide A k A k B k A k Omega B k AB k cut A k Omega B k B mul A k cut A k B k cut B k A k 1 A k 1 Delta Delta Delta A k r A k cut A k B exp A k 1 k A k ....

[Article contains additional citation context not shown here]

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1996. To appear.

....Elementary cut elimination. Light Linear Logic. Elementary Linear Logic. 1 Introduction The rationale for extending Gentzen s format for sequents is not unidimensional. It is often a blend of several issues that inspires the design of a particular system [11, 2, 3, 16, 1] The 2 sequent approach [12, 13, 14, 15] is not an exception. Its original goal was notational: providing symmetric and local (i.e. context free) rules for the minimal deontic logic KD. However, we discovered soon that 2 sequents could be used as a uniform tool for several logical systems. In particular, starting from a common ....

....could be used as a uniform tool for several logical systems. In particular, starting from a common core the logical rules we could shift from one system to another (e.g. from KD to S4) just changing the way in which syntactical objects are manipulated say, the structural rules. In [15], we applied this scalar approach to the modal logics in the K S4 range. Here, we shall apply the same methodology to Girard s Linear Logic (LL) This study started in [14] where we gave a natural deduction style presentation 1 Partially done while at Institute for Research in Cognitive Science ....

Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics, pages 213--241. Kluwer, 1996. volume 2 of Applied Logic Series.

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Martini, S. and Masini, A. (1994). A computational interpretation of modal proofs. In Wansing, H., editor, Proof Theory of Modal Logics. Kluwer. Workshop proceedings.

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Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. To appear.

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Martini, S. and Masini, A. 1996. A computational interpretation of modal proofs. In H. Wansing Ed., Proof theory of Modal Logics, pp. 213--241. Kluwer. Workshop proceedings.

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S. Martini and A. Masini, A computational interpretation of modal proofs, Proof theory of modal logics. workshop proceedings (H. Wansing, editor), Kluwer, 1994.

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Simone Martini and Andrea Masini. A computational interpretation of modal proofs. In H. Wansing, editor, Proof theory of Modal Logics. Kluwer, 1994. Workshop proceedings.

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A. Massini and S. Martini. A Computational Interpretation of Modal Proofs. In Proof theory of Modal Logic, H. Wansing (editor) , 213-241, 1996. 14

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