Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rule. GC] for example, is equivalent to require that f and g are both [#] functions, and that for all x, x v fgx, and x v gfx (here again, we just consider f and g as functions defined on the same poset) Galois connected operators have been also studied in the context of Linear Logic (see [16, 1, 13, 22]) and in related work by Lambek, e.g. 17] In contrast with these works, in which Galois properties are mixed with extra features guaranteeing, for example, a double negation law, we focus on the pure Galois properties and investigate the effects of adding Galois connected operators to the base ....

M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. The Journal of Symbolic Logic, 56(4):1403--1451, 1991.

....inequality in the characterization of residuation. When we cast this algebraic discussion in terms of categorial type logics the objects we will be considering are types, ordered by their derivability relation. Galois connected operators have been also studied in the context of Linear Logic [12,1,9,21] where they are intended to exhibit negation like behavior. This means that the Galois properties have to be mixed with extra features guaranteeing, for example, a double negation law ) A = In related work, Jim Lambek [13,14] considers algebraic structures he calls pregroups, where ....

Abrusci, V., Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic, The Journal of Symbolic Logic 56 (1991), pp. 1403-1451.

....a significant challenge: to get our results we use here for the first time some novel techniques, which constitute auniform and modular approach to cut elimination, contrary to what is possible in the sequent calculus. 1Introduction Non commutative logical operators have a long tradition [12, 22, 2, 13, 16, 3], and their proof theoretical properties havebeen studied in the sequent calculus [7] andinproof nets [8] Recent research has shown that the sequent calculus is not adequate to deal with very simple forms of non commutativity [9, 10, 21] On the other hand, proof nets are not ideal for dealing ....

V. Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linearpropositional logic. Journal of Symbolic Logic, 56(4):1403-- 1451, 1991.

.... science, like process calculi or concurrent constraint programming [3] In many cases, it appears that commutative linear logic is not enough expressive to specify processes having both asynchronous and synchronous behaviors and thus a possible solution can be the use of non commutative logic [2] or non commutative connectives to deal with sequentiality. Recently, a mixed version of classical propositional linear logic, which combines both commutative and non commutative connectives, has been proposed [1] with the motivation to de ne logical characterizations of synchronization mechanisms ....

V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403-1451, 1991.

....Our notation. Linear logic has been introduced by Girard [Gir] The inference rules of second order linear logic are represented on Table 1. Linear ane logic is linear logic with the weakening rule (see Table 2) T] Non commutative linear logic is linear logic without the permutation rule [Abr91]. Note than non commutative LL has two implications: right one and left one ( and ) We shall abbreviate second order linear logic and linear ane logic as LL2 and LLW2 correspondingly. And non commutative version of these logics as N LL2 and N LLW2. We shall use the abbreviations LL, LLW, N LL, ....

V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Jornal of Symbolic logic 56, 1403-1451. 1991.

....Several other connectives can be defined in this logic. The most popular ones are two linear implications, defined as A #B # # A # # B and B# A # # B # # A. Lemma 6.1 For any A # NFm the equalities # (A # ) A and ( # A) # = A hold true. Proof. Easy induction on the structure of A. In [1] V. M. Abrusci introduced a sequent calculus PNCL for the pure noncommutative classical linear propositional logic. In the same paper two one sided sequent calculi SPNCL and SPNCL # were introduced and it was proved that they are equivalent to PNCL. We shall use a slightly modified (but ....

....b A is derivable in SPNCL # . Proof. Both directions are proved using induction on derivation length. 7 Proof nets We define proof nets for the multiplicative fragment of the noncommutative classical linear propositional logic. The concept of proof net introduced here (an extension of that from [1]) appears to be mathematical folklore. We prove that a sequent is derivable if and only if there exists a proof net for this sequent. 12 Definition. For the purposes of this paper it is convenient to measure the length of a normal formula using the function : NFm # N defined in the ....

M. V. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic, The Journal of Symbolic Logic, 56(4):1403--1451, 1991.

....There is currently much interest in bringing together the tradition of categorial grammar, and especially the Lambek calculus [7] with the more recent paradigm of linear logic [5] to which it has strong ties. One active research area concerns the design of non commutative versions of linear logic [1, 8] which can be sensitive to word order while retaining the hypothetical reasoning capabilities of standard (commutative) linear logic that make it so well adapted to handling such phenomena as quantifier scoping [2] Some connections between the Lambek calculus and group structure have long been ....

V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4), 1991. 16

....is currently much interest in bringing together the tradition of categorial grammar, and especially the Lambek calculus [10] with the more recent paradigm of linear logic [8] to which it has strong ties. One active research area concerns the design of non commutative versions of linear logic [1, 14] which can be sensitive to word order while retaining the hypothetical reasoning capabilities of standard (commutative) linear logic that make it so well adapted to handling such phenomena as quantifier scoping [4] Some connections between the Lambek calculus and group structure have long been ....

Abrusci, V.: 1991, `Phase semantics and sequent calculus for pure noncommutative classical linear logic'. Journal of Symbolic Logic 56(4).

....and Information, ESSLLI 99, Utrecht, August 99 as part of the workshop on Resource Logics and Minimalist Grammars (C.Retore E. Stabler, organizers) 1 In this paper we propose a formal approach to lexical information in the perspective of the recent extensions of Non commutative Linear Logic [1, 2, 3, 4]. Following type logical grammars [17, 16] lexical items are treated as complex signs in which the phonological, syntactical and semantic dimensions are simultaneously considered and processed. We intend to show that the formal elegance and the appealing inferential properties of a type logical ....

....effect of cancelling a type A when it is preceded by its dual A or when it is followed by its dual A . We present some examples of the derivations that may be obtained in Non commutative Linear Logic for Wh movement (see the Appendix for the corresponding proof nets) 3 : 3 We refer to [1, 2, 4] for extended presentations of Non commutative Linear Logic, and particularly of the properties of its multiplicative fragment. 6 1. Whom did Elisa write a letter to 2. What did Elisa write 3. The student that admires that writer is registered in my course. 7 5 Appendix 5.1 Proof nets for ....

Abrusci, V. M. (1991), 'Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic', The Journal of Symbolic Logic, 56, IV, 1403-1451.

....logic into linear logic is for instance the local and optimal implementation of reduction which can be used for e.g. computing semantic recipes. The absence of structural rules enables the consideration of non commutative restrictions of linear logic, first introduced by Abrusci and Yetter [2, 99]. For instance one can rediscover the Lambek calculus as being exactly intuitionistic multiplicative linear logic [2, 81, 48, 75] Non commutative linear logic proofs i.e. parse structures can be viewed as linear logic proofs, even when the proof is lifted to the corresponding semantical types; ....

....computing semantic recipes. The absence of structural rules enables the consideration of non commutative restrictions of linear logic, first introduced by Abrusci and Yetter [2, 99] For instance one can rediscover the Lambek calculus as being exactly intuitionistic multiplicative linear logic [2, 81, 48, 75]. Non commutative linear logic proofs i.e. parse structures can be viewed as linear logic proofs, even when the proof is lifted to the corresponding semantical types; using the embedding of intuitionistic logic into linear logic, semantical terms which by Curry Howard isomorphism are ....

[Article contains additional citation context not shown here]

V. Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.

....and speci cation languages. Its proof theoretical study dates back to the Lambek calculus of 1958 [11] that is now recognised as one of the foundations of computational linguistics. The point is that the Lambek calculus is purely non commutative, as is Abrusci s noncommutative linear logic [2], which is a calculus obtained from linear logic by dropping the exchange rule. Therefore, the problem remains of nding calculi where commutative and non commutative connectives coexist (peacefully) and of getting the best of both worlds. Solutions exist already, and I will discuss them in the ....

Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56:1403-1451, 1991. 62 Alessio Guglielmi

....a distinct amount of information provided about the calculus, we show a Tarskian semantics based on, another sequential concept, monoids. This last issue is stronlgy influenced by the work of Girard [3] when providing a phase semantics for the classical linear logic, and the work of Abrusci [6], who investigate the sequent calculus and phase semantics of pure non commutative classical linear propositional logic. The name of our structure Quasi Phase Spaces is inherited from these works. 2 The Lambek Calculus The Lambek calculus was introduced by Lambek [2] in order to obtain ....

....) by application of the lemma. 8 Phase Semantics In this section we propose a phase semantics, that we called quasi phase semantics, based on Girard s phase semantics for the Classical Linear Logic [3] and Abrusci s phase semantics for Pure Non Commutative Classical Linear Propositional Logic [6], searching to implement a verificationistic approach for meaning in constructivism. To this end, we introduce our quasi phase space consisting of a non commutative monoid M, whose elements are called phases and the neutral element is called e (We should observe that we do not have the set of ....

V.M. Abrusci, Phase Semantics and Sequent Calculus for Pure NonCommutative Classical Linear Propositional Logic, in Journal of Symbolic Logic,1991, n. 56, p. 1403 - 1451.

....perturbation (or specialization) point of view. For instance, the non commutative linear logic, that seems suitable for various applications, is in fact linear logic without the commutativity property for the consequence relation for which speci c semantics and proof systems have been proposed [2]. But can proof search methods in this fragment be derived from the ones in commutative case or must tailored techniques be developed In the case of proof search based on proofs nets, one can naturally use the initial algorithm, designed for the commutative case, with a specialization of one ....

V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403-1451, 1991.

....this analogy, and suggests a general theory which we hope to explore in the future. The particular variant of linear logic that we will work with is the cyclic linear logic (CyLL) of Yetter [38] This variant is obtained by adding the cyclic exchange rule to the fully noncommutative logic of [3]. The corresponding version of proof net is also described in [38] This theory has subsequently been used substantially by Retor e in his work on linguistics [31] The Hopf algebra which provides our semantics is an example of the incidence algebras of [22, 33] It is also refered to as a shuffle ....

V.M. Abrusci, Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic, J. Symbolic Logic Vol. 56 (1991), pp. 1403-1456.

....calculus presentation of non commutative mLL without negation. Notice, in particular, the noncommutative restriction on cut: the usual cut rule separates into four rules. Since there is no negation we will be working with two sided sequents. This sequent calculus was first given by Abrusci in [A91]. A link between linearly distributive categories and linear logic is indicated by the resource sensitive character of the linear distributivities, as compared to an ordinary distribution. In [CS91] and [BCST] the connection is fully developed. To be more precise: there is an equivalence ....

....to the restriction to planar proof structures: that is, wires are not allowed to cross. To differentiate from the usual proof structures (as in [G87] we shall refer to our structures suggestively as circuit diagrams. Similar proof nets (without units or non logical axioms) were developed in [A91]. Although in this paper we shall primarily work with the graphical representation of circuit diagrams, it should be kept in mind that there is an underlying formalism supporting the proof structures. In particular the circuits are more than just graphs; there is a specific order to the ....

V.M. Abrusci, Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic. Journal of Symbolic Logic 56 (1991) 1403-- 1451.

....could naturally provide a connection (proof search) method for this logical fragment [7] In fact, automated theorem proving or verification can strongly benefit from such investigations on proof nets. In this paper, we consider the multiplicative fragment of Non Commutative Linear logic (NCMLL) [1,2] and we design an algorithm for automatic construction of non commutative proof nets that could be also viewed as a proof search method in this logical fragment. Such results are important for theorem proving in applications including planning, concurrency, sequentiality or computational ....

....commutativity of the multiplicative connectives. The aim to see what happens when we remove this rule from the linear sequent calculus leads to the study of the logic called non commutative linear logic. The sequent calculus and phase semantics in the propositional case have been investigated in [1] and other studies on non commutative classical linear logic were developed with a restricted form of exchange rule (cyclic exchange rule or schift rule) 12,21] The NMLL system [1] We present the multiplicative fragment of non commutative linear logic, called here NMLL, knowing that we can ....

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V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403--1451, 1991.

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Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.

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Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.

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Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.

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Vito Michele Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear logic. Journal of Symbolic Logic, 56(4):1403--1451, December 1991.

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V. M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403-1451, 1991.

No context found.

V. Michele Abrusci. Phase semantics and sequent calculus for pure non-commutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403-1451, December 1991.

No context found.

V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56(4):1403-1451, 1991.

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V.M. Abrusci, "Phase Semantics and Sequent Calculus for Pure NonCommutative Classical Linear Propositional Logic", The Journal of Symbolic Logic 56 No 4, 1991.

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V.M. Abrusci. Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic. Journal of Symbolic Logic, 56:1403--1451, 1991.

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