A. Barber, P. Gardner, M. Hasegawa, G. Plotkin. From action calculi to linear logic. In Annual Conference of the European Association for Computer Science Logic (CSL'97). Springer LNCS 1414, pages 78-97, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....a logic, though, the emphasis is on higher order features: The associated notion is that of controls, that is, of generalized parametric operators. Controls allow modeling some primitive operators of process calculi, such as prefixing . We recall here the presentation of action calculi given in [2,32]. It is slightly different from the original presentation in [64] regarding the choice of basic operators and axioms, but it is now considered standard, and it is better suited for our purposes. In addition, we freely use the notation introduced in Section 2 for monoidal and gs monoidal theories. ....

A. Barber, Ph. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. In M. Nielsen and W. Thomas, editors, Computer Science Logic, volume 1414 of Lect. Notes in Comp. Science, pages 78--97. Springer Verlag, 1998.

....a logic, though, the emphasis is on higher order features: The associated notion is that of controls, that is, of generalized parametric operators. Controls allow modeling some primitive operators of process calculi, such as prefixing . We recall here the presentation of action calculi given in [2,32]. It is slightly different from the original presentation in [64] regarding the choice of basic operators and axioms, but it is now considered standard, and it is better suited for our purposes. In addition, we freely use the notation introduced in Section 2 for monoidal and gs monoidal theories. ....

A. Barber, Ph. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. In M. Nielsen and W. Thomas, editors, Computer Science Logic, volume 1414 of LNCS, pages 78--97. Springer Verlag, 1998.

.... by Power and Robinson [54] These categories are extensively used as a semantical model in the community of action calculi, since they provide a categorical characterisation of some sort of data flow structures (see Hasegawa [26] and Jeffrey [34] we refer to Pavlovic [50] and to Barber et al..ii [3] for an overview. A combination of their approach and ours, with a specific application to term rewriting, is due to Miyoshi [49] 10 A multi algebra A over a one sorted signature Sigma has a carrier A and, for each operator f 2 Sigma n , a function fA : A n P(A) where P(A) is the ....

A. Barber, Ph. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. In M. Nielsen and W. Thomas, editors, Computer Science Logic, volume 1414 of LNCS, pages 78--97. Springer Verlag, 1998.

....related stream of research is the work on types for process calculi. It would be possible to encode interaction nets as terms in a process calculus, and then apply the corresponding type systems to the encodings. Alternatively, using the ideas of the type systems for process calculi (see e.g. [17, 8]) one could directly design a type system for interaction nets. For the reverse direction, we hope that the ideas presented in this paper can be used to define new type systems for process calculi, in particular for the ones that are based on graphs, like the calculus of Yoshida [30] which ....

A. Barber, P. Gardner, M. Hasegawa, and G. Plotkin. From Action Calculi to Linear Logic. In Proceedings of CSL'97, Lecture Notes in Computer Science, to appear. Springer-verlag, 1997.

....category C 0 to which C (fully and) faithfully embeds such C 0 is obtained, for example, by applying the Chu construction [8] to Set C op . In general this kind of model construction technique is useful for showing the conservativity of syntactic translations, see for example [18, 5]. A much harder property to show is that the translation is full, i.e. if an expression of the target theory has types which are definable in the source theory, it is already definable in the source theory. Equivalently it amounts to the fullness of I. We say that a translation between type ....

Barber, A., Gardner, P., Hasegawa, M., and Plotkin, G. (1998), From action calculi to linear logic, in "Computer Science Logic (CSL'97), Selected Papers", Lecture Notes in Computer Science, Vol. 1414, pp. 78--97, Springer-Verlag.

....from the action calculus to DILL. If we have only non parameterized constants, Basic Lemma applied to the canonical logical predicate implies that the translation is full. In fact we can deal with parameterized constants (control operators) as well (see [13] so together with the conservativity [4] we have the full completeness of DILL (LNL) over (static) action calculi. ut 6 Related Work, Further Work 6.1 Categorical Logical Predicates Our treatment of logical predicates in category theoretic framework is inspired by Hermida s work on brations and logical predicates [14] and also in ....

Barber, A., Gardner, P., Hasegawa, M., and Plotkin, G. (1998), From action calculi to linear logic, in \Computer Science Logic (CSL'97), Selected Papers", Springer LNCS 1414, pp. 78-97.

....j (s Omega t) Omega u for terms s, t and u (strict associativity) 2 The term K( x 1 )t 1 ; x r )t r ; t) binds the variables from sequence x i in t i . Plotkin has pointed out that this can be viewed as a variant of Aczel s general binding operators [Acz80] This issue is discussed in [BGHP98] and a concrete example is given in Example 3.12. The term let x be s in t binds the variables from sequence x in t. We write tfu=xg for the standard capture free substitution. Definition 3.2 A term is well typed if it can be shown to annotate a sequent using the rules: Gamma; x p x : p ....

.... y be t in u = let y be t in let x be s in u : n for Gamma s : m 1 , Gamma t : m 2 and Gamma; x m1 ; y m2 u : n 2 Remark 3.6 An alternative choice is to work with a two context type theory, whose sequents have the form Gamma; Sigma t : n. Such a type theory is introduced in [BGHP96, BGHP98]. It has a direct connection with the corresponding action calculus, in that the above sequent corresponds to an action a : j Sigmaj n with free names contained in Gamma. It also gives a simpler connection with intuitionistic linear type theories with two contexts [Ben95, BP98] From the ....

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Barber, A., Gardner, P., Hasegawa, M., and Plotkin, G. (1998), From action calculi to linear logic, in "Proceedings, Computer Science Logic (CSL'97)", Lecture Notes in Computer Science, Springer-Verlag, Berlin/New York.

....and those which cannot. Since action calculi are completely determined by the closed fragment [Gar95,Gar96] our type theories are as expressive as the corresponding action calculi. It is possible to work with type theories which relate to action calculi directly (see Remark 16, BGHP96] and [BGHP97]) the type theories given here are simpler. 3.1 First Order Theory Let us fix a signature K = P; K) as introduced in the last section. The type theory T(K) to be introduced below, consists of sequents of the form Gamma t : n, where the context Gamma = x p1 1 ; x pm m is a ....

....j (s Omega t) Omega u for terms s, t and u (strict associativity) 2 The term K( x 1 )t 1 ; x r )t r ; t) binds the variables from sequence x i in t i . Plotkin has pointed out that this can be viewed as a variant of Aczel s general binding operators [Acz80] and this issue is discussed in [BGHP97]. See Example 15 for a concrete example. The term let x be s in t binds the variables from sequence x in t. We allow ff conversion, and write tfu=xg for the standard capture free substitution. Definition 8. A term is well typed if it can be shown to annotate a sequent using the following rules. ....

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A. Barber, P. Gardner, M. Hasegawa and G. Plotkin, From action calculi to linear logic. Submitted, 1997.

....of computation . Hasegawa also extends the results to account for reflexion [Has97] In particular, he shows that the reflexion operator corresponds to adding a trace operator, due to Joyal, Street and Verity [JSV96] to the symmetric monoidal category S. Barber, Gardner, Hasegawa and Plotkin [BGHP97] have also given a direct type theoretic presentation of action calculi, with sequents of the form x; y t : n where the names x and y are kept separate: the x behave in an intuitionistic fashion, and the y in a linear fashion. This type theory has a sound translation in Benton s type ....

A. Barber, P. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. Presented at Computer Science Logic, Aarhus, and accepted for publication, 1997.

....for every tensor arity m. The indexing arity records the arity of the abstracted names. For example, the action combinator corresponding to the above term is in p Omega q (id p Omega q ) Further details are given in [Gar96] We also have the type theory and categorical models for action terms [BGHP98, GH97]. In particular, we have shown that the nested nodes correspond to the general binding operators of Aczel [Acz80] and equivalently Power has shown that they correspond to natural transformations in the category S mentioned in remark 3.7 [Pow96] 6 Symmetric Action Graphs Gardner and Wischik are ....

A. Barber, P. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. Computer Science Logic, Aarhus, 1998.

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A. Barber, P. Gardner, M. Hasegawa, G. Plotkin. From action calculi to linear logic. In Annual Conference of the European Association for Computer Science Logic (CSL'97). Springer LNCS 1414, pages 78-97, 1998.

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A. Barber, P. Gardner, M. Hasegawa, and G. Plotkin. From action calculi to linear logic. In CSL'97: Computer Science Logic, volume 1414 of Lecture Notes in Computer Science, pages 78-93. Springer-Verlag, 1997.

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Andrew Barber, Philippa Gardner, Masahito Hasegawa, and Gordon D. Plotkin. From action calculi to linear logic. In Mogens Nielsen and Wolfgang Thomas, editors, Computer Science Logic, 11 International Workshop, Annual Conference of the EACSL, Aarhus, Denmark, August 23-29, 1997.

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