S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers, using renaming operators for typed process composition [27] The operational structure of [5] follows his own p calculus encoding of proof nets [2], offering a process algebraic understanding of semantics of linear logic. The appeal of realisability lies in treating semantics and syntax uniformly on a logical basis. In the context of SN types for the p calculus, sharing of names and dynamic link creation in the p calculus would make it ....

Abramsky, S., Proofs as Processes, TCS, Vol. 135 (1994) 5--9, 1994.

....using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The operational structure of [5] follows his own p calculus encoding of proof nets [2]. The appeal of realisability lies in treating semantics and syntax uniformly on a logical basis. In the context of SN types for the p calculus, sharing of names and dynamic link creation would make the framework in [1, 5] hard to apply directly. In contrast, the present work offers a possibly ....

Abramsky, S., Proofs as Processes, TCS, Vol. 135 (1994) 5--9, 1994.

....using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The operational structure of [5] follows his own p calculus encoding of proof nets [2]. The appeal of realisability lies in treating semantics and syntax uniformly on a logical basis. In the context of SN types for the p calculus, sharing of names and dynamic link creation would make the framework in [1, 5] hard to apply directly. In contrast, the present work offers a possibly ....

Abramsky, S., Proofs as Processes, TCS, Vol. 135 (1994) 5--9, 1994.

....using proof expressions and proves SN [1] guiding our present usage of acyclicity in names. This programme is taken further with realisability semantics of linear logic in [5] where CCS processes act as realisers. The operational structure of [5] follows his own p calculus encoding of proof nets [2]. The appeal of realisability lies in treating semantics and syntax uniformly on a logical basis. In the context of SN types for the p calculus, sharing of names and dynamic link creation would make the framework in [1, 5] hard to apply directly. In contrast, the present work offers a possibly ....

Abramsky, S., Proofs as Processes, TCS, Vol. 135 (1994) 5--9, 1994.

....paper of Monteiro [29] gives a lock step simulation of proof nets for multiplicative linear logic into a version of CSP, although the author does not handle the additives or exponentials. This paper is the first example of the kind of analysis considered here (apparently independent of Abramsky [2]) of a sound and faithful translation of logical deduction in a language of concurrency. The paper of Dale Miller [23] is based on a logic programming view 4 of the calculus: Miller codes the calculus as a theory in linear logic, and discusses how techniques of logic programming (proof search, ....

....to is discussed in [25] and other references referred to there. The main feature (from our 8 viewpoint) is that distributive laws are needed to mirror the commutative reductions of linear logic (cf [13] 3 Proofs as Processes In this section we introduce a version of the Abramsky translation [2] mapping proofs in linear logic into process calculi, and discuss Soundness and Completeness Theorems for this translation. As mentioned in the Introduction, our treatment is an adaptation of the original Abramsky work to the synchronous calculus, a calculus better suited to this kind of ....

[Article contains additional citation context not shown here]

S. Abramsky. Proofs-as-processes. Seminar talk, Dept. of Computer Science, University of Edinburgh, 1992.

....paper of Monteiro [29] gives a lock step simulation of proof nets for multiplicative linear logic into a version of CSP, although the author does not handle the additives or exponentials. This paper is the first example of the kind of analysis considered here (apparently independent of Abramsky [2]) of a sound and faithful translation of logical deduction in a language of concurrency. The paper of Dale Miller [23] is based on a logic programming view 4 of the calculus: Miller codes the calculus as a theory in linear logic, and discusses how techniques of logic programming (proof search, ....

....to is discussed in [25] and other references referred to there. The main feature (from our 8 viewpoint) is that distributive laws are needed to mirror the commutative reductions of linear logic (cf [13] 3 Proofs as Processes In this section we introduce a version of the Abramsky translation [2] mapping proofs in linear logic into process calculi, and discuss Soundness and Completeness Theorems for this translation. As mentioned in the Introduction, our treatment is an adaptation of the original Abramsky work to the synchronous calculus, a calculus better suited to this kind of ....

[Article contains additional citation context not shown here]

S. Abramsky. Proofs-as-processes. Seminar talk, Dept. of Computer Science, University of Edinburgh, 1992.

....game, in which is the set of moves for Player, and the set of moves for Opponent. A resumption : is then a for Player. Note that we can represent such a strategy by its set of : One can then show that composition in ( is given by parallel composition plus hiding [Abr94, AJ94a, Abr96b]: where = and ( 1 The identities are the copycat strategies as in [AJ94a, Abr96b] We can then obtain the simple category of games described in [Abr96b] by applying a specification structure in the sense of [AGN96b] to ( in which the ....

S. Abramsky. Proofs as processes. , 135:5--9, 1994.

....and computation as cut elimination. Traditionally, the sequents used are of the form 1 0 G, where 1 is a set of propositions (generally typing judgments) and G is a single proposition. Such sequents are called single conclusion sequents. Following ideas of Girard presented in [Gir87] Abramsky [Abr90, Abr91] has extended this interpretation of computation to multiple conclusion sequents, that is, sequents of the form 1 0 0, where 1 and 0 are both sets (actually, multisets) of propositions. In this setting, cut elimination specifies concurrent programming. In particular, Abramsky presents a method ....

Samson Abramsky. Proofs as processes. Copy of transparencies, 1991.

.... [Girard 1989] long exemplified for functional languages by the simply typed lambda calculus, has grown to include a whole family of typed lambda calculi, linear calculi and more recently typed process calculi, through Abramsky s work on proofs asprocesses and linear realizability algebras [Abramsky 1994, 1993] However there is another relationship between logic and computation that one sees in concurrent constraint programming, and logic programming in general: not through the proofs as programs view but rather through programs as proof search [Miller 1994] In this paper we present yet ....

....indicate that key parts of our modelling extend to other concurrent languages that are not remotely like concurrent constraint programming. Our slogan Concurrency is Logic involves three identifications, to be contrasted with the realizability (or proofs as processes) viewpoint (for example see Abramsky [1994]) Concurrency is Logic Proofs as Processes ffl Processes are Formulas ffl Types are Formulas ffl Combinators are Connectives ffl Type Constructors are Connectives ffl Simulations are Proofs ffl Processes are Proofs Note the fundamental distinction: processes as formulas versus processes as ....

Abramsky, S. 1994. Proofs as Processes. Theoretical Computer Science 135, pp. 5--9.

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S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....6 1. Now, given oe : A B; B C, define oe; fsA; C j s 2 L(A; B; C) sA; B 2 oe; sB; C 2 g Here, sX; Y means the result of deleting all moves in s not in MX or M Y . Note that this definition clearly exhibits the Cut = Parallel Composition Hiding paradigm proposed by the first author [Abr91] as the correct computational interpretation of Cut in Classical Linear Logic, with respect to the CSP style trace semantics for parallel composition and hiding [Hoa85] What makes the game semantics so much richer than trace semantics is the explicit representation of the environment as the ....

....(i.e. internal communication) in the terminology of CSP [Hoa85] Proposition 2 G is a category. Proof: We define the identity morphism idA : A A as idA = fs 2 P A GammaffiA j s begins with an O move; 8t v s) jtj even ) tA = tA )g In process terms, this is a bidirectional one place buffer [Abr91]. In game terms, this is the copy cat strategy discussed previously. Next, we prove associativity. Given oe : A B; B C; AE : C D, we will show that (oe; AE = S, where S = ftA; D j t 2 L(A; B; C; D) tA; B 2 oe; tB; C 2 ; tC; D 2 AEg 11 A symmetrical argument shows that oe; AE) ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....of functions in the interpretation. For intuitionistic sequents Gamma A, whether in Intuitionistic or Intuitionistic Linear Logic, we have the usual functional interpretation. Gamma For sequents in Classical Linear Logic, the first author has proposed a process interpretation [Abr91b], where Gamma is an interface specification describing how the process P can be connected to its environment; the links are no longer directed since they correspond to a symmetric notion of communication. Axiom is interpreted by a communication buffer; and Cut by communicating parallel ....

....or r(u) Intuitively, for arguments of the right shape , the interpretation is well behaved. These ideas are developed in section 7. 6. 2 Linear Realizability Algebras As a preliminary to proving soundness, we review the formalism of Linear Realizability Algebras, introduced by the first author [Abr91b]. This provides a very convenient framework for proving soundness and allows some general lemmas to be factored out. Syntax We assume an infinite set of names N ranged over by ff; fi; fl. Names can be thought of as ports or channels as in various process formalisms; the closest analogy is in fact ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....categories [Abr94a, AGN96] for similar reasons. Much of what we will do here can be seen as a recasting of the work on interaction categories into a realizability framework. Indeed, the essential ideas on the process interpretation of proofs go back to a 1991 lecture on Proofs as Processes (see [Abr94b]) 2.1 Names, co names and actions As usual with CCS, we introduce two disjoint, countable sets N of names, and N of co names, with a bijection ( N = Gamma N , which we extend to an involution ( N N = Gamma N N : We use ff, fi, fl to range over names, and write ff, ....

S. Abramsky. Proofs as Processes. TCS vol. 135, 5--9, 1994.

....or Intuitionistic Linear Logic, we have the usual functional interpretation. Gamma Gamma Gamma Gamma Gamma Gamma fi CUT Delta AXIOM ff ff ff Gamma Gamma ff For sequents in Classical Linear Logic, the first author has proposed a process interpretation [Abr91b], where Gamma is an interface specification describing how the process P can be connected to its environment; the links are no longer directed since they correspond to a symmetric notion of communication. Axiom is interpreted by a communication buffer; and Cut by communicating parallel ....

....or r(u) Intuitively, for arguments of the right shape , the interpretation is well behaved. These ideas are developed in section 7. 6. 2 Linear Realizability Algebras As a preliminary to proving soundness, we review the formalism of Linear Realizability Algebras, introduced by the first author [Abr91b]. This provides a very convenient framework for proving soundness and allows some general lemmas to be factored out. Syntax We assume an infinite set of names N ranged over by ff; fi; fl. Names can be thought of as ports or channels as in various process formalisms; the closest analogy is in fact ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....given oe : A B; B C, define oe; fs A; C j s 2 L(A; B; C) s A; B 2 oe; s B; C 2 g Here, s X; Y means the result of deleting all moves in s not in MX or M Y . Note that this definition clearly exhibits the Cut = Parallel Composition Hiding paradigm proposed by the first author [Abr91] as the correct computational interpretation of Cut in Classical Linear Logic, with respect to the CSP style trace semantics for parallel composition and hiding [Hoa85] What makes the game semantics so much richer than trace semantics is the explicit representation of the environment as the ....

....(i.e. internal communication) in the terminology of CSP [Hoa85] Proposition 2 G is a category. Proof: We define the identity morphism idA : A A as idA = fs 2 P A GammaffiA j s begins with an O move; 8t v s) jtj even ) t A = t A )g In process terms, this is a bidirectional one place buffer [Abr91]. In game terms, this is the copy cat strategy discussed previously. Next, we prove associativity. Given oe : A B; B C; AE : C D, we will show that (oe; AE = S, where S = ft A; D j t 2 L(A; B; C; D) t A; B 2 oe; t B; C 2 ; t C; D 2 AEg A symmetrical argument shows that oe; ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....given oe : A B; B C, define oe; fs A; C j s 2 L(A; B; C) s A; B 2 oe; s B; C 2 g Here, s X; Y means the result of deleting all moves in s not in MX or M Y . Note that this definition clearly exhibits the Cut = Parallel Composition Hiding paradigm proposed by the first author [Abr91] as the correct computational interpretation of Cut in Classical Linear Logic, with respect to the CSP style trace semantics for parallel composition and hiding [Hoa85] What makes the game semantics so much richer than trace semantics is the explicit representation of the environment as the ....

....internal communication) in the terminology of CSP [Hoa85] Proposition 2 G is a category. Proof: We define the identity morphism idA : A A as idA = fs 2 P A GammaffiA j s begins with an O move; 8t v s) jtj even ) t A = t A )g In process terms, this is a bidirectional one place buffer [Abr91]. In game terms, this is the copy cat strategy discussed previously. Next, we prove associativity. Given oe : A B; B C; AE : C D, we will show that (oe; AE = S, where S = ft A; D j t 2 L(A; B; C; D) t A; B 2 oe; t B; C 2 ; t C; D 2 AEg A symmetrical argument shows that oe; ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

....: A B is a partial equivalence class [oe] 2 A Gamma ffiB The identity morphism [id A ] A A is de ned as follows. We use dioeerent subscripts on the A s to distinguish the occurrences. id A = fs 2 P even A1(A2 j s A 1 = s A 2 g In process terms, this is a bidirectional one place buoeer [1]. Composition The composition of (history free) strategies can be de ned in either of two ways: in terms of the set representation, or via the underlying functions on moves inducing the strategies. We begin with the set representation. Firstly, a preliminary de nition. Given a sequence of games ....

....MA j and s i 1 2 MAk implies that j is adjacent to k, i.e. jj Gamma kj 6 1. Now, given oe : A B; B C, de ne oe; fs A; C j s 2 L(A; B; C) s A; B 2 oe; s B; C 2 g even : This de nition clearly exhibits the iCut = Parallel Composition Hidingj paradigm proposed by the rst author [1] as the correct computational interpretation of Cut in Classical Linear Logic, with respect to the CSP style trace semantics for parallel composition and hiding [18] Proposition4. 1. Composition is compatible with : oe; oe 0 : A B; 0 : B C; oe oe 0 ; 0 = oe; oe ....

S. Abramsky. Proofs as processes. Unpublished Lecture, 1991.

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