Stable Computational Semantics of Conflict-Free Rewrite Systems (Partial Orders with Duplication)

We study orderings ⊴S on reductions in the style of Levy reflecting the growth of information w.r.t. (super)stable sets S of 'values' (such as head-normal forms or Bohm-trees). We show that sets of co-initial reductions ordered by ⊴S form finitary ω-algebraic complete lattices, and hence form computation and Scott domains. As a consequence, we obtain a relativized version of the computational semantics proposed by Boudol for term rewriting systems. Furthermore, we give a pure domain-theoretic characterization of the orderings ⊴S in the spirit of Kahn and Plotkin's concrete domains. These constructions are carried out in the framework of Stable Deterministic Residual Structures, which are abstract reduction systems with an axiomatized residual relations on redexes, that model all orthogonal (or conflict-free) reduction systems as well as many other interesting computation structures.

[1]  John R. W. Glauert,et al.  An Abstract Böhm-normalization , 2002, Electron. Notes Theor. Comput. Sci..

[2]  Jean-Jacques Lévy,et al.  Full abstraction for sequential algorithms: the state of the art , 1986 .

[3]  Fabio Gadducci,et al.  Relating Two Categorial Models of Term Rewriting , 1995, RTA.

[4]  J. Roger Hindley,et al.  To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism , 1980 .

[5]  Dov M. Gabbay,et al.  Handbook of logic in computer science (vol. 4): semantic modelling , 1995 .

[6]  Aart Middeldorp.  Call by need computations to root-stable form , 1997, POPL '97.

[7]  Gordon D. Plotkin,et al.  Concrete Domains , 1993, Theor. Comput. Sci..

[8]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[9]  Jean-Jacques Lévy,et al.  Full abstraction for sequential languages : The states of the art , 1983 .

[10]  J. R. Kennaway,et al.  Event structures and orthogonal term graph rewriting , 1993 .

[11]  Glynn Winskel,et al.  An Introduction to Event Structures , 1989 .

[12]  Glynn Winskel,et al.  Events in computation , 1980 .

[13]  Jean-Jacques Lévy,et al.  Minimal and optimal computations of recursive programs , 1977, JACM.

[14]  Carl A. Gunter.  Semantics of programming languages: structures and techniques , 1993, Choice Reviews Online.

[15]  Jean-Jacques Lévy,et al.  An abstract standardisation theorem , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[16]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[17]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[18]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[19]  John R. W. Glauert,et al.  Relating Conflict-Free Stable Transition and Event Models (Extended Abstract) , 1997, MFCS.

[20]  Terese.  Term rewriting systems , 2003, Cambridge tracts in theoretical computer science.

[21]  Paul-André Melliès.  A Factorisation Theorem in Rewriting Theory , 1997, Category Theory and Computer Science.

[22]  Paul-André Melliès.  A stability theorem in rewriting theory , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[23]  Eugene W. Stark,et al.  Concurrent Transition Systems , 1989, Theor. Comput. Sci..

[24]  Giuseppe Longo,et al.  Set-theoretical models of λ-calculus: theories, expansions, isomorphisms , 1983, Ann. Pure Appl. Log..

[25]  Cosimo LANEVE.  Distributive Evaluations of lambda-calculus , 1994, Fundam. Informaticae.

[26]  John C. Reynolds,et al.  Algebraic Methods in Semantics , 1985 .

[27]  Jean-Jacques Lévy,et al.  Computations in Orthogonal Rewriting Systems, II , 1991, Computational Logic - Essays in Honor of Alan Robinson.

[28]  Jean-Yves Girard,et al.  The System F of Variable Types, Fifteen Years Later , 1986, Theor. Comput. Sci..

[29]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

[30]  John R. W. Glauert,et al.  Stable results and relative normalization , 2000, J. Log. Comput..

[31]  G. Boudol.  Computational semantics of terms rewriting systems , 1983 .

[32]  Glynn Winskel,et al.  DI-Domains as a Model of Polymorphism , 1987, MFPS.

[33]  Glynn Winskel,et al.  Petri Nets, Event Structures and Domains , 1979, Semantics of Concurrent Computation.

[34]  Glynn Winskel,et al.  Petri Nets, Event Structures and Domains, Part I , 1981, Theor. Comput. Sci..

[35]  John R. W. Glauert,et al.  Relative Normalization in Deterministic Residual Structures , 1996, CAAP.

[36]  Glynn Winskel,et al.  Using Information Systems to Solve Recursive Domain Equations Effectively , 1984, Semantics of Data Types.

[37]  John R. W. Glauert,et al.  Relating conflict-free stable transition and event models via redex families , 2002, Theor. Comput. Sci..

[38]  Dana S. Scott,et al.  Some Domain Theory and Denotational Semantics in Coq , 2009, TPHOLs.

[39]  C.-H.L. Ong,et al.  Correspondence between operational and denotational semantics: the full abstraction problem for PCF , 1995, LICS 1995.

[40]  Jean-Jacques Lévy,et al.  Minimal and Optimal Computations of Recursive Programs , 1979, J. ACM.

[41]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[42]  Pierre-Louis Curien.  Categorical Combinators, Sequential Algorithms, and Functional Programming , 1993, Progress in Theoretical Computer Science.

[43]  G Boudol.  Computational semantics of term rewriting systems , 1986 .