On subexponentials, focusing and modalities in concurrent systems

Abstract In this work we present the focused proof system SELLF ⋒ , which extends intuitionistic linear logic with subexponentials with the ability of quantifying over them, hence allowing for the use of an arbitrary number of modalities. We show that the view of subexponentials as specific modalities is general enough to give a modular encoding of different flavors of Concurrent Constraint Programming (CCP), a simple and powerful model of concurrency. More precisely, we encode CCP calculi capturing time, spatial and epistemic behaviors into SELLF ⋒ , thus providing a proof theoretic foundation for those calculi and, at the same time, setting SELLF ⋒ as a general framework for specifying such systems.

[1]  Gerhard Gentzen,et al.  Investigations into Logical Deduction , 1970 .

[2]  Elaine Pimentel,et al.  Subexponential concurrent constraint programming , 2015, Theor. Comput. Sci..

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

[4]  Elaine Pimentel,et al.  A Proof Theoretic Study of Soft Concurrent Constraint Programming , 2014, Theory Pract. Log. Program..

[5]  Vivek Nigam,et al.  Exploiting non-canonicity in the sequent calculus , 2009 .

[6]  Elaine Pimentel,et al.  Dynamic Spaces in Concurrent Constraint Programming , 2014, LSFA.

[7]  Kaustuv Chaudhuri.  Classical and Intuitionistic Subexponential Logics Are Equally Expressive , 2010, CSL.

[8]  John C. Mitchell,et al.  Multiset rewriting and the complexity of bounded security protocols , 2004, J. Comput. Secur..

[9]  Chuck Liang,et al.  Focusing and polarization in linear, intuitionistic, and classical logics , 2009, Theor. Comput. Sci..

[10]  Gert Smolka.  A Foundation for Higher-order Concurrent Constraint Programming , 1994, CCL.

[11]  Dale Miller,et al.  A proof theory for generic judgments: an extended abstract , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[12]  Dale Miller,et al.  A Framework for Proof Systems , 2010, Journal of Automated Reasoning.

[13]  Dale Miller,et al.  Algorithmic specifications in linear logic with subexponentials , 2009, PPDP '09.

[14]  Francesca Rossi,et al.  Soft Concurrent Constraint Programming , 2002, ESOP.

[15]  Radha Jagadeesan,et al.  Timed Default Concurrent Constraint Programming , 1996, J. Symb. Comput..

[16]  Frank D. Valencia,et al.  Temporal Concurrent Constraint Programming: Denotation, Logic and Applications , 2002, Nord. J. Comput..

[17]  Vivek Nigam.  A framework for linear authorization logics , 2014, Theor. Comput. Sci..

[18]  François Fages,et al.  Linear Concurrent Constraint Programming: Operational and Phase Semantics , 2001, Inf. Comput..

[19]  Elaine Pimentel,et al.  A General Proof System for Modalities in Concurrent Constraint Programming , 2013, CONCUR.

[20]  Frank D. Valencia,et al.  Spatial and Epistemic Modalities in Constraint-Based Process Calculi , 2012, CONCUR.

[21]  JEAN-MARC ANDREOLI,et al.  Logic Programming with Focusing Proofs in Linear Logic , 1992, J. Log. Comput..

[22]  Elaine Pimentel,et al.  Proving Concurrent Constraint Programming Correct, Revisited , 2014, LSFA.

[23]  Vivek Nigam.  On the Complexity of Linear Authorization Logics , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

[24]  Frank Pfenning,et al.  A concurrent logical framework I & II , 2003 .

[25]  Prakash Panangaden,et al.  The semantic foundations of concurrent constraint programming , 1991, POPL '91.

[26]  Luca Cardelli,et al.  Mobile Ambients , 1998, FoSSaCS.

[27]  Lujo Bauer,et al.  A Linear Logic of Authorization and Knowledge , 2006, ESORICS.

[28]  Elaine Pimentel,et al.  Specifying Proof Systems in Linear Logic with Subexponentials , 2010, LSFA.

[29]  A. Troelstra.  Lectures on linear logic , 1992 .

[30]  Martín Abadi,et al.  A calculus for access control in distributed systems , 1991, TOPL.

[31]  Elaine Pimentel,et al.  Hybrid and Subexponential Linear Logics , 2016, LSFA.

[32]  Camilo Rueda,et al.  Models and emerging trends of concurrent constraint programming , 2013, Constraints.

[33]  Kaustuv Chaudhuri,et al.  A Hybrid Linear Logic for Constrained Transition Systems with Applications to Molecular Biology , 2013, TYPES.

[34]  Frank Pfenning,et al.  A hybrid logical framework , 2009 .

[35]  Dale Miller,et al.  From Proofs to Focused Proofs: A Modular Proof of Focalization in Linear Logic , 2007, CSL.

[36]  Vincent Danos,et al.  The Structure of Exponentials: Uncovering the Dynamics of Linear Logic Proofs , 1993, Kurt Gödel Colloquium.

[37]  Vijay A. Saraswat,et al.  Concurrent constraint programming , 1989, POPL '90.

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

[39]  Gérard Berry,et al.  The Esterel Synchronous Programming Language: Design, Semantics, Implementation , 1992, Sci. Comput. Program..

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