Algorithmic specifications in linear logic with subexponentials

The linear logic exponentials !,? are not canonical: one can add to linear logic other such operators, say !l,?1, which may or may not allow contraction and weakening, and where l is from some pre-ordered set of labels. We shall call these additional operators subexponentials and use them to assign locations to multisets of formulas within a linear logic programming setting. Treating locations as subexponentials greatly increases the algorithmic expressiveness of logic. To illustrate this new expressiveness, we show that focused proof search can be precisely linked to a simple algorithmic specification language that contains while-loops, conditionals, and insertion into and deletion from multisets. We also give some general conditions for when a focused proof step can be executed in constant time. In addition, we propose a new logical connective that allows for the creation of new subexponentials, thereby further augmenting the algorithmic expressiveness of logic.

[1]  Dale Miller,et al.  Cut-elimination for a logic with definitions and induction , 2000, Theor. Comput. Sci..

[2]  Dale Miller,et al.  Forum: A Multiple-Conclusion Specification Logic , 1996, Theor. Comput. Sci..

[3]  Daniel Le Métayer,et al.  Gamma and the chemical reaction model: ten years after , 1996 .

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

[5]  Dale Miller,et al.  Least and Greatest Fixed Points in Linear Logic , 2007, LPAR.

[6]  Dale Miller,et al.  Incorporating Tables into Proofs , 2007, CSL.

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

[8]  David McAllester A logical algorithm for ML type inference , 2003 .

[9]  Larry Wos,et al.  What Is Automated Reasoning? , 1987, J. Autom. Reason..

[10]  Harald Ganzinger,et al.  Logical Algorithms , 2002, ICLP.

[11]  Iliano Cervesato Typed MSR: Syntax and Examples , 2001, MMM-ACNS.

[12]  Dale Miller,et al.  Focusing and Polarization in Intuitionistic Logic , 2007, CSL.

[13]  Edsger W. Dijkstra,et al.  A Discipline of Programming , 1976 .

[14]  D. L. Métayer,et al.  Gamma and the chemical reaction , 1996 .

[15]  David Gelernter,et al.  Generative communication in Linda , 1985, TOPL.

[16]  Radha Jagadeesan,et al.  Testing Concurrent Systems: An Interpretation of Intuitionistic Logic , 2005, FSTTCS.

[17]  Harald Ganzinger,et al.  A New Meta-complexity Theorem for Bottom-Up Logic Programs , 2001, IJCAR.

[18]  Jean-Yves Girard,et al.  Light Linear Logic , 1998, Inf. Comput..

[19]  David A. McAllester On the complexity analysis of static analyses , 2002, JACM.

[20]  David A. McAllester On the complexity analysis of static analyses , 1999, JACM.

[21]  Frank Pfenning,et al.  Monadic concurrent linear logic programming , 2005, PPDP.

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

[23]  Robert J. Simmons,et al.  Linear Logical Algorithms , 2008, ICALP.