Tableaux for constructive concurrent dynamic logic

This is the first paper on constructive concurrent dynamic logic (CCDL). For the first time, either for concurrent or sequential dynamic logic, we give a satisfactory treatment of what statements are forced to be true by partial information about the underlying computer. Dynamic logic was developed by Pratt [V. Pratt, Semantical considerations on Floyd–Hoare logic, in: 17th Annual IEEE Symp. on Found. Comp. Sci., New York, 1976, pp. 109–121, V. Pratt, Applications of modal logic to programming, Studia Logica 39 (1980) 257–274] for nondeterministic sequential programs, and by Peleg [D. Peleg, Concurrent dynamic logic, Journal of the Association for Computing Machinery 34 (2) (1987), D. Peleg, Communication in concurrent dynamic logic, Journal of Computer and System Sciences 35 (1987)] for concurrent programs, for the purpose of proving properties of programs such as correctness. Here we define what it means for a dynamic logic formula to be forced to be true knowing only partial information about the results of assignments and tests. This informal CCDL semantics is formalized by intuitionistic Kripke frames modeling this partial information, and each such frame is interpreted as an idealized concurrent machine (a concurrent transition system). In CCDL, proofs and deductions are ω-height, ω-branching, well-founded labeled subtrees of ωω . These are a generalization of the signed tableaux of Nerode [A. Nerode, Some lectures ✩We acknowledge support from the Multiple University Research Initiative “An Integrated Approach to Intelligent Systems”, DAAH04-96-1-0341, joint between The University of California at Berkeley (Electronics Research Laboratory), Stanford University (Computer Science Department), and Cornell University (Center for Foundations of Intelligent Systems). This MURI was monitored by the U.S. Army Research Office. ∗ Corresponding author. E-mail addresses: [email protected] (D. Wijesekera), [email protected] (A. Nerode). 0168-0072/$ see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2004.12.001 2 D. Wijesekera, A. Nerode / Annals of Pure and Applied Logic 135 (2005) 1–72 in modal logic, Technical Report, M.S.I. Cornell University, 1989, CIME Logic and Computer Science Montecatini Volume, Springer-Verlag Lecture Notes, 1990, A. Nerode, Some lectures in intuitionistic logic, Technical Report, M.S.I. Cornell University, 1988, Marktoberdorf Logic and Computation NATO Summer School Volume, NATO Science Series, 1990 (in press)] stemming from the prefix tableaux of Fitting [M.C. Fitting, Proof Methods for Modal and Intuitionistic Logic, Reidel, 1983]. We demonstrate the correctness of our tableau proofs, define consistency properties, prove that consistency properties yield models, construct systematic tableaux, prove that systematic tableaux yield a consistency property, and conclude that CCDL is complete. This infinitary semantics and proof procedure will be the primary guide for defining, in a sequel, the correct finitary CCDL (FCCDL) based on induction principles. FCCDL is suitable for implementation in constructive logic software systems such as Constable’s NUPRL or Huet-Coquand’s CONSTRUCTIONS. Our goal is to develop a constructive logic programming tool for specification and modular verification of programs in any imperative concurrent language, and for the extraction of concurrent programs from constructive proofs. Subsequent papers will introduce analogous logics for declarative and functional concurrent languages. © 2005 Elsevier B.V. All rights reserved.