An axiomatization of PCTL*

A sound and complete axiomatization for a computation tree logic with past operators, PCTL*, is given. The logic extends the standard branching time logic CTL* of R-generable models via the use of past time operators and semantics based on a finite linear past leading back from any point in any fullpath. Furthermore, the valid formulas of CTL* are also valid in PCTL*. The past operators allow us to avoid use of any unusual rules of inference such as the ugly automata-motivated AA rule which is part of the existing complete axiomatization for CTL*.

[1]  Philippe Schnoebelen,et al.  A Hierarchy of Temporal Logics with Past (Extended Abstract) , 1994, STACS.

[2]  D. Gabbay,et al.  Handbook of Philosophical Logic, Volume II. Extensions of Classical Logic , 1986 .

[3]  Robert McNaughton,et al.  Counter-Free Automata (M.I.T. research monograph no. 65) , 1971 .

[4]  Philippe Schnoebelen,et al.  A Hierarchy of Temporal Logics with Past , 1995, Theor. Comput. Sci..

[5]  Uwe Mönnich,et al.  Aspects of Philosophical Logic , 1981 .

[6]  Amir Pnueli,et al.  Applications of Temporal Logic to the Specification and Verification of Reactive Systems: A Survey of Current Trends , 1986, Current Trends in Concurrency.

[7]  Amir Pnueli,et al.  The temporal logic of programs , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[8]  Amir Pnueli,et al.  A Deductive Proof System for CTL , 2002, CONCUR.

[9]  Colin Stirling,et al.  Modal and temporal logics , 1993, LICS 1993.

[10]  Zohar Manna,et al.  A hierarchy of temporal properties (invited paper, 1989) , 1990, PODC '90.

[11]  Mark Reynolds,et al.  An axiomatization of full Computation Tree Logic , 2001, Journal of Symbolic Logic.

[12]  Saharon Shelah,et al.  On the temporal analysis of fairness , 1980, POPL '80.

[13]  Alberto Zanardo,et al.  Ockhamist Computational Logic: Past-Sensitive Necessitation in CTL , 1993, J. Log. Comput..

[14]  John P. Burgess,et al.  Basic Tense Logic , 1984 .

[15]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic , 2002 .

[16]  R. McNaughton,et al.  Counter-Free Automata , 1971 .

[17]  Mark Reynolds.  More Past Glories , 2000, LICS 2000.

[18]  Dov M. Gabbay,et al.  Temporal logic (vol. 1): mathematical foundations and computational aspects , 1994 .

[19]  Willem P. de Roever,et al.  Real-time programming and asynchronous message passing , 1983, PODC '83.

[20]  Joseph Y. Halpern,et al.  “Sometimes” and “not never” revisited: on branching versus linear time temporal logic , 1986, JACM.

[21]  D. Gabbay,et al.  Temporal Logic Mathematical Foundations and Computational Aspects , 1994 .

[22]  Joseph Y. Halpern,et al.  "Sometimes" and "not never" revisited: on branching versus linear time (preliminary report) , 1983, POPL '83.

[23]  W. Hodges.  Elementary Predicate Logic , 1983 .

[24]  A. Prasad Sistla,et al.  Deciding Full Branching Time Logic , 1985, Inf. Control..

[25]  Amir Pnueli,et al.  The Glory of the Past , 1985, Logic of Programs.

[26]  D. Gabbay.  An Irreflexivity Lemma with Applications to Axiomatizations of Conditions on Tense Frames , 1981 .

[27]  Alberto Zanardo,et al.  Branching-time logic with quantification over branches: The point of view of modal logic , 1996, Journal of Symbolic Logic.

[28]  E. Allen Emerson,et al.  Temporal and Modal Logic , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[29]  John P. Burgess,et al.  Logic and time , 1979, Journal of Symbolic Logic.

[30]  R. Thomason.  Combinations of Tense and Modality , 2002 .

[31]  Max J. Cresswell,et al.  A New Introduction to Modal Logic , 1998 .

[32]  Moshe Y. Vardi.  A temporal fixpoint calculus , 1988, POPL '88.

[33]  Dov M. Gabbay,et al.  Temporal Logic: Mathematical Foundations and Computational Aspects: Volume 2 , 1994 .

[34]  François Laroussinie,et al.  Specification in CTL+Past for Verification in CTL , 1999, Inf. Comput..

[35]  Wolfgang Thomas,et al.  Computation Tree Logic CTL* and Path Quantifiers in the Monadic Theory of the Binary Tree , 1987, ICALP.

[36]  Amílcar Sernadas,et al.  Temporal aspects of logical procedure definiton , 1980, Inf. Syst..

[37]  Robert McNaughton,et al.  Testing and Generating Infinite Sequences by a Finite Automaton , 1966, Inf. Control..