Completeness

A conjecture of Gabbay (1981) states that any class of flows of time having the property known as finite H-dimension admits a finite set of expressively complete one-dimensional temporal connectives. Here we show that the class of ‘circular’ structures refutes the generalisation of this conjecture to Kripke frames. We then construct from this class, by a general method, a new class of irreflexive transitive flows of time that refutes the original conjecture.Our paper includes full descriptions of a method for establishing finite H-dimension for a class of structures and of the technique for extending finite H-dimension to other classes, and an introduction surveying the area of expressive completeness.

[1]  P. Cameron,et al.  Oligomorphic permutation groups , 1990 .

[2]  Dov M. Gabbay,et al.  EXPRESSIVE FUNCTIONAL COMPLETENESS IN TENSE LOGIC , 1981 .

[3]  Holger Schlingloff.  Expressive completeness of temporal logic of trees , 1992, J. Appl. Non Class. Logics.

[4]  Amihood Amir.  Separation in Nonlinear Time Models , 1985, Inf. Control..

[5]  J. van Benthem,et al.  Temporal logic , 1995 .

[6]  J. A. Makowsky.  Model theory and computer science: an appetizer , 1993, LICS 1993.

[7]  Dov M. Gabbay,et al.  Preservation of Expressive Completeness in Temporal Models , 1987, Inf. Comput..

[8]  Mark Reynolds,et al.  Axiomatisation and decidability ofF andP in cyclical time , 1994, J. Philos. Log..

[9]  D. Gabbay.  Expressive Functional Completeness in Tense Logic (Preliminary report) , 1981 .

[10]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[11]  Jaakko Hintikka,et al.  Time And Modality , 1958 .

[12]  Yde Venema,et al.  Expressiveness and Completeness of an Interval Tense Logic , 1990, Notre Dame J. Formal Log..

[13]  Johan Anthory Willem Kamp,et al.  Tense logic and the theory of linear order , 1968 .

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

[15]  D. Gabbay,et al.  Temporal expressive completeness in the presence of gaps , 1993 .

[16]  Peter Øhrstrøm,et al.  Temporal Logic , 1994, Lecture Notes in Computer Science.

[17]  Dov M. Gabbay,et al.  The Declarative Past and Imperative Future: Executable Temporal Logic for Interactive Systems , 1987, Temporal Logic in Specification.

[18]  George M. Reed,et al.  A Hierarchy of Domains for Real-Time Distributed Computing , 1990, Mathematical Foundations of Programming Semantics.