On complexity of propositional linear-time temporal logic with finitely many variables

It is known [4] that both satisfiability and model-checking problems for propositional Linear-time Temporal Logic, LTL, with only a single propositional variable in the language are PSPACE-complete, which coincides with the complexity of these problems for LTL with an arbitrary number of propositional variables [14]. In the present paper, we show that the same result can be obtained by modifying the original proof of PSPACE-hardness for LTL from [14]; i.e., we show how to modify the construction from [14] to model the computations of polynomially-space bound Turing machines using only formulas of one variable. We believe that our alternative proof of the results from [4] gives additional insight into the semantic and computational properties of LTL.

[1]  Alexander V. Chagrov,et al.  How Many Variables Does One Need to Prove PSPACE-hardness of Modal Logics , 2002, Advances in Modal Logic.

[2]  Edith Hemaspaandra.  The Complexity of Poor Man's Logic , 2001, J. Log. Comput..

[3]  Steven K. Thomason,et al.  The extensions of the modal logic K5 , 1985, Journal of Symbolic Logic.

[4]  Dmitry Shkatov,et al.  Complexity of finite-variable fragments of propositional modal logics of symmetric frames , 2018, Log. J. IGPL.

[5]  Philippe Schnoebelen,et al.  The Complexity of Propositional Linear Temporal Logics in Simple Cases , 1998, Inf. Comput..

[6]  Fred Kröger,et al.  Temporal Logic of Programs , 1987, EATCS Monographs on Theoretical Computer Science.

[7]  Vítezslav Svejdar,et al.  The decision problem of provability logic with only one atom , 2003, Arch. Math. Log..

[8]  Edith Hemaspaandra,et al.  A modal perspective on the computational complexity of attribute value grammar , 1993, J. Log. Lang. Inf..

[9]  Iwao Nishimura,et al.  On Formulas of One Variable in Intuitionistic Propositional Calculus , 1960, J. Symb. Log..

[10]  Dmitry Shkatov,et al.  Complexity and Expressivity of Branching- and Alternating-Time Temporal Logics with Finitely Many Variables , 2018, ICTAC.

[11]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[12]  Joseph Y. Halpern.  The Effect of Bounding the Number of Primitive Propositions and the Depth of Nesting on the Complexity of Modal Logic , 1995, Artif. Intell..

[13]  Richard Spencer-Smith,et al.  Modal Logic , 2007 .

[14]  Mark Ryan,et al.  Logic in Computer Science: Modelling and Reasoning about Systems , 2000 .

[15]  A. Prasad Sistla,et al.  The complexity of propositional linear temporal logics , 1982, STOC '82.

[16]  Michael Huth,et al.  Logic in computer science - modelling and reasoning about systems , 2000 .

[17]  Dmitry Shkatov,et al.  Complexity and expressivity of propositional dynamic logics with finitely many variables , 2018, Log. J. IGPL.