Small Model Property Research Articles

Reasoning about knowledge and probability

We provide a model for reasoning about knowledge and probability together. We allow explicit mention of probabilities in formulas, so that our language has formulas that essentially say «according to agent i, formula φ holds with probability at least b.» The language is powerful enough to allow reasoning about higher-order probabilities, as well as allowing explicit comparisons of the probabilities an agent places on distinct events. We present a general framework for interpreting such formulas, and consider various properties that might hold of the interrelationship between agents' probability assignments at different states. We provide a complete axiomatization for reasoning about knowledge and probability, prove a small model property, and obtain decision procedures

A probabilistic PDL

In this paper we give a probabilistic analog PPDL of Propositional Dynamic Logic. We prove a small model property and give a polynomial space decision procedure for formulas involving well-structured programs. We also give a deductive calculus and illustrate its use by calculating the expected running time of a simple random walk.

Decision procedures and expressiveness in the temporal logic of branching time

We consider the computation tree logic (CTL) proposed in ( Sci. Comput. Programming 2 (1982), 241–260) which extends the unified branching time logic (UB) of (“Proc. Ann. ACM Sympos. Principles of Programming Languages, 1981,” pp. 164–176) by adding an until operator. It is established that CTL has the small model property by showing that any satisfiable CTL formulae is satisfiable in a small finite model obtained from the small “pseudomodel” resulting from the Fischer-Ladner quotient construction. Then an exponential time algorithm is given for deciding satisfiability in CTL, and the axiomatization of UB given in ibid. is extended to a complete axiomatization for CTL. Finally, the relative expressive power of a family of temporal logics obtained by extending or restricting the syntax of UB and CTL is studied.

Results on the propositional μ-calculus

In this paper we define and study a propositional μ-calculus Lμ, which consists essentially of propositional modal logic with a least fixpoint operator. Lμ is syntactically simpler yet strictly more expressive than Propositional Dynamic Logic (PDL). For a restricted version we give an exponential-time decision procedure, small model property, and complete deductive system, theory subsuming the corresponding results for PDL.

Results on the Propositional µ-Calculus

We define a propositional version of the µ-calculus, and give an exponential-time decision procedure, small model property, and complete deductive system. We also show that it is strictly more expressive than PDL. Finally we give an algebraic semantics and prove a representation theorem.