We consider multi-agent systems where agents actions and beliefs are determined aleatorically, or "by the throw of dice". This system consists of possible worlds that assign distributions to independent random variables, and agents who assign probabilities to these possible worlds. We present a novel syntax and semantics for such system, and show that they generalise Modal Logic. We also give a sound and complete calculus for reasoning in the base semantics, and a sound calculus for the full modal semantics, that we conjecture to be complete. Finally we discuss some application to reasoning about game playing agents.
These approaches lose the simplicity of Boolean logics, as deductive systems must deal with propositions that are not independent. This limits their practicality as well defined semantics require the conditional probabilities of all atoms to be known. However, these approaches have been successfully combined with logic programming @cite_12 and machine learning @cite_14 . Feldman and Harel @cite_10 and Kozen @cite_6 gave a probabilistic variation of propositional dynamic logic for reasoning about the correctness of programs with random variables. Importantly, this work generalises a modal logic (PDL) as a many valued logic.
@cite_14 @cite_10 @cite_6 @cite_12
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The problem of deciding the satisfiability of a quantifier-free formula with respect to a background theory, also known as Satisfiability Modulo Theories (SMT), is gaining increasing relevance in verification: representation capabilities beyond propositional logic allow for a natural modeling of real-world problems (e.g., pipeline and RTL circuits verification, proof obligations in software systems). In this paper, we focus on the case where the background theory is the combination T1∪T2 of two simpler theories. Many SMT procedures combine a boolean model enumeration with a decision procedure for T1∪T2, where conjunctions of literals can be decided by an integration schema such as Nelson-Oppen, via a structured exchange of interface formulae (e.g., equalities in the case of convex theories, disjunctions of equalities otherwise). We propose a new approach for SMT(T1∪T2), called Delayed Theory Combination, which does not require a decision procedure for T1∪T2, but only individual decision procedures for T1 and T2, which are directly integrated into the boolean model enumerator. This approach is much simpler and natural, allows each of the solvers to be implemented and optimized without taking into account the others, and it nicely encompasses the case of non-convex theories. We show the effectiveness of the approach by a thorough experimental comparison. 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 program. Abstract 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. From a computational perspective, there is a close connection between various probabilistic reasoning tasks and the problem of counting or sampling satisfying assignments of a propositional theory. We consider the question of whether state-of-the-art satisfiability procedures, based on random walk strategies, can be used to sample uniformly or nearuniformly from the space of satisfying assignments. We first show that random walk SAT procedures often do reach the full set of solutions of complex logical theories. Moreover, by interleaving random walk steps with Metropolis transitions, we also show how the sampling becomes near-uniform.