Complete deductive systems for probability logic with application to harsanyi type spaces
Jon Michael Dunn, Lawrence S. Moss, Chunlai Zhou · 2007
These days, study of probabilistic systems is very popular not only in theoretical computer science but also in economics. There is a surprising concurrence between game theory and probabilistic programming. J.C. Harsanyi introduced notion of type spaces to give an implicit description of beliefs in games with incomplete information played by Bayesian players. Type functions on type spaces are same as stochastic kernels that are used to interpret probabilistic programs. In addition to this semantic approach to interactive epistemology, a syntactic approach was proposed by R.J. Aumann. It is of foundational importance to develop a deductive logic for his probabilistic belief logic. In first part of dissertation, we develop a sound and complete probability logic Σ+ for type spaces in a formal propositional language with operators Lir which means the agent i's belief is at least where index r is a rational number between 0 and 1. A crucial infinitary inference rule in system Σ+ captures Archimedean property about indices. By Fourier-Motzkin's elimination method in linear programming, we prove Professor Moss's conjecture that infinitary rule can be replaced by a finitary one. More importantly, our proof of completeness is in keeping with Henkin-Kripke style. Also we show through a probabilistic system with parameterized indices that it is decidable whether a formula p is derived from system Σ +. The second part is on its strong completeness. It is well-known that Σ + is not strongly complete, i.e., a set of formulas in language may be finitely satisfiable but not necessarily satisfiable. We show that even finitely satisfiable sets of formulas that are closed under Archimedean rule are not satisfiable. From these results, we develop a theory about probability logic that is parallel to relationship between explicit and implicit descriptions of belief types in game theory. Moreover, we use a linear system about probabilities over trees to prove that there is no strong completeness even for probability logic with finite indices. We conclude that lack of strong completeness does not depend on non-Archimedean property in indices but rather on use of explicit probabilities in syntax. We show completeness and some properties of probability logic for Harsanyi type spaces. By adding knowledge operators to our languages, we devise a sound and complete axiomatization for Aumann's semantic knowledge-belief systems. Its applications in labeled Markovian processes and semantics for programs are also discussed.