Probability Logic for Harsanyi Type Spaces

Probability logic has contributed to significant developments in belief types for game-theoretical economics. We present a new probability logic for Harsanyi Type spaces, show its completeness, and prove both a de-nesting property and a unique extension theorem. We then prove that multi-agent interactive epistemology has greater complexity than its single-agent counterpart by showing that if the probability indices of the belief language are restricted to a finite set of rationals and there are finitely many propositional letters, then the canonical space for probabilistic beliefs with one agent is finite while the canonical one with at least two agents has the cardinality of the continuum. Finally, we generalize the three notions of definability in multimodal logics to logics of probabilistic belief and knowledge, namely implicit definability, reducibility, and explicit definability. We find that S5-knowledge can be implicitly defined by probabilistic belief but not reduced to it and hence is not explicitly definable by probabilistic belief

Complete Deductive Systems for Probability Logic with Application in Harsanyi Type Spaces

Thesis (PhD) - Indiana University, Mathematics, 2007These days, the 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 the 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 the same as the 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 the first part of the dissertation, we develop a sound and complete probability logic $\Sigma_+$ for type spaces in a formal propositional language with operators $L_r^i$ which means ``the agent $i$'s belief is at least $r$" where the index $r$ is a rational number between 0 and 1. A crucial infinitary inference rule in the system $\Sigma_+$ captures the Archimedean property about indices. By the Fourier-Motzkin's elimination method in linear programming, we prove Professor Moss's conjecture that the infinitary rule can be replaced by a finitary one. More importantly, our proof of completeness is in keeping with the Henkin-Kripke style. Also we show through a probabilistic system with parameterized indices that it is decidable whether a formula $\phi$ is derived from the system $\Sigma_+$. The second part is on its strong completeness. It is well-known that $\Sigma_+$ is not strongly complete, i.e., a set of formulas in the language may be finitely satisfiable but not necessarily satisfiable. We show that even finitely satisfiable sets of formulas that are closed under the Archimedean rule are not satisfiable. From these results, we develop a theory about probability logic that is parallel to the 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 the lack of strong completeness does not depend on the non-Archimedean property in indices but rather on the use of explicit probabilities in the syntax. We show the completeness and some properties of the 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

Approximate equivalence relations

Generalizing results for approximate subgroups, we study approximate equivalence relations up to commensurability, in the presence of a definable measure. As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover’s normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim–Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations. We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov–Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric. For sequences of approximate equivalence relations with an “approximately unique” probability logic 1-type, we obtain a structure theorem generalizing the “Lie model” theorem for approximate subgroups (Theorem 5.5). The models here are Riemannian homogeneous spaces, fibered over a locally finite graph. Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. This generalizes the main lemma for strong approximation of groups. For NIP theories, pursuing a question of Pillay’s, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. This can be seen as an archimedean analogue of results of Macpherson and Tent on NIP profinite groups

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type