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

Non-simultaneous blow-up for a reaction-diffusion system with absorption and coupled boundary flux

This paper deals with non-simultaneous blow-up for a reaction-diffusion system with absorption and nonlinear boundary flux. We establish necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data

Coexistence of a diffusive predator–prey model with Holling type-II functional response and density dependent mortality

AbstractIn this paper, we consider a two competitor–one prey diffusive model in which both competitors exhibit Holling type-II functional response and one of the competitors exhibits density dependent mortality rate. First, we study the local and global existence of strong solution by using the C0 analytic semigroup. Then, we consider the local and global stability of the positive constant equilibrium by using the linearization method and Laypunov functional method, respectively. Furthermore, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain species is small or large. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system

Momentum return in Chinese capital market and the influences of size, liquidity and historical return

1 online resource (iv, 25 leaves)Includes abstract.Includes bibliographical references (leaves 24-25).This paper investigates short term to intermediate-horizon momentum effect in Chinese capital market. The result of the research supports the assertion that momentum effect exists in Chinese capital market. Using momentum strategies could create return in excess of market average return. This paper also examines influence of firm size and average trading volume on the effectiveness of momentum strategies. We found that firm size has a negative relationship with momentum return and that relationship is statistically significant. On the other hand, our results confirm a negative relationship between trading volume and momentum return and that relationship is not as significant as firm size effect. The regression analysis also conclude that historical returns contribute the most to momentum return, indicating that momentum effect is not subsumed by size and liquidity effect

Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition

This paper deals with a nonlinear p-Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatis x=1 are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results

The total belief theorem

In this paper, motivated by the treatment of conditional constraints in the data association problem, we state and prove the generalisation of the law of total probability to belief functions, as finite random sets. Our results apply to the case in which Dempster’s conditioning is employed. We show that the solution to the resulting total belief problem is in general not unique, whereas it is unique when the a-priori belief function is Bayesian. Examples and case studies underpin the theoretical contributions. Finally, our results are compared to previous related work on the generalisation of Jeffrey’s rule by Spies and Smets

Properties of Intuitionistic Provability Logics

Abstract. We study the modal properties of intuitionistic modal logics that belong to the provability logic or the preservativity logic of Heyting Arithmetic. We describe thefragment of some preservativity logics and we present fixed point theorems for the logics iL and iP L, and show that they imply the Beth property. These results imply that the fixed point theorem and the Beth property hold for both the provability and preservativity logic of Heyting Arithmetic. We present a frame correspondence result for the preservativity principle W p that is related to an extension of Löb's principle