Inspired from a joint work by A. Beckmann, S. Buss and S. Friedman, we propose a class of set-theoretic functions, predicatively computable set functions. Each function in this class is polynomial time computable when we restrict to finite binary strings.

In this paper we show that the intuitionistic fixed point theory FiXi over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.

In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system$OT$is introduced based on$\psi $-functions. Provable$\Sigma _{1}$-sentences on$L_{\omega _{1}^{CK}}$are bounded through cut-elimination on operator controlled derivations.

In this paper, we discuss guidelines for a reward design problem that defines when and what amount of reward should be given to the agent/s, within the context of reinforcement learning approach. We would like to take keepaway soccer as a standard task of the multiagent domain which requires skilled teamwork. The difficulties of designing reward for this task are due to its features as follows: i) since it belongs to the continuing task which has no explicit goal to achieve, (...) it is hard to tell when reward should be given to the agent/s. ii) since it is a multiagent cooperative task, it is hard to decide what is a fair share of reward for each agent's contribution to achieve the goal. Through some experiments, we show that the reward design have a major effect on the agent's behavior, and introduce the successful reward function that makes agents perform keepaway better and more interesting than the conventional one does. Finally, we explore the relationship between `reward design' and `acquired behaviors' from the viewpoint of teamwork. (shrink)

In this paper we formulate epsilon substitution method for a theory $\Pi _{2}^{0}$-FIX for non-monotonic $\Pi _{2}^{0}$ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].

McEntire (1984) proved that, for a portfolio problem with independent assets, the generalized harmonic mean plays the role of a risk-free threshold. Based upon this property, he developed a criterion for including or excluding assets in an optimal portfolio, and he proved an ordering theorem showing that an optimal portfolio always consists of positive amounts of the assets with the largest mean values. Also, some commonly used utility functions were shown to satisfy the property that the dominance of an asset (...) over another is unaffected by the addition of other assets. In this paper we extend these results to the case where assets are dependent. (shrink)

The method of analytic tableaux is employed in many introductory texts and has also been used quite extensively as a basis for automated theorem proving. In this paper, we discuss the complexity of the system as a method for refuting contradictory sets of clauses, and resolve several open questions. We discuss the three forms of analytic tableaux: clausal tableaux, generalized clausal tableaux, and binary tableaux. We resolve the relative complexity of these three forms of tableaux proofs and also resolve the (...) relative complexity of analytic tableaux versus resolution. We show that there is a quasi-polynomial simulation of tree resolution by analytic tableaux; this simulation is close to optimal, since we give a matching lower bound that is tight to within a polynomial. (shrink)

This paper deals with a proof theory for a theory T22 of recursively Mahlo ordinals in the form of Π2-reflecting on Π2-reflecting ordinals using a subsystem Od of the system O of ordinal diagrams in Arai 353). This paper is the first published one in which a proof-theoretic analysis à la Gentzen–Takeuti of recursively large ordinals is expounded.

In this note the well-ordering principle for the derivative \ of normal functions \ on ordinals is shown to be equivalent to the existence of arbitrarily large countable coded \-models of the well-ordering principle for the function \.

In Arai (An ordinal analysis of a single stable ordinal, submitted) it is shown that an ordinal \(\sup _{N is an upper bound for the proof-theoretic ordinal of a set theory \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma _{1}}V)\). In this paper we show that a second order arithmetic \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) proves the wellfoundedness up to \(\psi _{\varOmega _{1}}(\varepsilon _{\varOmega _{{\mathbb {S}}+N+1}})\) for each _N_. It is easy to interpret \(\Sigma ^{1-}_{2}{\mathrm {-CA}}+\Pi ^{1}_{1}{\mathrm {-CA}}_{0}\) in \(\mathsf {KP}\ell ^{r}+(M\prec _{\Sigma (...) _{1}}V)\). (shrink)

This paper deals with a proof theory for a theory T3 of Π3-reflecting ordinals using the system O of ordinal diagrams in Arai 1375). This is a sequel to the previous one 1) in which a theory for recursively Mahlo ordinals is analyzed proof-theoretically.