Hilbert proposed the epsilon substitution method as a basis for consistency proofs. Hilbert’s Ansatz for finding a solving substitution for any given finite set of transfinite axioms is, starting with the null substitution S0, to correct false values step by step and thereby generate the process S0,S1,…. The problem is to show that the approximating process terminates. After Gentzen’s innovation, Ackermann [W. Ackermann, Zur Widerspruchsfreiheit der Zahlentheorie, Math. Ann. 117 162–194] succeeded in proving the termination of the process for the (...) first order arithmetic. In this note we report recent progress on the subject, and expound basic ideas of the epsilon substitution method à la Ackermann for the theory -FIX of non-monotonic inductive definitions. (shrink)

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].

The Correspondence of Tiglath-pileser III and Sargon II from Calah/Nimrud. By Mikko Luukko. State Archives of Assyria, vol. 19. Helsinki: Neo-Assyrian Text Corpus Project, 2012. Pp. lxxiv + 287, 3 plts., illus. $89.50.

人間存在の総体性の回復、人称的対立における和解と統合の試み。日々の風景、断想、聖書研究等も収録。.

日本初の“日英対訳版”登場!今こそ日本精神の源流を学ぶ時。「武士道」は、人類最高の指導理念です。本書は「武士道」の全てが分かる珠玉の一冊。ぜひ、世界の人々に「日本精神」を知って欲しい。.

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.

Hilbert proposed the epsilon substitution method as a basis for consistency proofs. Hilbert's Ansatz for finding a solving substitution for any given finite set of transfinite axioms is, starting with the null substitution S0, to correct false values step by step and thereby generate the process S0,S1,… . The problem is to show that the approximating process terminates. After Gentzen's innovation, Ackermann 162) succeeded to prove termination of the process for first order arithmetic. Inspired by G. Mints as an Ariadne's (...) thread we formulate the epsilon substitution method for the theory ID1 of non-iterated inductive definitions for disjunctions of simply universal and existential operators, and give a termination proof of the H-process based on Ackermann 162). The termination proof is based on transfinite induction up to the Howard ordinal. (shrink)

In referring to biographies of Edison as examples, the following are shown: the image of a scientist or an engineer in biographies has dramatically changed over time; the images produced anew in each period fitted well to the social milieu of the day; biographies therefore acquired a large readership and contributed to informing to the public of the value of science and technology and the necessity of promoting them. It is also pointed out that a new image of scientist or (...) engineer is to be created which is surely attractive to people today if we expect biographies to remain effective in promoting public understanding of science as before. (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.

In this, journal, W. Buchholz gave an elegant proof of a characterization theorem for provably total recursive functions in the theory IDv for the v-times iterated inductive definitions . He characterizes the classes of functions by Hardy functions. In this note we will show that a slow growing analogue to the theorem can be obtained by a slight modification of Buchholz' proof.

We formulate epsilon substitution method for theories (H)α0 of absolute jump hierarchies, and give two termination proofs of the H-process: The first proof is an adaption of Mints M, Mints-Tupailo-Buchholz MTB, i.e., based on a cut-elimination of a specially devised infinitary calculus. The second one is an adaption of Ackermann Ack. Each termination proof is based on transfinite induction up to an ordinal θ(α0+ ω)0, which is best possible.

We gather the following miscellaneous results in proof theory from the attic.1. 1. A provably well-founded elementary ordering admits an elementary order preserving map.2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21.3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic.4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE.5. 5. Intuitionistic fixed (...) point theories which are conservative extensions of HA.6. 6. Proof theoretic strengths of classical fixed points theories.7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣn.8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions.Each section can be read separately in principle. (shrink)

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 this paper, we prove the wellfoundedness of recursive notation systems for reflecting ordinals up to Π₃-reflection by relevant inductive definitions.