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)

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].

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].

Previous research reported that in processing structurally ambiguous sentences comprehenders often preserve an initial incorrect analysis even after adopting a correct analysis following structural disambiguation. One criticism is that the sentences tested in previous studies involved referential ambiguity and allowed comprehenders to make inferences about the initial interpretation using pragmatic information, suggesting the possibility that the initial analysis persisted due to comprehenders' pragmatic inference but not to their failure to perform complete reanalysis of the initial misanalysis. Our study investigated this (...) by testing locally ambiguous relative clause sentences in Japanese, in which the initial misinterpretation contradicts the correct interpretation. Our study using a self-paced reading technique demonstrated evidence for the persistence of the initial analysis with this structure. The results from an eye-tracking study further suggested that the phenomenon directly reflected the amount of support given to the initial incorrect analysis prior to disambiguating information: The more supported the incorrect main clause analysis was, the more likely comprehenders were to preserve the analysis even after the analysis was falsified. Our results thus demonstrated that the preservation of the initial analysis occurs not due to referential ambiguities but to comprehenders' difficulty to fully revise the highly supported initial interpretation. (shrink)

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)

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.

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 introduce a recursive notation system $O(\mu)$ of ordinals. An element of the notation system is called an ordinal diagram following G. Takeuti [25]. The system is designed for proof theoretic study of theories of recursively Mahlo universes. We show that for each $\alpha<\Omega$ in $O(\mu)$ KPM proves that the initial segment of $O(\mu)$ determined by $\alpha$ is a well ordering. Proof theoretic study for such theories will be reported in [9].

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

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

In this paper we introduce a recursive notation system O(Π 3 ) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π 3 -reflection. We show that for each $\alpha in O(Π 3 ) a set theory KP Π 3 for Π 3 -reflection proves that the initial segment of O(Π 3 ) determined by α is a well ordering. Proof theoretic study for such theories (...) will be reported in [4]. (shrink)

This study addressed the question of whether L2 learners are able to utilize verb’s argument structure information in online structural analysis. Previous L2 research has shown that L2 learners have difficulty in using verb’s intransitive information to guide online syntactic processing. This is true even though L2 learners have grammatical knowledge that is correct and similar to that of native speakers. In the present study, we contrasted three hypotheses, the initial inaccessibility account, the intransitivity overriding account, and the fuzzy subcategorization (...) frame account, to investigate whether L2 learner’s knowledge of intransitive verbs is in fact ignored in L2 online structural analysis. The initial inaccessibility account and the fuzzy subcategorization frame account predicted that L2 learners cannot access intransitivity information in building syntactic structures in any situation. The intransitivity overriding account predicted that intransitivity information is accessed in L2 parsing, but this process is overridden by the strong transitivity preference when a verb is followed by a noun phrase. Importantly, the intransitivity overriding account specifically predicted that L2 learners would be able to use intransitive information in online syntactic processing when a noun phrase does not appear immediately following a verb. We tested the three accounts in an eye-tracking reading experiment using filler-gap dependency structures. We manipulated verb’s transitivity information and lexically based plausibility information and tested English native speakers as a control L1 group and Japanese-English L2 participants. The results showed that L2 learners as well as native speakers processed sentences differently depending on the subcategorization information of the verb, and adopted transitive analysis only when the verb was optionally transitive, providing support for the intransitivity overriding. The results further demonstrated that L2 learners had strong expectations for the transitive structure, which is consistent with the view proposed by the hyper-active gap-filling hypothesis. In addition, the results showed that the semantic mismatch in the incorrect transitive analysis facilitated native speaker’s processing but caused difficulty for L2 learners. Together, the current study provides evidence that L2 learners use intransitive information of the verbs to guide their structural analysis when there are no overriding constraints. (shrink)

We show that the consistency of the first order arithmetic $PA$ follows from the pointwise induction up to the Howard ordinal. Our proof differs from U. Schmerl [Sc]: We do not need Girard's Hierarchy Comparison Theorem. A modification on the ordinal assignment to proofs by Gentzen and Takeuti [T] is made so that one step reduction on proofs exactly corresponds to the stepping down $\alpha\mapsto\alpha [1]$ in ordinals. Also a generalization to theories $ID_q$ of finitely iterated inductive definitions is proved.

In this paper we introduce a system AID of bounded arithmetic. The main feature of AID is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems F in Buss 3–29). We show that AID proves the soundness of F , and conversely any Σ 0 b -theorem in AID yields boolean sentences of which F has polysize proofs. Further we define Σ 1 b -faithful interpretations between AID+Σ 0 b -CA and (...) a quantified theory QALV of an equational system ALV in Clote 57–106). Hence ALV also proves the soundness of F. (shrink)

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.

For α < ε0, Nα denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number-theoretic function f let B denote the length n of the longest descending chain of ordinals <ε0 such that for all i < n, Nαi ≤ f . Simpson [2] called ε0 as slowly well ordered when B is totally defined for f = K · . Let |n| denote the binary length of the (...) natural number n, and |n|k the k-times iterate of the logarithmic function |n|. For a unary function h let L denote the function B ) with h0 = K + |i| · |i|h. In this note we show, inspired from Weiermann [4], that, under a reasonable condition on h, the functionL is primitive recursive in the inverse h–1 and vice versa. (shrink)

Weiermann [18] introduces a new method to generate fast growing functions in order to get an elegant and perspicuous proof of a bounding theorem for provably total recursive functions in a formal theory, e.g., in PA. His fast growing function θαn is described as follows. For each ordinal α and natural number n let T α n denote a finitely branching, primitive recursive tree of ordinals, i.e., an ordinal as a label is attached to each node in the tree so (...) that the labelling is compatible with the tree ordering. Then the tree T α n is well founded and hence finite by Konig's lemma. Define θαn=the depth of the tree T α n =the length of the longest branch in T α n . We introduce new fast and slow growing functions in this mode of definitions and show that each of these majorizes provably total recursive functions in PA. (shrink)

Speech is intelligible even when the temporal envelope of speech is distorted. The current study investigates how native and non-native speakers perceptually restore temporally distorted speech. Participants were native English speakers (NS), and native Japanese speakers who spoke English as a second language (NNS). In Experiment 1, participants listened to “locally time-reversed speech” where every x-ms of speech signal was reversed on the temporal axis. Here, the local time reversal shifted the constituents of the speech signal forward or backward from (...) the original position, and the amplitude envelope of speech was altered as a function of reversed segment length. In Experiment 2, participants listened to “modulation-filtered speech” where the modulation frequency components of speech were low-pass filtered at a particular cut-off frequency. Here, the temporal envelope of speech was altered as a function of cut-off frequency. The results suggest that speech becomes gradually unintelligible as the length of reversed segments increases (Experiment 1), and as a lower cut-off frequency is imposed (Experiment 2). Both experiments exhibit the equivalent level of speech intelligibility across six levels of degradation for native and non-native speakers respectively, which poses a question whether the regular occurrence of local time reversal can be discussed in the modulation frequency domain, by simply converting the length of reversed segments (ms) into frequency (Hz). (shrink)

In this paper we show that the intuitionistic theory for finitely many iterations of strictly positive operators is a conservative extension of Heyting arithmetic. The proof is inspired by the quick cut-elimination due to G. Mints. This technique is also applied to fragments of Heyting arithmetic.

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.

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.

We formulate epsilon substitution method for a theory [Π0 1, Π0 1]-FIX for two steps non-monotonic Π0 1 inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].