On bar recursion of types 0 and 1

For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by Tait [9]. The main tools of the proof are (i) the normalization theorem for (notations for) infinite terms and (ii) valuation functionals. Both are elaborated in [6]; for brevity some familiarity with this paper is assumed here. Using (i) and (ii) we reduce BRo.1 to ';-recursion with'; < co. From this the result follows by work of Tait [10], who gave a reduction of 2E-recursion to ';-recursion at a higher type. At the end of the paper we discuss a perhaps more natural variant of bar recursion introduced by Kreisel in [4]. Related results are due to rKeisel (in his appendix to [8]), who obtains results which imply, using the reduction given by Howard [2] of the constant of bar recursion of type '0 to the rule of bar recursion of type (0 ~ '0) ~ '0, that T is not closed under the rule of bar recursion of a type of level ~ 2, to Diller [1], who gave a reduction of BRo.1 to ';-recursion with'; bounded by the least (V-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type O. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions. Recall that a functional F of type 0 ~ (0 ~ '0) ~ (J is said to be defined by (the rule of) bar recursion of type '0 from Yand functionals G, H of the proper types i

A bound for Dickson's lemma

We consider a special case of Dickson's lemma: for any two functions $f,g$ on the natural numbers there are two numbers $i<j$ such that both $f$ and $g$ weakly increase on them, i.e., $f_i\le f_j$ and $g_i \le g_j$. By a combinatorial argument (due to the first author) a simple bound for such $i,j$ is constructed. The combinatorics is based on the finite pigeon hole principle and results in a descent lemma. From the descent lemma one can prove Dickson's lemma, then guess what the bound might be, and verify it by an appropriate proof. We also extract (via realizability) a bound from (a formalization of) our proof of the descent lemma. Keywords: Dickson's lemma, finite pigeon hole principle, program extraction from proofs, non-computational quantifiers

Logic for exact real arithmetic

Continuing earlier work of the first author with U. Berger, K. Miyamoto and H. Tsuiki, it is shown how a division algorithm for real numbers given as a stream of signed digits can be extracted from an appropriate formal proof. The property of being a real number represented as a stream is formulated by means of coinductively defined predicates, and formal proofs involve coinduction. The proof assistant Minlog is used to generate the formal proofs and extract their computational content as terms of the underlying theory, a form of type theory for finite or infinite data. Some experiments with running the extracted term are described, after its translation to Haskell