The Omega Rule is $mathbf{Pi_{1}^{1}}$-Complete in the $lambdabeta$-Calculus

Logical Methods in Computer Science Vol. 5 (2:6) 2009, pp. 1–21 www.lmcs-online.org Submitted Feb. 21, 2008 Published Apr. 27, 2009 THE OMEGA RULE IS Π1 1-COMPLETE IN THE λβ-CALCULUS BENEDETTO INTRIGILA a AND RICHARD STATMAN b a Universit`a di Roma ”Tor Vergata”, Rome, Italy e-mail address: intrigil@mat.uniroma2.it b Carnegie-Mellon University, Pittsburgh, PA, USA e-mail address: rs31@andrew.cmu.edu Abstract. In a functional calculus, the so called ω-rule states that if two terms P and Q applied to any closed term N return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the λβ-calculus the ω-rule does not hold, even when the η-rule (weak extensionality) is added to the calculus. A long-standing problem of H. Barendregt (1975) concerns the determination of the logical power of the ω-rule when added to the λβ-calculus. In this paper we solve the problem, by showing that the resulting theory is Π1 1-Complete. Introduction In a functional calculus, the so called ω-rule states that if two terms P and Q applied to any closed term N return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the λβ-calculus the ω-rule does not hold, even when the η-rule (weak extensionality) is added to the calculus. It is therefore natural to investigate the logical status of the ω-rule in λ-theories. We have first considered constructive forms of such rule in [7], obtaining r.e. λ-theories which are closed under the ω-rule. This gives the counterintuitive result that closure under the ω-rule does not necessarily give rise to non constructive λ-theories, thus solving a problem of A. Cantini (see [3]). Then we have considered the ω-rule with respect to the highly non constructive λ-theory H. The theory H is obtained extending β-conversion by identifying all closed unsolvables. Hω is the closure of this theory under the ω-rule (and β-conversion). A long-standing conjecture of H. Barendregt ([1], Conjecture 17.4.15) stated that the provable equations of Hω form a Π1 1-Complete set. In [8], we solved in the affirmative the problem. Of course the most important problem is to determine the logical power of ω-rule when added to the pure λβ-calculus. As in [1], we call λω the theory that results from adding the ω-rule to the pure λβ- calculus. In [6], we showed that the λω is not recursively enumerable, by giving a many-one 1998 ACM Subject Classification: F.4.1. Key words and phrases: lambda calculus; omega rule; lambda theories. LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.2168/LMCS-5 (2:6) 2009 c ⃝ B. Intrigila and R. Statman CC ⃝ Creative Commons 2 B. INTRIGILA AND R. STATMAN reduction of the set of true Π0 2 sentences to the set of closed equalities provable in λω, thus solving a problem originated with H. Barendregt and re-raised in [4]. The problem of the logical upper bound to λω remained open. That this bound is Π1 1 has been conjectured again by H. Barendregt in the well known Open Problems List, which ends the 1975 Conference on ”λ-Calculus and Computer Science Theory”, edited by C. B¨ohm [2]. Here we solve in the affirmative this conjecture. The celebrated Plotkin terms (introduced in [10]) furnish the main technical tool. 0.1. Remarks on the Structure of the Proof. The present paper is a revised and improved version of [9]. It is self-contained, with the exception of some specific points where we use results and methods from [6]. Such points will be precisely indicated in Section 3 and in Section 4. The authors are working to a comprehensive formalism to give a unified presentation of all the results. At present, however, this could not have been done without great complications. To help the reader, we now describe in an informal way the general idea of the proof. As already for the result in [6], the proof relies on suitable modifications of the men- tioned Plotkin terms. Roughly speaking, Plotkin’s construction gives rise, in the usual λβη- calculus, to pairs of closed terms P0 and P1 such that for every closed term M, P0M and P1M are βη-convertible. On the other hand, P0 and P1 are not themselves βη-convertible (see [1], 17.3.26). When we add the ω-rule to the λβ-calculus, such terms - suitably modified - become a way to express various forms of universal quantification. Intuitively, P0 and P1 are equal if and only if for all M belonging to some given set of terms, P0M and P1M are equal. There are two points that must be stressed. • First, different quantifiers require different specific constructions of suitable Plotkin terms. • Second, to properly use equality between P0 and P1 as a test for quantification, one must exclude that P0M = P1M holds for some M not belonging to the set of interest. Focusing on the second problem, the technical tool that we have used - both in [6] and in the present paper - is to cast proofs in the λβ-calculus with the ω-rule, in some kind of ”normal form”. (Observe that, in presence of the ω-rule, proofs become infinitary

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