Equational theory of positive numbers with exponentiation

A. Tarski asked if all true identities involving 1, addition, multiplication, and exponentiation can be derived from certain so-called "high-school" identities (and a number of related questions). I prove that equational theory of (N, 1, +, , T) is decidable (a T b means ac' for positive a, b) and that entailment relation in this theory is decidable (and present a similar result for inequalities). A. J. Wilkie found an identity not derivable from Tarski's axioms with a difficult proof-theoretic argument of nonderivability. I present a model of Tarski's axioms consisting of 59 elements in which Wilkie's identity fails. 1. This note is related to "Tarski's high school algebra problem" and a number of other model-theoretic questions concerning exponentiation of positive real numbers and positive integers (see e.g. [1]). Let a T b = a b for positive a, b, and L = the set of terms in signature (1, +, *, T). As always, R+ is the set of positive reals. We give proofs of decidability for two problems about identities, and we also present a 59-element model in which Tarski's "high school algebra" identities are true, while Wilkie's identity is false. Our first result gives a new proof of a theorem of A. Macintyre [3] (proved for terms in one variable by Richardson [4]). THEOREM 1. Let X be any subset of R+ containing 1 and closed under addition, multiplication, and exponentiation. Then the set of valid equalities T = { t1 = t2l tl, t2 E L, X W= t1 = t2 } is decidable and does not depend on X. The proof is based on the following lemma, which is proved in ??2 and 3. LEMMA 1. There is a recursive function M: L X L -N such that, for any t1(r, s), t2(r, s) E L, for any positive real (values of ) -Fif card{s E R+ It1(r, s) = t2Qr, s)} > M(t1, t2) then Vs E R+: t1(r, s) = t2(r, s). PROOF OF THEOREM 1. Proceed by induction on the number of variables: Vs E X: t1(r, s) = t2(r, s) is equivalent to &m 1t1(r, k) = t2(r-, k), where M = M(t1, t2). Received by the editors February 4, 1983 and, in revised form, April 2, 1984. 1980 Mathematics Subject Classification. Primary 03B25, 03C05, 03C13. Kev words and phrases. Exponentiation of positive reals, exponentiation of positive integers, Tarski's high school algebra problem, decidability of equational theory, decidability of entailment relation, differential ring, finite model of Tarski's axioms. ?01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page