The theory of computational complexity deals with those functions which can be computed subject to certain restrictions on the resources (for instance, time or memory) available for computation. Blum [5] gave an axiomatic characterization of some of the properties which should be possessed by a measure of computational complexity and established the existence of speed-upable functions—computable functions which fail to possess optimal programs in a particularly strong sense. Recursion theorists tend to like such functions, and people concerned with the specifics of real computing tend to consider such functions somewhat pathological. In Theorem 2 we show that such pathology is rampant: there is a great diversity of behavior among the collections of “run-times” of different functions which do not possess optimal programs, where such diversity is gauged by certain algebraic criteria which have computational significance. Roughly speaking, these algebraic criteria concern the ways in which various functions can be intermixed to satisfy requirements that certain functions can or cannot be computed more easily than certain other functions. (More detailed motivation for the relevance of these algebraic notions is given later.) Specifically, in Theorem 2 we generalize the embeddability theorem of McCreight and Meyer [12] (discussed below) by making speed-upable functions responsible for the embedding.

The object of this note is to close a slight gap, hitherto seemingly over-looked, in Nicod's derivation from his postulates of the primitives of Principia mathematica. The gap consists in this: Nicod's propositions corresponding to *1·1−*1·6 of the Principia, though they bear the same names and have the same symbol-forms as the Principia propositions, are not precisely *1·1−1·6. Indeed, if we denote Nicod's “p ⊃ q” and “p · q” by “p < q” and “pq” respectively, and if we write p′ for p∣p and p″ for (p′)′ then in terms of the “stroke,” we have

But p ⊃ q and p · q of the Principia are

Accordingly, the Nicod-Principia propositions are not *1·1−*1·6. They are *1·1−*1·6 in which < takes the place of ⊃, and Nicod has not shown us explicitly how to pass from < to ⊃.

It is true that Nicod also proves the following:

i.e.,

This yields: If ⊢· p < q then ⊢· p ⊃ q, and conversely. This would seem to allow passing from < to ⊃ throughout a proposition. But, it is clear, T1 permits this change only in the case of the principal <. The change in the case of a minor < still has to be justified.

A system formalized within the first order predicate calculus either with or without the identity but without predicate or function variables we call simply a theory. We say that a theory T has a recursively enumerable complement (r.e. Comp.) if the set of all sentences of T not valid in T is recursively enumerable. This is equivalent to saying that there exists an effective, purely mechanical procedure for establishing the non-validity in T of precisely those sentences of T which are in fact not theorems of T. If in addition T is recursively enumerable, that is, if there is an effective procedure for establishing the validity in T of its theorems, then T is decidable. It will be shown here that several well-known results concerning decidable theories can be extended to cover theories with r.e. comps. In this respect it should be observed that there exist theories having r.e. comps. which are not decidable. An example is the theory which has as valid sentences just those which contain a given two-place predicate and which are valid in all non-void finite universes [3], [4].

In [1], Bemays gives an axiomatization of set-theory which represents the furthest development to date of the Zermelo-Frankel-Gödel tradition. The purpose of this note is to show that one of his axioms is redundant.

Primitives are ∈, set-abstraction (denoted [x/A(x)]) and the Hilbert selector (denoted σ). Different styles are used for free variables (a, b,c, …) and bound variables (x, y, z, …); both kinds of variables range over both sets and classes. Quantifiers and connectives are classical.

In 1984, Henson and Rubel [2] proved the following theorem: If p(x1,…, xn) is an exponential polynomial with coefficients in ℂ with no zeroes in ℂ, then p(x1,…, xn) = eg(x1,…, xn) where g(x1,…, xn) is some exponential polynomial over C. In this paper, I will prove the analog of this theorem for Zilber's Pseudoexponential fields directly from the axioms. Furthermore, this proof relies only on the existential closedness axiom without any reference to Schanuel's conjecture.

We prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.

Two applications come immediately from this result:

First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.

The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.

Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals cT, lT, ϕT, and μT: roughly, cT is the “tallness” of T: lT is the “width” of T, ϕ is the number of “points of finite branching” in T: and μ is the number of “sections of no branching” in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by a pseudo-tree? We show that the possible inequalities among cT, lT, ϕT, and μT attained by pseudo-trees T are closely related to the existence of generalized Suslin trees.

Let P be the set of prime numbers. Theorem 1 of [1] shows that the linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,P〉 and therefore that the first-order theory of this structure is undecidable. Let m be any fixed natural number >2, let R be the set of natural numbers <m which are prime to m, and let r be any fixed element of R. The set

is infinite (Dirichlet). Theorem 1 of [1] can be improved as follows:

Proposition. The linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,Pm,r〉 and therefore that the first-order theory of this structure is undecidable.

Proof. We follow [1] with the following new ingredients. Let k be the number of elements of R, i.e. k = ϕ(m) where ϕ is Euler's totient function. Since k is even, the polynomial g(n) = nk + n satisfies g(0) = g(−1) = 0, so (by Lemma 1 of [1]) it follows from the linear case of (H) that there are natural numbers al (l ϵ ω) such that al+g(0), al+g(1),…, al+g(l) are consecutive primes. Since R is finite, we may assume that all the al's have the same residue t in R, so that al+g(i) ≡ t+1+i (mod m) for i ϵ R. This implies that the function t+1+i (reduced mod m) gives a permutation of R, so we can find s ϵ R such that al+g(s) ≡ r (mod m). Consider the polynomial h(n) = g(s + mn) and let bl = as+ml. Then bl + h(0), bl + h(1),…, bl + h(l) are elements of Pm,r. They are not necessarily consecutive elements of Pm,r, but they are separated by a fixed number of elements of Pm,r. This implies that {h(n) ∣ n ϵ ω} is definable in 〈ω,+,Pm,r〉(by adapting the proof of Theorem 1 of [1]), and the result follows.