Presents a new framework for the complexity of algorithms, for all readers interested in the theory of computation.

This little gem is stated unbilled and proved in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong form, it reads as follows:Second Recursion Theorem. Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of arguments with values in V so that the standard (...) assumptions and hold with. Every n-ary recursive partial function with values in V is for some e. For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let.We will abuse notation and write ž; rather than ž() when m = 0, so that takes the simpler formin this case ). (shrink)

We show that several theorems on ordinal bounds in different parts of logic are simple consequences of a basic result in the theory of global inductive definitions.

By language we understand a lower predicate calculus with identity and (perhaps) relation and function symbols. It is convenient to allow for more than one sort of variable. Now each individual constant (if there are any) is of a specified sort, the formal expressions R(t1, … tn), f(t1,…, tn) are well formed only if the terms t1, …, tn are of specified sorts determined by the relation symbol R and the function symbol f, and the term f(t1, …, tn) (if (...) well formed) is of a sort determined by f. We admit s = t as well formed, no matter what the sorts of s and t. (shrink)

The Suslin–Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation.

of L ATEX2 and essentially1 all commands of the AMS-L ATEX documentclass amsart are recognized by the documentclass asl, and so manuscripts prepared for those versions of TEX can be processed by the ASL Typesetting Office practically as-they-are.2 This brief guide is aimed at those authors who are reasonably..

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from the remainder function rem, meaning that in computing its time complexity function cε, we assume that the values rem are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch (...) more is known about cε, but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd from rem with time complexity cα, then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα > r log2a. (shrink)

We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The "pure," propositional fragment FLR 0 turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models.

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive programwhere rem is the remainder in the division of a by b, the unique natural number r such that for some natural number q,It is an algorithm from the remainder function rem, meaning that in computing its time complexity function cε, we assume that the values rem are provided on demand by some “oracle” in one “time unit”. It is easy to prove thatMuch (...) more is known about cε, but this simple-to-prove upper bound suggests the proper formulation of the Euclidean's optimality among its peers—algorithms from rem:Conjecture. If an algorithm α computes gcd from rem with time complexity cα, then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα > r log2a. (shrink)