Kirby and Paris have exhibited combinatorial algorithms whose computations always terminate, but for which termination is not provable in elementary arithmetic. However, termination of these computations can be proved by adding an axiom first introduced by Goodstein in 1944. Our purpose is to investigate this axiom of Goodstein, and some of its variants, and to show that these are potentially adequate to prove termination of computations of a wide class of algorithms. We prove that many variations of Goodstein's axiom are (...) equivalent, over elementary arithmetic, and contrast these results with those recently obtained for Kruskal's theorem. (shrink)

We describe KAT-ML, an implementation of an interactive theorem prover for Kleene algebra with tests. The system is designed to reflect the natural style of reasoning with KAT that one finds in the literature. One can also use the system to reason about properties of simple imperative programs using schematic KAT. We explain how the system works and illustrate its use with some examples, including an extensive scheme equivalence proof.

Nondeterministic programs occurring in recently developed programming languages define nondeterminate partial functions. Formulas (Boolean expressions) of such nondeterministic languages are interpreted by a nonempty subset of {T (true), F (false), U (undefined)}. As a semantic basis for the propositional part of a corresponding nondeterministic three-valued logic we study the notion of a truth-function over {T, F, U} which is computable by a nondeterministic evaluation procedure. The main result is that these truth-functions are precisely the functions satisfying four basic properties, called (...) -isotonic, –-isotonic, hereditarily guarded, and hereditarily guard-using, and that a function satisfies these properties iff it is explicitly definable (in a certain normal form) from if..then..else..fi, binary choice, and constants. (shrink)