Michael J. Beeson

Proving Programs and Programming Proofs

Publisher Summary Progress in applications to large-scale computer systems, depends on the design of new formal systems. This chapter raises some issues and makes a technical contribution by considering some theories of Feferman from the viewpoint of computer science, and comparing them with the theories of Martin-Lőf. The command language and assertion language of proving programs are discussed in the chapter. By “proving programs” is meant more explicitly: proving properties of programs or proving properties of the execution of programs. People speak of the “correctness” of a program with respect to its “specifications”—by this they mean that if the program gets an input of the kind it is designed for, it will produce an output having certain specified relations with the input. This notion is divided into the two notions of “total correctness” (an output is always produced and it is correct) and “partial correctness” (if an output is produced then it is correct).

Recursive models for constructive set theories

Abstract We define recursive models of Martin-Lofs (type or) set theories. These models are a sort of recursive realizability; in fact, we show that for implication-free formulae of HAω, satisfaction in the model coincides with mr-HEO realizability. Using an idea of Aczel, we extend the model to a recursive model of the constructive set theories of Myhill and Friedman. Our models can be described without presupposing any knowledge of Martin-Lofs theories, and may make them seem less mysterious. We use our models to obtain several metamathematical results, for example consistency and independence results concerning continuity of functions on compact metric spaces. On the other hand, Martin-Lofs (latest) theories refute continuity of functions from NN to N, as well as Churchs thesis, although a show that all provably well-defined functions are continuous.

Towards a computation system based on set theory

Abstract An axiomatic theory of sets and rules is formulated, which permits the use of sets as data structures and allows rules to operate on rules, numbers, or sets. We might call it a “polymorphic set theory”. Our theory combines the λ-calculus with traditional set theories. A natural set-theoretic model of the theory is constructed, establishing the consistency of the theory and bounding its proof-theoretic strength, and giving in a sense its denotational semantics. Another model, a natural recursion-theoretic model, is constructed, in which only recursive operations from integers to integers are represented, even though the logic can be classical. Some related philosophical considerations on the notions of set, type, and data structure are given in an appendix.

Continuity in Intuitionistic Set Theories

Publisher Summary This chapter examines the connections between the two concepts of continuity and constructivity. A connection between constructivity and continuity has long been perceived in Hadamards criteria for a well-posed problem in differential equations. One of these criteria is that the solution should depend continuously on the parameters of the problem. One way of making the principle of continuity precise is as a derived rule of inference: if the problem can be proved in a constructive formal system T to have a solution, then the solution depends continuously on the parameters of the problem. The chapter describes the principal intuitionistic set theories and the formulation of nonextensional set theories.

Logic and computation in MATHPERT: an expert system for learning mathematics

MATHPERT (as in “Math Expert”) is an expert system in mathematics explicitly designed to support the learning of algebra, trigonometry, and first semester calculus. This paper gives an overview of the design of MATHPERT and goes into detail about some connections it has with automated theorem proving. These connections arise at the borderline between logic and computation, which is to be found when computational “operators” have logical side conditions that must be satisfied before they are applicable. The paper also explains how MATHPERT maintains and uses an internal model of its user to produce individually tailored explanations, and how it dynamically generates individualized and helpful error messages by comparing user errors to its own internal solution of the problem.

Some applications of Gentzen's proof theory in automated deduction

Reasoning and problem-solving programs must eventually allow the full use of quantifiers and sets, and have strong enough control methods to use them without combinatorial explosion. J. McCarthy

Some results on finiteness in Plateau's problem, part II

In this paper we attack the problem, whether any Jordan curve in R 3 may bound infinitely many surfaces (of the topological type of the disk) which furnish relative minima of the area functional. In Part I, we have reduced this problem to the study of certain one-parameter families of minimal surfaces, terminating in a minimal surface with a branch point; and in Part I, we dealt with the case of an interior branch point. In this paper, we take up the study of the boundary branch point case. We make several computations based on the eigenvalue problem associated with the second variation of the area functional. The result of these computations is that a one-parameter family of the sort described must satisfy certain conditions. If these conditions were contradictory, we would show that no real-analytic Jordan curve can bound infintely many relative minima of area; but instead, our computations led to the discovery of some one-parameter families that do satisfy these conditions, although they are bounded by a straight line instead of a Jordan curve. We are able to obtain a partial result formulated in geometric terms, showing that under certain conditions on a real-analytic Jordan curve F, F cannot bound infinitely many relative minima of area. Namely, this is so if every plane which contains a point not on the convex hull of F, meets F in at most 8 points. Note that this generalizes the result of Part I, where F was required to lie on the boundary of a convex body. If it is indeed true that no real-analytic Jordan curve bounds infinitely many relative minima of area, it must be proved by some method which (unlike those used here) would not apply to the examples mentioned (with straight-line boundaries). We now give a more general discussion of the context of this problem. If F is a Jordan curve in R 3, the classical problem of Plateau calls for a minimal surface (of the type of the disk) bounded by F. These minimal surfaces may be relative minima of area (in some suitable class of surfaces bounded by F and some suitable topology on this class of surfaces), or they may be unstable, i.e. not relative minima. Those minimal surfaces which are the mathematical

A constructive version of Tarski's geometry

Abstract Constructivity, in this context, refers to a theory of geometry whose axioms and language are closely related to ruler and compass constructions. It may also refer to the use of intuitionistic (or constructive) logic, but the reader who is interested in ruler and compass geometry but not in constructive logic will still find this work of interest. Euclids reasoning is essentially constructive (in both senses). Tarskis elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive (in both senses), even if we restrict attention (as we do here) to the theory with line–circle continuity in place of first-order Dedekind completeness. Hilberts axiomatization has a much more elaborate language and many more axioms, but it contains no essential non-constructivities. Here we exhibit three constructive versions of Tarskis theory. One, like Tarskis theory, has existential axioms and no function symbols. We then consider a version in which function symbols are used instead of existential quantifiers. This theory is quantifier-free and proves the continuous dependence on parameters of the terms giving the intersections of lines and circles, and of circles and circles. The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Paschs axiom and the parallel axiom; this choice of function symbols connects directly to ruler and compass constructions. All three versions have this in common: the axioms have been modified so that the points they assert to exist are unique and depend continuously on parameters. This modification of Tarskis axioms, with classical logic, has the same theorems as Tarskis theory, but we obtain results connecting it with ruler and compass constructions as well. In particular, we show that constructions involving the intersection points of two circles are justified, even though only line–circle continuity is included as an axiom. We obtain metamathematical results based on the Godel double-negation interpretation, which permit the wholesale importation of proofs of negative theorems from classical to constructive geometry, and of proofs of existential theorems where the object asserted to exist is constructed by a single construction (as opposed to several constructions applying in different cases). In particular, this enables us to import the proofs of correctness of the geometric definitions of addition and multiplication, once these can be given by a uniform construction. We also show, using cut-elimination, that objects proved to exist can be constructed by ruler and compass. (This was proved in [3] for a version of constructive geometry based on Hilberts axioms.) Since these theories are interpretable in the theory of Euclidean fields, the independence results about different versions of the parallel postulate given in [5] apply to them; and since addition and multiplication can be defined geometrically, their models are exactly the planes over (constructive) Euclidean fields.

Automatic Generation of Epsilon-Delta Proofs of Continuity

As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilon-delta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, trigonometry, and some calculus. At the border of logic and computation, they involve several types of arguments involving inequalities, including transitivity chaining and several types of bounding arguments, in which bounds are sought that do not depend on certain variables. Control mechanisms have been developed for intermixing logical deduction steps with computational steps and with inequality reasoning. Problems discussed here as examples involve the continuity and uniform continuity of various specific functions.

Triangles with vertices on lattice points

A triangle is called embeddable in Zn if it is similar to a triangle whose vertices have integer coordinates in Rn. It was already known that a triangle is embeddable in Z2 if and only if all its angles have rational tangents. We show that a triangle is embeddable in some Zn if and only if it is embeddable in Z5, and if and only if all its angles have tangents with rational squares. We reduce the problem of embeddability to a certain Diophantine equation. We give a complete characterization of the triangles embeddable in Zn for every n. In particular, there are triangles embeddable in Z5 but not Z4, and in Z3 but not Z2, but surprisingly, the same triangles are embeddable in Z3 as are embeddable in Z4. A triangle is embeddable in Z3 if and only if the tangents of its angles are all rational multiples of v4 for some integer k which is a sum of three squares. The proofs use only elementary number theory. The simplest question concerning embeddability is this: is the equilateral triangle embeddable in Z2? That is, are there lattice points in the plane forming the vertices of an equilateral triangle? As it turns out, there are not. Of course, the equilateral triangle is embeddable in Z3 , with vertices at the points one unit along each of the three. axes. This illustrates that more triangles may be embeddable if more dimensions are allowed. The general problem addressed in this paper is to characterize the triangles embeddable in Zn for each n. We give a complete solution of this problem, as described in the preceding abstract. The problem solved in this paper has a surprisingly long history, and is connected to the work of several other authors. These points are discussed in a separate section near the end of the paper. Dimension Two. The following proposition is included as an introduction to the subject. (In the proposition, infinity counts as a rational tangent.)