Automatic Synthesis of Numerical Computer Programs

This paper is about rule-based problem solving. It has a motto: The less a rule can do, the better the effects of using it can be predicted. This motto suggests the key to obtaining coherent behavior by avoiding exponential behavior (combinatorial explosion) in a problem solving program for the domain of synthesizing numerical computer programs from their non-procedural specifications Introduction To construct an automatic programming system, several separate but related sub-domains and their corresponding experts are required. These include program understanding, specification acquisition, program verification, and program synthesis. The last is the domain of expertise of the system described by this paper. A program synthesis system takes a specification (for example, a predicate calculus description of the relationships between the inputs and the outputs) of a program and produces a program meets those specifications. Diagram "Bernoulli Example" shows an input specification (in mathematical notation) and the LISP code produced by the system. The fundamental difficulty in constructing problem-solving systems is the so-called "combinatorial explosion" problem. That is to say, given a reasonable measure of how hard the problem at hand should be, a "combinatorially explosive" problem solver takes an amount of time proportional to an exponential (or worse!) function of that measure In general, the size of program (and therefore certainly the time needed to write it) is an unbounded function of the size of its specification For purposes of measuring the coherence of a problem solver, a measure of "how hard a problem is" could be the number of rule applications that were needed (as opposed to the larger number actually performed) in the solution found. The result of this research is a collection of techniques (embedded in a synthesis system) that synthesize numerical computer programs. These techniques do not depend strongly on the fact that the task is synthesizing programs, so there is hope that they could be adapted to problem-solving in other domains. I conjecture that these techniques in fact run in time proportional to a polynomial function of the number of rules required to solve a synthesis problem A working program was obtained that (on a relatively unloaded PDP-IO) can solve several examples faster than an expert human programmer.