Automatic program synthesis via synthesis

Work on theorem-proving-based automatic program synthesis (see Lee, e t ai.,1 for example) has been neglected iately. In their 1971 paper, Manna and Waldinger8 covered one of the main reasons why-the difficulty of synthesizing program loops within the current state of the art of automatic theorem-proving.· However, there is a great deal of continuing work in theorem-proving, and it is important that motivating work in related areas such as program synthesis not be neglected. Most theorem-proving-based synthesis systems attempt to constructively prove theorems of the form, "for all input values satisfying the desired input predicate I, there exists a corresponding set of values of output variables which satisfy the desired result predicate, R." The resulting program (if any) is correct with respect to the input and output predicates. In general, inductive proofs are needed to synthesize the loops within a program. This causes difficulty, and in fact previous work has not been notably successful with loops. To quote Manna and Waldinger, " ... these systems have been fairly limited; for example, either they have been completely unable to produce programs with loops or they have introduced loops by underhanded methods." Since most interesting programs contain loops in some form, this is a crucial problem for successful synthesis. We will limit our discussion to iterative loops. Manna and Waldinger outlined an approach to automatic synthesis which attacked the problem of iterative loops, and discussed the use of various forms of induction for reducing synthesis to the proving of loop-free (i.e., induction-free) lemmas. However, they were dissatisfied with the large number of equivalent induction principles required by their approach. In fact, only a single rule is needed. Loop invariants can be used to provide a single, general rule for expressing the synthesis of loops in terms of loop-free lemmas. We consider only input and output predicate pairs, I and R, which make sense in the context of program synthesis; i.e., they must be decidable and "define" a nontrivial recursive function, in the sense that 3F{I(v )~R(v,F(v»} is true. If a recursive function is "defined" by an I,R pair,