On the border between functional programming and program synthesis

The importance of compositionality in program construction is being accepted quite well. With this respect, the relational programming has clear advantages over the functional programming. Unfortunately, there is no general technique of relational programming efficient enough to compete with the existing functional programming techniques. Here we discuss structural synthesis of programs a method of synthesis of functional programs explainable in terms of higher-order functional constraint nets, simple types or intuitionistic logic. This method has been used in implementation of declarative languages that allow us to specify concepts as relations and use them in specifications more flexibly than functions. 1. About the SSP Structural synthesis of programs (SSP) has been known for quite a number of years and has been used at least in two commercial systems: XpertPriz and PRIZ (4). During the years, SSP has been applied and extended in several ways. Here we summarize the recent extensions, and refer to (7) for some earlier extensions. SSP is a deductive program synthesis method based on the idea that we can construct programs taking into account only their structural properties. We use this idea for constructing programs from small as well as large modules whose behavior we don ́t describe in detail. Each preprogrammed module is supplied with a specification used as an axiom telling under which conditions it can be applied and which values it computes. However, the specification doesn ́t specify the relation between the input and output values. Instead of the general form of a program module specification .(P(x) -> .R(x,y)) with precondition P and postcondition R we use .(P(x) -> .R(y)) which tells us that a correct value (satisfying R) can be computed, but doesn ́t show the relation between this value and the input. The latter formula has the following equivalent form .P(x) -> .R(y). As a consequence, we can present axioms in a propositional language, considering the closed formulae .P(x) and .R(y) as propositions whose internal structure is inessential. This allows us to use logic in which we are able to handle theories (specifications for the synthesis of programs) with large amount (up to tens of thousands) of axioms. x ∀ y ∃ x ∀ y ∃