Complete functional synthesis

Abstract

Synthesis of program fragments from specifications can make programs easier to write and easier to reason about. To integrate synthesis into programming languages, synthesis algorithms should behave in a predictable way-they should succeed for a well-defined class of specifications. They should also support unbounded data types such as numbers and data structures. We propose to generalize decision procedures into predictable and complete synthesis procedures. Such procedures are guaranteed to find code that satisfies the specification if such code exists. Moreover, we identify conditions under which synthesis will statically decide whether the solution is guaranteed to exist, and whether it is unique. We demonstrate our approach by starting from decision procedures for linear arithmetic and data structures and transforming them into synthesis procedures. We establish results on the size and the efficiency of the synthesized code. We show that such procedures are useful as a language extension with implicit value definitions, and we show how to extend a compiler to support such definitions. Our constructs provide the benefits of synthesis to programmers, without requiring them to learn new concepts or give up a deterministic execution model.

Reductions for Synthesis Procedures

“…Note, however, that the inference rules that we refer to are from the previous section and therefore also specify how to compute the corresponding program. Unlike the procedure in [9] the procedure below does not perform efficient solving of equations, but could be refined to do so by adapting the description in [9] to our inference rules.…”

“…In principle, we can rewrite t 1 = t 2 into t 1 ≤ t 2 ∧ t 2 ≤ t 1 (see [9] for more efficient approaches). We rewrite t 1 ≤ t 2 into t 1 < t 2 + 1.…”

“…The synthesized functions are guaranteed to satisfy a given input/output relation expressed in a decidable logic. We call this approach complete functional synthesis [8,9]. The appeal of this direction is that it can synthesize functions over unbounded domains, and that the produced code is guaranteed to satisfy the specification for the entire unbounded range of inputs.…”

“…Previous work described synthesis procedures for linear arithmetic and sets [8,9] as well as extensions to unbounded bitvector constraints [4,18]. In this paper we make further steps towards systematic derivation of synthesis procedures by showing how inference rules that describe decision procedure steps (possibly for a combination of theories) can be generalized to synthesis procedures.…”

“…As remarked in [9], compiled synthesis procedures could be viewed as a result of partial evaluation of the execution of a constraint solver at run time. This is a useful observation from a methodological point of view.…”