Exposition: Synthesis via Functional Interpretation

The aim of this short paper is to give a practical introduction to functional interpretation of proofs in arithmetic for computer scientists interested in synthesis. Towards this, we will define our own notion of functional interpretation which differs (sometimes only inessentially) from those used in the literature, but has the advantage (in the opinion of the author) of being very natural. Note that we only show functional interpretation by definition and example — it is quite possible that proving the correctness of this formalism is cumbersome (or even impossible). Still, as can be witnessed below, our formalism allows extraction of a correct program from a non-trivial proof in a systematic way, hopefully elucidating a few central ideas common to all notions of functional interpretation. With this disclaimer in mind, we can start setting up our machinery for functional interpretation. Towards extracting a program from a proof, we have to fix (1) a programming language, (2) a proof system, and (3) a translation from (2) to (1). The programming language. Choosing a suitable programming language for functional extraction involves some design decisions. Usually, one takes a functional programming language, since this often induces a simple and natural translation from proofs to programs. Some other decisions are more inessential, e.g. how to represent boolean values in the programming language (e.g. by constants K,J, or by numerals 0, 1). For simplicity, we choose the untyped λ-calculus extended with some constants for arithmetic, pairs, and program control. More precisely, we assume existence of a countable set of variables V , and a fixed set of constant symbols t0, 1,`, ́,pair, left, right, isZero, IfThenElse,Ru. Variables and constants are λ-terms, and if s, t are λ-terms and x a variable then st and λx.t are λ-terms. We use infix notation for `, ́ and write IfThenElset1t2t3 as If t1 Then t2 Else t3 and pairt1t2 as pt1, t2q). The intended semantics of the symbols are clear except maybe for R, which will be the recursion operator. Variable-free terms consisting only of 0, 1,` may denote numbers (i.e. 1` p1` 1q and p1` 1q` 1 both denote 3 P N, while ``` does not denote a number). Such terms are called arithmetical, and we will often not distinguish between a number α P N and the arithmetical terms that denote it. In particular, if α P N and t is a λ-term, then by tα we denote the λ-term which applies t to the numeral representing α.