A Layered Approach to Extracting Programs From Proofs With an Application in Graph Theory

Abstract

In this paper we describe our system for automatically extracting \correct" programs from proofs using a development of the Curry-Howard process. Although program extraction has been developed by many authors (see 5, ?,?]), our system has a number of novel features designed to make it very easy to use and as close as possible to ordinary mathematical terminology and practice. These features include 1. the use of Henkin's technique 6] to reduce higher-order logic to many-sorted (rst-order) logic 2. the free use of new rules for induction subject to certain conditions 3. the extensive use of previously programmed (primitive) recursive functions. 4. the use of templates to make the reasoning much closer to normal mathematical proofs. 5. an extension of the technique of the use of Harrop formulae to classically true formulae (cf. the footnote on p. 101 in Kreisel 9]); As an example of our system we give a constructive proof of the well-known theorem that every graph of even parity, which is non-trivial in the sense that it does not consist of isolated vertices, has a cycle. Given such a graph as input, the extracted program produces a cycle as promised. 1 The word \correct" in this paper means \meeting its speci cation". 2 It is convenient to call the entities \sorts" rather than \types" as there are many other \types" in this paper. In fact for our present purposes we could easily reduce everything to rst order. To do this we should just use a predicate, In(x; y), say, to represent \x is in the list y" and similarly for lists of lists. The technique is described in Henkin 6]. However we write our expressions in the conventional way and they therefore sometimes appear as involving higher order expressions. 3 The axioms one would normally employ in (constructive) mathematics are Harrop formulae (de ned below). The restriction is a natural one and also has a signi cant e ect on reducing the size of our extracted programs. Harrop axioms are axioms that are Harrop formulae and Harrop formulae are de ned as follows: 1. An atomic formula or ? is a Harrop formula. 2. If and are Harrop, then so is ^. 3. If is a Harrop formula and is any formula, then ! is a Harrop formula. 4. If is a Harrop formula, then 8x is a Harrop formula.

Protocols between Programs and Proofs

“…The function computes the nth member of the list. The definitions for the cases α = N at, List(N at) are given in [13].…”

“…We set a predicate Graph(l) to mean that a list l : List (List(N at)) represents a graph. The formula Graph(l) is defined in Fred by the conjunction of four Harrop formulae (see [13]). A graph has even parity, which is represented by the predicate Evenparity(l), if the number of vertices adjacent to each vertex is even.…”

“…We next sketch a proof that every even parity graph can be decomposed into a list of disjoint cycles. (More details of the proof may be found in [13] and [14].) From this proof we obtain a program that gives a list of the non-trivial disjoint cycles.…”

“…The theorem to be proved is Remark 6. See [13] for details of the function start which takes as its argument a list, l, of lists of numbers and returns the head of the first list in l that has length greater than 1. If there is no list in l (with length > 1) then the default 0 is returned.…”

“…In [13] we showed how to extract a program, from a Curry-Howard term t, for finding a cycle (represented by the predicate Cycle) in an even parity graph (represented by the predicate Evenparity):…”