Abstract
There are many attempts to extract programs from formal proofs of theories of constructive mathematics, e. g. [3] [4] [5] [9] [11]. This paper is one of such attempts. The origin of the problem is the so called deductive or theorem-proving approach to the problem of automatic program synthesis. We explain the problem briefly according to [9] and [11]. Let D be a set of data and let A(x, y) be a binary predicate. Then a specification is given by the formula V^e D3yA(x, y). A program P is called a solution of the specification, and the predicate A is called the output predicate. The problem of automatic program synthesis is the problem to find a solution automatically from a given specification. A program-synthesis system using the theorem-proving approach finds an appropriate proof of the given specification and extracts a solution from the proof. However, there are no sufficiently powerful theorem-proving systems at the present time. On the other hand, it is relatively easy to construct a solution mechanically from a given proof. Even if the proof is constructed by a person, there is an advantage. If the proof has been checked mechanically, the resulting program does not require debugging or verification. First we will introduce a formal system called LM, which is a modification of T^~ } of Feferman [2]. The system T^~ is based on combinatory logic, on the other hand, LM is based on a variant of Lisp, which is called Lisp*. Our intended interpretation of the universe of LM is the set of S-expressions of Lisp*, on the other hand, the universe of T^~ does not have a fixed intended interpretation. These are the main difference between LM and To'. We will define a universal function of Lisp in LM, and using it define a modified <?-realizability for LM. By the aid of this realizability interpretation, we can extract Lisp programs with formal verifications in LM from constructive formal
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