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Proofs in the Nuprl system, an implementation of a constructive type theory, yield "correct-by-construction" programs. In

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In this paper, we take an abstract view of search by describing search procedures via particular kinds of proofs in type theory. We rely on the proofs-as-programs interpretation to extract programs from our proofs. Using these techniques we explore, in depth, a large family of search problems by parameterizing the speciÿcation of the problem. A constructive proof is presented which has as its computational content a correct search procedure for these problems. We show how a classical extension to an otherwise constructive system can be used to describe a typical use of the nonlocal control operator call/cc. Using the classical typing of nonlocal control we extend our purely constructive proof to incorporate a sophisticated backtracking technique known as 'con ict-directed backjumping' (CBJ). A variant of this proof is formalized in Nuprl yielding a correct-by-construction implementation of CBJ. The extracted program has been translated into Scheme and serves as the basis for an implementation of a new solution to the Hamiltonian circuit problem. This paper demonstrates a nontrivial application of the proofs-as-programs paradigm by applying the technique to the derivation of a sophisticated search algorithm; also, it shows the generality of the resulting implementation by demonstrating its application in a new problem domain for CBJ.

Abstract. This paper highlights a methodology of Nuprl proof that results in e cient programs that are more readable than those produced by other established methods for extracting programs from proofs. We d escribe a formal constructive proof of the decidability of a sequent calculus for classical propositional logic. The proof is implemented in the Nuprl system and the resulting proof object yields a "correct-by-construction" program for deciding propositional sequents. If the sequent i s v alid, the program reports that fact otherwise, the program returns a counterexample in the form of a falsifying assignment. We employ Kleene's strong three-valued logic to give more informative c o u n ter-examples, it is also shown how t h i s semantics agrees with the standard two-valued presentation.