Cause-effect relationships and partially defined Boolean functions

This paper investigates the use of Boolean techniques in a systematic study of cause-effect relationships. The model uses partially defined Boolean functions. Procedures are provided to extrapolate from limited observations, concise and meaningful theories to explain the effect under study, and to prevent (or provoke) its occurrenc

A proof-producing decision procedure for real arithmetic

Abstract. We present a fully proof-producing implementation of a quantifierelimination procedure for real closed fields. To our knowledge, this is the first generally useful proof-producing implementation of such an algorithm. Whilemany problems within the domain are intractable, we demonstrate convincing examples of its value in interactive theorem proving. 1 Overview and related work Arguably the first automated theorem prover ever written was for a theory of lineararithmetic [8]. Nowadays many theorem proving systems, even those normally classified as `interactive ' rather than `automatic', contain procedures to automate routinearithmetical reasoning over some of the supported number systems like N, Z, Q, R and C. Experience shows that such automated support is invaluable in relieving users ofwhat would otherwise be tedious low-level proofs. We can identify several very common limitations of such procedures:- Often they are restricted to proving purely universal formulas rather than dealingwith arbitrary quantifier structure and performing general quantifier elimination.- Often they are not complete even for the supported class of formulas; in partic-ular procedures for the integers often fail on problems that depend inherently on divisibility properties (e.g. 8x y 2 Z. 2x + 1 6 = 2y)- They seldom handle non-trivial nonlinear reasoning, even in such simple cases as 8x y 2 R. x> 0 ^ y> 0) xy> 0, and those that do [18] tend to use heuristicsrather than systematic complete methods.- Many of the procedures are standalone decision algorithms that produce no certifi-cate of correctness and do not produce a `proof ' in the usual sense. The earliest serious exception is described in [4]. Many of these restrictions are not so important in practice, since subproblems aris-ing in interactive proof can still often be handled effectively. Indeed, sometimes the restrictions are unavoidable: Tarski's theorem on the undefinability of truth implies thatthere cannot even be a complete semidecision procedure for nonlinear reasoning ove