Logic Programming as a Basis for Lean Deduction: Achieving Maximal Efficiency from Minimal Means

Researchers in Automated Reasoning often complain that there are sparse applications of the techniques they develop. One reason might be that implementation-oriented research favors huge and highly complex systems and that this does not suit the needs of many applications.1 It is hard to see how to apply these systems – besides using them as a black box. Adaptability, however, is an important criterion for applying techniques; this discrepancy can be overcome by using lean theorem provers. The idea of lean deduction is to achieve maximal efficiency from minimal means. Every possible effort is made to eliminate overhead; based on experience in implementing (complex) deduction systems, only the most important and efficient techniques and methods are implemented. Logic programming languages provide an ideal tool for implementing lean deduction, as they offer a level of abstraction that is close to the needs for building first-order deduction systems. The Prolog program shown in Figure 1, called leanTAP (Beckert & Posegga, 1994b), is an instance of such a lean deduction system: it implements a complete and sound theorem prover for first-order logic in Skolemized negation normal form. The underlying calculus is based on free-variable semantic tableaux (Fitting, 1990) (we shall explain the program in Section 2). Our approach surely does not lead to deduction systems that are superior to highly sophisticated theorem provers like Otter (McCune, 1990) or Setheo (Letz et al., 1992); these are better on solving difficult problems. However, many applications do not require deduction which is as complex as the state of the art in automated theorem proving. Furthermore, there are often strong constraints on the time allowed for deduction. In such areas our approach can be extremely useful: it offers high inference rates on simple to moderately complex problems and a high degree of adaptability. Another important argument for lean deduction is safety: It is easily possible to verify the couple of lines of standard Prolog implementing leanTAP (Beckert & Posegga, 1994a; Posegga & Schmitt, 1995); verifying thousands of lines of C code, however, is hard—if not impossible—in practice. 2 leanTAP : The Program We will briefly describe the basic working principle of the tableau-based theorem prover leanTAP shown in Figure 1.2 We assume familiarity with free-variable tableaux (e.g. (Fitting, 1990)) and the basics of programming in Prolog. For the sake of simplicity, we restrict our considerations to first-order formulæ in Skolemized negation normal form. This is not a strong restriction; the prover can easily be extended to full first-order logic by adding the standard tableau rules. However, Skolemization has to be implemented carefully to achieve the highest possible performance (Beckert et al., 1993).3 Proc. Workshop on Logic Programming, University of Zürich, Switzerland. Oct. 1994 1Empirical evidence for this claim can easily be given: Hardware and Software Verification are usually listed as the most important application areas with practical relevance of Automated Deduction. However, when looking closer, one realizes that the implemented systems in this area either use interactive provers, or application-specific developments. The classical, stand-alone theorem provers seem to be not flexible enough to be integrated into such systems. 2The interested reader is might wish to consult (Beckert & Posegga, 1994a) for a more detailed explanation. 3A Prolog program for computing an optimized negation normal form, as well as leanTAP ’s source code, is available upon request from the authors.