Automated Deduction with Shannon Graphs

Binary Decision Diagrams (BDDs) are a well-known tool for representing Boolean functions. We show how BDDs can be extended to full rst-order logic by integrating means for representing quantiiers. The resulting structures are called Shannon graphs. A calculus based on these Shannon graphs is set up, and its soundness and completeness proofs are outlined. A comparison of deduction with rst-order BDDs and semantic tableaux shows that both calculi are closely related. From a practical perspective, however, BDDs have advantages over tableaux: they provide a more compact representation, since BDDs can be understood as a linear, graphical representation of a fully expanded tableaux. Furthermore, BDDs represent not only the models of a formula, but also its counter models: this ooers a very eecient way to represent lemmata during the proof search. The last part of the paper introduces a compilation-based approach to implementing deduction systems based on Shannon graphs. The idea is to compile the graphs into programs that carry out the proof search at run time.