Matrix-based Constructive Theorem Proving

Formal methods for program verification, optimization, and synthesis rely on complex mathematical proofs, which often involve reasoning about computations. Because of that there is no single automated proof procedure that can handle all the reasoning problems occurring during a program derivation or verification. Instead, one usually relies on proof assistants like NuPRL (Constable et al., 1986), Coq (Dowek and et. al, 1991), Alf (Altenkirch et al., 1994) etc., which are based on very expressive logical calculi and support interactive and tactic controlled proof and program development. Proof assistants, however, suffer from a very low degree of automation, since all their inferences must eventually be based on sequent or natural deduction rules. Even proof parts that rely entirely on predicate logic can seldomly be found automatically, as there are no complete proof search procedures embedded into these systems. It is therefore desirable to extend the reasoning power of proof assistants by integrating well-understood techniques from automated theorem proving. Matrix-based proof search procedures (Bibel, 1981; Bibel, 1987) can be understood as compact representations of tableaux or sequent proof techniques. They avoid the usual redundancies contained in these calculi and are driven by complementary connections, i.e. pairs of atomic formulae that may become leaves in a sequent proof, instead of the logical connectives of a proof goal. Although originally developed for classical logic, the connection method has recently been extended to a variety of non-classical logics such as intuitionistic logic (Otten and Kreitz, 1995), modal logics (Kreitz and Otten, 1999), and fragments of linear logic (Kreitz et al., 1997; Mantel and Kreitz, 1998). Furthermore, algorithms for converting matrix proofs into sequent proofs have been developed (Schmitt and Kreitz, 1995; Schmitt and Kreitz, 1996), which makes it possible to view matrix proofs as plans for predicate logic proofs that can be executed within a proof assistant (Bibel et al., 1996; Kreitz et al., 1996). Viewing matrix proofs as proof plans also suggests the integration of additional proof planning techniques into the connection method. Rewrite techniques such as rippling (Bundy et al., 1993), for instance, have successfully been used as proof planners