Proof finding algorithms for implicational logics

AbstractThe work in Bunder (Theoret. Comput. Sci. 169 (1996) 3–21) shows that for each one of many implicational logics the set of all lambda terms, that represent proofs in that logic, can be specified. This paper gives, for most of these logics, algorithms which produce, for any given formula, a form of minimal proof within a fixed number of steps or otherwise a guarantee of unprovability. For the remaining logics there are similar algorithms that produce proofs, but not within a fixed number of steps. The new algorithms have been implemented in Oostdijk (Lambda Cal2)