LPAR-05 Workshop: Empirically Successfull Automated Reasoning in Higher-Order Logic (ESHOL)

Otter-lambda is a theorem-prover based on an untyped logic with lambda calculus, called Lambda Logic. Otter-lambda is built on Otter, so it uses resolution proof search, supplemented by demodulation and paramodulation for equality reasoning, but it also uses a new algorithm, lambda unification, for instantiating variables for functions or predicates. The basic idea of a typed interpretation of a proof is to “type” the function and predicate symbols by specifying the legal types of their arguments and return values. The idea of “implicit typing” is that if the axioms can be typed in this way then the consequences should be typable too. This is not true in general if unrestricted lambda unification is allowed, but for a restricted form of “type-safe” lambda unification it is true. The main theorem of the paper shows that the ability to type proofs if the axioms can be typed works for the rules of inference used by Otter-lambda, if type-safe lambda unification is used, and if demodulation and paramodulation from or into variables are not allowed. All the interesting proofs obtained with Otter-lambda, except those explicitly involving untypable constructions such as fixed-points, are covered by this theorem.