y Abstract What is an algorithm? The interest in this foundational problem is not only theoretical; applications include specification, validation and verification of software and hardware systems. We describe the quest to understand and define the notion of algorithm. We start with the Church-Turing thesis and contrast Church's and Turing's approaches, and we finish with some recent investigations.

Most current research on automated theorem proving is concerned with proving theorems of first-order logic. We discuss two examples which illustrate the advantages of using higher-order logic in certain contexts. For some purposes type theory is a much more convenient vehicle for formalizing mathematics than axiomatic set theory. Even theorems of first-order logic can sometimes be proven more expeditiously in higher-order logic than in first-order logic. We also note the need to develop automatic theorem-proving methods which may produce proofs which do not have the subformula property. 1. Introduction In some ways it appears that the field of automated deduction is ahead of its time. We have increasingly good methods for proving theorems, but the hardware available is still not adequate for many of the problems to which we would like to apply our techniques. However, this will change. Radically new computers based on exotic technologies such as DNA computing, quantum computing, and ...

Numerous classical and non-classical logics can be elegantly embedded in Church’s simple type theory, also known as classical higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.