2 3 Objectives 2 4 Background and State of the Art 3 5 Project Organization 6 5.1 Technical Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.1 Logical Representation of the Specification Paradigms . . . . 6 5.1.2 Integration with the Proof Assistant Systems . . . . . . . . . 8 5.1.3 Publications and Cooperation . . . . . . . . . . . . . . . . . . 8 5.2 Task Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6 Work Planning 13 7 Deliverables 14 8 Project Management 14 9 Publication of Results 14 10 Ethical, Social and Environmental Impact 14 11 Team Member Experience 15 12 Available and Requested Resources 19 Research and Development medium size project proposal submitted to the PRAXIS XXI Program, in the area of Information Technology and Telecommunications, in August 1995 1 1 Title LOGCOMP-- Logic and Computation Integration of Proof Assistants with Symbolic Systems for Specification and Prototyping. 2 Abstract The project aims to integra...

We propose a new way to reason about general recursive functional programs in the dependently typed programming language Agda, which is based on Martin-Löf’s intuitionistic type theory. We show how to embed an external programming logic, Aczel’s Logical Theory of Constructions (LTC) inside Agda. To this end we postulate the existence of a domain of untyped functional programs and the conversion rules for these programs. Furthermore, we represent the inductive notions in LTC (intuitionistic predicate logic and totality predicates) as inductive notions in Agda. To illustrate our approach we specify an LTC-style logic for PCF, and show how to prove the termination and correctness of a general recursive algorithm for computing the greatest common divisor of two numbers. Categories and Subject Descriptors F.3.1 [Logics and meanings of programs]: Specifying and Verifying and Reasoning about Programs–Logics of programs; D.2.4 [Software Engineering]:

ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semi-inductive definitions, which require non-trivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of anti-foundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of non-classical logical systems, from contraction-free and many-valued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of self-referential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.