Combining Deduction~Modulo and Logics of Fixed-Point Definitions

Coplas Talk by Gopalan Nadathur, University of Minnesota and IT University of Copenhagen


Inductive and coinductive specifications are widely used in formalizing computational systems. Such specifications have a natural rendition in logics that support fixed-point definitions. Another useful formalization device is that of recursive specifications. These specifications are not directly complemented by fixed-point reasoning techniques and, correspondingly, do not have to satisfy strong monotonicity restrictions. We show how to incorporate a rewriting capability into logics of fixed-point definitions towards additionally supporting recursive specifications. Specifically, we describe a natural deduction calculus that adds a form of ``closed-world'' equality---a key ingredient to supporting fixed-point definitions---to deduction modulo a framework for extending a logic with a rewriting layer operating on formulas. Our calculus enjoys strong normalizability when the rewrite system satisfies general properties and we have shown its usefulness in formalizing arguments such as those based on logical relations. Our integration of closed-world equality into deduction modulo uses an elimination principle for this form of equality that, for the first time, allows us to require finiteness of proofs without sacrificing stability under reduction.


I am interested in the design, use and implementation of programming languages. I am also interested in logic, especially as it underlies our understanding of programming formalisms and as it informs our construction of general reasoning systems.


I have developed a higher-order logic programming language called λProlog in collaboration with Dale Miller.


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