Relational descriptions have been used in formalizing diverse computational notions, including, for example, operational semantics, typing, and acceptance by non-deterministic machines. We therefore propose a (restricted) logical theory over relations as a language for specifying such notions. Our specification logic is further characterized by an ability to explicitly treat binding in object languages. Once such a logic is fixed, a natural next question is how we might prove theorems about specifications written in it. We propose to use a second logic, called a reasoning logic, for this purpose. A satisfactory reasoning logic should be able to completely encode the specification logic. Associated with the specification logic are various notions of binding: for quantifiers within formulas, for eigenvariables within sequents, and for abstractions within terms. To provide a natural treatment of these aspects, the reasoning logic must encode binding structures as well as their associated notions of scope, free and bound variables, and capture-avoiding substitution. Further, to support arguments about provability, the reasoning logic should possess strong mechanisms for constructing proofs by induction and co-induction. We provide these capabilities here by using a logic called G which represents relations over a;-terms via definitions of atomic judgments, contains inference rules for induction and co-induction, and includes a special generic quantifier. We show how provability in the specification logic can be transparently encoded in G. We also describe an interactive theorem prover called Abella that implements G and this two-level logic approach and we present several examples that demonstrate the efficacy of Abella in reasoning about computations.
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Acknowledgements This work has been supported by the National Science Foundation grants CCR-0429572 and CCF-0917140 and by INRIA through the “Equipes Associées” Slimmer. Opinions, findings, and conclusions or recommendations expressed in this papers are those of the authors and do not necessarily reflect the views of the National Science Foundation.
- Generic judgments
- Nominal abstraction
- Two-level logic
- λ-tree syntax
- ∇- quantification