A Two-Level Logic Approach to Reasoning About Computations

• Published in: Journal of automated reasoning 2012-08, Vol.49 (2), p.241-273
• Main Authors: Gacek, Andrew, Miller, Dale, Nadathur, Gopalan
• Format: Article
• Language: English
• Subjects: Artificial Intelligence; Binding; Computation; Computer Science; Inference; Logic; Logic in Computer Science; Mathematical analysis; Mathematical Logic and Formal Languages; Mathematical Logic and Foundations; Mathematical models; Programming Languages; Reasoning; Specifications; Symbolic and Algebraic Manipulation
• ISSN: 0168-7433; 1573-0670
• DOI: 10.1007/s10817-011-9218-1

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 which represents relations over λ -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 . We also describe an interactive theorem prover called Abella that implements and this two-level logic approach and we present several examples that demonstrate the efficacy of Abella in reasoning about computations.