A Two-Level Logic Approach to Reasoning About Computations

Abstract

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 specif ication 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 λ-terms via definitions of atomic judgments, contains inference rules for induction and coinduction, 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 242 A. Gacek et al.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.

The Next 700 Challenge Problems for Reasoning with Higher-Order Abstract Syntax Representations

“…In this section we briefly review the two-level architecture that Hybrid and Abella share, referring to [16] and [22] for more extensive explanations.…”

“…the pr_b case. Specialization is just function application in Coq, while it is a rather deep property of the nabla quantifier and SL-derivability [22] in Abella.…”

Automation of Higher-Order Logic

“…As a result, some systems-such as the Teyjus implementation of λProlog and the interactive theorem provers Minlog (Benl et al, 1998) and Abella (Gacek et al, 2012)-only implement the pattern fragment since this makes their design and implementation easier.…”

“…Related provers include the general-purpose, interactive, type-free, equational higher-order theorem prover Watson (Holmes and Alves-Foss, 2001) and the fully automated theorem prover Otter-λ (Beeson, 2006) for λ-logic (a combination of λ-calculus and first-order logic). Abella (Gacek et al, 2012) is a recently implemented interactive theorem prover for an intuitionistic, predicative higher-order logic with inference rules for induction and co-induction. ACL2 (Kaufmann and Moore, 1997) and KeY (Beckert et al, 2007) are prominent first-order interactive proof assistants that integrate induction.…”

Reasoning about Computations Using Two-Levels of Logic

“…This paper is a summary of some existing papers (particularly [16]) and is structured as follows. Section 2 presents a specific reasoning language G and Section 3 presents a specific specification logic hH 2 .…”