A Concurrent Logical Framework I: Judgments and Properties

Abstract

The Concurrent Logical Framework, or CLF, is a new logical framework in which concurrent computations can be represented as monadic objects, for which there is an intrinsic notion of concurrency. It is designed as a conservative extension of the linear logical framework LLF with the synchronous connectives ⊗, 1, !, and ∃ of intuitionistic linear logic, encapsulated in a monad. LLF is itself a conservative extension of LF with the asynchronous connectives −•, & and. The present report, the first of two technical reports describing CLF, presents the framework itself and its meta-theory. A novel, algorithmic formulation of the underlying type theory concentrating on canonical forms leads to a simple notion of definitional equality for concurrent computations in which the order of independent steps cannot be distinguished. The new formulation of the framework constitutes an original contribution even for the LF fragment. For many additional examples illustrating the use of the framework to specify and reason about object systems of interest, the reader is referred to the companion technical report on applications [CPWW02].

Linear Logical Voting Protocols

“…To assign a logic programming interpretation, restrictions must be placed on the ways in which these connectives fit together so that proof search becomes more deterministic. Thus, in Celf [14] and its predecessor, LolliMon [13], the formulas of linear logic are polarized into positive and negative classes [1] and a monad is used to prevent interference between the two classes [16]. For the examples in this paper, only the following fragment of polarized monadic linear logic is needed:…”

Refinement Types as Proof Irrelevance

“…Here, we briefly recapitulate some of the details of the formal development to set the stage for what is to come. LF with refinement types, or LFR, is specified using the methodology of canonical forms, pioneered by Watkins, et al [16] in the definition of the Concurrent Logical Framework, CLF. Following this methodology, we consider only canonical forms, or terms that are β-normal and η-long.…”

“…An essential guiding principle was to restrict attention to canonical forms using bidirectional typing [16]. Under the canonical forms methodology, features which typically complicate a type system's metatheory could be expressed cleanly and simply.…”

Focusing on Binding and Computation

“…The canonical forms of simply-typed LF (STLF) are summarized in Figure 3; see Watkins et al [2002] for an introduction to canonical-forms presentations of logical frameworks. In this section, we show that the STLF terms exist as closed patterns, and therefore as values, in our type theory.…”