M-LISP: a representation-independent dialect of LISP with reduction semantics

Abstract

In this paper we introduce M-LISP, a dialect of LISP designed with an eye toward reconciling LISP's metalinguistics power with the structural style of operational semantics advocated by Plotkin [28]. We begin by reviewing the original defimtion of LISP [20] in an attempt to clarify the source of its metalinguistics power. We find that it arises from a problematic clause in this definition. We then define the abstract syntax and operational semantics of M-LISP, essentially a hybrid of M-expression LISP and Scheme. Next, we tie the operational semantics to the corresponding equational logic. As usual, provable equality in the logic implies operational equality. Having established this framework, we then extend M-LISP with the metalinguistics eual and rezfy operators (the latter is a nonstrict operator that converts its argument to its metalanguage representation). These operators encapsulate the metalinguistics representation conversions that occur globally in S-expression LISP. We show that the naive versions of these operators render LISPS equational logic inconsistent. On the positive side, we show that a naturally restricted form of the eual operator is confluent and therefore a conservative extension of M-LISP.Unfortunately, we must weaken the logic considerably to obtain a consistent theory of reification.

Binary Lambda Calculus and Combinatory Logic

“…Frandsen and Sturtivant [10] amply demonstrate the efficiency of λ calculus with a linear time implementation of k-tree Turing Machines. Clear semantics should be a primary concern, and Lisp is somewhat lacking in this regard [4]. This paper thus develops the approach suggested but discarded by Chaitin.…”

Inter-language reflection: A conceptual model and its implementation

“…Muller [28] has argued that the quotation form in the original definition of LISP is essentially flawed, and hence that the apply in LISP and descendants like 2-LISP and 3-LISP or even Scheme crosses levels. Moreover, it is this levelcrossing that allows much of the meta-circular capabilities of LISP.…”

Diagrams for Meaning Preservation

“…At the same time, Plotkin proved that evaluation via rewriting was equivalent to evaluation via abstract machine. Subsequently, this approach has been applied to many systems, including systems with imperative features such as assignments and continuations (examples include [10,11,21,3,19,12,26,17]). Warning 1.1 (Not Quite Same as Observational Equivalence) What we call meaning preservation is related to observational equivalence (sometimes called observational soundness [18], operational equivalence, consistency [25], etc.…”