On the Recursive Enumerability of Fixed-Point Combinators
TL;DR: It is shown that the set of fixed-point combinators forms a recursively-enumerable subset of a larger set of terms that is not Recursively enumerable, and the terms of which are observationally equivalent to fixed- point combinators in any computable context.
Abstract: We show that the set of fixed-point combinators forms a recursively-enumerable subset of a larger set of terms that is (A) not recursively enumerable, and (B) the terms of which are observationally equivalent to fixed-point combinators in any computable context.
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TL;DR: The paper explains the crucial differences between SOS and MSOS, and illustrates how MSOS descriptions are written, and discusses standard notions of semantic equivalence based on MSOS.
226 citations
TL;DR: The construction of a refocus function shows how to mechanically obtain an abstract machine out of a reduction semantics, which was done previously on a case-by-case basis.
Abstract: The evaluation function of a reduction semantics (i.e., a small-step operational semantics with an explicit representation of the reduction context) is canonically defined as the transitive closure of (1) decomposing a term into a reduction context and a redex, (2) contracting this redex, and (3) plugging the contractum in the context. Directly implementing this evaluation function therefore yields an interpreter with a worst-case overhead, for each step, that is linear in the size of the input term. We present sufficient conditions over the constituents of a reduction semantics to circumvent this overhead, by replacing the composition of (3) plugging and (1) decomposing by a single ``refocus'' function mapping a contractum and a context into a new context and a new redex, if any. We also show how to construct such a refocus function, we prove the correctness of this construction, and we analyze the complexity of the resulting refocus function. The refocused evaluation function of a reduction semantics implements the transitive closure of the refocus function, i.e., a ``pre-abstract machine.'' Fusing the refocus function with the trampoline function computing the transitive closure gives a state-transition function, i.e., an abstract machine. This abstract machine clearly separates between the transitions implementing the congruence rules of the reduction semantics and the transitions implementing its reduction rules. The construction of a refocus function therefore shows how to mechanically obtain an abstract machine out of a reduction semantics, which was done previously on a case-by-case basis. We illustrate refocusing by mechanically constructing Felleisen et al.'s CK machine from a call-by-value reduction semantics of the lambda-calculus, and by constructing a substitution-based version of Krivine's machine from a call-by-name reduction semantics of the lambda-calculus. We also mechanically construct three one-pass CPS transformers from three quadratic context-based CPS transformers for the lambda-calculus.
124 citations
105 citations
TL;DR: The correspondence between evaluators and abstract machines is extended to the impure setting of the λ-calculus, and the tail-recursive stack inspection presented by Clements and Felleisen is characterized as a lifted state monad, which enables to combine this stack-inspection monad with other monads and to construct abstract machines for languages with properly tail- Recursion stack inspection and other computational effects.
82 citations
TL;DR: In this article, an abstract machine and a reduction semantics for the lambda-calculus extended with control operators that give access to delimited continuations in the CPS hierarchy are presented.
Abstract: We present an abstract machine and a reduction semantics for the lambda-calculus extended with control operators that give access to delimited continuations in the CPS hierarchy. The abstract machine is derived from an evaluator in continuation-passing style (CPS); the reduction semantics (i.e., a small-step operational semantics with an explicit representation of evaluation contexts) is constructed from the abstract machine; and the control operators are the shift and reset family. We also present new applications of delimited continuations in the CPS hierarchy: finding list prefixes and normalization by evaluation for a hierarchical language of units and products.
73 citations
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TL;DR: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular and generalizations of recursion theory.
3,665 citations
Book•
30 Apr 2012
TL;DR: In this article, the Lambda-Calculus has been studied as a theory of composition and reduction, and the theory of reduction has been used to construct models of Lambda Theories.
Abstract: Towards the Theory. Introduction. Conversion. Reduction. Theories. Models. Conversion. Classical Lambda Calculus. The Theory of Combinators. Classical Lambda Calculus (Continued). The Lambda-Calculus. Bohm Trees. Reduction. Fundamental Theorems. Strongly Equivalent Reductions. Reduction Strategies. Labelled Reduction. Other Notions of Reduction. Theories. Sensible Theories. Other Lambda Theories. Models. Construction of Models. Local Structure of Models. Global Structure of Models. Combinatory Groups. Appendices: Typed Lambda Calculus. Illative Combinatory Logic. Variables. References.
2,632 citations
Book•
22 Apr 1987TL;DR: In this paper, the authors discuss related theories of recursively enumerable sets, degree of un-solvability and turing degrees in particular, and generalizations of recursion theory.
Abstract: Central concerns of the book are related theories of recursively enumerable sets, of degree of un-solvability and turing degrees in particular. A second group of topics has to do with generalizations of recursion theory. The third topics group mentioned is subrecursive computability and subrecursive hierarchies
1,779 citations
Book•
01 Sep 1981
TL;DR: Treats various kinds of languages, beginning with the pure-lambda-calculus and progressing through languages with states, commands, jumps, and assignments.
Abstract: From the Publisher:
"First book-length exposition of the denotational (or `mathematical' or `functional') approach to the formal semantics of programming languages (in contrast to `operational' and `axiomatic' approaches) Treats various kinds of languages, beginning with the pure-lambda-calculus and progressing through languages with states, commands, jumps, and assignments This somewhat discursive account is a valuable compilation of results not otherwise available in a single source"
-- American Mathematical Monthly
1,311 citations
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1,124 citations