A Denotational Account of Untyped Normalization by Evaluation

We show that the standard normalization-by-evaluation construction for the simply-typed lambda_{beta eta}-calculus has a natural counterpart for the untyped lambda_beta-calculus, with the central type-indexed logical relation replaced by a "recursively defined'' invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation. In the untyped setting, not all terms have normal forms, so the normalization function is necessarily partial. We establish its correctness in the senses of soundness (the output term, if any, is beta-equivalent to the input term); standardization ( beta-equivalent terms are mapped to the same result); and completeness (the function is defined for all terms that do have normal forms). We also show how the semantic construction enables a simple yet formal correctness proof for the normalization algorithm, expressed as a functional program in an ML-like call-by-value language.

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Denotational Aspects of Untyped Normalization by Evaluation

BRICS Report Series ◽

10.7146/brics.v12i4.21870 ◽

2005 ◽

Vol 12 (4) ◽

Cited By ~ 1

Author(s):

Andrzej Filinski ◽

Henning Korsholm Rohde

Keyword(s):

Normal Form ◽

Normal Forms ◽

Logical Relation ◽

Invariant Relation ◽

Correctness Proof ◽

Functional Program ◽

Central Type ◽

Natural Counterpart ◽

Normalization Function

We show that the standard normalization-by-evaluation construction for the simply-typed lambda_{beta eta}-calculus has a natural counterpart for the untyped lambda_beta-calculus, with the central type-indexed logical relation replaced by a "recursively defined'' invariant relation, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation. In the untyped setting, not all terms have normal forms, so the normalization function is necessarily partial. We establish its correctness in the senses of soundness (the output term, if any, is in normal form and beta-equivalent to the input term); identification ( beta-equivalent terms are mapped to the same result); and completeness (the function is defined for all terms that do have normal forms). We also show how the semantic construction enables a simple yet formal correctness proof for the normalization algorithm, expressed as a functional program in an ML-like call-by-value language. Finally, we generalize the construction to produce an infinitary variant of normal forms, namely Böhm trees. We show that the three-part characterization of correctness, as well as the proofs, extend naturally to this generalization.

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A First-Order One-Pass CPS Transformation

BRICS Report Series ◽

10.7146/brics.v8i49.21709 ◽

2001 ◽

Vol 8 (49) ◽

Author(s):

Olivier Danvy ◽

Lasse R. Nielsen

Keyword(s):

Control Flow ◽

Logical Relation ◽

Correctness Proof ◽

First Order ◽

Structural Induction ◽

Flow Information

We present a new transformation of call-by-value lambda-terms into continuation-passing style (CPS). This transformation operates in one pass and is both compositional and first-order. Because it operates in one pass, it directly yields compact CPS programs that are comparable to what one would write by hand. Because it is compositional, it allows proofs by structural induction. Because it is first-order, reasoning about it does not require the use of a logical relation. This new CPS transformation connects two separate lines of research. It has already been used to state a new and simpler correctness proof of a direct-style transformation, and to develop a new and simpler CPS transformation of control-flow information.

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McCammond’s normal forms for free aperiodic semigroups revisited

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000448 ◽

2015 ◽

Vol 18 (1) ◽

pp. 130-147 ◽

Cited By ~ 3

Author(s):

J. Almeida ◽

J. C. Costa ◽

M. Zeitoun

Keyword(s):

Word Problem ◽

Normal Forms ◽

Regular Languages ◽

Correctness Proof ◽

Original Proof ◽

Proof Of Correctness ◽

New Applications

AbstractThis paper revisits the solution of the word problem for ${\it\omega}$-terms interpreted over finite aperiodic semigroups, obtained by J. McCammond. The original proof of correctness of McCammond’s algorithm, based on normal forms for such terms, uses McCammond’s solution of the word problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness proof of McCammond’s algorithm, based on properties of certain regular languages associated with the normal forms. This method leads to new applications.

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