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Spring 1995 NP-Completeness of a Combinator Optimization Problem
M. S. Joy, V. J. Rayward-Smith
Notre Dame J. Formal Logic 36(2): 319-335 (Spring 1995). DOI: 10.1305/ndjfl/1040248462


We consider a deterministic rewrite system for combinatory logic over combinators $S,K,I,B,C,S',B'$, and $C'$. Terms will be represented by graphs so that reduction of a duplicator will cause the duplicated expression to be "shared" rather than copied. To each normalizing term we assign a weighting which is the number of reduction steps necessary to reduce the expression to normal form. A lambda-expression may be represented by several distinct expressions in combinatory logic, and two combinatory logic expressions are considered equivalent if they represent the same lambda-expression (up to $\beta $-$\eta $-equivalence). The problem of minimizing the number of reduction steps over equivalent combinator expressions (i.e., the problem of finding the "fastest running" combinator representation for a specific lambda-expression) is proved to be NP-complete by reduction from the "Hitting Set" problem.


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M. S. Joy. V. J. Rayward-Smith. "NP-Completeness of a Combinator Optimization Problem." Notre Dame J. Formal Logic 36 (2) 319 - 335, Spring 1995.


Published: Spring 1995
First available in Project Euclid: 18 December 2002

zbMATH: 0837.03015
MathSciNet: MR1345752
Digital Object Identifier: 10.1305/ndjfl/1040248462

Primary: 03B40
Secondary: 68Q25, 68Q55

Rights: Copyright © 1995 University of Notre Dame


Vol.36 • No. 2 • Spring 1995
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