Notre Dame Journal of Formal Logic

NP-Completeness of a Combinator Optimization Problem

M. S. Joy and V. J. Rayward-Smith


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.

Article information

Notre Dame J. Formal Logic, Volume 36, Number 2 (1995), 319-335.

First available in Project Euclid: 18 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03B40: Combinatory logic and lambda-calculus [See also 68N18]
Secondary: 68Q25: Analysis of algorithms and problem complexity [See also 68W40] 68Q55: Semantics [See also 03B70, 06B35, 18C50]


Joy, M. S.; Rayward-Smith, V. J. NP-Completeness of a Combinator Optimization Problem. Notre Dame J. Formal Logic 36 (1995), no. 2, 319--335. doi:10.1305/ndjfl/1040248462.

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