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Faster Linear Unification Algorithm

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Abstract

The Robinson unification algorithm has exponential worst case behavior. This triggered the development of (semi-)linear versions around 1976 by Martelli and Montanari as well as by Paterson and Wegman (J Comput Syst Sci 16(2):158–167, 1978, https://doi.org/10.1016/0022-0000(78)90043-0). Another version emerged by Baader and Snyder around 2001. While these versions are distinctly faster on larger input pairs, the Robinson version still does better than them on small-sized inputs. This paper describes yet another (semi-)linear version that is faster and challenges also the Robinson version on small-sized inputs. All versions have been implemented and compared against each other on different types and sizes of input pairs.

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  1. San Jose, USA

    Dennis de Champeaux

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  1. Dennis de Champeaux
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Correspondence to Dennis de Champeaux.

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Appendix: Eight Generators

Appendix: Eight Generators

Eight generators were used for obtaining argument pairs to compare the performance of the versions discussed. These generators take as argument the size for a pair. To get an impression of the size of these problems, we show for each generator the instances for the sizes 1 and 2. The generators produce unifiable or non-unifiable pairs. The generator names that end with “f” produce non-unifiable pairs.

Generator

Size 1

Size 2

gen1

P(h(x1 x1) y2 x2)

P(x2 h(y1  y1) y2)

P(h(x1 x1) h(x2 x2) y2 y3 x3)

P(x2 x3 h(y1 y1) h(y2 y2) y3

gen1f

P(h(x1 x1) y2 aa)

P(x2 h(y1 y1) y2)

P(h(x1 x1) h(x2 x2) y2 y3 aa)

P(x2 x3 h(y1 y1) h(y2 y2) y3

gen3

P(x0 f(x1 x1) x1 f(x2 x2))

P(f(y0 y0) y0 f(y1 y1) y2)

P(x0 f(x1 x1) x1 f(x2 x2) x2 f(x3 x3))

P(f(y0 y0) y0 f(y1 y1) y1 f(y2 y2) y3)

gen3f

P(x0 f(x1 x1) x1 f(x2 x2))

P(f(y0 y0) y0 f(x0 x0) y2)

P(x0 f(x1 x1) x1 f(x2 x2) x2 f(x3 x3))

P(f(y0 y0) y0 f(y1 y1) y1 f(x0 x0) y3)

gen4

P(x1 y1)

P(g(y1 y1) f(x2))

P(x1 y1 x2 y2)

P(g(y1 y1) f(x2) g(y2 y2) f(x3))

gen4f

P(x1 y1)

P(g(y1 y1) x1)

P(x1 y1 x2 y2)

P(g(y1 y1) f(x2) g(y2 y2) x1)

gen2

P(x1)

P(f(y)))

P(x1 f(x2))

P(f(x2) f(f(y)))

gen2f

P(x1)

P(f(x1))

P(x1 f(x2))

P(f(x2) f(f(x1)))

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de Champeaux, D. Faster Linear Unification Algorithm. J Autom Reasoning 66, 845–860 (2022). https://doi.org/10.1007/s10817-022-09635-1

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