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What Is Essential Unification?

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Martin Davis on Computability, Computational Logic, and Mathematical Foundations

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 10))

Abstract

A unifier of two terms s and t is a substitution \(\sigma \) such that \(s\sigma =t\sigma \). For first-order terms there exists a most general unifier \(\sigma \) in the sense that any other unifier \(\tau \) can be composed from \(\sigma \) with some substitution \(\lambda \) such that \(\tau =\sigma \circ \lambda \). For many practical applications it turned out to be useful to generalize this notion to E-unification, where E is an equational theory , \(=_{E}\) is equality under E and \(\sigma \) is an E-unifier if \(s\sigma =_{E}t\sigma \). Depending on the equational theory E, the set of most general unifiers is always a singleton (as above) or it may have more than one unifier, either finitely or infinitely many unifiers and for some theories it may not even exist, in which case we call the theory of type nullary. The set of most general unifiers is denoted as \(\mu \mathscr {U}\Sigma _{E}(\varGamma )\) for a unification problem \(\varGamma \), which is a system of equations and an equational theory E. Unfortunately the set \(\mu \mathscr {U}\Sigma _{E}(\varGamma )\) may be very large in general—even if it is finite—and for all practical purposes not really useful. For this and other reasons there is hence (i) a strong interest to compute a much smaller generating set of minimal unifiers and then (ii) to find efficient engineering solutions to handle these sets. Essential unifiers, as introduced by Hoche and Szabo, generalize the notion of a most general unifier and they have a dramatically pleasant effect: the set of essential unifiers is often much smaller than the set of most general unifiers . Essential unification may even reduce an infinitary theory to an essentially finitary theory. For example the one variable string unification problem is essentially finitary whereas it is infinitary in the usual sense. The most drastic reduction known so far is obtained for idempotent semigroups, or bands as they are called in computer science, which are of type nullary: there exist two unifiable terms s and t, but the set of most general unifiers does not exist. This is in stark contrast to essential unification: the set of essential unifiers for bands always exists and is finite. The key idea for essential unification is to base the notion of generality not on the standard subsumption order for terms with the associated subsumption order for substitutions , but on the encompassment order for terms and substitutions. Hence we propose the encompassment order as a more natural order relation for minimal and complete sets of E-unifiers and call these sets essential unifiers, denoted as \(e\mathscr {U}\Sigma _{E}(\varGamma )\). This paper introduces essential unification, provides a definitional framework based on order relations and surveys what is presently known. We conclude with a list of some of the more important open problems, including the main open problem, namely how to build essential unification into an automated reasoning system.

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Notes

  1. 1.

    First workshop in Val d’Ajol in 1987 and since then annually. Since 1997, there is a website UNIF1997, UNIF1998, UNIF1999 up to UNIF2005 in Japan and UNIF2006 at the FLOC conference in Seattle, UNIF2007 and UNIF2008 at the Schloss Hagenberg, Linz, Austria. The current UNIF’s can be found at UNIF2013, UNIF2014 and UNIF2015.

  2. 2.

    http://www.math.uwaterloo.ca/~snburris/htdocs/WWW/PDF/e_unif.pdf.

  3. 3.

    Actually several other logicians of the time had this idea and it is not known who came first.

  4. 4.

    Signs and notation are still not uniform in all related fields, in particular our notation is used more often in the literature on automated theorem proving and unification theory [6], whereas term rewriting systems usually prefer notational conventions such as \(\lessdot \)and \(\gtrdot \); see [22, 23].

  5. 5.

    Unfortunately we do not know if the axioms H1 to H4 can be directed into a canonical rewrite system as the axioms F1 to F4 in ([2]). So we make a little detour and look at the actual derivation, instead of the more elegant proof by Franz Baader for F1 to F4, based on the canonical rewrite system for F, in ([2]). Thanks to Franz Baader for this hint.

  6. 6.

    See also http://www.springer.com/article/10.1007%2FBF03024472#page-1. There is actually a nice film about the three and how John McCarthy informed Martin about the result, see http://www.zalafilms.com/films/jrbackground4.html.

  7. 7.

    Google scholar finds 62,600,000 entries in 0.21 s for word equations this year (not all of which is relevant for our topic of course, but narrowing it down to “word equations” still leads to 1500 entries in 0.16 s) and several 100,000 more entries if one is patient enough to continue the search and to filter gold from garbage. In the year 2008, at the unification workshop, where we published a preliminary result, we asked Dr. Google and “he” found 70,300 entries for “word equations” in 0.13 s—so what are we to make of this fact?

  8. 8.

    See http://www.math.uwaterloo.ca/~snburris/htdocs/WWW/PDF/e_unif.pdf, example 15.

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Authors and Affiliations

  1. Kurt-Schumacher-Str. 13, 75180, Pforzheim, Germany

    Peter Szabo

  2. Saarland University/DFKI, Stuhlsatzenhausweg, 66123, Saarbrücken, Germany

    Jörg Siekmann

  3. Airbus Defense and Space, Claude-Dornier-Str., 88090, Immenstaad, Germany

    Michael Hoche

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Correspondence to Peter Szabo .

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  1. University of Trieste, Trieste, Italy

    Eugenio G. Omodeo

  2. University of Udine, Udine, Italy

    Alberto Policriti

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Szabo, P., Siekmann, J., Hoche, M. (2016). What Is Essential Unification?. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_11

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