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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.

References

  1. Adian, S. I., & Durnev, V. G. (2000). Decision problems for groups and semigroups. Russian Mathematical Surveys, 55(2), 207.

    Google Scholar 

  2. Baader, F. (1986). Unification in idempotent semigroups is of type zero. Journal of Automated Reasoning, 2(3), 283–286.

    Article  Google Scholar 

  3. Baader, F. (1988). A note on unification type zero. Information Processing Letters, 27, 91–93.

    Article  Google Scholar 

  4. Baader, F., Binh, N. Th., Borgwardt, S., & Morawska, B. (2015). Deciding unifiability and computing local unifiers in the description logic \(\cal {EL} \) without top constructor. Notre Dame Journal of Formal Logic.

    Google Scholar 

  5. Baader, F., & Ghilardi, S. (2011). Unification in modal and description logics. Logic Journal of the IGPL, 19(6), 705–730. http://jigpal.oxfordjournals.org/content/19/6/705.abstract.

  6. Baader, F., & Nipkow, T. (1998). Term rewriting and all that. Cambridge University Press.

    Google Scholar 

  7. Baader, F., & Siekmann, J. (1994). General unification theory. In D. Gabbay, C. Hogger & J. Robinson (Eds.), Handbook of logic in artificial intelligence and logic programming (pp. 41–126). Oxford University Press.

    Google Scholar 

  8. Baader, F., & Snyder, W. (2001). Unification theory. In A. Robinson & A. Voronkov (Eds.), Handbook of automated reasoning (Vol. 1). Elsevier Science Publishers.

    Google Scholar 

  9. Baader, F., Borgwardt, S., & Morawska, B. (2015). Dismatching and local disunification in \(\cal {EL}\). In M. Fernández (Ed.), Proceedings of the 26th International Conference on Rewriting Techniques and Applications (RTA’15). Volume 36 of Leibniz International Proceedings in Informatics, Warsaw, Poland. Dagstuhl Publishing.

    Google Scholar 

  10. Baader, F., Gil, O. F., & Morawska, B. (2013). Hybrid unification in the description logic \(\cal {EL}\). In B. Morawska & K. Korovin (Eds.), Proceedings of the 27th International Workshop on Unification (UNIF’13). The Netherlands: Eindhoven.

    Google Scholar 

  11. Baader, F., & Morawska, B. (2014). Matching with respect to general concept inclusions in the description logic \(\cal {EL}\). In C. Lutz & M. Thielscher (Eds.), Proceedings of the 37th German Conference on Artificial Intelligence (KI’14). Volume 8736 of Lecture Notes in Artificial Intelligence (pp. 135–146). Springer.

    Google Scholar 

  12. Buerckert, H. J., Herold, A., Kapur, D., Siekmann, J., Stickel, M., Tepp, M., et al. (1988). Opening the AC-unification race. Journal of Automated Reasoning, 4(4), 465–474.

    Google Scholar 

  13. Bulitko, V. K. (1970). Equations and inequalities in a free group and semigroup. Geometr. i Algebra Tul. Gos. Ped. Inst. Ucen. Zap. Mat. Kafedr., 2, 242–252.

    Google Scholar 

  14. Charatonik, W., & Pacholski, L. (1993). Word equations with two variables. Word Equations and Related Topics, IWWERT (pp. 43–56). Berlin: Springer.

    Google Scholar 

  15. Choffrut, Ch., & Karhumaeki, J. (1997). Combinatorics of words. In Handbook of formal languages (pp. 329–438). Berlin: Springer.

    Google Scholar 

  16. Dabrowski, R., & Plandowski, W. (2002, 2011). On word equations in one variable. Lecture Notes in Computer Science, 2420, 212–220 and In Algorithmica, 60(4), 819–828.

    Google Scholar 

  17. Davis, M. (1983). The prehistory and early history of automated deduction. In J. Siekmann & G. Wrightson (Eds.), Automation of reasoning 1: Classical papers on computational logic 1957–1966 (pp. 1–28). Berlin: Springer.

    Chapter  Google Scholar 

  18. Davis, M. (1973). Hilbert’s tenth problem is unsolvable. American Mathematical Monthly 233–269.

    Google Scholar 

  19. Davis, M., & Hersh, R. (1984). Hilbert’s 10th problem. Mathematics: People, Problems, Results, 2, 136–148.

    Google Scholar 

  20. Davis, M., & Putnam, H. (1960). A computing procedure for quantification theory. Journal of the ACM (JACM), 7(3), 201–215.

    Article  Google Scholar 

  21. Davis, M., Putnam, H., & Robinson, J. (1961). The decision problem for exponential diophantine equations. Annals of Mathematics, 74, 425–436.

    Article  Google Scholar 

  22. Dershowitz, N., & Jouannaud, J.-P. (1990). Rewrite systems. In J. van Leeuwen (Ed.), Handbook of theoretical computer science (pp. 244–320). North-Holland: Elsevier Science Publishers.

    Google Scholar 

  23. Dershowitz, N., & Jouannaud, J.-P. (1991). Notations for rewriting. Bulletin of the EATCS, 43, 162–174.

    Google Scholar 

  24. Diekert, V. (2002). Makanin’s algorithm. In M. Lothaire (Ed.), Algebraic combinatorics on words (Chap. 12, pp. 387–442). Cambridge University Press.

    Google Scholar 

  25. Domenjoud, E. (1992). A technical note on AC-unification: The number of minimal unifiers of the AC equation. Journal of Automated Reasoning, 8(1), 39–44.

    Article  Google Scholar 

  26. Dougherty, D. J., & Johann, P. (1994). An improved general E-unification method. Journal of Symbolic Computation, 11, 1–19.

    Google Scholar 

  27. Durnev, V. (1997). Studying algorithmic problems for free semi-groups and groups. In Logical foundations of computer science (pp. 88–101). Berlin: Springer.

    Google Scholar 

  28. Durnev, V. G. (1974). On equations in free semigroups and groups. Mathematical Notes of the Academy of Sciences of the USSR, 16(5), 1024–1028.

    Article  Google Scholar 

  29. Erne, M. (2009). Closure. Journal of the American Mathematical Society. In F. Mynard & E. Pearl (Ed.), Beyond topology, contemporary mathematics (pp. 163–283). American Mathematical Society.

    Google Scholar 

  30. Fages, F., & Huet, G. (1986). Complete sets of unifiers and matchers in equational theories. Theoretical Computer Science, 43(1), 189–200.

    Article  Google Scholar 

  31. Gallier, J. H. (1991). Unification procedures in automated deduction methods based on matings: A survey. Technical Report CIS-436, University of Pensylvania, Department of Computer and Information Science.

    Google Scholar 

  32. Ganzinger, H., & Korovin, K. (2003). New directions in instantiation-based theorem proving. In Proceedings. 18th Annual IEEE Symposium on Logic in Computer Science, 2003 (pp. 55–64). IEEE.

    Google Scholar 

  33. Gilmore, P. C. (1960). A proof method for quantification theory: Its justification and realization. IBM Journal of research and development, 4(1), 28–35.

    Article  Google Scholar 

  34. Hmelevskij, J. I. (1964). The solution of certain systems of word equations. Dokl. Akad. Nauk SSSR Soviet Math., 5, 724.

    Google Scholar 

  35. Hmelevskij, J. I. (1966). Word equations without coefficients. Soviet Math. Dokl., 7, 1611–1613.

    Google Scholar 

  36. Hmelevskij, J. I. (1967). Solution of word equations in three unknowns. Dokl. Akad. Nauk SSSR Soviet Math., 5, 177.

    Google Scholar 

  37. Hoche, M., Siekmann, J., & Szabo, P. (2008). String unification is essentially infinitary. In M. Marin (Ed.), The 22nd International Workshop on Unification (UNIF’08) (pp. 82–102). Hagenberg, Austria.

    Google Scholar 

  38. Hoche, M., Siekmann, J., & Szabo, P. (2016). String unification is essentially infinitary. IFCoLog Journal of Logics and their Applications.

    Google Scholar 

  39. Hoche, M., & Szabo, P. (2006). Essential unifiers. Journal of Applied Logic, 4(1), 1–25.

    Article  Google Scholar 

  40. Howie, J. M. (1976). An introduction to semigroup theory. Academic Press.

    Google Scholar 

  41. Huet, G. (2002). Higher order unification 30 years later. In Theorem proving in higher order logics (Vol. 2410, pp. 3–12). Springer LNCS.

    Google Scholar 

  42. Jacobs, S., & Waldmann, U. (2007). Comparing instance generation methods for automated reasoning. Journal of Automated Reasoning, 38(1–3), 57–78.

    Article  Google Scholar 

  43. Jaffar, J. (1990). Minimal and complete word unification. JACM, 27(1), 47–85.

    Article  Google Scholar 

  44. Jez, A. (2013). One-variable word equation in linear time. ICALP, Lecture Notes in Computer Science, 7966, 324–335.

    Article  Google Scholar 

  45. Kanger, S. (1963). A simplified proof method for elementary logic. Studies in Logic and the Foundations of Mathematics, 35, 87–94.

    Article  Google Scholar 

  46. Kapur, D., & Narendran, P. (1992). Complexity of unification problems with associative-commutative operators. Journal of Automated Reasoning, 9(2), 261–288.

    Article  Google Scholar 

  47. Kapur, D., & Narendran, P. (1992). Double-exponential complexity of computing a complete set of AC-unifiers. In Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, LICS92.

    Google Scholar 

  48. Kirchner, C., & Kirchner, H. (2006). Rewriting solving proving. http://www.loria.fr/~ckirchne/=rsp/rsp.pdf.

  49. Korovin, K. (2009). An invitation to instantion-based reasoning: From theory to practice. In Proceedings of CADE 22, Springer Lecture Notes on AI (Vol. 5663, pp. 163–166). Springer.

    Google Scholar 

  50. Knight, K. (1989). Unification: A multidisciplinary survey. ACM Computing Surveys (CSUR), 21(1), 93–124.

    Article  Google Scholar 

  51. Laine, M., & Plandowski, W. (2011). Word equations with one unknown. International Journal of Foundations of Computer Science, 122(2), 345–375.

    Article  Google Scholar 

  52. Lee, S.-J., & Plaisted, D. A. (1992). Eliminating duplication with the hyper-linking strategy. Journal of Automated Reasoning, 9(1), 25–42.

    Google Scholar 

  53. Lentin, A. (1972). Equations in free monoids. In M. Nivat (Ed.), Automata, languages and programming. North Holland.

    Google Scholar 

  54. Lentin, A., & Schuetzenberger, M. P. (1967). A combinatorial problem in the theory of free monoids. In R. C. Bose & T. E. Dowling (Eds.), Combinatorial mathematics (pp. 112–144). University of North Carolina Press.

    Google Scholar 

  55. Livesey, M., & Siekmann, J. (1975). Termination and decidability results for string unification. Report Memo CSM-12, Computer Centre, Essex University, England.

    Google Scholar 

  56. Lothaire, M. (1997). Combinatorics on words. Volume 17 of Encyclopedia of Mathematics. Addison-Wesley. (Reprinted from Cambridge University Press, Cambridge Mathematical Library (1983)).

    Google Scholar 

  57. Lothaire, M. (2002). Algebraic combinatorics on words. Cambridge University Press.

    Google Scholar 

  58. Makanin, G. S. (1977). The problem of solvability of equations in a free semigroup. Original in Russian: Mathematicheskii Sbornik, 103(2), 147–236 (Math. USSR Sbornik, 32, 129–198).

    Google Scholar 

  59. Markov, A. A. (1954). Theory of algorithms. Trudy Mathematicheskogo Instituta Imeni VA Steklova, Izdat.Akad. Nauk SSSR, 17, 1038.

    Google Scholar 

  60. Matijasevich, Ju. V. (1970). Enumerable sets are diophantine. Dokl. Akad. Nauk SSSR 191=Soviet Math. Dokl., 11, 354-358, 279–282.

    Google Scholar 

  61. Matiyasevich, Y. (1970). The connection between Hilbert’s 10th problem and systems of equations between words and lengths. Seminars in Mathematics, 8, 61–67.

    Google Scholar 

  62. Plandowski, W. (2004). Satisfiability of word equations with constants is in Pspace. JACM, 51(3), 483–496.

    Article  Google Scholar 

  63. Plotkin, G. (1972). Building-in equational theories. In B. Meltzer & D. Michie (Eds.), Machine Intelligence (Vol. 7, pp. 73–90). Edinburgh University Press.

    Google Scholar 

  64. Prawitz, D. (1960). An improved proof procedure. Theoria, 26, 102–139.

    Article  Google Scholar 

  65. Raulefs, P., Siekmann, J., Szabo, P., & Unvericht, E. (1979). A short survey on the state of the art in matching and unification problems. ACM Sigsam Bulletin, 13(2), 14–20.

    Article  Google Scholar 

  66. Raulefs, P., & Siekmann, J. H. (1978). Unification of idempotent functions. Technical report, Fachbereich Informatik, Universität Kaiserslautern.

    Google Scholar 

  67. Robinson, J. A. (1965). A machine-oriented logic based on the resolution principle. Journal of the ACM, 12(1), 23–41.

    Article  Google Scholar 

  68. RTA list of open problems (2008). http://rtaloop.pps.jussieu.fr.

  69. Schmidt-Schauss, M. (1986). Unification under associativity and idempotence is of type nullary. Journal of Automated Reasoning, 2(3), 277–281.

    Article  Google Scholar 

  70. Schulz, K. (1993). Word unification and transformations of generalized equations. Journal of Automated Reasoning, 11, 149–184.

    Article  Google Scholar 

  71. Schulz, K. U. (1992). Makanin’s algorithm for word equations: Two improvements and a generalization. In Proceedings of the First International Workshop on Word Equations and Related Topics (IWWERT90) (Vol. 572, pp. 85–150). Springer LNCS.

    Google Scholar 

  72. Siekmann, J. (1975). String unification. Report CSM-7, Computer Centre, University of Essex.

    Google Scholar 

  73. Siekmann, J. (1976). Unification and matching problems. PhD thesis, Essex University, Computer Science.

    Google Scholar 

  74. Siekmann, J. (1979). Unification of commutative terms. In Notes in Computer Science. Volume 72 of Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (pp. 531–545). EUROSAM’79, Springer.

    Google Scholar 

  75. Siekmann, J. (1984). Universal unification. In Proceedings of the 7th International Conference on Automated Deduction (pp. 1–42). London: Springer.

    Google Scholar 

  76. Siekmann, J. (1989). Unification theory. Journal of Symbolic Computation, 7(3 & 4), 207–274.

    Article  Google Scholar 

  77. Siekmann, J., & Szabo, P. (1982). A noetherian and confluent rewrite system for idempotent semigroups. Semigroup Forum, 25, 83–100.

    Article  Google Scholar 

  78. Siekmann, J., & Wrightson, G. (1983). Automation of reasoning: Classical papers on computational logic (Vols. 1 & 2). Berlin: Springer.

    Google Scholar 

  79. Szabo, P., & Unvericht, E. (1982). D-unification has infinitely many mgus. Technical report, University of Karlsruhe, Inst. f. Informatik I.

    Google Scholar 

  80. Szabo, P. (1982). Unifikationstheorie erster Ordnung. PhD thesis, University Karlsruhe.

    Google Scholar 

  81. Urban, C., Pitts, A. M., & Gabbay, M. J. (2004). Nominal unification. Theoretical Computer Science, 323(1), 473–497.

    Article  Google Scholar 

  82. Veenker, G. (1967). Beweisalgorithmen für die prädikatenlogik. Computing, 2(3), 263–283.

    Article  Google Scholar 

  83. Wang, H. (1960). Toward mechanical mathematics. IBM Journal of Research and Development, 4(1), 2–22.

    Article  Google Scholar 

  84. Wirth, C.-P., Siekmann, J., Benzmüller, C., & Autexier, S. (2009). Jacques herbrand: Life, logic, and automated deduction. In D. Gabbay & J. Woods (Eds.), Handbook of the History of Logic, Volume 5—Logic from Russell to Church. Elsevier.

    Google Scholar 

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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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  1. Peter Szabo
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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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