Formal Aspects of Computing

, Volume 29, Issue 3, pp 423–452 | Cite as

Equational formulas and pattern operations in initial order-sorted algebras

Original Article
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Abstract

A pattern t, i.e., a term possibly with variables, denotes the set (language) \({\llbracket t \rrbracket}\) of all its ground instances. In an untyped setting, symbolic operations on finite sets of patterns can represent Boolean operations on languages. But for the more expressive patterns needed in declarative languages supporting rich type disciplines such as subtype polymorphism, untyped pattern operations and algorithms break down. We show how they can be properly defined by means of a signature transformation \({\Sigma \mapsto \Sigma^{\#}}\) that enriches the types of \({\Sigma}\). We also show that this transformation allows a systematic reduction of the first-order logic properties of an initial order-sorted algebra supporting subtype-polymorphic functions to equivalent properties of an initial many-sorted (i.e., simply typed) algebra. This yields a new, simple proof of the known decidability of the first-order theory of an initial order-sorted algebra.

Keywords

Pattern operations Initial decidability Order-sorted logic 

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References

  1. AEE14.
    Alpuente M, Escobar S, Espert J, Meseguer J (2014) A modular order-sorted equational generalization algorithm. Inf Comput 235: 98–136MathSciNetCrossRefMATHGoogle Scholar
  2. CD94.
    Comon H, Delor C (1994) Equational formulae with membership constraints. Inf Comput, 112(2): 167–216MathSciNetCrossRefMATHGoogle Scholar
  3. CDE07.
    Clavel M, Durán F, Eker S, Meseguer J, Lincoln P, Martí-Oliet N, Talcott C (2007) All about Maude. In: LNCS, vol 4350. Springer, BerlinGoogle Scholar
  4. CDG07.
    Comon H, Dauchet M, Gilleron R, Löding C, Jacquemard F, Lugiez D, Tison S, Tommasi M (2007) Tree automata techniques and applications. http://www.grappa.univ-lille3.fr/tata. Release 12 Oct 2007
  5. CL89.
    Comon H, Lescanne P (1989) Equational problems and disunification. J Symb Comput 7: 371–425MathSciNetCrossRefMATHGoogle Scholar
  6. Com90.
    Comon H (1990) Equational formulas in order-sorted algebras. In: Proceedings of the ICALP’90. LNCS, vol 443. Springer, Berlin, pp 674–688Google Scholar
  7. EM85.
    Ehrig H, Mahr B (1985) Fundamentals of algebraic specification 1. Springer, BerlinCrossRefMATHGoogle Scholar
  8. FD98.
    Futatsugi K, Diaconescu R (1998) CafeOBJ Report. World Scientific, SingaporeMATHGoogle Scholar
  9. Fer98.
    Fernández M (1998) Negation elimination in empty or permutative theories. J Symb Comput 26(1): 97–133MathSciNetCrossRefMATHGoogle Scholar
  10. GB92.
    Goguen J, Burstall R (1992) Institutions: abstract model theory for specification and programming. J. ACM 39(1): 95–146MathSciNetCrossRefMATHGoogle Scholar
  11. GM92.
    Goguen J, Meseguer J (1992) Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor Comput Sci 105: 217–273MathSciNetCrossRefMATHGoogle Scholar
  12. Gut75.
    Guttag J (1975) The specification and application to programming of abstract data types. Ph.D. thesis, University of Toronto. Computer Science Department, Report CSRG-59Google Scholar
  13. GWM00.
    Goguen J, Winkler T, Meseguer J, Futatsugi K, Jouannaud JP (2000) Introducing OBJ. In: Software engineering with OBJ: algebraic specification in action. Kluwer, Dordrecht, pp 3–167Google Scholar
  14. HHJ07.
    Hudak P, Hughes J, Peyton Jones SL, Wadler P (2007) A history of haskell: being lazy with class. In: Proceedings of the third ACM SIGPLAN history of programming languages conference (HOPL-III). ACM, New york, pp 1–55Google Scholar
  15. LMM91.
    Lassez JL, Maher M, Marriott K (1991) Elimination of negation in term algebras. In: Mathematical foundations of computer science. Springer, Berlin, pp 1–16Google Scholar
  16. LM87.
    Lassez JL, Marriott K (1987) Explicit representation of terms defined by counter examples. J Autom Reason 3(3): 301–317CrossRefMATHGoogle Scholar
  17. Mah88a.
    Maher MJ (1988) Complete axiomatizations of the algebras of finite, rational and infinite trees. In: Proceedings of the LICS’88. IEEE Computer Society, Silver Spring, pp 348–357Google Scholar
  18. Mah88b.
    Maher MJ (1988) Complete axiomatizations of the algebras of finite, rational and infinite trees. Technical report, IBM T. J. Watson Research CenterGoogle Scholar
  19. Mes92.
    Meseguer J (1992) Conditional rewriting logic as a unified model of concurrency. Theor Comput Sci 96(1): 73–155MathSciNetCrossRefMATHGoogle Scholar
  20. Mes98.
    Meseguer J (1998) Membership algebra as a logical framework for equational specification. In: Proceedings of the WADT’97. In: LNCS, vol 1376. Springer, Berlin, pp 18–61Google Scholar
  21. Mes16.
    Meseguer J (2016) Variant-based satisfiability in initial algebras. In: Artho C, Ölveczky PC (eds), proc. FTSCS 2015. Springer CCIS 596, pp 1–32Google Scholar
  22. MGS.
    José Meseguer, Joseph Goguen, Gert Smolka (1989) Order-sorted unification. J Symb Comput 8: 383–413MathSciNetCrossRefMATHGoogle Scholar
  23. MPM08.
    Meseguer J, Palomino M, Martí-Oliet N (2008) Equational abstractions. Theor Comput Sci 403(2–3):239–264Google Scholar
  24. MS15.
    Meseguer J, Skeirik S (2015) Equational formulas and pattern operations in initial order-sorted algebras. In: Falaschi M (ed) Proceedings of the LOPSTR 2015. LNCS, vol 9527. Springer, Berlin, pp 36–53Google Scholar
  25. Pic03.
    Pichler R (2003) Explicit versus implicit representations of subsets of the Herbrand universe. Theor Comput Sci 290(1): 1021–1056MathSciNetCrossRefMATHGoogle Scholar
  26. Taj93.
    Tajine M (1993) The negation elimination from syntactic equational formula is decidable. In: Proceedings of the RTA-93. LNCS, vol 690. Springer, Berlin, pp 316–327Google Scholar
  27. DHK96.
    van Deursen A, Heering J, Klint P (1996) Language prototyping: an algebraic specification approach. World Scientific, SingaporeCrossRefMATHGoogle Scholar

Copyright information

© British Computer Society 2017

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignChampaignUSA

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