Abstract
Terminating functional programs should be deterministic, i.e., should evaluate to a unique result, regardless of the evaluation order. For equational functional programs such determinism is exactly captured by the ground confluence property. For terminating equations this is equivalent to ground local confluence, which follows from local confluence. Checking local confluence by computing critical pairs is the standard way to check ground confluence. The problem is that some perfectly reasonable equational programs are not locally confluent and it can be very hard or even impossible to make them so by adding more equations. We propose a three-step strategy to prove that an equational program as is is ground confluent: First: apply the strategy proposed in [9] to use non-joinable critical pairs as completion hints to either achieve local confluence or reduce the number of critical pairs. Second: use the inductive inference system proposed in this paper to prove the remaining critical pairs ground joinable. Third: to show ground confluence of the original specification, prove also ground joinable the equations added. These methods apply to order-sorted and possibly conditional equational programs modulo axioms such as, e.g., Maude functional modules.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For combinations of associativity, commutativity, and identity axioms, this last condition only rules out identity axioms. However, both for termination and confluence analysis purposes, identity axioms can always be turned into convergent rewrite rules modulo associativity and/or commutativity axioms, as explained in [8].
References
Aoto, T., Yoshida, J., Toyama, Y.: Proving confluence of term rewriting systems automatically. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 93–102. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02348-4_7
Bouhoula, A.: Simultaneous checking of completeness and ground confluence for algebraic specifications. ACM Trans. Comput. Log. 10(3), 1–33 (2009)
Bruni, R., Meseguer, J.: Semantic foundations for generalized rewrite theories. Theor. Comput. Sci. 360(1–3), 386–414 (2006)
Clavel, M., et al.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71999-1
Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–320. North-Holland, Amsterdam (1990)
Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving operational termination of membership equational programs. High.-Order Symb. Comput. 21(1–2), 59–88 (2008)
Durán, F., Lucas, S., Meseguer, J.: MTT: the Maude termination tool (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 313–319. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71070-7_27
Durán, F., Lucas, S., Meseguer, J.: Termination modulo combinations of equational theories. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 246–262. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04222-5_15
Durán, F., Meseguer, J.: On the Church-Rosser and coherence properties of conditional order-sorted rewrite theories. J. Log. Algebr. Program. 81(7–8), 816–850 (2012)
Durán, F., Meseguer, J., Rocha, C.: Proving ground confluence of equational specifications modulo axioms. Technical report 2142/99548, University of Illinois, Urbana, USA, March 2018
Durán, F., Rocha, C., Álvarez, J.M.: Towards a Maude formal environment. In: Agha, G., Danvy, O., Meseguer, J. (eds.) Formal Modeling: Actors, Open Systems, Biological Systems. LNCS, vol. 7000, pp. 329–351. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24933-4_17
Goguen, J.A., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theor. Comput. Sci. 105(2), 217–273 (1992)
Hrbacek, K., Jech, T.J.: Introduction to Set Theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 45, 3rd edn. M. Dekker, New York (1999)
Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Presicce, F.P. (ed.) WADT 1997. LNCS, vol. 1376, pp. 18–61. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-64299-4_26
Meseguer, J.: Strict coherence of conditional rewriting modulo axioms. Theor. Comput. Sci. 672, 1–35 (2017)
Nagele, J., Felgenhauer, B., Middeldorp, A.: CSI: new evidence – a progress report. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 385–397. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_24
Nakamura, M., Ogata, K., Futatsugi, K.: Incremental proofs of termination, confluence and sufficient completeness of OBJ specifications. In: Iida, S., Meseguer, J., Ogata, K. (eds.) Specification, Algebra, and Software. LNCS, vol. 8373, pp. 92–109. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54624-2_5
Rocha, C., Meseguer, J.: Constructors, sufficient completeness, and deadlock freedom of rewrite theories. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 594–609. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16242-8_42
Acknowledgments
The authors would like to thank the anonymous referees for their helpful comments that helped us improve the paper. The first author was partially supported by Spanish MINECO/FEDER project TIN2014-52034-R and Univ. Málaga, Campus de Excelencia Internacional Andalucía Tech. The second author was partially supported by NRL under contract number N00173-17-1-G002. The third author was partially supported by CAPES, Colciencias, and INRIA via the STIC AmSud project “EPIC: EPistemic Interactive Concurrency” (Proc. No 88881.117603/2016-01).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Durán, F., Meseguer, J., Rocha, C. (2018). Proving Ground Confluence of Equational Specifications Modulo Axioms. In: Rusu, V. (eds) Rewriting Logic and Its Applications. WRLA 2018. Lecture Notes in Computer Science(), vol 11152. Springer, Cham. https://doi.org/10.1007/978-3-319-99840-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-99840-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-99839-8
Online ISBN: 978-3-319-99840-4
eBook Packages: Computer ScienceComputer Science (R0)