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Inductive Reasoning with Equality Predicates, Contextual Rewriting and Variant-Based Simplification

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Rewriting Logic and Its Applications (WRLA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12328))

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Abstract

We present an inductive inference system for proving validity of formulas in the initial algebra \(T_{\mathcal {E}}\) of an order-sorted equational theory \(\mathcal {E}\) with 17 inference rules, where only 6 of them require user interaction, while the remaining 11 can be automated as simplification rules and can be combined together as a limited, yet practical, automated inductive theorem prover. The 11 simplification rules are based on powerful equational reasoning techniques, including: equationally defined equality predicates, constructor variant unification, variant satisfiability, order-sorted congruence closure, contextual rewriting and recursive path orderings. For \(\mathcal {E} = (\varSigma , E \uplus B)\), these techniques work modulo B, with B a combination of associativity and/or commutativity and/or identity axioms.

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Notes

  1. 1.

    As explained in [22], there is no real loss of generality in assuming that all atomic formulas are equations: predicates can be specified by equational formulas using additional function symbols of a fresh new sort Pred with a constant tt, so that a predicate \(p(t_{1},\ldots , t_{n})\) becomes \(p(t_{1},\ldots , t_{n})={tt}\).

  2. 2.

    If \(B = B_{0} \uplus U\), with \(B_{0}\) associativity and/or commutativity axioms, and U identity axioms, the B-preregularity notion can be broadened by requiring only that: (i) \(\varSigma \) is \(B_{0}\)-preregular in the standard sense that \({ls}(u\rho )={ls}(v\rho )\) for all \(u=v \in B_{0}\) and substitutions \(\rho \); and (ii) the axioms U oriented as rules \(\vec {U}\) are sort-decreasing in the sense explained in Sect. 2.2.

  3. 3.

    In [13] the equality predicate is denoted \(\_\sim \_\), instead of the standard notation \(\_=\_\). Here we use \(\_=\_\) throughout. This has the pleasant effect that a QF formula \(\varphi \) is both a formula and a Boolean expression, which of course amounts to mechanizing by equational rewriting the Tarskian semantics of QF formulas in first-order-logic for initial algebras.

  4. 4.

    That is, there is a subtheory inclusion \(\mathcal {B} \subseteq \mathcal {E}\), with \(\mathcal {B}\) having signature \(\varSigma _{\mathcal {B}}\) and only sort NewBool such that: (i) \(T_{\mathcal {B}}\) the initial algebra of the Booleans, and (ii) \(T_{\mathcal {E}^{=}}|_{\varSigma _{\mathcal {B}}} \cong T_{\mathcal {B}}\).

  5. 5.

    An \(\vec {\mathcal {E}}_{1}\)-variant (or \(\vec {E}_{1},B_{1}\)-variant) of a \(\varSigma _{1}\)-term t is a pair \((v,\theta )\), where \(\theta \) is a substitution in canonical form, i.e., \(\theta = \theta !_{\vec {\mathcal {E}}_{1}}\), and \(v =_{B_{1}} (t\theta )!_{\vec {\mathcal {E}}_{1}}\). \(\vec {\mathcal {E}_{1}}\) is FVP iff any such t has a finite set of variants \(\{(u_{1},\alpha _{1}),\ldots , (u_{n},\alpha _{n})\}\) which are “most general possible” in the precise sense that for any variant \((v,\theta )\) of t there exist i, \({1 \leqslant i \leqslant n}\), and substitution \(\gamma \) such that: (i) \(v =_{B_{1}} u_{i}\gamma \), and (ii) \(\theta =_{B_{1}} \alpha _{i}\gamma \).

  6. 6.

    Even when, say, an induction hypothesis in H might originally be a superclause \(\varGamma \rightarrow \bigwedge _{l\in L}\varDelta _l\), for executability reasons we will always decompose it into its corresponding set of clauses \(\{\varGamma \rightarrow \varDelta _l\}_{l\in L}\).

  7. 7.

    Recall that \(\varGamma \) is a conjunction and \(\varLambda \) a conjunction of disjunctions. Therefore, the equality predicate rewrite rules together with \(\vec {H}_{e_U}\) may have powerful “cascade effects.” For example, if either \(\varGamma !\,_{\vec {\mathcal {E}}_{\overline{X}_U}^=\cup \,\vec {H}_{e_U}} = \bot \) or \(\varLambda !\,_{\vec {\mathcal {E}}_{\overline{X}_U}^=\cup \,\vec {H}_{e_U}} = \top \), then \((\varGamma \rightarrow \varLambda ) !\,_{\vec {\mathcal {E}}_{\overline{X}_U}^=\cup \,\vec {H}_{e_U}}\) is a tautology and the goal is proved.

  8. 8.

    The net effect is not only that (EPS) both subsumes (ERL) and (ERR) and becomes more powerful: by adding such extra rules to \(\vec {\mathcal {E}}_{U}^=\), the ICC simplification rule discussed next, which also performs simplification with equality predicates, also becomes more powerful.

  9. 9.

    More generally, the equality predicate theory \(\vec {\mathcal {E}}_{U}^=\) can be extended by adding to it conditional rewrite rules that orient inductive theorems of \(\mathcal {E}\) or \(\mathcal {E}_{U}^=\), are executable, and keep \(\vec {\mathcal {E}}_{U}^=\) operationally terminating. For example, if c and \(c'\) are different constructors whose sorts belong to the same connected component having a top sort, say, s, then the conditional rewrite rule \(x=c(x_{1},\ldots , x_{n})\wedge x=c'(y_{1},\ldots , y_{m})\rightarrow \bot \), where x has sort s orients an inductively valid lemma, clearly terminates, and can thus be added to \(\vec {\mathcal {E}}_{U}^=\). In particular, if p is a Boolean-valued predicate and \(u_{i}=_{B_{0}}v_{i}\), \(p(u_{1},\ldots , u_{n})= {true}\wedge p(v_{1},\ldots , v_{n})= {false}\) rewrites to \(\bot \).

  10. 10.

    A cover set for s is a finte set of \(\varOmega \)-terms such that \({{ls}(u_i) \leqslant s}\), \({1\leqslant i \leqslant n}\), and generating all constructor ground terms of sort s modulo \(B_{\varOmega }\), i.e., .

References

  1. Bouhoula, A., Rusinowitch, M.: SPIKE: a system for automatic inductive proofs. In: Alagar, V.S., Nivat, M. (eds.) AMAST 1995. LNCS, vol. 936, pp. 576–577. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60043-4_79

    Chapter  Google Scholar 

  2. Boyer, R., Moore, J.: A Computational Logic. Academic Press, San Diego (1980)

    MATH  Google Scholar 

  3. 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

    Book  MATH  Google Scholar 

  4. Clavel, M., Palomino, M.: The ITP tool’s manual, universidad Complutense, Madrid, April 2005. http://maude.sip.ucm.es/itp/

  5. Comon, H., Nieuwenhuis, R.: Induction = i - axiomatization + first-order consistency. Inf. Comput. 159(1–2), 151–186 (2000)

    Article  MathSciNet  Google Scholar 

  6. Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–320. North-Holland (1990)

    Google Scholar 

  7. 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

    Chapter  MATH  Google Scholar 

  8. Escobar, S., Meadows, C., Meseguer, J.: Maude-NPA: cryptographic protocol analysis modulo equational properties. In: Aldini, A., Barthe, G., Gorrieri, R. (eds.) FOSAD 2007-2009. LNCS, vol. 5705, pp. 1–50. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03829-7_1

    Chapter  MATH  Google Scholar 

  9. Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Algebraic Logic Programm. 81, 898–928 (2012)

    Article  MathSciNet  Google Scholar 

  10. Găină, D., Lucanu, D., Ogata, K., Futatsugi, K.: On automation of OTS/CafeOBJ method. In: Iida, S., Meseguer, J., Ogata, K. (eds.) Specification, Algebra, and Software. LNCS, vol. 8373, pp. 578–602. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54624-2_29

    Chapter  Google Scholar 

  11. Goguen, J., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoret. Comput. Sci. 105, 217–273 (1992)

    Article  MathSciNet  Google Scholar 

  12. Gutiérrez, R., Meseguer, J., Skeirik, S.: The Maude termination assistant. In: Pre-Proceedings of WRLA 2018

    Google Scholar 

  13. Gutiérrez, R., Meseguer, J., Rocha, C.: Order-sorted equality enrichments modulo axioms. Sci. Comput. Program. 99, 235–261 (2015)

    Article  Google Scholar 

  14. Hendrix, J.D.: Decision procedures for equationally based reasoning. Ph.D. thesis, University of Illinois at Urbana-Champaign (2008). http://hdl.handle.net/2142/10967

    Google Scholar 

  15. Kapur, D., Zhang, H.: An overview of rewrite rule laboratory (RRL). In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 559–563. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-51081-8_138

    Chapter  Google Scholar 

  16. Kaufmann, M., Manolios, P., Moore, J.: Computer-Aided Reasoning: An Approach. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

  17. Lucas, S., Meseguer, J.: Normal forms and normal theories in conditional rewriting. J. Log. Algebr. Meth. Program. 85(1), 67–97 (2016)

    Article  MathSciNet  Google Scholar 

  18. Meier, S., Schmidt, B., Cremers, C., Basin, D.: The TAMARIN prover for the symbolic analysis of security protocols. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 696–701. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_48

    Chapter  Google Scholar 

  19. Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theoret. Comput. Sci. 96(1), 73–155 (1992)

    Article  MathSciNet  Google Scholar 

  20. 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

    Chapter  Google Scholar 

  21. Meseguer, J.: Order-sorted rewriting and congruence closure. In: Jacobs, B., Löding, C. (eds.) FoSSaCS 2016. LNCS, vol. 9634, pp. 493–509. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49630-5_29

    Chapter  MATH  Google Scholar 

  22. Meseguer, J.: Variant-based satisfiability in initial algebras. Sci. Comput. Program. 154, 3–41 (2018)

    Article  Google Scholar 

  23. Meseguer, J., Skeirik, S.: Inductive reasoning with equality predicates, contextual rewriting and variant-based simplification. Technical report, University of Illinois at Urbana-Champaign, Computer Science Department, July 2020. http://hdl.handle.net/2142/107774

  24. Rocha, C., Meseguer, J.: Proving safety properties of rewrite theories. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 314–328. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22944-2_22

    Chapter  Google Scholar 

  25. Rocha, C.: Symbolic reachability analysis for rewrite theories. Ph.D. thesis, University of Illinois at Urbana-Champaign (2012)

    Google Scholar 

  26. Rubio, A.: Automated deduction with constrained clauses. Ph.D. thesis, Universitat Politècnica de Catalunya (1994)

    Google Scholar 

  27. Rubio, A.: A fully syntactic AC-RPO. Inf. Comput. 178(2), 515–533 (2002)

    Article  MathSciNet  Google Scholar 

  28. Sasse, R.: Security models in rewriting logic for cryptographic protocols and browsers. Ph.D. thesis, University of Illinois at Urbana-Champaign (2012). http://hdl.handle.net/2142/34373

  29. Sasse, R., King, S.T., Meseguer, J., Tang, S.: IBOS: a correct-by-construction modular browser. In: Păsăreanu, C.S., Salaün, G. (eds.) FACS 2012. LNCS, vol. 7684, pp. 224–241. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35861-6_14

    Chapter  Google Scholar 

  30. Skeirik, S., Meseguer, J., Rocha, C.: Verification of the IBOS browser security properties in reachability logic. In: Escobar, S., Martí-Oliet, N. (eds.) WRLA 2020, LNCS 12328, pp. 176–196 (2020)

    Google Scholar 

  31. Skeirik, S.: Rewriting-based symbolic methods for distributed system verification. Ph.D. thesis, University of Illinois at Urbana-Champaign (2019)

    Google Scholar 

  32. Skeirik, S., Meseguer, J.: Metalevel algorithms for variant satisfiability. J. Log. Algebr. Meth. Program. 96, 81–110 (2018)

    Article  MathSciNet  Google Scholar 

  33. Skeirik, S., Stefanescu, A., Meseguer, J.: A constructor-based reachability logic for rewrite theories. Fundam. Inform. 173(4), 315–382 (2020)

    Article  MathSciNet  Google Scholar 

  34. Tang, S.: Towards secure web browsing. Ph.D. thesis, University of Illinois at Urbana-Champaign (2011), 25 May 2011. http://hdl.handle.net/2142/24307

    Google Scholar 

  35. Tang, S., Mai, H., King, S.T.: Trust and protection in the Illinois browser operating system. In: Proceedings of the 9th USENIX Symposium on Operating Systems Design and Implementation, OSDI 2010, Vancouver, BC, Canada, pp. 17–32, 4–6 October 2010. USENIX Association (2010)

    Google Scholar 

  36. Zhang, H.: Contextual rewriting in automated reasoning. Fundam. Inform. 24(1/2), 107–123 (1995)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We cordially thank the referees for their very helpful suggestions to improve the paper. Work partially supported by NRL under contract N00173-17-1-G002.

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  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, USA

    José Meseguer & Stephen Skeirik

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  1. José Meseguer
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Correspondence to José Meseguer .

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Editors and Affiliations

  1. Valencian Research Institute for Artificial Intelligence, Universitat Politècnica de València, Valencia, Spain

    Santiago Escobar

  2. Facultad de Informática, Universidad Complutense de Madrid, Madrid, Spain

    Narciso Martí-Oliet

A The Natural Numbers Theory \(\mathcal {N}\)

A The Natural Numbers Theory \(\mathcal {N}\)

Fig. 1.
figure 1

Natural number theory specification.

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Note that we have a “sandwich” of theories \(\mathcal {N}_{\varOmega } \subseteq \mathcal {N}_{1} \subseteq \mathcal {N}\), where \(\mathcal {N}_{\varOmega }\) is given by the operators marked as ctor and the associativity-commutativity of \(+\), and \(\mathcal {N}_{1}\) is the FVP theory extending \(\mathcal {N}_{\varOmega }\) with the other symbols for \(+\) and the equation for as identity element for \(+\) (Fig. 1).

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Meseguer, J., Skeirik, S. (2020). Inductive Reasoning with Equality Predicates, Contextual Rewriting and Variant-Based Simplification. In: Escobar, S., Martí-Oliet, N. (eds) Rewriting Logic and Its Applications. WRLA 2020. Lecture Notes in Computer Science(), vol 12328. Springer, Cham. https://doi.org/10.1007/978-3-030-63595-4_7

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