Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints

  • Andreas TeuckeEmail author
  • Christoph Weidenbach
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)


The monadic shallow linear Horn fragment is well-known to be decidable and has many application, e.g., in security protocol analysis, tree automata, or abstraction refinement. It was a long standing open problem how to extend the fragment to the non-Horn case, preserving decidability, that would, e.g., enable to express non-determinism in protocols. We prove decidability of the non-Horn monadic shallow linear fragment via ordered resolution further extended with dismatching constraints and discuss some applications of the new decidable fragment.



We thank the reviewers as well as Konstantin Korovin and Giles Reger for a number of important remarks.


  1. 1.
    Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. Logic Comput. 4(3), 217–247 (1994). Revised version of Max-Planck-Institut für Informatik Technical report, MPI-I-91-208 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H.: Ordered chaining calculi for first-order theories of transitive relations. J. ACM 45(6), 1007–1049 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baeten, J.C.M., Bergstra, J.A., Klop, J., Weijland, W.P.: Term-rewriting systems with rule priorities. Theor. Comput. Sci. 67(2&3), 283–301 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baumgartner, P., Fuchs, A., Tinelli, C.: Implementing the model evolution calculus. Int. J. Artif. Intell. Tools 15(1), 21–52 (2006)CrossRefzbMATHGoogle Scholar
  5. 5.
    Baumgartner, P., Tinelli, C.: The model evolution calculus. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 350–364. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45085-6_32 CrossRefGoogle Scholar
  6. 6.
    Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007). Accessed 12 Oct 2007
  7. 7.
    Goubault-Larrecq, J.: Deciding \(\cal{H}_1\) by resolution. Inf. Process. Lett. 95(3), 401–408 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Jacquemard, F., Meyer, C., Weidenbach, C.: Unification in extensions of shallow equational theories. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 76–90. Springer, Heidelberg (1998). doi: 10.1007/BFb0052362 CrossRefGoogle Scholar
  9. 9.
    Korovin, K.: iProver – an instantiation-based theorem prover for first-order logic (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS, vol. 5195, pp. 292–298. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-71070-7_24 CrossRefGoogle Scholar
  10. 10.
    Korovin, K.: Inst-Gen – a modular approach to instantiation-based automated reasoning. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics. LNCS, vol. 7797, pp. 239–270. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-37651-1_10 CrossRefGoogle Scholar
  11. 11.
    Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 1–35. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_1 CrossRefGoogle Scholar
  12. 12.
    Seidl, H., Reuß, A.: Extending H1-clauses with disequalities. Inf. Process. Lett. 111(20), 1007–1013 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Seidl, H., Reuß, A.: Extending \({\cal{H}_1}\)-clauses with path disequalities. In: Birkedal, L. (ed.) FoSSaCS 2012. LNCS, vol. 7213, pp. 165–179. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-28729-9_11 Google Scholar
  14. 14.
    Seidl, H., Verma, K.N.: Cryptographic protocol verification using tractable classes of horn clauses. In: Reps, T., Sagiv, M., Bauer, J. (eds.) Program Analysis and Compilation, Theory and Practice. LNCS, vol. 4444, pp. 97–119. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-71322-7_5 CrossRefGoogle Scholar
  15. 15.
    Slaney, J.K., Surendonk, T.: Combining finite model generation with theorem proving: problems and prospects. In: Baader, F., Schulz, K.U. (eds.), Frontiers of Combining Systems, First International Workshop FroCoS 1996, Munich, Germany, March 26-29, 1996, Proceedings, vol. 3. Applied Logic Series, pp. 141–155. Kluwer Academic Publishers (1996)Google Scholar
  16. 16.
    Suda, M., Weidenbach, C., Wischnewski, P.: On the saturation of YAGO. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS (LNAI), vol. 6173, pp. 441–456. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14203-1_38 CrossRefGoogle Scholar
  17. 17.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure: the FOF and CNF Parts, v3.5.0. J. Autom. Reasoning 43(4), 337–362 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Teucke, A., Weidenbach, C.: First-order logic theorem proving and model building via approximation and instantiation. In: Lutz, C., Ranise, S. (eds.) FroCoS 2015. LNCS (LNAI), vol. 9322, pp. 85–100. Springer, Cham (2015). doi: 10.1007/978-3-319-24246-0_6 CrossRefGoogle Scholar
  19. 19.
    Teucke, A., Weidenbach, C.: Ordered resolution with straight dismatching constraints. In: Fontaine, P., Schulz, S., Urban, J. (eds.) Proceedings of the 5th Workshop on Practical Aspects of Automated Reasoning Co-located with International Joint Conference on Automated Reasoning (IJCAR 2016), Coimbra, Portugal, 2 July 2016, vol. 1635. CEUR Workshop Proceedings, pp. 95–109 (2016).
  20. 20.
    Teucke, A., Weidenbach, C.: Decidability of the monadic shallow linear first-order fragment with straight dismatching constraints (2017).
  21. 21.
    Voronkov, A.: AVATAR: the architecture for first-order theorem provers. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 696–710. Springer, Cham (2014). doi: 10.1007/978-3-319-08867-9_46 Google Scholar
  22. 22.
    Weidenbach, C.: Towards an automatic analysis of security protocols in first-order logic. CADE 1999. LNCS, vol. 1632, pp. 314–328. Springer, Heidelberg (1999). doi: 10.1007/3-540-48660-7_29 CrossRefGoogle Scholar
  23. 23.
    Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 140–145. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02959-2_10 CrossRefGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Max-Planck Institut für InformatikSaarbrückenGermany
  2. 2.Graduate School of Computer ScienceSaarbrückenGermany

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