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Semantically-Guided Goal-Sensitive Reasoning: Decision Procedures and the Koala Prover

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Abstract

The main topic of this article are SGGS decision procedures for fragments of first-order logic without equality. SGGS (Semantically-Guided Goal-Sensitive reasoning) is an attractive basis for decision procedures, because it generalizes to first-order logic the Conflict-Driven Clause Learning (CDCL) procedure for propositional satisfiability. As SGGS is both refutationally complete and model-complete in the limit, SGGS decision procedures are model-constructing. We investigate the termination of SGGS with both positive and negative results: for example, SGGS decides Datalog and the stratified fragment (including Effectively PRopositional logic) that are relevant to many applications. Then we discover several new decidable fragments, by showing that SGGS decides them. These fragments have the small model property, as the cardinality of their SGGS-generated models can be upper bounded, and for most of them termination tools can be applied to test a set of clauses for membership. We also present the first implementation of SGGS—the Koala theorem prover—and we report on experiments with Koala.

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Notes

  1. The name of the procedure in [63] is DPLL(\({{\mathcal {T}}}\)), but the recent literature calls it CDCL(\({{\mathcal {T}}}\)), since the DPLL (Davis-Putnam-Logemann-Loveland) [27] and CDCL procedures have been recognized as distinct. The same remark applies to DPLL(\(\varGamma \!+\!{{\mathcal {T}}}\)) [21].

  2. In this paper stratified is used in the sense of sort-stratified [1, 47, 57], not in the sense of stratified logic programs (e.g., [26]).

  3. An SGGS-splitting is trivial if it produces a singleton partition, such as when trying to split a ground clause or trying to split a clause by a more general one.

  4. The SGGS-derivation with \(I^-\) given for this set in [17, Ex. 11] is incorrect.

  5. For PVD also the finite basis approach applies implying the small model property [17].

  6. Lemma 5 and Theorem 7 were proved for ordered resolution assuming > ensures that \(L^\prime \vee D\) is positive [17, Lem. 5 and Thm. 6]; it is better to work with PO-resolution.

  7. SGGS and Koala do not have a built-in treatment of equality.

  8. The average TPTP ratings of the discovered restrained, sort-restrained, and sort-refined-PVD problems are 0.06, 0.08, and 0.08, respectively.

  9. Koala is available at https://github.com/bytekid/koala.

  10. The experimental data are posted at http://cl-informatik.uibk.ac.at/users/swinkler/koala/, http://profs.sci.univr.it/~bonacina/sggs.html or https://github.com/bytekid/koala.

  11. Ordered resolution decides \(\textsf{FO}^2\) via a reduction to the Gödel fragment [41, 75] that is unlikely to be implemented in provers.

References

  1. Abadi, A., Rabinovich, A., Sagiv, M.: Decidable fragments of many-sorted logic. J. Symb. Comput. 45(2), 153–172 (2010). https://doi.org/10.1016/j.jsc.2009.03.003

    Article  MathSciNet  MATH  Google Scholar 

  2. Ackermann, W.: Solvable Cases of the Decision Problem. North Holland, Amsterdam (1954). https://doi.org/10.1007/BFb0022557

  3. Alagi, G., Weidenbach, C.: NRCL – a model building approach to the Bernays-Schönfinkel fragment. In: C. Lutz, S. Ranise (eds.) Proceedings of FroCoS-10, Lecture Notes in Artificial Intelligence, vol. 9322, pp. 69–84. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-24246-0_5

  4. Andréka, H., van Benthem, J., Nemeti, I.: Modal logics and bounded fragments of predicate logic. J. Phil. Log. 27(3), 217–274 (1998). https://doi.org/10.1023/A:1004275029985

    Article  MATH  Google Scholar 

  5. Bachmair, L., Tiwari, A., Vigneron, L.: Abstract congruence closure. J. Autom. Reason. 31(2), 129–168 (2003). https://doi.org/10.1023/B:JARS.0000009518.26415.49

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbosa, H., Barrett, C.W., Brain, M., Kremer, G., Lachnitt, H., Mann, M., Mohamed, A., Mohamed, M., Niemetz, A., Nötzli, A., Ozdemir, A., Preiner, M., Reynolds, A., Sheng, Y., Tinelli, C., Zohar, Y.: CVC5: A versatile and industrial-strength SMT solver. In: D. Fisman, G. Rosu (eds.) Proceedings of TACAS-28, Lecture Notes in Computer Science, vol. 13243, pp. 415–442. Springer, Berlin (2022). https://doi.org/10.1007/978-3-030-99524-9_24

  7. Barrett, C.W., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Splitting on demand in SAT modulo theories. In: M. Hermann, A. Voronkov (eds.) Proceedings of LPAR-13, Lecture Notes in Artificial Intelligence, vol. 4246, pp. 512–526. Springer, Berlin (2006). https://doi.org/10.1007/11916277_35

  8. Baumgartner, P.: Hyper tableaux – the next generation. In: H. de Swart (ed.) Proceedings of TABLEAUX-7, Lecture Notes in Artificial Intelligence, vol. 1397, pp. 60–76. Springer, Berlin (1998)

  9. Baumgartner, P., Schmidt, R.A.: Blocking and other enhancements for bottom-up model generation methods. J. Autom. Reason. 64, 197–251 (2020). https://doi.org/10.1007/s10817-019-09515-1

    Article  MathSciNet  MATH  Google Scholar 

  10. Baumgartner, P., Tinelli, C.: The model evolution calculus as a first-order DPLL method. Artif. Intell. 172(4–5), 591–632 (2008). https://doi.org/10.1016/j.artint.2007.09.005

    Article  MathSciNet  MATH  Google Scholar 

  11. Baumgartner, P., Furbach, U., Niemelä, I.: Hyper tableaux. In: J.J. Alferes, L.M. Pereira, E. Orłowska (eds.) Proceedings of JELIA-5, Lecture Notes in Artificial Intelligence, vol. 1126, pp. 1–17. Springer, Berlin (1996)

  12. Baumgartner, P., Fuchs, A., Tinelli, C.: Implementing the model evolution calculus. Int. J. Artif. Intell. Tools 15(1), 21–52 (2006). https://doi.org/10.1142/S0218213006002552

    Article  MATH  Google Scholar 

  13. Baumgartner, P., Furbach, U., Pelzer, B.: The hyper tableaux calculus with equality and an application to finite model computation. J. Log. Comput. 20(1), 77–109 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bernays, P., Schönfinkel, M.: Zum Entscheidungsproblem der mathematischen Logik. Math. Ann. 99, 342–372 (1928). https://doi.org/10.1007/BF01459101

    Article  MathSciNet  MATH  Google Scholar 

  15. Bonacina, M.P.: On conflict-driven reasoning. In: N. Shankar, B. Dutertre (eds.) Proceedings of the 6th Workshop on Automated Formal Methods (AFM) May 2017, Kalpa Publications, vol. 5, pp. 31–49. EasyChair (2018). https://doi.org/10.29007/spwm

  16. Bonacina, M.P., Dershowitz, N.: Canonical ground Horn theories. In: A. Voronkov, C. Weidenbach (eds.) Programming Logics: Essays in Memory of H. Ganzinger, Lecture Notes in Computer Science, vol. 7797, pp. 35–71. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-37651-1_3

  17. Bonacina, M.P., Winkler, S.: SGGS decision procedures. In: N. Peltier, V. Sofronie-Stokkermans (eds.) Proceedings of IJCAR-10, Lecture Notes in Artificial Intelligence, vol. 12166, pp. 356–374. Springer, Berlin (2020). https://doi.org/10.1007/978-3-030-51074-9_20

  18. Bonacina, M.P., Hsiang, J.: On the modelling of search in theorem proving - towards a theory of strategy analysis. Inf. Comput. 147, 171–208 (1998). https://doi.org/10.1006/inco.1998.2739

    Article  MathSciNet  MATH  Google Scholar 

  19. Bonacina, M.P., Plaisted, D.A.: Semantically-guided goal-sensitive reasoning: model representation. J. Autom. Reason. 56(2), 113–141 (2016). https://doi.org/10.1007/s10817-015-9334-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Bonacina, M.P., Plaisted, D.A.: Semantically-guided goal-sensitive reasoning: inference system and completeness. J. Autom. Reason. 59(2), 165–218 (2017). https://doi.org/10.1007/s10817-016-9384-2

    Article  MathSciNet  MATH  Google Scholar 

  21. Bonacina, M.P., Lynch, C.A., de Moura, L.: On deciding satisfiability by theorem proving with speculative inferences. J. Autom. Reason. 47(2), 161–189 (2011). https://doi.org/10.1007/s10817-010-9213-y

    Article  MathSciNet  MATH  Google Scholar 

  22. Bonacina, M.P., Furbach, U., Sofronie-Stokkermans, V.: On first-order model-based reasoning. In: N. Martí-Oliet, P. Olveczky, C. Talcott (eds.) Logic, Rewriting, and Concurrency: Essays Dedicated to José Meseguer, Lecture Notes in Computer Science, vol. 9200, pp. 181–204. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-23165-5_8

  23. Bonacina, M.P., Graham-Lengrand, S., Shankar, N.: Conflict-driven satisfiability for theory combination: transition system and completeness. J. Autom. Reason. 64(3), 579–609 (2020). https://doi.org/10.1007/s10817-018-09510-y

    Article  MathSciNet  MATH  Google Scholar 

  24. Bonacina, M.P., Graham-Lengrand, S., Shankar, N.: Conflict-driven satisfiability for theory combination: lemmas, modules, and proofs. J. Autom. Reason. 66(1), 43–91 (2022). https://doi.org/10.1007/s10817-021-09606-y

    Article  MathSciNet  MATH  Google Scholar 

  25. Caferra, R., Leitsch, A., Peltier, N.: Automated Model Building. Kluwer Academic Publishers, Oxford (2004). https://doi.org/10.1007/978-1-4020-2653-9

    Book  MATH  Google Scholar 

  26. Ceri, S., Gottlob, G., Tanca, L.: What you always wanted to know about Datalog (and never dared to ask). IEEE Trans. Knowl. Data Eng. 1(1), 146–166 (1989). https://doi.org/10.1109/69.43410

    Article  Google Scholar 

  27. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. C. ACM 5(7), 394–397 (1962). https://doi.org/10.1145/368273.368557

    Article  MathSciNet  MATH  Google Scholar 

  28. de Moura, L., Jovanović, D.: A model-constructing satisfiability calculus. In: R. Giacobazzi, J. Berdine, I. Mastroeni (eds.) Proceedings of VMCAI-14, Lecture Notes in Computer Science, vol. 7737, pp. 1–12. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-35873-9_1

  29. de Nivelle, H., de Rijke, M.: Deciding the guarded fragments by resolution. J. Symb. Comput. 35(1), 21–58 (2003). https://doi.org/10.1016/S0747-7171(02)00092-5

    Article  MathSciNet  MATH  Google Scholar 

  30. Dershowitz, N.: Orderings for term-rewriting systems. Theoret. Comput. Sci. 17(3), 279–301 (1982). https://doi.org/10.1016/0304-3975(82)90026-3

    Article  MathSciNet  MATH  Google Scholar 

  31. Dershowitz, N., Plaisted, D.A.: Rewriting. In: J.A. Robinson, A. Voronkov (eds.) Handbook of Automated Reasoning, vol. 1, chap. 9, pp. 535–610. Elsevier (2001). https://doi.org/10.1016/b978-044450813-3/50011-4

  32. Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the common subexpression problem. J. ACM 27(4), 758–771 (1980). https://doi.org/10.1145/322217.322228

    Article  MathSciNet  MATH  Google Scholar 

  33. Duarte, A., Korovin, K.: Implementing superposition in iProver (system description). In: N. Peltier, V. Sofronie-Stokkermans (eds.) Proceedings of IJCAR-10, Lecture Notes in Computer Science, vol. 12167, pp. 388–397. Springer, Berlin (2020). https://doi.org/10.1007/978-3-030-51054-1_24

  34. Fermüller, C.G., Leitsch, A.: Model building by resolution. In: E. Börger, G. Jäger, H. Kleine Büning, S. Martini (eds.) Proceedings of CSL-6, Lecture Notes in Computer Science, vol. 702, pp. 134–148. Springer, Berlin (1993). https://doi.org/10.1007/3-540-56992-8_10

  35. Fermüller, C.G., Salzer, G.: Ordered paramodulation and resolution as decision procedure. In: A. Voronkov (ed.) Proceedings of LPAR-4, Lecture Notes in Artificial Intelligence, vol. 698, pp. 122–133. Springer, Berlin (1993). https://doi.org/10.1007/3-540-56944-8_47

  36. Fermüller, C.G., Leitsch, A., Hustadt, U., Tammet, T.: Resolution decision procedures. In: Handbook of Automated Reasoning, pp. 1791–1849. Elsevier and MIT Press, Amsterdam and Cambridge (2001). https://doi.org/10.1016/b978-044450813-3/50027-8

  37. Fiori, A., Weidenbach, C.: SCL clause learning from simple models. In: P. Fontaine (ed.) Proceedings of CADE-27, Lecture Notes in Artificial Intelligence, vol. 11716, pp. 233–249. Springer, Berlin (2019). https://doi.org/10.1007/978-3-030-29436-6_14

  38. Fontaine, P.: Combinations of theories for decidable fragments of first-order logic. In: S. Ghilardi, R. Sebastiani (eds.) Proceedings of FroCoS-7, Lecture Notes in Artificial Intelligence, vol. 5749, pp. 263–278. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-04222-5_16

  39. Ganzinger, H., Korovin, K.: New directions in instantiation-based theorem proving. In: Proceedings of LICS-18, pp. 55–64. IEEE (2003)

  40. Giesl, J., Aschermann, C., Brockschmidt, M., Emmes, F., Frohn, F., Fuhs, C., Hensel, J., Otto, C., Plücker, M., Schneider-Kamp, P., Ströder, T., Swiderski, S., Thiemann, R.: Analyzing program termination and complexity automatically with AProVE. J. Autom. Reason. 58(1), 3–31 (2017). https://doi.org/10.1007/s10817-016-9388-y

    Article  MathSciNet  MATH  Google Scholar 

  41. Grädel, E., Kolaitis, P.G., Vardi, M.Y.: On the decision problem for two-variable first-order logic. Bull. Symb. Log. 3, 53–69 (1997). https://doi.org/10.2307/421196

    Article  MathSciNet  MATH  Google Scholar 

  42. Hsiang, J., Rusinowitch, M.: Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. J. ACM 38(3), 559–587 (1991). https://doi.org/10.1145/116825.116833

    Article  MathSciNet  MATH  Google Scholar 

  43. Hustadt, U., Schmidt, R.A., Georgieva, L.: A survey of decidable first-order fragments and description logics. J. of Relational Methods in Computer Science 1, 251–276 (2004). https://doi.org/10.1007/978-3-642-37651-1_15

    Article  Google Scholar 

  44. Joyner, W.H., Jr.: Resolution strategies as decision procedures. J. ACM 23(3), 398–417 (1976). https://doi.org/10.1145/321958.321960

    Article  MathSciNet  MATH  Google Scholar 

  45. Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: J. Leech (ed.) Proceedings of the Conference on Computational Problems in Abstract Algebras, pp. 263–298. Pergamon Press (1970). https://doi.org/10.1016/B978-0-08-012975-4

  46. Korovin, K.: Inst-Gen – a modular approach to instantiation-based automated reasoning. In: A. Voronkov, C. Weidenbach (eds.) Programming Logics: Essays in Memory of H. Ganzinger, Lecture Notes in Computer Science, vol. 7797, pp. 239–270. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-37651-1_10

  47. Korovin, K.: Non-cyclic sorts for first-order satisfiability. In: P. Fontaine, C. Ringeissen, R.A. Schmidt (eds.) Proceedings of FroCoS-9, Lecture Notes in Artificial Intelligence, vol. 8152, pp. 214–228. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-40885-4_15

  48. Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean Termination Tool 2. In: R. Treinen (ed.) Proceedings of RTA-20, Lecture Notes in Computer Science, vol. 5595, pp. 295–304. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-02348-4_21

  49. Kounalis, E., Rusinowitch, M.: On word problems in Horn theories. In: E. Lusk, R. Overbeek (eds.) Proceedings of CADE-9, Lecture Notes in Computer Science, vol. 310, pp. 527–537. Springer, Berlin (1988). https://doi.org/10.1007/BFb0012854

  50. Kounalis, E., Rusinowitch, M.: On word problems in Horn theories. J. Symb. Comput. 11(1–2), 113–128 (1991). https://doi.org/10.1016/S0747-7171(08)80134-4

    Article  MathSciNet  MATH  Google Scholar 

  51. Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: N. Sharygina, H. Veith (eds.) Proceedings of CAV-25, Lecture Notes in Computer Science, vol. 8044, pp. 1–35. Springer, Berlin (2013). https://doi.org/10.1007/978-3-642-39799-8_1

  52. Lamotte-Schubert, M., Weidenbach, C.: BDI: a new decidable clause class. J. Log. Comput. 27(2), 441–468 (2017). https://doi.org/10.1093/logcom/exu074

    Article  MathSciNet  MATH  Google Scholar 

  53. Lee, S.J., Plaisted, D.A.: Eliminating duplication with the hyperlinking strategy. J. Autom. Reason. 9, 25–42 (1992)

    Article  MATH  Google Scholar 

  54. Ludwig, M., Waldmann, U.: An extension of the Knuth-Bendix ordering with LPO-like properties. In: N. Dershowitz, A. Voronkov (eds.) Proceedings of LPAR-14, Lecture Notes in Artificial Intelligence, vol. 4790, pp. 348–362. Springer, Berlin (2007). https://doi.org/10.1007/978-3-540-75560-9_26

  55. Manthey, R., Bry, F.: SATCHMO: a theorem prover implemented in Prolog. In: E. Lusk, R. Overbeek (eds.) Proceedings of CADE-9, Lecture Notes in Computer Science, vol. 310, pp. 415–434. Springer, Berlin (1988). https://doi.org/10.1007/BFb0012847

  56. Marques Silva, J., Lynce, I., Malik, S.: Conflict-driven clause learning SAT solvers. In: A. Biere, M. Heule, H. Van Maaren, T. Walsh (eds.) Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 131–153. IOS Press (2009). https://doi.org/10.3233/978-1-58603-929-5-131

  57. McMillan, K.L.: Developing distributed protocols with Ivy. Slides from http://vmcaischool19.tecnico.ulisboa.pt/ (2019)

  58. Mei, H., Qin, G., Xu, M., Esiner, J.: Neural Datalog through time: informed temporal modeling via logical specification. In: H. Daumé III, A. Singh (eds.) Proceedings of ICML-37, Proceedings of Machine Learning Research, vol. 119, pp. 6808–6819 (2020)

  59. Navarro, J.A., Voronkov, A.: Proof systems for effectively propositional logic. In: A. Armando, P. Baumgartner, G. Dowek (eds.) Proceedings of IJCAR-4, Lecture Notes in Artificial Intelligence, vol. 5195, pp. 426–440. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-71070-7_36

  60. Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27(2), 356–364 (1980). https://doi.org/10.1145/322186.322198

    Article  MathSciNet  MATH  Google Scholar 

  61. Nicolas, J.M.: Logic for improving integrity checking in relational databases. Acta Infor. 18(3), 227–253 (1982). https://doi.org/10.1145/322186.322198

    Article  MATH  Google Scholar 

  62. Nieuwenhuis, R., Oliveras, A.: Fast congruence closure and extensions. Inf. Comput. 205(4), 557–580 (2007). https://doi.org/10.1016/j.ic.2006.08.009

    Article  MathSciNet  MATH  Google Scholar 

  63. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. Padon, O., McMillan, K.L., Panda, A., Sagiv, M., Shoham, S.: Ivy: Safety verification by interactive generalization. SIGPLAN Notices 51(6), 614–630 (2016). https://doi.org/10.1145/2980983.2908118

    Article  Google Scholar 

  65. Piskac, R., de Moura, L., Bjørner, N.: Deciding effectively propositional logic using DPLL and substitution sets. J. Autom. Reason. 44(4), 401–424 (2010). https://doi.org/10.1007/978-3-540-71070-7_35

    Article  MathSciNet  MATH  Google Scholar 

  66. Plaisted, D.A., Zhu, Y.: The Efficiency of Theorem Proving Strategies. Friedr. Vieweg & Sohn, Berlin (1997)

    Book  MATH  Google Scholar 

  67. Plaisted, D.A., Zhu, Y.: Ordered semantic hyper linking. J. Autom. Reason. 25, 167–217 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  68. Ramsey, F.P.: On a problem in formal logic. Proc. Lond. Math. Soc. 30, 264–286 (1930). https://doi.org/10.1112/plms/s2-30.1.264

    Article  MATH  Google Scholar 

  69. Reger, G., Suda, M., Voronkov, A.: Playing with AVATAR. In: A.P. Felty, A. Middeldorp (eds.) Proceedings of CADE-25, Lecture Notes in Artificial Intelligence, vol. 9195, pp. 399–415. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-21401-6_28

  70. Riazanov, A.: Implementing an efficient theorem prover. Ph.D. thesis, Department of Computer Science, The University of Manchester (2003)

  71. Robinson, J.A.: Automatic deduction with hyper-resolution. Int. J. of Computer Mathematics 1, 227–234 (1965). https://doi.org/10.2307/2272384

    Article  MathSciNet  MATH  Google Scholar 

  72. Robinson, J.A.: A machine oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965). https://doi.org/10.1145/321250.321253

    Article  MathSciNet  MATH  Google Scholar 

  73. Rubio, A.: A fully syntactic AC-RPO. Inf. Comput. 178(2), 515–533 (2002). https://doi.org/10.1006/inco.2002.3158

    Article  MathSciNet  MATH  Google Scholar 

  74. Schulz, S., Cruanes, S., Vukmirovic, P.: Faster, higher, stronger: E 2.3. In: P. Fontaine (ed.) Proceedings of CADE-27, Lecture Notes in Computer Science, vol. 11716, pp. 495–507. Springer, Berlin (2019). https://doi.org/10.1007/978-3-030-29436-6_29

  75. Scott, D.: A decision method for validity of sentences in two variables. J. Symb. Log. 27, 377–377 (1962)

    Google Scholar 

  76. Slagle, J.R.: Automatic theorem proving with renamable and semantic resolution. J. ACM 14(4), 687–697 (1967). https://doi.org/10.1145/321420.321428

    Article  MathSciNet  MATH  Google Scholar 

  77. Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure. From CNF to TH0, TPTP v6.4.0. J. Autom. Reason. 59(4), 483–502 (2017). https://doi.org/10.1007/s10817-009-9143-8

    Article  MathSciNet  MATH  Google Scholar 

  78. van Gelder, A., Topor, R.W.: Safety and translation of relational calculus queries. ACM Trans. Database Syst. 16(2), 235–278 (1991). https://doi.org/10.1145/114325.103712

    Article  MathSciNet  Google Scholar 

  79. Waldmann, U., Schmidt, R.A.: Modal tableau systems with blocking and congruence closure. In: H. de Nivelle (ed.) Proceedings of TABLEAUX-24, Lecture Notes in Artificial Intelligence, vol. 9323, pp. 38–53. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-24312-2_4

  80. Weidenbach, C.: Combining superposition, sorts and splitting. In: A. Robinson, A. Voronkov (eds.) Handbook of Automated Reasoning, vol. 2, pp. 1965–2012. Elsevier, Amsterdam (2001). https://doi.org/10.1016/b978-044450813-3/50029-1

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Acknowledgements

We thank Konstantin Korovin for the iProver (v2.8) code for basic data structures, term indexing, and type inference, imported in Koala. Parts of this work were done while the first author was visiting the Simons Institute for the Theory of Computing, the Leibniz Zentrum für Informatik at Schloss Dagstuhl, and the Computer Science Laboratory of SRI International, whose support is greatly appreciated.

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    Maria Paola Bonacina

  2. Free University of Bozen-Bolzano, Piazza Domenicani 3, 39100, Bolzano/Bozen, Italy

    Sarah Winkler

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Bonacina, M.P., Winkler, S. Semantically-Guided Goal-Sensitive Reasoning: Decision Procedures and the Koala Prover. J Autom Reasoning 67, 6 (2023). https://doi.org/10.1007/s10817-022-09656-w

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  • Published:

  • DOI: https://doi.org/10.1007/s10817-022-09656-w

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