Abstract
In designing a large-scale computerized proof system, one is often confronted with issues of two kinds: issues regarding an underlying logical calculus, and issues that refer to theories, either specified axiomatically or characterized by indication of either a privileged model or a family of intended models. Proof services related to the theories most often take the form of satisfiability decision or semi-decision procedures (in a sense, polyadic inference rules), while some of the services offered by the calculus (e.g., the Davis-Putnam propositional satisfiability checker) provide low-level mechanisms for integrating services of the former kind. Integration among services can ensure speed-up (i.e., lower number of steps) in the proofs, but it must always be legitimatized by a conservativeness result. Interoperability among proof checkers and autonomous theorem provers is another key point of integration.
In discussing these and related issues, this paper refers to Set Theory as the unifying background, and to a specific proof-checker based on a slightly unorthodox formalization of it as an arena for experimentation.
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References
Bundy, A. (ed.): CADE 1994. LNCS (LNAI), vol. 814, pp. 238–251. Springer, Heidelberg (1994); The QED Manifesto
Andrews, P.B.: Resolution in type theory. The J. of Symbolic Logic 36, 414–432 (1971)
Andrews, P.B., Longini Cohen, E.: Theorem proving in type theory. In: Proc. of IJCAI 1977, pp. 566–566 (1977)
Arenas-Sánchez, P., Dovier, A.: Minimal set unification. In: Alpuente, M., Sessa, M.I. (eds.) GULP-PRODE 1995, Marina di Vietri, Italy, September 11-14, pp. 447–458 (1995)
Asperti, A., Geuvers, H., Natarajan, R.: Social processes, program verification, and all that. Math. Struct. in Comp. Science 19(5), 877–896 (2009)
Baader, F., Tinelli, C.: Combining equational theories sharing non-collapse-free constructors. In: Kirchner, H., Ringeissen, C. (eds.) FroCos 2000. LNCS (LNAI), vol. 1794, pp. 260–274. Springer, Heidelberg (2000)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Algorithms and computation in mathematics, vol. 10. Springer, Heidelberg (2006)
Behmann, H.: Beiträge zur Algebra der Logik, insbesondere zum Entscheidungsproblem. Math. Annalen 86, 163–220 (1922)
Belinfante, J.G.F.: Reasoning about iteration in Gödel’s class theory. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 228–242. Springer, Heidelberg (2003)
Bellè, D., Parlamento, F.: Decidability and completeness for open formulas of membership theories. Notre Dame J. of Formal Logic 36 (1995)
Bellè, D., Parlamento, F.: The decidability of the ∀ * ∃ class and the axiom of foundation. Notre Dame J. of Formal Logic 42 (2001)
Bellè, D., Parlamento, F.: Truth in V for ∃ * ∀ ∀-sentences is decidable. J. of Symbolic Logic 71 (2006)
Bledsoe, W.W.: Non-resolution theorem proving. Artificial Intelligence 9, 1–35 (1977)
Breban, M., Ferro, A., Omodeo, E.G., Schwartz, J.T.: Decision Procedures for Elementary Sublanguages of Set Theory II. Formulas involving Restricted Quantifiers, together with Ordinal, Integer, Map, and Domain Notions. Comm. Pure Appl. Math. 34, 177–195 (1981)
Buchberger, B., Winkler, F.: Gröebner bases and Applications. London Mathematical Society Lecture Note Series, vol. 251. Cambridge University Press, Cambridge (1998)
Burstall, R., Goguen, J.: Putting theories together to make specifications. In: Reddy, R. (ed.) Proc. 5th International Joint Conference on Artificial Intelligence, Cambridge, MA, pp. 1045–1058 (1977)
Cantone, D., Chiaruttini, C., Nicolosi Asmundo, M., Omodeo, E.G.: Cumulative hierarchies and computability over universes of sets. Le Matematiche 63, 31–84 (2008)
Cantone, D., Cincotti, G.: Decision algorithms for some fragments of analysis and related areas. Comm. Pure Appl. Math. 40, 281–300 (1987)
Cantone, D., Cincotti, G., Gallo, G.: Decision algorithms for fragments of real analysis. I. Continuous functions with strict convexity and concavity predicates. J. of Symbolic Computation 41(7), 763–789 (2006)
Cantone, D., Cutello, V., Ferro, A.: Decision procedures for elementary sublanguages of set theory. XIV. Three languages involving rank related constructs. In: Gianni, P. (ed.) ISSAC 1988. LNCS, vol. 358, pp. 407–422. Springer, Heidelberg (1989)
Cantone, D., Cutello, V., Schwartz, J.T.: Decision problems for Tarski’s and Presburger’s arithmetics extended with sets. In: Schönfeld, W., Börger, E., Kleine Büning, H., Richter, M.M. (eds.) CSL 1990. LNCS, vol. 533, pp. 95–109. Springer, Heidelberg (1991)
Cantone, D., Ferro, A.: Some recent decidability results in set theory. Atti degli incontri di Logica Matematica III, 383–387 (1985)
Cantone, D., Ferro, A., Omodeo, E.G.: Computable set theory, Vol.1. Oxford Science Publications of International Series of Monographs on Computer Science, vol. no.6. Clarendon Press (1989)
Cantone, D., Ferro, A., Omodeo, E.G., Policriti, A.: Scomposizione sillogistica disgiuntiva. In: Mello [78], pp. 199–209
Cantone, D., Ferro, A., Schwartz, J.T.: Decision procedures for elementary sublanguages of set theory. V. Multilevel syllogistic extended by the general union operator. J. of Computer and System Sciences 34(1), 1–18 (1987)
Cantone, D., Formisano, A., Omodeo, E.G., Schwartz, J.T.: Various commonly occurring decidable extensions of multi-level syllogistic. In: Ranise, S., Tinelli, C. (eds.) Pragmatics of Decision Procedures in Automated Reasoning, PDPAR 2003 (CADE-19), Electronic proceedings, Miami, USA (2003)
Cantone, D., Nicolosi Asmundo, M.: On the satisfiability problem for a 3-level quantified syllogistic. In: Complexity, Expressibility, and Decidability in Automated Reasoning – CEDAR 2008, Sydney, Australia, pp. 31–46 (2008)
Cantone, D., Omodeo, E.G.: On the decidability of formulae involving continuous and closed functions. In: Sridharan, N.S. (ed.) Proc. of the 11th International Joint Conference on Artificial Intelligence, pp. 425–430. Morgan Kaufmann, San Francisco (1989)
Cantone, D., Omodeo, E.G., Policriti, A.: Set Theory for Computing. From Decision Procedures to Declarative Programming with Sets. Monographs in Computer Science. Springer, Heidelberg (2001)
Cantone, D., Omodeo, E.G., Schwartz, J.T., Ursino, P.: Notes from the logbook of a proof-checker’s project. In: Dershowitz (ed.) [45], pp. 182–207
Cantone, D., Schwartz, J.T., Zarba, C.G.: Decision procedures for fragments of set theory with monotone and additive functions. In: Rossi, Jayaraman [106], pp. 1–8
Cantone, D., Ursino, P., Omodeo, E.G.: Formative processes with applications to the decision problem in set theory: I. Powerset and singleton operators. Inf. Comput. 172(2), 165–201 (2002); Appeared as Transitive Venn diagrams with applications to the decision problem in set theory. In: [79]
Cantone, D., Zarba, C.G.: A new fast tableau-based decision procedure for an unquantified fragment of set theory. In: Caferra, R., Salzer, G. (eds.) FTP 1998. LNCS (LNAI), vol. 1761, pp. 126–136. Springer, Heidelberg (2000)
Cantone, D., Zarba, C.G.: A tableau-based decision procedure for a fragment of set theory involving a restricted form of quantification. In: Murray, N.V. (ed.) TABLEAUX 1999. LNCS (LNAI), vol. 1617, pp. 97–112. Springer, Heidelberg (1999)
Caviness, B.F., Johnson, J.R.: Quantifier elimination and cylindrical algebraic decomposition. Texts and Monographs in Computer Science. Springer, Heidelberg (1998)
Chou, S.C.: Mechanical Geometry Theorem Proving. Reidel Publ. Comp., Dordrecht (1988)
Collins, G.E.: Quantifier elimination for real closed fields by cylindric algebra decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
D’Agostino, G., Omodeo, E.G., Schwartz, J.T., Tomescu, A.I.: Self-applied proof verification (Extended abstract). In: Cordón-Franco, A., Fernández-Margarit, A., Lara-Martin, F.F. (eds.) JAF, 26èmes Journées sur les Arithmétiques Faibles, pp. 113–117. Fénix Editora, Sevilla, Spain (2007), http://www.cs.us.es/glm/jaf26
Davis, M.: A program for Presburger’s algorithm. In: Summary of talks presented at the Summer Institute for Symbolic Logic, pp. 215–233. Cornell University (1957); In: [112]
Davis, M.: Eliminating the irrelevant from mechanical proofs. In: Proc. of Symposia in Applied Mathematics, vol. 15, pp. 15–30. AMS (1963); Reprinted in [112]
Davis, M.: The early history of automated deduction. In: Handbook of Automated Reasoning, pp. 3–13. Elsevier, Amsterdam (2001)
Davis, M., Putnam, H.: A computing procedure for quantification theory. J. of the ACM 7(3), 201–215 (1960)
Davis, M., Schwartz, J.T.: Correct-program technology / Extensibility of verifiers – Two papers on Program Verification with Appendix of Edith Deak. Technical Report No. NSO-12, Courant Institute of Mathematical Sciences, New York University (1977)
Davis, M., Schwartz, J.T.: Metatheoretic extensibility for theorem verifiers and proof-checkers. Computers and Mathematics with Applications 5, 217–230 (1979)
Dershowitz, N. (ed.): International symposium on verification (Theory and Practice) celebrating Zohar Manna’s 10000002 th birthday. LNCS, vol. 2772. Springer, Heidelberg (2003)
Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Semantics, vol. B, pp. 243–320. Elsevier and MIT Press (1990)
Dovier, A., Formisano, A., Omodeo, E.G.: Provable ∃ ∗ ∀-sentences about sets with atoms. In: Rossi, Jayaraman [106], pp. 9–17
Dovier, A., Formisano, A., Omodeo, E.G.: Decidability results for sets with atoms. ACM Transactions on Computational Logic 7(2), 269–301 (2006)
Dovier, A., Formisano, A., Policriti, A.: On T-logic programming. In: Falaschi, M., Navarro, M., Policriti, A. (eds.) Joint Conference on Declarative Programming, AGP 1997, Grado, Italy, June 16-19, pp. 457–466 (1997)
Dovier, A., Formisano, A., Policriti, A.: On T-logic programming. In: Proc. of ILPS 1997, pp. 323–337 (1997); A preliminary version appeared in [49]
Dovier, A., Omodeo, E.G., Policriti, A.: Solvable set/hyperset contexts: II. A goal-driven unification algorithm for the blended case. Appl. Algebra Eng. Commun. Comput. 9(4), 293–332 (1999)
Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: {log}: A logic programming language with finite sets. In: Furukawa, K. (ed.) ICLP 1991, pp. 111–124. MIT Press, Cambridge (1991)
Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: {log}: A logic programming language with finite sets. In: Asirelli, P. (ed.) Sesto convegno nazionale di programmazione logica, GULP 1991, Pisa, pp. 241–355 (1991)
Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: Embedding finite sets in a logic programming language. In: Lamma, E., Mello, P. (eds.) ELP 1992. LNCS (LNAI), vol. 660, pp. 150–167. Springer, Heidelberg (1993)
Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: A language for programming in logic with finite sets. J. of Logic Programming 28(1), 1–44 (1996); See also [52, 54, 53]
Dovier, A., Piazza, C., Rossi, G.: Narrowing the gap between set-constraints and CLP(SET)-constraints. In: Freire-Nistal, J.L., Falaschi, M., Ferro, M.V. (eds.) Joint Conference on Declarative Programming, AGP 1998, A Coruña, Spain, July 20-23, pp. 43–56 (1998)
Dovier, A., Pontelli, E., Rossi, G.: Set unification. Theory and Practice of Logic Programming 6(6), 645–701 (2006)
Downey, P.J.: Undecidability of Presburger arithmetic with a single monadic predicate letter. Technical Report 18-72, Harvard University Center for Research in Computing Technology (1972)
Dreben, B., Goldfarb, W.D.: The Decision Problem. Solvable classes of quantificational formulas. Addison-Wesley, Reading (1979)
Ershov, Y.L., Lavrov, I.A., Taimanov, A.D., Taitslin, M.A.: Elementary theories. Russ. Math. Survey 20, 35–106 (1965)
Farmer, W.M., Guttman, J.D., Thayer, F.J.: IMPS: An interactive mathematical proof system. J. Automated Reasoning 11, 213–248 (1993)
Ferro, A., Omodeo, E.G., Schwartz, J.T.: Decision procedures for some fragments of set theory. In: Bibel, W., Kowalski, R. (eds.) CADE 1980. LNCS, vol. 87, pp. 88–96. Springer, Heidelberg (1980)
Fisher, M.J., Rabin, M.O.: Super-exponential complexity of Presburger arithmetic. In: Complexity and computation, vol. VII, pp. 27–41. SIAM-AMS, Philadelphia (1974)
Formisano, A., Omodeo, E.G.: An equational re-engineering of set theories. In: Caferra, R., Salzer, G. (eds.) FTP 1998. LNCS (LNAI), vol. 1761, pp. 175–190. Springer, Heidelberg (2000)
Formisano, A., Omodeo, E.G., Temperini, M.: Instructing equational set-reasoning with Otter. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 152–167. Springer, Heidelberg (2001)
Formisano, A., Policriti, A.: T-resolution: Refinements and model elimination. J. Automated Reasoning 22(4), 433–483 (1999)
Geuvers, H.: Proof assistants: History, ideas and future. Sādhanā 34, 3–25 (2009)
Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decision procedures for extensions of the theory of arrays. Ann. Math. Artif. Intell. 50(3-4), 231–254 (2007)
Grigoriev, D.: Complexity of deciding Tarski algebra. J. of Symbolic Computation 5(1/2), 65–108 (1988)
Gurevich, Y.: Elementary properties of ordered Abelian groups. Translations of AMS 46, 165–192 (1965)
Jaffar, J., Maher, M.J.: Constraint logic programming: a survey. J. of Logic Programming (19/20), 503–581 (1994)
Kharlampovich, O., Myasnikov, A.: Elementary theory of free non-Abelian groups. J. of Algebra 302(2), 451–552 (2006)
Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–267. Pergamon Press, Oxford (1970)
Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer, Heidelberg (2008)
Kroening, D., Strichman, O.: Decision procedures: an algorithmic point of view. Texts in Theoretical Computer Science. Springer, Heidelberg (2008)
Loveland, D.W.: Automated theorem proving: A quarter century review. In: Bledsoe, W.W., Loveland, D.W. (eds.) Contemporary Mathematics: Automated Theorem Proving - After 25 Years, pp. 1–45. AMS (1984)
Manna, Z., Zarba, C.G.: Combining decision procedures. In: Aichernig, B.K., Maibaum, T. (eds.) Formal Methods at the Cross Roads: From Panacea to Foundational Support. LNCS, vol. 2757, pp. 381–422. Springer, Heidelberg (2003)
Mello, P. (ed.): Quarto convegno nazionale di programmazione logica. In: GULP 1989, Bologna (1989)
Meo, M.C., Vilares Ferro, M. (eds.): Joint Conference on Declarative Programming, AGP 1999, L’Aquila, Italy, September 6-9. GTE (1999)
Montagna, F., Mancini, A.: A minimal predicative set theory. Notre Dame J. of Formal Logic 35(2), 186–203 (1994)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Transaction on Programming Languages and Systems 1(2), 245–257 (1979)
Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. of the ACM 27(2), 356–364 (1980)
Nicolini, E., Ringeissen, C., Rusinowitch, M.: Satisfiability procedures for combination of theories sharing integer offsets. In: Kowalewski, S., Philippou, A. (eds.) TACAS-ETAPS 2009. LNCS, vol. 5505, pp. 428–442. Springer, Heidelberg (2009); Also in CILC 2009: 24-esimo Convegno Italiano di Logica Computazionale
Niemelä, I.: Stable models and difference logic. Ann. Math. Artif. Intell. 53(1-4), 313–329 (2008)
Omodeo, E.G.: The Linked Conjunct method for automatic deduction and related search techniques. Computers and Mathematics with Applications 8, 185–203 (1982)
Omodeo, E.G., Bossi, A., Sambin, G.: Tre possibili orientamenti per una programmazione dichiarativa basata sulla teoria degli insiemi. In: Demo, B. (ed.) Secondo convegno nazionale di programmazione logica, GULP 1987, Torino, pp. 265–276 (1987)
Omodeo, E.G., Cantone, D., Policriti, A., Schwartz, J.T.: A computerized Referee. In: Stock, O., Schaerf, M. (eds.) Reasoning, Action and Interaction in AI Theories and Systems. LNCS (LNAI), vol. 4155, pp. 117–139. Springer, Heidelberg (2006)
Omodeo, E.G., Parlamento, F., Policriti, A.: A derived algorithm for evaluating ε-expressions over abstract sets. J. of Symbolic Computation 15(5-6), 673–704 (1993)
Omodeo, E.G., Parlamento, F., Policriti, A.: Decidability of ∃ ∗ ∀-sentences in membership theories. Mathematical Logic Quarterly (formerly Zeitschrift für Mathematische Logik und Grundlagen der Mathematik) 42 (1996)
Omodeo, E.G., Policriti, A.: Solvable set/hyperset contexts: I. Some decision procedures for the pure, finite case. Comm. Pure Appl. Math. 48(9-10), 1123–1155 (1995); Special Issue in honor of J.T. Schwartz
Omodeo, E.G., Policriti, A.: The Bernays-Schönfinkel-Ramsey class for set theory: semidecidability. J. of Symbolic Logic (2010)
Omodeo, E.G., Policriti, A., Rossi, G.: Che genere di insiemi/multi-insiemi/iper-insiemi incorporare nella programazione logica? In: Saccà, D. (ed.) GULP 1993, pp. 55–70 (1993)
Omodeo, E.G., Schwartz, J.T.: A ‘Theory’ mechanism for a proof-verifier based on first-order set theory. In: Kakas, A.C., Sadri, F. (eds.) Computational Logic: Logic Programming and Beyond. LNCS (LNAI), vol. 2408, pp. 214–230. Springer, Heidelberg (2002)
Omodeo, E.G., Tomescu, A.I.: Using ÆtnaNova to formally prove that the Davis-Putnam satisfiability test is correct. Le Matematiche 63, 85–105 (2008); A preliminary version was presented at CILC 2007 (Messina)
Papadimitriou, C.: On the complexity of integer programming. J. of the ACM 28 (1981)
Parlamento, F., Policriti, A.: Decision procedures for elementary sublanguages of set theory. IX: Unsolvability of the decision problem for a restricted subclass of the Δ0-formulas in set theory. Comm. Pure Appl. Math. XLI, 221–251 (1988)
Parlamento, F., Policriti, A.: Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability. J. Automated Reasoning 7(2), 271–284 (1991)
Paulson, L.C.: Set Theory for Verification. II: Induction and Recursion. J. Automated Reasoning 15(2), 167–215 (1995)
Policriti, A., Schwartz, J.T.: T-theorem proving. I. J. of Symbolic Computation 20(3), 315–342 (1995)
Prawitz, D., Prawitz, H., Voghera, N.: A mechanical proof procedure and its realization in an electronic computer. J. of the ACM 7, 102–128 (1960); Reprinted in [112]
Presburger, M.: Über die vollständigkeit eines gewissen systems der aritmethik ganzer zahlen, in welchem die addition als einzige operation hervortritt. In: Comptes Rendus du premier Congrès des Mathématiciens des Pays slaves, Warsaw, pp. 92–101 (1929)
Quaife, A.: Automated Deduction in von Neumann-Bernays-Gödel Set Theory. J. Automated Reasoning 8(1), 91–147 (1992)
Renegar, J.: A faster PSPACE algorithm for deciding the existential theory of the reals. In: 29th Annual Symposium on Foundations of Computer Science (FOCS 1988), Los Angeles, Ca., USA, pp. 291–295. IEEE Computer Society Press, Los Alamitos (1988)
Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. of the ACM 12(1), 23–41 (1965); Reprinted in [112]
Robu, J.: Geometry Theorem Proving in the Frame of the Theorema Project. Technical Report 02-23, RISC Report Series, University of Linz, Austria. PhD Thesis (2002)
Rossi, G., Jayaraman, B. (eds.): Proc. of the Workshop on Declarative Programming with Sets, DPS 1999, Paris. Technical Report N. 200, Dipartimento di Matematica, Università di Parma, Italy (1999)
Schwartz, J.T.: Instantiation and decision procedures for certain classes of quantified set-theoretic formulae. Technical Report 78-10, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia (1978)
Schwartz, J.T., Dewar, R.K.B., Dubinsky, E., Schonberg, E.: Programming with sets: An introduction to SETL. Texts and Monographs in Computer Science. Springer, Heidelberg (1986)
Shankar, N., Rueß, H.: Combining Shostak theories. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 1–18. Springer, Heidelberg (2002)
Shostak, R.E.: A practical decision procedure for arithmetic with function symbols. J. of the ACM 26(2), 351–360 (1979)
Shostak, R.E.: Deciding combinations of theories. J. of the ACM 31, 1–12 (1984)
Siekmann, J., Wrightson, G.: Automation of Reasoning I and II. Springer, Heidelberg (1983)
Siekmann, J.H.: Unification theory. J. of Symbolic Computation 7(3-4), 207–274 (1989)
Sigal, R.: Desiderata for logic programming with sets. In: Mello [78], pp. 127–141
Stickel, M.E.: Automated deduction by theory resolution. J. Automated Reasoning 1(4), 333–355 (1985)
Szmielew, W.: Elementary properties of Abelian groups. Fundamenta Mathematicae 41, 203–271 (1954)
Tarski, A.: Sur les ensembles fini. Fundamenta Mathematicae VI, 45–95 (1924)
Tarski, A.: A decision method for elementary algebra and geometry. Berkeley University Press (1951)
Tarski, A.: What is elementary geometry? In: Hintikka, J. (ed.) The philosophy of mathematics — Oxford readings in philosophy, pp. 164–175. Oxford University Press, Oxford (1969); First published in 1959
Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theoretical Computer Science 290(1), 291–353 (2003)
Tinelli, C., Zarba, C.G.: Combining nonstably infinite theories. J. Automated Reasoning 34(3), 209–238 (2005)
Vaught, R.L.: On a theorem of Cobham concerning undecidable theories. In: Nagel, E., Suppes, P., Tarski, A. (eds.) Proc. of the 1960 International Congress on Logic, Methodology, and Philosophy of Science, pp. 14–25. Stanford University Press (1962)
Wos, L.: The problem of finding an inference rule for set theory. J. Automated Reasoning 5(1), 93–95 (1989)
Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Scientia Sinica 21(2), 159–172 (1978); Also in Selected works of Wen-Tsün Wu. World Scientific Publishing, Singapore (2008)
Zarba, C.G.: Combining lists with integers. In: Goré, R., Leitsch, A., Nipkov, T. (eds.) International Joint Conference on Automated Reasoning, IJCAR 2001 (Short Papers), Technical Report DII 11/01, pp. 180–189. University of Siena, Italy (2001)
Zarba, C.G.: Combining multisets with integers. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, p. 363. Springer, Heidelberg (2002)
Zarba, C.G.: Combining sets with integers. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, p. 103. Springer, Heidelberg (2002)
Zarba, C.G.: A tableau calculus for combining non-disjoint theories. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 315–329. Springer, Heidelberg (2002)
Zarba, C.G.: Combining sets with elements. In: Dershowitz [45], pp. 762–782
Zarba, C.G.: Combining sets with cardinals. J. Automated Reasoning 34(1), 1–29 (2005)
Zarba, C.G., Cantone, D., Schwartz, J.T.: A decision procedure for a sublanguage of set theory involving monotone, additive, and multiplicative functions, I: The two-level case. J. Automated Reasoning 33(3-4), 251–269 (2004)
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Formisano, A., Omodeo, E.G. (2010). Theory-Specific Automated Reasoning. In: Dovier, A., Pontelli, E. (eds) A 25-Year Perspective on Logic Programming. Lecture Notes in Computer Science, vol 6125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14309-0_3
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