Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories

[EN] In program analysis, the synthesis of models of logical theories representing the program semantics is often useful to prove program properties. We use order-sorted first- order logic as an appropriate framework to describe the semantics and properties of programs as given theories. Then we investigate the automatic synthesis of models for such theories. We use convex polytopic domains as a flexible approach to associate different domains to different sorts. We introduce a framework for the piecewise definition of functions and predicates. We develop its use with linear expressions (in a wide sense, including linear transformations represented as matrices) and inequalities to specify functions and predicates. In this way, algorithms and tools from linear algebra and arithmetic constraint solving (e.g., SMT) can be used as a backend for an efficient implementation.Partially supported by the EU (FEDER), projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/ 013. R. Gutiérrez also supported by Juan de la Cierva Fellowship JCI-2012-13528.Lucas Alba, S.; Gutiérrez Gil, R. (2018). Automatic Synthesis of Logical Models for Order-Sorted First-Order Theories. Journal of Automated Reasoning. 60(4):465-501. https://doi.org/10.1007/s10817-017-9419-3S465501604Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. In: Proceedings of AMAST’10. LNCS, vol. 6486, pp. 201–208 (2011)Alarcón, B., Lucas, S., Navarro-Marset, R.: Using matrix interpretations over the reals in proofs of termination. In: Proceedings of PROLE’09, pp. 255–264 (2009)Albert, E., Genaim, S., Gutiérrez, R.: A Transformational Approach to Resource Analysis with Typed-Norms. Revised Selected Papers from LOPSTR’13. LNCS, vol. 8901, pp 38–53 (2013)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Proving correctness of imperative programs by linearizing constrained Horn clauses. Theory Pract. Log. Program. 15(4–5), 635–650 (2015)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Semantics-based generation of verification conditions by program specialization. In: Proceedings of PPDP’15, pp. 91–102. ACM Press, New York (2015)Aoto, T.: Solution to the problem of zantema on a persistent property of term rewriting systems. J. Funct. Log. Program. 2001(11), 1–20 (2001)Barwise, J.: An Introduction to First-Order Logic. In: Barwise, J. (ed.) Handbook of Mathematical Logic. North-Holland, Amsterdam (1977)Barwise, J.: Axioms for Abstract Model Theory. Ann. Math. Log. 7, 221–265 (1974)Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)Birkhoff, G., Lipson, J.D.: Heterogeneous algebras. J. Comb. Theory 8, 115–133 (1970)Bofill, M., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., Rubio, A.: The Barcelogic SMT Solver. In: Proceedings of CAV’08. LNCS, vol. 5123, pp. 294–298 (2008)Bjørner, N., Gurfinkel, A., McMillan, K., Rybalchenko, A.: Horn-clause solvers for program verification. In: Fields of Logic and Computation II—Essays Dedicated to Yuri Gurevich on the Occasion of His 75th Birthday. LNCS, vol. 9300, pp. 24–51 (2015)Bjørner, N., McMillan, K., Rybalchenko, A.: On solving universally quantified horn-clauses. In: Proceedings of SAS’13. LNCS vol. 7935, pp. 105–125 (2013)Bjørner, N., McMillan, K., Rybalchenko, A.: Program verification as satisfiability modulo theories. In: Proceedings of SMT’12, EPiC Series in Computing, vol. 20, pp. 3–11 (2013)Bliss, G.A.: Algebraic Functions. Dover (2004)Bonfante, G., Marion, J.-Y., Moyen, J.-Y.: On Lexicographic Termination Ordering With Space Bound Certifications. Revised Papers from PSI 2001. LNCS, vol. 2244, pp. 482–493 (2001)Boolos, G.S., Burgess, J.P., Jeffrey, R.C.: Computability and Logic, 4th edn. Cambridge University Press, Cambridge (2002)Borralleras, C., Lucas, S., Oliveras, A., Rodríguez, E., Rubio, A.: SAT modulo linear arithmetic for solving polynomial constraints. J. Autom. Reason. 48, 107–131 (2012)Bürckert, H.-J., Hollunder, B., Laux, A.: On Skolemization in constrained logics. Ann. Math. Artif. Intell. 18, 95–131 (1996)Burstall, R.M., Goguen, J.A.: Putting Theories together to make specifications. In: Proceedings of IJCAI’77, pp. 1045–1058. William Kaufmann (1977)Caplain, M.: Finding invariant assertions for proving programs. In: Proceedings of the International Conference on Reliable Software, pp. 165–171. ACM Press, New York (1975)Chang, C.L., Lee, R.C.: Symbolic Logic and Mechanical Theorem Proving. Academic Press, Orlando (1973)Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude—A High-Performance Logical Framework. LNCS 4350, (2007)Cohn, A.G.: Improving the expressiveness of many sorted logic. In: Proceedings of the National Conference on Artificial Intelligence, pp. 84–87. AAAI Press, Menlo Park (1983)Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. J. Autom. Reason. 34(4), 325–363 (2006)Cooper, D.C.: Programs for mechanical program verification. Mach. Intell. 6, 43–59 (1971). Edinburgh University PressCooper, D.C.: Theorem proving in arithmetic without multiplication. Mach. Intell. 7, 91–99 (1972)Courtieu, P., Gbedo, G., Pons, O.: Improved matrix interpretations. In: Proceedings of SOFSEM’10. LNCS, vol. 5901, pp. 283–295 (2010)Cousot, P., Cousot, R., Mauborgne, L.: Logical abstract domains and interpretations. In: The Future of Sofware Engineering, pp. 48–71. Springer, New York (2011)Cousot, P., Halbwachs, N.: Automatic Discovery of linear restraints among variables of a program. In: Conference Record of POPL’78, pp. 84–96. ACM Press, New York (1978)Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)Elspas, B., Levitt, K.N., Waldinger, R.J., Waksman, A.: An assessment of techniques for proving program correctness. Comput. Surv. 4(2), 97–147 (1972)van Emdem, M.H., Kowalski, R.A.: The semantics of predicate logic as a programming language. J. ACM 23(4), 733–742 (1976)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. In: Proceedings of IJCAR’06. LNCS, vol. 4130, pp. 574–588 (2006)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40(2–3), 195–220 (2008)Floyd, R.W.: Assigning meanings to programs. Math. Asp. Comput. Sci. 19, 19–32 (1967)Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal termination. In: Proceedings of RTA’08. LNCS, vol. 5117, pp. 110–125 (2008)Fuhs, C., Giesl, J., Parting, M., Schneider-Kamp, P., Swiderski, S.: Proving termination by dependency pairs and inductive theorem proving. J. Autom. Reason. 47, 133–160 (2011)Fuhs, C., Kop, C.: Polynomial interpretations for higher-order rewriting. In: Proceedings of RTA’12. LIPIcs, vol. 15, pp. 176–192 (2012)Futatsugi, K., Diaconescu, R.: CafeOBJ Report. World Scientific, AMAST Series, (1998)Gaboardi, M., Péchoux, R.: On bounding space usage of streams using interpretation analysis. Sci. Comput. Program. 111, 395–425 (2015)Giesl, J., Mesnard, F., Rubio, A., Thiemann, R., Waldmann, J.: Termination competition (termCOMP 2015). In: Proceedings of CADE’15. LNCS, vol. 9195, pp. 105–108 (2015)Giesl, J., Ströder, T., Schneider-Kamp, P., Emmes, F., Fuhs, C.: Symbolic evaluation graphs and term rewriting—a general methodology for analyzing logic programs. In: Proceedings of the PPDP’12, pp. 1–12. ACM Press (2012)Giesl, J., Raffelsieper, M., Schneider-Kamp, P., Swiderski, S., Thiemann, R.: Automated termination proofs for haskell by term rewriting. ACM Trans. Program. Lang. Syst. 33(2), 7 (2011)Gnaedig, I.: Termination of Order-sorted Rewriting. In: Proceedings of ALP’92. LNCS, vol. 632, pp. 37–52 (1992)Goguen, J.A.: Order-Sorted Algebra. Semantics and Theory of Computation Report 14, UCLA (1978)Goguen, J.A., Burstall, R.M.: Some fundamental algebraic tools for the semantics of computation. Part 1: comma categories, colimits, signatures and theories. Theoret. Comput. Sci. 31, 175–209 (1984)Goguen, J.A., Burstall, R.M.: Some fundamental algebraic tools for the semantics of computation. Part 2 signed and abstract theories. Theoret. Comput. Sci. 31, 263–295 (1984)Goguen, J., Meseguer, J.: Models and equality for logical programming. In: Proceedings of TAPSOFT’87. LNCS, vol. 250, pp. 1–22 (1987)Goguen, J.A., Thatcher, J.W., Wagner, E.G.: An initial algebra approach to the specification, correctness and implementation of abstract data types. In: Current Trends in Programming Methodology, pp. 80–149. Prentice Hall (1978)Goguen, J.A., Meseguer, J.: Remarks on remarks on many-sorted equational logic. Sigplan Notices 22(4), 41–48 (1987)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)Goguen, J.A., Winkler, T., Meseguer, J., Futatsugi, K., Jouannaud, J.-P.: Introducing OBJ. In: Goguen, J., Malcolm, G. (eds.) Software Engineering with OBJ: Algebraic Specification in Action. Kluwer, Boston (2000)Grebenshikov, S., Lopes, N.P., Popeea, C., Rybalchenko, A.: Synthesizing software verifiers from proof rules. In: Proceedings of PLDI’12, pp. 405–416. ACM Press (2012)Gulwani, S., Tiwari, A.: Combining Abstract Interpreters. In: Proceedings of PLDI’06, pp. 376–386. ACM Press (2006)Gurfinkel, A., Kahsai, T., Komuravelli, A., Navas, J.A.: The seahorn verification framework. In: Proceedings of CAV’15, Part I. LNCS, vol. 9206, pp. 343–361 (2015)Gutiérrez, R., Lucas, S., Reinoso, P.: A tool for the automatic generation of logical models of order-sorted first-order theories. In: Proceedings of PROLE’16, pp. 215–230 (2016). http://zenon.dsic.upv.es/ages/Hantler, S.L., King, J.C.: An introduction to proving the correctness of programs. ACM Comput. Surv. 8(3), 331–353 (1976)Hayes, P.: A logic of actions. Mach. Intell. 6, 495–520 (1971). Edinburgh University Press, EdinburghHeidergott, B., Olsder, G.J., van der Woude, J.: Max plus at work. A course on max-plus algebra and its applications. In: Modeling and Analysis of Synchronized Systems, Princeton University Press (2006)Hirokawa, N., Moser, G.: Automated complexity analysis based on the dependency pair method. In: Proceedings of IJCAR 2008. LNCS, vol. 5195, pp. 364–379 (2008)Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576–583 (1969)Hodges, W.: Elementary Predicate Logic. Handbook of Philosophical Logic, vol. 1, pp. 1–131. Reidel Publishing Company (1983)Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)Hofbauer, D.: Termination proofs by context-dependent interpretation. In: Proceedings of RTA’01. LNCS, vol. 2051, pp. 108–121 (2001)Hofbauer, D.: Termination proofs for ground rewrite systems. interpretations and derivational complexity. Appl. Algebra Eng. Commun. Comput. 12, 21–38 (2001)Hofbauer, D., Lautemann, C.: Termination proofs and the length of derivations. In: Proceedings of RTA’89. LNCS, vol. 355, pp. 167–177 (1989)Hull, T.E., Enright, W.H., Sedgwick, A.E.: The correctness of numerical algorithms. In: Proceedings of PAAP’72, pp. 66–73 (1972)Igarashi, S., London, R.L., Luckham, D.: Automatic program verification I: a logical basis and its implementation. Acta Inform. 4, 145–182 (1975)Iwami, M.: Persistence of termination of term rewriting systems with ordered sorts. In: Proceedings of 5th JSSST Workshop on Programming and Programming Languages, Shizuoka, Japan, pp. 47–56. (2003)Iwami, M.: Persistence of termination for non-overlapping term rewriting systems. In: Proceedings of Algebraic Systems, Formal Languages and Conventional and Unconventional Computation Theory, Kokyuroku RIMS, University of Kyoto, vol. 1366, pp. 91–99 (2004)Katz, S., Manna, Z.: Logical analysis of programs. Commun. ACM 19(4), 188–206 (1976)Langford, C.H.: Review: Über deduktive Theorien mit mehreren Sorten von Grunddingen. J. Symb. Log. 4(2), 98 (1939)Lankford, D.S.: Some approaches to equality for computational logic: a survey and assessment. Memo ATP-36, Automatic Theorem Proving Project, University of Texas, Austin, TXLondon, R.L.: The current state of proving programs correct. In: Proceedings of ACM’72, vol. 1, pp. 39–46. ACM (1972)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theor. Inform. Appl. 39(3), 547–586 (2005)Lucas, S.: Synthesis of models for order-sorted first-order theories using linear algebra and constraint solving. Electron. Proc. Theor. Comput. Sci. 200, 32–47 (2015)Lucas, S.: Use of logical models for proving operational termination in general logics. In: Selected Papers from WRLA’16. LNCS, vol. 9942, pp. 1–21 (2016)Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inform. Proces. Lett. 95, 446–453 (2005)Lucas, S., Meseguer, J.: Models for logics and conditional constraints in automated proofs of termination. In: Proceedings of AISC’14. LNAI, vol. 8884, pp. 7–18 (2014)Lucas, S., Meseguer, J.: Order-sorted dependency pairs. In: Proceedings of PPDP’08 , pp. 108–119. ACM Press (2008)Lucas, S., Meseguer, J.: Proving operational termination of declarative programs in general logics. In: Proceedings of PPDP’14, pp. 111–122. ACM Digital Library (2014)Lucas, S., Meseguer, J.: Dependency pairs for proving termination properties of conditional term rewriting systems. J. Log. Algebr. Methods Program. 86, 236–268 (2017)Manna, Z.: The correctness of programs. J. Comput. Syst. Sci. 3, 119–127 (1969)Manna, Z.: Properties of programs and the first-order predicate calculus. J. ACM 16(2), 244–255 (1969)Manna, Z.: Termination of programs represented as interpreted graphs. In: Proceedings of AFIPS’70, pp. 83–89 (1970)Manna, Z., Ness, S.: On the termination of Markov algorithms. In: Proceedings of the Third Hawaii International Conference on System Science, pp. 789–792 (1970)Manna, Z., Pnueli, A.: Formalization of properties of functional programs. J. ACM 17(3), 555–569 (1970)Marion, Y.-I., Péchoux, R.: Sup-interpretations, a semantic method for static analysis of program resources. ACM Trans. Comput. Log. 10(4), 27 (2009)Martí-Oliet, N., Meseguer, J., Palomino, M.: Theoroidal maps as algebraic simulations. Revised Selected Papers from WADT’04. LNCS, vol. 3423, pp. 126–143 (2005)McCarthy, J.: Recursive functions of symbolic expressions and their computation by machine. Part I. Commun. ACM 3(4), 184–195 (1960)Meseguer, J.: General logics. In: Ebbinghaus, H.D., et al. (eds.) Logic Colloquium’87, pp. 275–329. North-Holland (1989)Meseguer, J., Skeirik, S.: Equational formulas and pattern operations in initial order-sorted algebras. Revised Selected Papers from LOPSTR’15. LNCS, vol. 9527, pp. 36–53 (2015)Middeldorp, A.: Matrix interpretations for polynomial derivational complexity of rewrite systems. In: Proceedings of LPAR’12. LNCS, vol. 7180, p. 12 (2012)Monin, J.-F.: Understanding Formal Methods. Springer, London (2003)Montenegro, M., Peña, R., Segura, C.: Space consumption analysis by abstract interpretation: inference of recursive functions. Sci. Comput. Program. 111, 426–457 (2015)de Moura, L., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69–77 (2011)Naur, P.: Proof of algorithms by general snapshots. Bit 6, 310–316 (1966)Neurauter, F., Middeldorp, A.: Revisiting matrix interpretations for proving termination of term rewriting. In: Proceedings of RTA’11. LIPICS, vol. 10, pp. 251–266 (2011)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, New York (2002)Ölveczky, P.C., Lysne, O.: Order-sorted termination: the unsorted way. In: Proceedings of ALP’96. LNCS, vol. 1139, pp. 92–106 (1996)Otto, C., Brockschmidt, M., von Essen, C., Giesl, J.: Automated termination analysis of java bytecode by term rewriting. In: Proceedings of RTA’10. LIPICS, vol. 6, pp. 259–276 (2010)Péchoux, R.: Synthesis of sup-interpretations: a survey. Theoret. Comput. Sci. 467, 30–52 (2013)Podelski, A., Rybalchenko, A.: Transition invariants. In: IEEE Computer Society Proceedings of LICS’04, pp. 32–41 (2004)Prestel, A., Delzell, C.N.: Positive Polynomials. From Hilbert’s 17th Problem to Real Algebra. Springer, Berlin (2001)Robinson, D.J.S.: A Course in Linear Algebra with Applications, 2nd edn. World Scientific Publishing, Co, Singapore (2006)Rümmer, P., Hojjat, H., Kuncak, V.: Disjunctive interpolants for horn-clause verification. In: Proceedings of CAV’13, vol. 8044, pp. 347–363 (2013)Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Amsterdam (1986)Schmidt, A.: Über deduktive Theorien mit mehreren Sorten von Grunddingen. Matematische Annalen 115(4), 485–506 (1938)Schmidt-Schauss, M.: Computational Aspects Of An Order-Sorted Logic With Term Declarations. PhD Thesis, Fachbereich Informatik der Universität Kaiserslautern (1988)Shapiro, S.: Foundations without Foundationalism: A Case for Second-Order Logic. Clarendon Press, New York (1991)Shostak, R.E.: A practical decision procedure for arithmetic with function symbols. J. ACM 26(2), 351–360 (1979)Smullyan, R.M.: Theory of Formal Systems. Princeton University Press, Princeton (1961)Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1951)Toyama, Y.: Counterexamples to termination for the direct sum of term rewriting systems. Inform. Process. Lett. 25, 141–143 (1987)Turing, A.M.: Checking a large routine. In: Report of a Conference on High Speed Automatic Calculating Machines, University Mathematics Laboratory, Cambridge, pp. 67–69 (1949)Urban, C.: The abstract domain of segmented ranking functions. In: Proceeding of SAS’13. LNCS, vol. 7935, pp. 43–62 (2013)Urban, C., Gurfinkel, A., Kahsai, T.: Synthesizing ranking functions from bits and pieces. In: Proceedings of TACAS’16. LNCS, vol. 9636, pp. 54–70 (2016)Waldmann, J.: Matrix interpretations on polyhedral domains. In: Proceedings of RTA’15. LIPICS, vol. 26, pp. 318–333 (2015)Waldmann, J., Bau, A., Noeth, E.: Matchbox termination prover. http://github.com/jwaldmann/matchbox/ (2014)Walther, C.: A mechanical solution of schubert’s steamroller by many-sorted resolution. Aritif. Intell. 26, 217–224 (1985)Wang, H.: Logic of many-sorted theories. J. Symb. Logic 17(2), 105–116 (1952)Zantema, H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17, 23–50 (1994

Proving Correctness and Completeness of Normal Programs - a Declarative Approach

We advocate a declarative approach to proving properties of logic programs. Total correctness can be separated into correctness, completeness and clean termination; the latter includes non-floundering. Only clean termination depends on the operational semantics, in particular on the selection rule. We show how to deal with correctness and completeness in a declarative way, treating programs only from the logical point of view. Specifications used in this approach are interpretations (or theories). We point out that specifications for correctness may differ from those for completeness, as usually there are answers which are neither considered erroneous nor required to be computed. We present proof methods for correctness and completeness for definite programs and generalize them to normal programs. For normal programs we use the 3-valued completion semantics; this is a standard semantics corresponding to negation as finite failure. The proof methods employ solely the classical 2-valued logic. We use a 2-valued characterization of the 3-valued completion semantics which may be of separate interest. The presented methods are compared with an approach based on operational semantics. We also employ the ideas of this work to generalize a known method of proving termination of normal programs.Comment: To appear in Theory and Practice of Logic Programming (TPLP). 44 page

Correctness and completeness of logic programs

We discuss proving correctness and completeness of definite clause logic programs. We propose a method for proving completeness, while for proving correctness we employ a method which should be well known but is often neglected. Also, we show how to prove completeness and correctness in the presence of SLD-tree pruning, and point out that approximate specifications simplify specifications and proofs. We compare the proof methods to declarative diagnosis (algorithmic debugging), showing that approximate specifications eliminate a major drawback of the latter. We argue that our proof methods reflect natural declarative thinking about programs, and that they can be used, formally or informally, in every-day programming.Comment: 29 pages, 2 figures; with editorial modifications, small corrections and extensions. arXiv admin note: text overlap with arXiv:1411.3015. Overlaps explained in "Related Work" (p. 21

Proving theorems by program transformation

In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming (CLP), and we present a set of transformation rules (including the familiar unfolding and folding rules) which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: (i) the proof of predicate equivalences, applied to the verification of equality between CCS processes, (ii) the proof of first order formulas via an extension of the quantifier elimination method, and (iii) the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization