2 research outputs found
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. GutiĂ©rrez also supported by Juan de la Cierva Fellowship JCI-2012-13528.Lucas Alba, S.; GutiĂ©rrez 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/s10817-017-9419-3S465501604AlarcĂłn, B., GutiĂ©rrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. In: Proceedings of AMASTâ10. LNCS, vol. 6486, pp. 201â208 (2011)AlarcĂłn, B., Lucas, S., Navarro-Marset, R.: Using matrix interpretations over the reals in proofs of termination. In: Proceedings of PROLEâ09, pp. 255â264 (2009)Albert, E., Genaim, S., GutiĂ©rrez, R.: A Transformational Approach to Resource Analysis with Typed-Norms. Revised Selected Papers from LOPSTRâ13. LNCS, vol. 8901, pp 38â53 (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(4â5), 635â650 (2015)de Angelis, E., Fioravante, F., Pettorossi, A., Proietti, M.: Semantics-based generation of verification conditions by program specialization. In: Proceedings of PPDPâ15, pp. 91â102. 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), 1â20 (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, 221â265 (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, 115â133 (1970)Bofill, M., Nieuwenhuis, R., Oliveras, A., RodrĂguez-Carbonell, E., Rubio, A.: The Barcelogic SMT Solver. In: Proceedings of CAVâ08. LNCS, vol. 5123, pp. 294â298 (2008)BjĂžrner, N., Gurfinkel, A., McMillan, K., Rybalchenko, A.: Horn-clause solvers for program verification. In: Fields of Logic and Computation IIâEssays Dedicated to Yuri Gurevich on the Occasion of His 75th Birthday. LNCS, vol. 9300, pp. 24â51 (2015)BjĂžrner, N., McMillan, K., Rybalchenko, A.: On solving universally quantified horn-clauses. In: Proceedings of SASâ13. LNCS vol. 7935, pp. 105â125 (2013)BjĂžrner, N., McMillan, K., Rybalchenko, A.: Program verification as satisfiability modulo theories. In: Proceedings of SMTâ12, EPiC Series in Computing, vol. 20, pp. 3â11 (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. 482â493 (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., RodrĂguez, E., Rubio, A.: SAT modulo linear arithmetic for solving polynomial constraints. J. Autom. Reason. 48, 107â131 (2012)BĂŒrckert, H.-J., Hollunder, B., Laux, A.: On Skolemization in constrained logics. Ann. Math. Artif. Intell. 18, 95â131 (1996)Burstall, R.M., Goguen, J.A.: Putting Theories together to make specifications. In: Proceedings of IJCAIâ77, pp. 1045â1058. William Kaufmann (1977)Caplain, M.: Finding invariant assertions for proving programs. In: Proceedings of the International Conference on Reliable Software, pp. 165â171. ACM Press, New York (1975)Chang, C.L., Lee, R.C.: Symbolic Logic and Mechanical Theorem Proving. Academic Press, Orlando (1973)Clavel, M., DurĂĄn, F., Eker, S., Lincoln, P., MartĂ-Oliet, N., Meseguer, J., Talcott, C.: All About MaudeâA 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. 84â87. AAAI Press, Menlo Park (1983)Contejean, E., MarchĂ©, C., TomĂĄs, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. J. Autom. Reason. 34(4), 325â363 (2006)Cooper, D.C.: Programs for mechanical program verification. Mach. Intell. 6, 43â59 (1971). Edinburgh University PressCooper, D.C.: Theorem proving in arithmetic without multiplication. Mach. Intell. 7, 91â99 (1972)Courtieu, P., Gbedo, G., Pons, O.: Improved matrix interpretations. In: Proceedings of SOFSEMâ10. LNCS, vol. 5901, pp. 283â295 (2010)Cousot, P., Cousot, R., Mauborgne, L.: Logical abstract domains and interpretations. In: The Future of Sofware Engineering, pp. 48â71. Springer, New York (2011)Cousot, P., Halbwachs, N.: Automatic Discovery of linear restraints among variables of a program. In: Conference Record of POPLâ78, pp. 84â96. 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), 97â147 (1972)van Emdem, M.H., Kowalski, R.A.: The semantics of predicate logic as a programming language. J. ACM 23(4), 733â742 (1976)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. In: Proceedings of IJCARâ06. LNCS, vol. 4130, pp. 574â588 (2006)Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40(2â3), 195â220 (2008)Floyd, R.W.: Assigning meanings to programs. Math. Asp. Comput. Sci. 19, 19â32 (1967)Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal termination. In: Proceedings of RTAâ08. LNCS, vol. 5117, pp. 110â125 (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, 133â160 (2011)Fuhs, C., Kop, C.: Polynomial interpretations for higher-order rewriting. In: Proceedings of RTAâ12. LIPIcs, vol. 15, pp. 176â192 (2012)Futatsugi, K., Diaconescu, R.: CafeOBJ Report. World Scientific, AMAST Series, (1998)Gaboardi, M., PĂ©choux, R.: On bounding space usage of streams using interpretation analysis. Sci. Comput. Program. 111, 395â425 (2015)Giesl, J., Mesnard, F., Rubio, A., Thiemann, R., Waldmann, J.: Termination competition (termCOMP 2015). In: Proceedings of CADEâ15. LNCS, vol. 9195, pp. 105â108 (2015)Giesl, J., Ströder, T., Schneider-Kamp, P., Emmes, F., Fuhs, C.: Symbolic evaluation graphs and term rewritingâa general methodology for analyzing logic programs. In: Proceedings of the PPDPâ12, pp. 1â12. 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 ALPâ92. LNCS, vol. 632, pp. 37â52 (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, 175â209 (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, 263â295 (1984)Goguen, J., Meseguer, J.: Models and equality for logical programming. In: Proceedings of TAPSOFTâ87. LNCS, vol. 250, pp. 1â22 (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. 80â149. Prentice Hall (1978)Goguen, J.A., Meseguer, J.: Remarks on remarks on many-sorted equational logic. Sigplan Notices 22(4), 41â48 (1987)Goguen, J., Meseguer, J.: Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoret. Comput. Sci. 105, 217â273 (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 PLDIâ12, pp. 405â416. ACM Press (2012)Gulwani, S., Tiwari, A.: Combining Abstract Interpreters. In: Proceedings of PLDIâ06, pp. 376â386. ACM Press (2006)Gurfinkel, A., Kahsai, T., Komuravelli, A., Navas, J.A.: The seahorn verification framework. In: Proceedings of CAVâ15, Part I. LNCS, vol. 9206, pp. 343â361 (2015)GutiĂ©rrez, R., Lucas, S., Reinoso, P.: A tool for the automatic generation of logical models of order-sorted first-order theories. In: Proceedings of PROLEâ16, pp. 215â230 (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), 331â353 (1976)Hayes, P.: A logic of actions. Mach. Intell. 6, 495â520 (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. 364â379 (2008)Hoare, C.A.R.: An axiomatic basis for computer programming. Commun. ACM 12(10), 576â583 (1969)Hodges, W.: Elementary Predicate Logic. Handbook of Philosophical Logic, vol. 1, pp. 1â131. 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 RTAâ01. LNCS, vol. 2051, pp. 108â121 (2001)Hofbauer, D.: Termination proofs for ground rewrite systems. interpretations and derivational complexity. Appl. Algebra Eng. Commun. Comput. 12, 21â38 (2001)Hofbauer, D., Lautemann, C.: Termination proofs and the length of derivations. In: Proceedings of RTAâ89. LNCS, vol. 355, pp. 167â177 (1989)Hull, T.E., Enright, W.H., Sedgwick, A.E.: The correctness of numerical algorithms. In: Proceedings of PAAPâ72, pp. 66â73 (1972)Igarashi, S., London, R.L., Luckham, D.: Automatic program verification I: a logical basis and its implementation. Acta Inform. 4, 145â182 (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. 47â56. (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. 91â99 (2004)Katz, S., Manna, Z.: Logical analysis of programs. Commun. ACM 19(4), 188â206 (1976)Langford, C.H.: Review: Ăber 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 ACMâ72, vol. 1, pp. 39â46. ACM (1972)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theor. Inform. Appl. 39(3), 547â586 (2005)Lucas, S.: Synthesis of models for order-sorted first-order theories using linear algebra and constraint solving. Electron. Proc. Theor. Comput. Sci. 200, 32â47 (2015)Lucas, S.: Use of logical models for proving operational termination in general logics. In: Selected Papers from WRLAâ16. LNCS, vol. 9942, pp. 1â21 (2016)Lucas, S., MarchĂ©, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inform. Proces. Lett. 95, 446â453 (2005)Lucas, S., Meseguer, J.: Models for logics and conditional constraints in automated proofs of termination. In: Proceedings of AISCâ14. LNAI, vol. 8884, pp. 7â18 (2014)Lucas, S., Meseguer, J.: Order-sorted dependency pairs. In: Proceedings of PPDPâ08 , pp. 108â119. ACM Press (2008)Lucas, S., Meseguer, J.: Proving operational termination of declarative programs in general logics. In: Proceedings of PPDPâ14, pp. 111â122. 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, 236â268 (2017)Manna, Z.: The correctness of programs. J. Comput. Syst. Sci. 3, 119â127 (1969)Manna, Z.: Properties of programs and the first-order predicate calculus. J. ACM 16(2), 244â255 (1969)Manna, Z.: Termination of programs represented as interpreted graphs. In: Proceedings of AFIPSâ70, pp. 83â89 (1970)Manna, Z., Ness, S.: On the termination of Markov algorithms. In: Proceedings of the Third Hawaii International Conference on System Science, pp. 789â792 (1970)Manna, Z., Pnueli, A.: Formalization of properties of functional programs. J. ACM 17(3), 555â569 (1970)Marion, Y.-I., PĂ©choux, R.: Sup-interpretations, a semantic method for static analysis of program resources. ACM Trans. Comput. Log. 10(4), 27 (2009)MartĂ-Oliet, N., Meseguer, J., Palomino, M.: Theoroidal maps as algebraic simulations. Revised Selected Papers from WADTâ04. LNCS, vol. 3423, pp. 126â143 (2005)McCarthy, J.: Recursive functions of symbolic expressions and their computation by machine. Part I. Commun. ACM 3(4), 184â195 (1960)Meseguer, J.: General logics. In: Ebbinghaus, H.D., et al. (eds.) Logic Colloquiumâ87, pp. 275â329. North-Holland (1989)Meseguer, J., Skeirik, S.: Equational formulas and pattern operations in initial order-sorted algebras. Revised Selected Papers from LOPSTRâ15. LNCS, vol. 9527, pp. 36â53 (2015)Middeldorp, A.: Matrix interpretations for polynomial derivational complexity of rewrite systems. In: Proceedings of LPARâ12. 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, 426â457 (2015)de Moura, L., BjĂžrner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69â77 (2011)Naur, P.: Proof of algorithms by general snapshots. Bit 6, 310â316 (1966)Neurauter, F., Middeldorp, A.: Revisiting matrix interpretations for proving termination of term rewriting. In: Proceedings of RTAâ11. LIPICS, vol. 10, pp. 251â266 (2011)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, New York (2002)Ălveczky, P.C., Lysne, O.: Order-sorted termination: the unsorted way. In: Proceedings of ALPâ96. LNCS, vol. 1139, pp. 92â106 (1996)Otto, C., Brockschmidt, M., von Essen, C., Giesl, J.: Automated termination analysis of java bytecode by term rewriting. In: Proceedings of RTAâ10. LIPICS, vol. 6, pp. 259â276 (2010)PĂ©choux, R.: Synthesis of sup-interpretations: a survey. Theoret. Comput. Sci. 467, 30â52 (2013)Podelski, A., Rybalchenko, A.: Transition invariants. In: IEEE Computer Society Proceedings of LICSâ04, pp. 32â41 (2004)Prestel, A., Delzell, C.N.: Positive Polynomials. From Hilbertâs 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)RĂŒmmer, P., Hojjat, H., Kuncak, V.: Disjunctive interpolants for horn-clause verification. In: Proceedings of CAVâ13, vol. 8044, pp. 347â363 (2013)Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Amsterdam (1986)Schmidt, A.: Ăber deduktive Theorien mit mehreren Sorten von Grunddingen. Matematische Annalen 115(4), 485â506 (1938)Schmidt-Schauss, M.: Computational Aspects Of An Order-Sorted Logic With Term Declarations. PhD Thesis, Fachbereich Informatik der UniversitĂ€t 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), 351â360 (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, 141â143 (1987)Turing, A.M.: Checking a large routine. In: Report of a Conference on High Speed Automatic Calculating Machines, University Mathematics Laboratory, Cambridge, pp. 67â69 (1949)Urban, C.: The abstract domain of segmented ranking functions. In: Proceeding of SASâ13. LNCS, vol. 7935, pp. 43â62 (2013)Urban, C., Gurfinkel, A., Kahsai, T.: Synthesizing ranking functions from bits and pieces. In: Proceedings of TACASâ16. LNCS, vol. 9636, pp. 54â70 (2016)Waldmann, J.: Matrix interpretations on polyhedral domains. In: Proceedings of RTAâ15. LIPICS, vol. 26, pp. 318â333 (2015)Waldmann, J., Bau, A., Noeth, E.: Matchbox termination prover. http://github.com/jwaldmann/matchbox/ (2014)Walther, C.: A mechanical solution of schubertâs steamroller by many-sorted resolution. Aritif. Intell. 26, 217â224 (1985)Wang, H.: Logic of many-sorted theories. J. Symb. Logic 17(2), 105â116 (1952)Zantema, H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17, 23â50 (1994
Order-Sorted Equational Computation
The expressive power of many-sorted equational logic can be greatly enhanced by allowing for subsorts and multiple function declarations. In this paper we study some computational aspects of such a logic. We start with a self-contained introduction to order-sorted equational logic including initial algebra semantics and deduction rules. We then present a theory of order-sorted term rewriting and show that the key results for unsorted rewriting extend to sort decreasing rewriting. We continue with a review of order-sorted uniïŹcation and prove the basic results.
In the second part of the paper we study hierarchical order-sorted speciïŹcations with strict partial functions. We deïŹne the appropriate homomorphisms for strict algebras and show that every strict algebra is base isomorphic to a strict algebra with at most one error element. For strict speciïŹcations, we show that their categories of strict algebras have initial objects. We validate our approach to partial functions by proving that completely deïŹned total functions can be deïŹned as partial without changing the initial algebra semantics. Finally, we provide decidable sufïŹcient criteria for the consistency and strictness of ground conïŹuent rewriting systems