731 research outputs found
Models for logics and conditional constraints in automated proofs of termination
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-13770-4_3Reasoning about termination of declarative programs, which are described by means of a computational logic, requires the definition of appropriate abstractions as semantic models of the logic, and also handling the conditional constraints which are often obtained. The formal treatment of such constraints in automated proofs, often using numeric interpretations and (arithmetic) constraint solving can greatly benefit from appropriate techniques to deal with the conditional (in)equations at stake. Existing results from linear algebra or real algebraic geometry are useful to deal with them but have received only scant attention to date. We investigate the definition and use of numeric models for logics and the resolution of linear and algebraic conditional constraints as unifying techniques for proving termination of declarative programs.Developed during a sabbatical year at UIUC. Supported by projects NSF CNS13-19109, MINECO TIN2010-21062-C02-02 and TIN2013-45732-C4-1-P, and GV BEST/2014/026 and PROMETEO/2011/052.Lucas Alba, S.; Meseguer, J. (2014). Models for logics and conditional constraints in automated proofs of termination. En Artificial Intelligence and Symbolic Computation. Springer Verlag (Germany). 9-20. https://doi.org/10.1007/978-3-319-13770-4_3S920AlarcĂłn, B., GutiĂ©rrez, R., Lucas, S., Navarro-Marset, R.: Proving Termination Properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201â208. Springer, Heidelberg (2011)AlarcĂłn, B., Lucas, S., Navarro-Marset, R.: Using Matrix Interpretations over the Reals in Proofs of Termination. In: Proc. of PROLE 2009, pp. 255â264 (2009)Clavel, M., DurĂĄn, F., Eker, S., Lincoln, P., MartĂ-Oliet, N., Meseguer, J., Talcott, C. (eds.): All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)Contejean, E., MarchĂ©, C., TomĂĄs, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. J. of Aut. Reas. 34(4), 325â363 (2006)Endrullis, J., Waldmann, J., Zantema, H.: Matrix Interpretations for Proving Termination of Term Rewriting. J. of Aut. Reas. 40(2-3), 195â220 (2008)Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal Termination. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 110â125. Springer, Heidelberg (2008)Futatsugi, K., Diaconescu, R.: CafeOBJ Report. AMAST Series. World Scientific (1998)Hudak, P., Peyton-Jones, S.J., Wadler, P.: Report on the Functional Programming Language Haskell: a nonâstrict, purely functional language. Sigplan Notices 27(5), 1â164 (1992)Lucas, S.: Context-sensitive computations in functional and functional logic programs. Journal of Functional and Logic Programming 1998(1), 1â61 (1998)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theoretical Informatics and Applications 39(3), 547â586 (2005)Lucas, S., MarchĂ©, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Information Processing Letters 95, 446â453 (2005)Lucas, S., Meseguer, J.: Proving Operational Termination of Declarative Programs in General Logics. In: Proc. of PPDP 2014, pp. 111â122. ACM Digital Library (2014)Lucas, S., Meseguer, J.: 2D Dependency Pairs for Proving Operational Termination of CTRSs. In: Proc. of WRLA 2014. LNCS, vol. 8663 (to appear, 2014)Lucas, S., Meseguer, J., GutiĂ©rrez, R.: Extending the 2D DP Framework for CTRSs. In: Selected papers of LOPSTR 2014. LNCS (to appear, 2015)Meseguer, J.: General Logics. In: Ebbinghaus, H.-D., et al. (eds.) Logic Colloquium 1987, pp. 275â329. North-Holland (1989)Nguyen, M.T., de Schreye, D., Giesl, J., Schneider-Kamp, P.: Polytool: Polynomial interpretations as a basis for termination of logic programs. Theory and Practice of Logic Programming 11(1), 33â63 (2011)Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer (April 2002)Prestel, A., Delzell, C.N.: Positive Polynomials. In: From Hilbertâs 17th Problem to Real Algebra. Springer, Berlin (2001)Podelski, A., Rybalchenko, A.: A Complete Method for the Synthesis of Linear Ranking Functions. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 239â251. Springer, Heidelberg (2004)Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons (1986)Zantema, H.: Termination of Context-Sensitive Rewriting. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 172â186. Springer, Heidelberg (1997
Polytool: polynomial interpretations as a basis for termination analysis of Logic programs
Our goal is to study the feasibility of porting termination analysis
techniques developed for one programming paradigm to another paradigm. In this
paper, we show how to adapt termination analysis techniques based on polynomial
interpretations - very well known in the context of term rewrite systems (TRSs)
- to obtain new (non-transformational) ter- mination analysis techniques for
definite logic programs (LPs). This leads to an approach that can be seen as a
direct generalization of the traditional techniques in termination analysis of
LPs, where linear norms and level mappings are used. Our extension general-
izes these to arbitrary polynomials. We extend a number of standard concepts
and results on termination analysis to the context of polynomial
interpretations. We also propose a constraint-based approach for automatically
generating polynomial interpretations that satisfy the termination conditions.
Based on this approach, we implemented a new tool, called Polytool, for
automatic termination analysis of LPs
CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates
Termination is an important property of programs; notably required for
programs formulated in proof assistants. It is a very active subject of
research in the Turing-complete formalism of term rewriting systems, where many
methods and tools have been developed over the years to address this problem.
Ensuring reliability of those tools is therefore an important issue. In this
paper we present a library formalizing important results of the theory of
well-founded (rewrite) relations in the proof assistant Coq. We also present
its application to the automated verification of termination certificates, as
produced by termination tools
Polynomial Interpretations over the Natural, Rational and Real Numbers Revisited
Polynomial interpretations are a useful technique for proving termination of
term rewrite systems. They come in various flavors: polynomial interpretations
with real, rational and integer coefficients. As to their relationship with
respect to termination proving power, Lucas managed to prove in 2006 that there
are rewrite systems that can be shown polynomially terminating by polynomial
interpretations with real (algebraic) coefficients, but cannot be shown
polynomially terminating using polynomials with rational coefficients only. He
also proved the corresponding statement regarding the use of rational
coefficients versus integer coefficients. In this article we extend these
results, thereby giving the full picture of the relationship between the
aforementioned variants of polynomial interpretations. In particular, we show
that polynomial interpretations with real or rational coefficients do not
subsume polynomial interpretations with integer coefficients. Our results hold
also for incremental termination proofs with polynomial interpretations.Comment: 28 pages; special issue of RTA 201
Automated verification of termination certificates
In order to increase user confidence, many automated theorem provers provide
certificates that can be independently verified. In this paper, we report on
our progress in developing a standalone tool for checking the correctness of
certificates for the termination of term rewrite systems, and formally proving
its correctness in the proof assistant Coq. To this end, we use the extraction
mechanism of Coq and the library on rewriting theory and termination called
CoLoR
Polynomial Interpretations for Higher-Order Rewriting
The termination method of weakly monotonic algebras, which has been defined
for higher-order rewriting in the HRS formalism, offers a lot of power, but has
seen little use in recent years. We adapt and extend this method to the
alternative formalism of algebraic functional systems, where the simply-typed
lambda-calculus is combined with algebraic reduction. Using this theory, we
define higher-order polynomial interpretations, and show how the implementation
challenges of this technique can be tackled. A full implementation is provided
in the termination tool WANDA
Synthesis of sup-interpretations: a survey
In this paper, we survey the complexity of distinct methods that allow the
programmer to synthesize a sup-interpretation, a function providing an upper-
bound on the size of the output values computed by a program. It consists in a
static space analysis tool without consideration of the time consumption.
Although clearly related, sup-interpretation is independent from termination
since it only provides an upper bound on the terminating computations. First,
we study some undecidable properties of sup-interpretations from a theoretical
point of view. Next, we fix term rewriting systems as our computational model
and we show that a sup-interpretation can be obtained through the use of a
well-known termination technique, the polynomial interpretations. The drawback
is that such a method only applies to total functions (strongly normalizing
programs). To overcome this problem we also study sup-interpretations through
the notion of quasi-interpretation. Quasi-interpretations also suffer from a
drawback that lies in the subterm property. This property drastically restricts
the shape of the considered functions. Again we overcome this problem by
introducing a new notion of interpretations mainly based on the dependency
pairs method. We study the decidability and complexity of the
sup-interpretation synthesis problem for all these three tools over sets of
polynomials. Finally, we take benefit of some previous works on termination and
runtime complexity to infer sup-interpretations.Comment: (2012
12th International Workshop on Termination (WST 2012) : WST 2012, February 19â23, 2012, Obergurgl, Austria / ed. by Georg Moser
This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19â23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
Using Representation Theorems for Proving Polynomials Non-negative
Proving polynomials non-negative when variables range on a
subset of numbers (e.g., [0, +â)) is often required in many applications
(e.g., in the analysis of program termination). Several representations for
univariate polynomials P that are non-negative on [0, +â) have been
investigated. They can often be used to characterize the property, thus
providing a method for checking it by trying a match of P against the
representation. We introduce a new characterization based on viewing
polynomials P as vectors, and find the appropriate polynomial basis B
in which the non-negativeness of the coordinates [P]B representing P in
B witnesses that P is non-negative on [0, +â). Matching a polynomial
against a representation provides a way to transform universal sentences
âx â [0, +â) P(x) â„ 0 into a constraint solving problem which can be
solved by using efficient methods. We consider different approaches to
solve both kind of problems and provide a quantitative evaluation of
performance that points to an early result by PÂŽolya and Szegšoâs as an
appropriate basis for implementations in most cases.Lucas Alba, S. (2014). Using Representation Theorems for Proving Polynomials Non-negative. En Artificial Intelligence and Symbolic Computation: 12th International Conference, AISC 2014, Seville, Spain, December 11-13, 2014. Proceedings. Springer Verlag (Germany). 21-33. doi:10.1007/978-3-319-13770-4_4S2133AlarcĂłn, B., GutiĂ©rrez, R., Lucas, S., Navarro-Marset, R.: Proving Termination Properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201â208. Springer, Heidelberg (2011)Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Berlin (2006)Bernstein, S.: DĂ©monstration du thĂ©orĂšme de Weierstrass fondĂ©e sur le calcul des probabilitĂ©s. Communic. Soc. Math. de Kharkow 13(2), 1â2 (1912)Bernstein, S.: Sur la rĂ©presentation des polynĂŽmes positifs. Communic. Soc. Math. de Kharkow 14(2), 227â228 (1915)Borralleras, C., Lucas, S., Oliveras, A., RodrĂguez, E., Rubio, A.: SAT Modulo Linear Arithmetic for Solving Polynomial Constraints. Journal of Automated Reasoning 48, 107â131 (2012)Boudaoud, F., Caruso, F., Roy, M.-F.: Certificates of Positivity in the Bernstein Basis. Discrete Computational Geometry 39, 639â655 (2008)Choi, M.D., Lam, T.Y., Reznick, B.: Sums of squares of real polynomials. In: Proc. of the Symposium on Pure Mathematics, vol. 4, pp. 103â126. American Mathematical Society (1995)Contejean, E., MarchĂ©, C., TomĂĄs, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 32(4), 315â355 (2006)Hilbert, D.: Ăber die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen 32, 342â350 (1888)Hong, H., JakuĆĄ, D.: Testing Positiveness of Polynomials. Journal of Automated Reasoning 21, 23â38 (1998)Karlin, S., Studden, W.J.: Tchebycheff systems: with applications in analysis and statistics. Interscience, New York (1966)Lucas, S.: Polynomials over the reals in proofs of termination: from theory to practice. RAIRO Theoretical Informatics and Applications 39(3), 547â586 (2005)Polya, G., Szegö, G.: Problems and Theorems in Analysis II. Springer (1976)Powers, V., Reznick, B.: Polynomials that are positive on an interval. Transactions of the AMS 352(10), 4677â4692 (2000)Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Algebra 127, 99â104 (1998
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