Higher-Order Termination: from Kruskal to Computability

Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination). In computer science, termination has always been an important issue for showing programs correct. In the early days of logic, strong normalization was usually shown by assigning ordinals to expressions in such a way that eliminating a cut would yield an expression with a smaller ordinal. In the early days of verification, computer scientists used similar ideas, interpreting the arguments of a program call by a natural number, such as their size. Showing the size of the arguments to decrease for each recursive call gives a termination proof of the program, which is however rather weak since it can only yield quite small ordinals. In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied the strong normalization property for that expression. The predicate being defined by induction on types, or even as a fixpoint, this method could yield much larger ordinals. Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi..

Using Representation Theorems for Proving Polynomials Non-negative

Proving polynomials non-negative when variables range on a subset of numbers (e.g., [0, +∞)) 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, +∞) 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, +∞). Matching a polynomial against a representation provides a way to transform universal sentences ∀x ∈ [0, +∞) P(x) ≥ 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 P´olya and Szeg¨o’s 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_4S2133Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving Termination Properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201–208. Springer, Heidelberg (2011)Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Berlin (2006)Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Communic. Soc. Math. de Kharkow 13(2), 1–2 (1912)Bernstein, S.: Sur la répresentation des polynômes positifs. Communic. Soc. Math. de Kharkow 14(2), 227–228 (1915)Borralleras, C., Lucas, S., Oliveras, A., Rodríguez, E., Rubio, A.: SAT Modulo Linear Arithmetic for Solving Polynomial Constraints. Journal of Automated Reasoning 48, 107–131 (2012)Boudaoud, F., Caruso, F., Roy, M.-F.: Certificates of Positivity in the Bernstein Basis. Discrete Computational Geometry 39, 639–655 (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. 103–126. American Mathematical Society (1995)Contejean, E., Marché, C., Tomás, A.-P., Urbain, X.: Mechanically proving termination using polynomial interpretations. Journal of Automated Reasoning 32(4), 315–355 (2006)Hilbert, D.: Über die Darstellung definiter Formen als Summe von Formenquadraten. Mathematische Annalen 32, 342–350 (1888)Hong, H., Jakuš, D.: Testing Positiveness of Polynomials. Journal of Automated Reasoning 21, 23–38 (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), 547–586 (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), 4677–4692 (2000)Powers, V., Wörmann, T.: An algorithm for sums of squares of real polynomials. Journal of Pure and Applied Algebra 127, 99–104 (1998

Speeding up the constraint-based method in difference logic

"The final publication is available at http://link.springer.com/chapter/10.1007%2F978-3-319-40970-2_18"Over the years the constraint-based method has been successfully applied to a wide range of problems in program analysis, from invariant generation to termination and non-termination proving. Quite often the semantics of the program under study as well as the properties to be generated belong to difference logic, i.e., the fragment of linear arithmetic where atoms are inequalities of the form u v = k. However, so far constraint-based techniques have not exploited this fact: in general, Farkas’ Lemma is used to produce the constraints over template unknowns, which leads to non-linear SMT problems. Based on classical results of graph theory, in this paper we propose new encodings for generating these constraints when program semantics and templates belong to difference logic. Thanks to this approach, instead of a heavyweight non-linear arithmetic solver, a much cheaper SMT solver for difference logic or linear integer arithmetic can be employed for solving the resulting constraints. We present encouraging experimental results that show the high impact of the proposed techniques on the performance of the VeryMax verification systemPeer ReviewedPostprint (author's final draft

Satisfiability Checking and Symbolic Computation

Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly connected. We introduce a new project SC-square to build a joint community in this area, supported by a newly accepted EU (H2020-FETOPEN-CSA) project of the same name. We aim to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap. This abstract and accompanying poster describes the motivation and aims for the project, and reports on the first activities.Comment: 3 page Extended Abstract to accompany an ISSAC 2016 poster. Poster available at http://www.sc-square.org/SC2-AnnouncementPoster.pd

Heterozygous deletion of the Williams-Beuren syndrome critical interval in mice recapitulates most features of the human disorder

Williams-Beuren syndrome is a developmental multisystemic disorder caused by a recurrent 1.55-1.83 Mb heterozygous deletion on human chromosome band 7q11.23. Through chromosomal engineering with the cre-loxP system, we have generated mice with an almost complete deletion (CD) of the conserved syntenic region on chromosome 5G2. Heterozygous CD mice were viable, fertile and had a normal lifespan, while homozygotes were early embryonic lethal. Transcript levels of most deleted genes were reduced 50% in several tissues, consistent with gene dosage. Heterozygous mutant mice showed postnatal growth delay with reduced body weight and craniofacial abnormalities such as small mandible. The cardiovascular phenotype was only manifested with borderline hypertension, mildly increased arterial wall thickness and cardiac hypertrophy. The neurobehavioral phenotype revealed impairments in motor coordination, increased startle response to acoustic stimuli and hypersociability. Mutant mice showed a general reduction in brain weight. Cellular and histological abnormalities were present in the amygdala, cortex and hippocampus, including increased proportion of immature neurons. In summary, these mice recapitulate most crucial phenotypes of the human disorder, provide novel insights into the pathophysiological mechanisms of the disease such as the neural substrates of the behavioral manifestations, and will be valuable to evaluate novel therapeutic approaches.This work was supported by the Spanish Ministry of Ecomomy and Competitivity to V.C. (grant SAF2012-40036) and to L.P.J. (FIS PM002512 and SAF2004-06382), the European AnEuploidy project to L.P.J., M.D. and Y.H. The Rare Diseases CIBER (CIBERER) Fellowship supported M.S-P. and C.B

CIBERER : Spanish national network for research on rare diseases: A highly productive collaborative initiative

Altres ajuts: Instituto de Salud Carlos III (ISCIII); Ministerio de Ciencia e Innovación.CIBER (Center for Biomedical Network Research; Centro de Investigación Biomédica En Red) is a public national consortium created in 2006 under the umbrella of the Spanish National Institute of Health Carlos III (ISCIII). This innovative research structure comprises 11 different specific areas dedicated to the main public health priorities in the National Health System. CIBERER, the thematic area of CIBER focused on rare diseases (RDs) currently consists of 75 research groups belonging to universities, research centers, and hospitals of the entire country. CIBERER's mission is to be a center prioritizing and favoring collaboration and cooperation between biomedical and clinical research groups, with special emphasis on the aspects of genetic, molecular, biochemical, and cellular research of RDs. This research is the basis for providing new tools for the diagnosis and therapy of low-prevalence diseases, in line with the International Rare Diseases Research Consortium (IRDiRC) objectives, thus favoring translational research between the scientific environment of the laboratory and the clinical setting of health centers. In this article, we intend to review CIBERER's 15-year journey and summarize the main results obtained in terms of internationalization, scientific production, contributions toward the discovery of new therapies and novel genes associated to diseases, cooperation with patients' associations and many other topics related to RD research

Tsukuba termination tool

We present a tool for automatically proving termination of first-order rewrite systems. The tool is based on the dependency pair method of Arts and Giesl [1]. It incorporates several new ideas that make the method more efficient. The tool produces high-quality output and has a convenient web interface. If T T T succeeds in proving termination, it outputs a proof script which explains in considerable detail how termination was proved. This script is available in both HTML and L ATEX format. In the latter, the approximated dependency graph is visualized using the dot tool of the Graphviz toolkit. T T T is written in Objective Caml. We tested the various options of T T T on numerous examples. The results, as well as a comparison with other tools that implement the dependency pair method and some implementation details, can be found in [2, 3]. We describe some of the features of the tool (T T T in the sequel) by means of its web interface, displayed in Fig. 1. TRS The user inputs a TRS by typing the rules into the upper right text area or by uploading a file via the browse button. The exact input format is obtained by clicking the TRS link. Comment and Bibtex Anything typed into the upper right text area will appear as a footnote in the generated L ATEX code. This is useful to identify TRSs. L ATEX \cite commands may be included. In order for this to work correctly, a corresponding bibtex entry should be supplied. This can be done by typing the entry into the appropriate text area or by uploading an appropriate bibtex file via the browse button. Base Order The current version of T T T supports the following three base orders: LPO with strict precedence, LPO with quasi-precedence, and KBO with strict precedence. Dependency Pairs T T T supports the basic features of the dependency pair technique (argument filtering, dependency graph, cycle analysis) described below. Advanced features like narrowing, rewriting, and instantiation are not yet available. Also innermost termination analysis is not yet implemented

A Monotonic Higher-Order Semantic Path Ordering

. There is an increasing use of (first- and higher-order) rewrite rules in many programming languages and logical systems. The recursive path ordering (RPO) is a well-known tool for proving termination of such rewrite rules in the first-order case [Der82]. However, RPO has some weaknesses. For instance, since it is a simplification ordering, it can only handle simply terminating systems. Several techniques have been developed for overcoming these weaknesses of RPO. A very recent such technique is the monotonic semantic path ordering (MSPO) [BFR00], a simple and easily automatizable ordering which generalizes other more ad-hoc methods. Another recent extension of RPO is its higher-order version HORPO [JR99]. HORPO is an ordering on terms of a typed lambda-calculus generated by a signature of higher-order function symbols. Althgough many interesting examples can be proved terminating using HORPO, it inherits the weaknesses of the first-order RPO. Therefore, there is an obvious need for higher-order termination orderings without these weaknesses. Here we define the first such ordering, the monotonic higher-order semantic path ordering (MHOSPO), which is still automatizable like MSPO. We give evidence of its power by means of several natural and non-trivial examples which cannot be handled by HORPO.