A representation of a compressed de Bruijn graph for pan-genome analysis that enables search

Recently, Marcus et al. (Bioinformatics 2014) proposed to use a compressed de Bruijn graph to describe the relationship between the genomes of many individuals/strains of the same or closely related species. They devised an $O(n \log g)$ time algorithm called splitMEM that constructs this graph directly (i.e., without using the uncompressed de Bruijn graph) based on a suffix tree, where $n$ is the total length of the genomes and $g$ is the length of the longest genome. In this paper, we present a construction algorithm that outperforms their algorithm in theory and in practice. Moreover, we propose a new space-efficient representation of the compressed de Bruijn graph that adds the possibility to search for a pattern (e.g. an allele - a variant form of a gene) within the pan-genome.Comment: Submitted to Algorithmica special issue of CPM201

Edge minimization in de Bruijn graphs

This paper introduces the de Bruijn graph edge minimization problem, which is related to the compression of de Bruijn graphs: find the order-k de Bruijn graph with minimum edge count among all orders. We describe an efficient algorithm that solves this problem. Since the edge minimization problem is connected to the BWT compression technique called "tunneling", the paper also describes a way to minimize the length of a tunneled BWT in such a way that useful properties for sequence analysis are preserved. Although being a restriction, this is significant progress towards a solution to the open problem of finding optimal disjoint blocks that minimize space, as stated in Alanko et al. (DCC 2019).Comment: Accepted for Data Compression Conference 202

Lightweight Lempel-Ziv Parsing

We introduce a new approach to LZ77 factorization that uses O(n/d) words of working space and O(dn) time for any d >= 1 (for polylogarithmic alphabet sizes). We also describe carefully engineered implementations of alternative approaches to lightweight LZ77 factorization. Extensive experiments show that the new algorithm is superior in most cases, particularly at the lowest memory levels and for highly repetitive data. As a part of the algorithm, we describe new methods for computing matching statistics which may be of independent interest.Comment: 12 page

Efficient chaining of seeds in ordered trees

We consider here the problem of chaining seeds in ordered trees. Seeds are mappings between two trees Q and T and a chain is a subset of non overlapping seeds that is consistent with respect to postfix order and ancestrality. This problem is a natural extension of a similar problem for sequences, and has applications in computational biology, such as mining a database of RNA secondary structures. For the chaining problem with a set of m constant size seeds, we describe an algorithm with complexity O(m2 log(m)) in time and O(m2) in space

Conversion to Tail Recursion in Term Rewriting

NOTICE: this is the author’s version of a work that was accepted for publication in "Journal of Logic and Algebraic Programming". Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Logic and Algebraic Programming , vol. 83 issue 1, 2014 DOI 10.1016/j.jlap.2013.07.001Tail recursive functions are a special kind of recursive functions where the last action in their body is the recursive call. Tail recursion is important for a number of reasons (e.g., they are usually more efficient). In this article, we introduce an automatic transformation of first-order functions into tail recursive form. Functions are defined using a (first-order) term rewrite system. We prove the correctness of the transformation for constructor-based reduction over constructor systems (i.e., typical first-order functional programs). © 2013 Elsevier Inc. All rights reservedThis work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion (Secretaria de Estado de Investigation) under grant TIN2008-06622-C03-02, by the Generalitat Valenciana under grant PROMETEO/2011/052, and by MEXT KAKENHI #21700011.Nishida, N.; Vidal Oriola, GF. (2014). Conversion to Tail Recursion in Term Rewriting. Journal of Logic and Algebraic Programming. 83(1):53-63. https://doi.org/10.1016/j.jlap.2013.07.001S536383

Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over an extended signature. They are not sound w.r.t. reduction for every CTRS, while they are complete w.r.t. reduction. Here, soundness w.r.t. reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch's unraveling for a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled TRS is left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies that of Ohlebusch's unraveling. Finally, we show that soundness of Ohlebusch's unraveling is the weakest in soundness of the other unravelings and a transformation, proposed by Serbanuta and Rosu, for (normal) deterministic CTRSs, i.e., soundness of them respectively implies that of Ohlebusch's unraveling.Comment: 49 pages, 1 table, publication in Special Issue: Selected papers of the "22nd International Conference on Rewriting Techniques and Applications (RTA'11)

On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped \lambda-calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of M\"uller and Dougherty for unconditional rewriting. Two cases are considered, whether \beta-reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty's result is improved from the assumption of strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules

Use of logical models for proving infeasibility in term rewriting

[EN] Given a (Conditional) Rewrite System R and terms s and t, we consider the following problem: is there a substitution a instantiating the variables in s and t such that the reachability test sigma(s) -> *(R) sigma(t) succeeds? If such a substitution does not exist, we say that the problem is infeasible; otherwise, we call it feasible. Similarly, we can consider reducibility, involving a single rewriting step. In term rewriting, a number of important problems involve such infeasibility tests (e.g., confluence and termination analysis). We show how to recast infeasibility tests into the problem of finding a model of a set of (first-order) sentences representing the operational semantics of R together with some additional sentences representing the considered property which is formulated as an infeasibility test. (C) 2018 Elsevier B.V. All rights reserved.Partially supported by the EU (FEDER) which is co-funding the project TIN2015-69175-C4-1-R, Spanish MINECO project TIN2015-69175-C4-1-R and Generalitat Valenciana (GV) project PROMETEOII/2015/013.Lucas Alba, S.; Gutiérrez Gil, R. (2018). Use of logical models for proving infeasibility in term rewriting. Information Processing Letters. 136:90-95. https://doi.org/10.1016/j.ipl.2018.04.002S909513

Automatically Proving and Disproving Feasibility Conditions

[EN] In the realm of term rewriting, given terms s and t, a reachability condition s>>t is called feasible if there is a substitution O such that O(s) rewrites into O(t) in zero or more steps; otherwise, it is called infeasible. Checking infeasibility of (sequences of) reachability conditions is important in the analysis of computational properties of rewrite systems like confluence or (operational) termination. In this paper, we generalize this notion of feasibility to arbitrary n-ary relations on terms defined by first-order theories. In this way, properties of computational systems whose operational semantics can be given as a first-order theory can be investigated. We introduce a framework for proving feasibility/infeasibility, and a new tool, infChecker, which implements it.Supported by EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098, and SP20180225. Also by INCIBE program "Ayudas para la excelencia de los equipos de investigación avanzada en ciberseguridad" (Raul Gutiérrez).Gutiérrez Gil, R.; Lucas Alba, S. (2020). Automatically Proving and Disproving Feasibility Conditions. Springer Nature. 416-435. https://doi.org/10.1007/978-3-030-51054-1_27S416435Andrianarivelo, N., Réty, P.: Over-approximating terms reachable by context-sensitive rewriting. In: Bojańczyk, M., Lasota, S., Potapov, I. (eds.) RP 2015. LNCS, vol. 9328, pp. 128–139. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24537-9_12Dershowitz, N.: Termination of rewriting. J. Symb. Comput. 3(1/2), 69–116 (1987). https://doi.org/10.1016/S0747-7171(87)80022-6Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reasoning 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7Goguen, J.A., Meseguer, J.: Models and equality for logical programming. In: Ehrig, H., Kowalski, R., Levi, G., Montanari, U. (eds.) TAPSOFT 1987. LNCS, vol. 250, pp. 1–22. Springer, Heidelberg (1987). https://doi.org/10.1007/BFb0014969Gutiérrez, R., Lucas, S.: Automatic generation of logical models with AGES. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 287–299. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_17Kojima, Y., Sakai, M.: Innermost reachability and context sensitive reachability properties are decidable for linear right-shallow term rewriting systems. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 187–201. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70590-1_13Kojima, Y., Sakai, M., Nishida, N., Kusakari, K., Sakabe, T.: Context-sensitive innermost reachability is decidable for linear right-shallow term rewriting systems. Inf. Media Technol. 4(4), 802–814 (2009)Kojima, Y., Sakai, M., Nishida, N., Kusakari, K., Sakabe, T.: Decidability of reachability for right-shallow context-sensitive term rewriting systems. IPSJ Online Trans. 4, 192–216 (2011)Lucas, S.: Analysis of rewriting-based systems as first-order theories. In: Fioravanti, F., Gallagher, J.P. (eds.) LOPSTR 2017. LNCS, vol. 10855, pp. 180–197. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94460-9_11Lucas, S.: Context-sensitive computations in functional and functional logic programs. J. Funct. Logic Program. 1998(1) (1998). http://danae.uni-muenster.de/lehre/kuchen/JFLP/articles/1998/A98-01/A98-01.htmlLucas, S.: Proving semantic properties as first-order satisfiability. Artif. Intell. 277 (2019). https://doi.org/10.1016/j.artint.2019.103174Lucas, S.: Using well-founded relations for proving operational termination. J. Autom. Reasoning 64(2), 167–195 (2019). https://doi.org/10.1007/s10817-019-09514-2Lucas, S., Gutiérrez, R.: Use of logical models for proving infeasibility in term rewriting. Inf. Process. Lett. 136, 90–95 (2018). https://doi.org/10.1016/j.ipl.2018.04.002Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005). https://doi.org/10.1016/j.ipl.2005.05.002Lucas, S., Meseguer, J.: Proving operational termination of declarative programs in general logics. In: Chitil, O., King, A., Danvy, O. (eds.) Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming, Kent, Canterbury, United Kingdom, 8–10 September 2014, pp. 111–122. ACM (2014). https://doi.org/10.1145/2643135.2643152Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems. Part I: definition and basic processors. J. Comput. Syst. Sci. 96, 74–106 (2018). https://doi.org/10.1016/j.jcss.2018.04.002Lucas, S., Meseguer, J., Gutiérrez, R.: The 2D dependency pair framework for conditional rewrite systems—Part II: advanced processors and implementation techniques. J. Autom. Reasoning (2020, in press)McCune, W.: Prover9 and Mace4. https://www.cs.unm.edu/~mccune/mace4/Meßner, F., Sternagel, C.: nonreach – a tool for nonreachability analysis. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 337–343. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_19Middeldorp, A., Nagele, J., Shintani, K.: Confluence competition 2019. In: Beyer, D., Huisman, M., Kordon, F., Steffen, B. (eds.) TACAS 2019. LNCS, vol. 11429, pp. 25–40. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17502-3_2Nishida, N., Maeda, Y.: Narrowing trees for syntactically deterministic conditional term rewriting systems. In: Kirchner, H. (ed.) Proceedings of the 3rd International Conference on Formal Structures for Computation and Deduction. FSCD 2018. Leibniz International Proceedings in Informatics (LIPIcs), vol. 108, pp. 26:1–26:20. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018). https://doi.org/10.4230/LIPIcs.FSCD.2018.26Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Heidelberg (2002). http://www.springer.com/computer/swe/book/978-0-387-95250-5Prawitz, D.: Natural Deduction: A Proof-Theoretical Study. Dover, New York (2006)Sternagel, C., Sternagel, T., Middeldorp, A.: CoCo 2018 Participant: ConCon 1.5. In: Felgenhauer, B., Simonsen, J. (eds.) Proceedings of the 7th International Workshop on Confluence. IWC 2018, p. 66 (2018). http://cl-informatik.uibk.ac.at/events/iwc-2018/Sternagel, C., Yamada, A.: Reachability analysis for termination and confluence of rewriting. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 262–278. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_15Winkler, S., Moser, G.: MædMax: a maximal ordered completion tool. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 472–480. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_3