Ternary Constraints Research Articles

Correct approximation of IEEE 754 floating-point arithmetic for program verification

Verification of programs using floating-point arithmetic is challenging on several accounts. One of the difficulties of reasoning about such programs is due to the peculiarities of floating-point arithmetic: rounding errors, infinities, non-numeric objects (NaNs), signed zeroes, denormal numbers, different rounding modes, etc. One possibility to reason about floating-point arithmetic is to model a program computation path by means of a set of ternary constraints of the form and use constraint propagation techniques to infer new information on the variables’ possible values. In this setting, we define and prove the correctness of algorithms to precisely bound the value of one of the variables x, y or z, starting from the bounds known for the other two. We do this for each of the operations and for each rounding mode defined by the IEEE 754 binary floating-point standard, even in the case the rounding mode in effect is only partially known. This is the first time that such so-called filtering algorithms are defined and their correctness is formally proved. This is an important slab for paving the way to formal verification of programs that use floating-point arithmetics.

Automatic classification of protein structures relying on similarities between alignments

BackgroundIdentification of protein structural cores requires isolation of sets of proteins all sharing a same subset of structural motifs. In the context of an ever growing number of available 3D protein structures, standard and automatic clustering algorithms require adaptations so as to allow for efficient identification of such sets of proteins.ResultsWhen considering a pair of 3D structures, they are stated as similar or not according to the local similarities of their matching substructures in a structural alignment. This binary relation can be represented in a graph of similarities where a node represents a 3D protein structure and an edge states that two 3D protein structures are similar. Therefore, classifying proteins into structural families can be viewed as a graph clustering task. Unfortunately, because such a graph encodes only pairwise similarity information, clustering algorithms may include in the same cluster a subset of 3D structures that do not share a common substructure. In order to overcome this drawback we first define a ternary similarity on a triple of 3D structures as a constraint to be satisfied by the graph of similarities. Such a ternary constraint takes into account similarities between pairwise alignments, so as to ensure that the three involved protein structures do have some common substructure. We propose hereunder a modification algorithm that eliminates edges from the original graph of similarities and gives a reduced graph in which no ternary constraints are violated. Our approach is then first to build a graph of similarities, then to reduce the graph according to the modification algorithm, and finally to apply to the reduced graph a standard graph clustering algorithm. Such method was used for classifying ASTRAL-40 non-redundant protein domains, identifying significant pairwise similarities with Yakusa, a program devised for rapid 3D structure alignments.ConclusionsWe show that filtering similarities prior to standard graph based clustering process by applying ternary similarity constraints i) improves the separation of proteins of different classes and consequently ii) improves the classification quality of standard graph based clustering algorithms according to the reference classification SCOP.

A Stronger Consistency for Soft Global Constraints in Weighted Constraint Satisfaction

Weighted Constraint Satisfaction is made practical by powerful consistency techniques, such as AC*, FDAC* and EDAC*, which reduce search space effectively and efficiently during search, but they are designed for only binary and ternary constraints. To allow soft global constraints, usually of high arity, to enjoy the same benefits, Lee and Leung give polynomial time algorithms to enforce generalized AC* (GAC*) and FDAC* (FDGAC*) for projection-safe soft non-binary constraints. Generalizing the stronger EDAC* is less straightforward. In this paper, we first reveal the oscillation problem when enforcing EDAC* on constraints sharing more than one variable. To avoid oscillation, we propose a weak version of EDAC* and generalize it to weak EDGAC* for non-binary constraints. Weak EDGAC* is stronger than FDGAC* and GAC*, but weaker than VAC and soft k-consistency for k > 2. We also show that weak EDGAC* can be enforced in polynomial time for projection-safe constraints. Extensive experimentation confirms the efficiency of our proposal.

Lifshitz topological black holes

We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries.

Branchwidth of chordal graphs

This paper revisits the ‘branchwidth territories’ of Kloks, Kratochvíl and Müller [T. Kloks, J. Kratochvíl, H. Müller, New branchwidth territories, in: 16th Ann. Symp. on Theoretical Aspect of Computer Science, STACS, in: Lecture Notes in Computer Science, vol. 1563, 1999, pp. 173–183] to provide a simpler proof, and a faster algorithm for computing the branchwidth of an interval graph. We also generalize the algorithm to the class of chordal graphs, albeit at the expense of exponential running time. Compliance with the ternary constraint of the branchwidth definition is facilitated by a simple new tool called k -troikas: three sets of size at most k each are a k -troika of set S , if any two have union S . We give a straightforward O ( m + n + q 2 ) algorithm, computing branchwidth for an interval graph on m edges, n vertices and q maximal cliques. We also prove a conjecture of Mazoit [F. Mazoit, A general scheme for deciding the branchwidth, Technical Report RR2004-34, LIP — École Normale Supérieure de Lyon, 2004. http://www.ens-lyon.fr/LIP/Pub/Rapports/RR/RR2004/RR2004-34.pdf], by showing that branchwidth can be computed in polynomial time for a chordal graph given with a clique tree having a polynomial number of subtrees.

Trimoraic Feet in Gilbertese

Trimoraic Feet In Gilbertese1 Juliette Blevins and Sheldon P. Harrison Abstract In this paper we present evidence for ternary metrical constituents in Gilbertese (Kiribati), a Micronesian language. The terminal elements of stress feet in Gilbertese are moras, not syllables. Further, the typical foot in Gilbertese contains three moras. These trimoraic constituents are units of stress in Gilbertese, and also define minimal prosodic word size, where possible. Ternary metrical constituents of the sort found in Gilbertese are quite rare cross-linguistically, and as far as we know, Gilbertese is the only language in the world reported to have a ternary constraint on prosodic word size. We present a constraint-based analysis of Gilbertese prosody that also makes use of a language-specific template-mapping algorithm. 1. Introduction. In this paper we present evidence for ternary metrical constituents in Gilbertese, a Micronesian language spoken in Kiribati by approximately 55,000 people.2 The terminal elements of stress feet in Gilbertese are moras, not syllables. Further, the typical foot in Gilbertese contains three moras. These trimoraic constituents are units of stress in Gilbertese, and also define prosodic word size and shape. Ternary metrical constituents of the sort found in Gilbertese are quite rare cross-linguistically, and as far as we know, Gilbertese is the only language in the world reported to have a ternary constraint on prosodic word size. This study offers two fundamental contributions to linguistic theories of stress. It provides the first in-depth study of prosody for a Micronesian language, thereby [End Page 203] contributing to the overall empirical basis of stress typology. Second, it provides a principled constraint-based analysis of Gilbertese prosody. The occurrence of trimoraic feet where possible in Gilbertese but bimoraic feet elsewhere leads us to propose that ternarity is the consequence of a grammar of ranked and violable metrical constraints. Essentially, we argue that ternarity results from the interaction of a binary requirement on foot size with a binary requirement on foot-head size. The failure of Gilbertese words to obey syllable integrity (Prince 1976; Hayes 1995), the principle that demands that a syllable never be split by foot structure, is a straightforward indication of the low ranking or absence of the syllable integrity constraint in this language. Languages with purported ternary constituents and violations of syllable integrity are not new. Levin (1988a), Everett (1988), Dresher and Lahiri (1991), and Rice (1992) present arguments for ternary metrical constituents in Cayuvava, Pirahã, Old English, and Chuvach Alutiiq, respectively. Hayes (1995) summarizes descriptions of ternary patterns for Estonian, Finnish, Karelian, Hungarian, Sen-tani, Winnebago, and Auca as well. Data from Banawá (Buller, Buller, and Everett 1993) demonstrate violations of syllable integrity. What is unique to Gilbertese is the convergent evidence for ternary metrical constituents in the stress domain and with respect to a minimal prosodic word constraint. A novel aspect of this analysis is the account of minimal prosodic word effects. Though ultimately minimal word effects follow from the prosodic hierarchy in which word dominates foot, in Gilbertese, specific violations of the prosodic-word minimum forces one to the conclusion that a language-specific rule is responsible for bulking words of two moras up to three moras. Before presenting the Gilbertese facts in sections 3 and 4 and our analysis in section 5, we begin with a summary of earlier work on Micronesian prosody. 2. Micronesian Prosody. Rehg (1993) is the first compilation of prosodic descriptions of Micronesian languages, descriptions that have tended to be incomplete, contradictory, and impressionistic. Nevertheless, Rehg is able to glean overarching generalizations from earlier descriptions of Pulo Annian, Puluwat, Woleaian, Ponapean, Marshallese, Gilbertese, and Kosraean. One generalization is that in all of these languages, it is the mora, not the syllable, that is the prosody-bearing unit. We will demonstrate this point for Gilbertese in section 4. Another generalization is that wherever high pitch and stress are described, they fall on adjacent moras. Based on these observations, and the reconstitution of historical final vowels, Rehg is able to reconstruct Proto-Micronesian primary accent in unmarked terminal prosodic contours. His proposal for Proto-Micronesian accent is shown in (1). (1) Proto-Micronesian accent (Rehg 1993) [End Page 204] The unmarked terminal contour...