Cyclic resultants

We characterize polynomials having the same set of nonzero cyclic resultants. Generically, for a polynomial f of degree d, there are exactly 2^d−1 distinct degree d polynomials with the same set of cyclic resultants as f. However, in the generic monic case, degree d polynomials are uniquely determined by their cyclic resultants. Moreover, two reciprocal (“palindromic”) polynomials giving rise to the same set of nonzero cyclic resultants are equal. In the process, we also prove a unique factorization result in semigroup algebras involving products of binomials. Finally, we discuss how our results yield algorithms for explicit reconstruction of polynomials from their cyclic resultants.     

Expressing Priorities and External Probabilities in Process Algebra via Mixed Open/Closed Systems

Defining operational semantics for a process algebra is often based either on labeled transition systems that account for interaction with a context or on the so-called reduction semantics: we assume to have a representation of the whole system and we compute unlabeled reduction transitions (leading to a distribution over states in the probabilistic case). In this paper we consider mixed models with states where the system is still open (towards interaction with a context) and states where the system is already closed. The idea is that (open) parts of a system “P” can be closed via an operator “P↑G” that turns already synchronized actions whose “handle” is specified inside “G” into prioritized reduction transitions (and, therefore, states performing them into closed states). We show that we can use the operator “P↑G” to express multi-level priorities and external probabilistic choices (by assigning weights to handles inside G), and that, by considering reduction transitions as the only unobservable τ transitions, the proposed technique is compatible, for process algebra with general recursion, with both standard (probabilistic) observational congruence and a notion of equivalence which aggregates reduction transitions in a (much more aggregating) trace based manner. We also observe that the trace-based aggregated transition system can be obtained directly in operational semantics and we present the “aggregating” semantics. Finally, we discuss how the open/closed approach can be used to also express discrete and continuous (exponential probabilistic) time and we show that, in such timed contexts, the trace-based equivalence can aggregate more with respect to traditional lumping based equivalences over Markov Chains.     

Languages Defined with Modular Counting Quantifiers

We prove that a regular language defined by a boolean combination of generalized Σ₁-sentences built using modular counting quantifiers can be defined by a boolean combination of Σ₁-sentences in which only regular numerical predicates appear. The same statement, with “Σ₁” replaced by “first-order,” is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC¹. The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case.     

A Novel Stochastic Game Via the Quantitative μ-calculus

The quantitative μ-calculus qMμ extends the applicability of Kozen's standard μ-calculus [D. Kozen, Results on the propositional μ-calculus, Theoretical Computer Science 27 (1983) 333–354] to probabilistic systems. Subsequent to its introduction [C. Morgan, and A. McIver, A probabilistic temporal calculus based on expectations, in: L. Groves and S. Reeves, editors, Proc. Formal Methods Pacific '97 (1997), available at [PSG, Probabilistic Systems Group: Collected reports, http://web.comlab.ox.ac.uk/oucl/research/areas/probs/bibliography.html]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 9], M. Huth, and M. Kwiatkowska, Quantitative analysis and model checking, in: Proceedings of 12th annual IEEE Symposium on Logic in Computer Science, 1997] it has been developed by us [A. McIver, and C. Morgan, Games, probability and the quantitative μ-calculus qMu, in: Proc. LPAR, LNAI 2514 (2002), pp. 292–310, revised and expanded at [A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL]; also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap. 11], A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, A. McIver, and C. Morgan, Results on the quantitative μ-calculus qMμ (2005), to appear in ACM TOCL] and by others [L. de Alfaro, and R. Majumdar, Quantitative solution of omega-regular games, Journal of Computer and System Sciences 68 (2004) 374–397]. Beyond its natural application to define probabilistic temporal logic [C. Morgan, and A. McIver, An expectation-based model for probabilistic temporal logic, Logic Journal of the IGPL 7 (1999), pp. 779–804, also appears at [A. McIver, and C. Morgan, “Abstraction, Refinement and Proof for Probabilistic Systems,” Technical Monographs in Computer Science, Springer, New York, 2005, Chap.10]], there are a number of other areas that benefit from its use.One application is stochastic two-player games, and the contribution of this paper is to depart from the usual notion of “absolute winning conditions” and to introduce a novel game in which players can “draw”.The extension is motivated by examples based on economic games: we propose an extension to qMμ so that they can be specified; we show that the extension can be expressed via a reduction to the original logic; and, via that reduction, we prove that the players can play optimally in the extended game using memoryless strategies.     

On the extension of solutions of linear constant-coefficient PDE

We solve the following over-determined boundary value problem (the “extension problem”): Let R(∂) be a matrix whose entries are linear partial differential operators, with constant coefficients. Let Ω be a non-empty, open, bounded, convex set. We consider the homogeneous system R(∂)f=0 in a neighborhood of , subject to the boundary condition f=g in a neighborhood of ∂Ω. For a given g, we give a criterion for the (unique) existence of a smooth solution f to this problem. There are two obvious necessary conditions: g is smooth and R(∂)g=0 in a neighborhood of ∂Ω. We characterize the class of differential operators R(∂) for which the problem is solvable for any g satisfying the necessary conditions. Finally, in the case where the solution is non-unique, we consider the possibility of obtaining uniqueness by fixing several components of the desired solution.     

On the Jacopini Technique

The general concern of the Jacopini technique is the question: “Is it consistent to extend a given lambda calculus with certain equations?” The technique was introduced by Jacopini in 1975 in his proof that in the untyped lambda calculusΩis easy, i.e.,Ωcan be assumed equal to any other (closed) term without violating the consistency of the lambda calculus. The presentations of the Jacopini technique that are known from the literature are difficult to understand and hard to generalise. In this paper we generalise the Jacopini technique for arbitrary lambda calculi. We introduce the concept ofproof-replaceabilityby which the structure of the technique is simplified considerably. We illustrate the simplicity and generality of our formulation of the technique with some examples. We apply the Jacopini technique to theλμ-calculus, and we prove a general theorem concerning the consistency of extensions of theλμ-calculus of a certain form. Many well known examples (e.g., the easiness ofΩ) are immediate consequences of this general theorem.     

On the Construction of Classes of Suffix Trees for Square Matrices: Algorithms and Applications

We provide a uniform framework for the study of index data structures for a two-dimensional matrixTEXT[1:n, 1:n] whose entries are drawn from an ordered alphabetΣ. An index forTEXTcan be informally seen as the two-dimensional analog of the suffix tree for a string. It allows on-line searches and statistics to be performed onTEXTby representing compactly theΘ(n³) square submatrices ofTEXTin optimalO(n²) space. We identify 4ⁿ⁻¹families of indices forTEXT, each containing ∏ⁿ_i=1 (2i−1)! isomorphic data structures. We also develop techniques leading to a single algorithm that efficiently builds any index in any family inO(n² log n) time andO(n²) space. Such an algorithm improves in various respects the algorithms for the construction of the PAT tree and the Lsuffix tree. The framework and the algorithm easily generalize tod>2 dimensions. Moreover, as part of our algorithm, we provide new algorithmic tools that yield a space-efficient implementation of the “naming scheme” of R. Karpet al.(in“Proceedings, Fourth Symposium on Theory of Computing,” pp. 125–136) for strings and matrices.     

A simple algorithm for the unique characterization of convex polygons

A simple method for specifying the shape and orientation of a convex polygon is described. The method utilizes eigenvalues and eigenvectors of a “moment of inertia” or mass matrix computed from the nodal coordinates of the polygon. The “shape” is characterized then by the parameter ( , where ξ₁ and ξ₂ are the eigenvalues (ξ₁ ξ₂), and the orientation by the axial direction of the first eigenvector. FORTRAN subroutines are provided for this algorithm.     

Completeness of Continuation Models for λμ-Calculus

We show that a certain simple call-by-name continuation semantics of Parigot's λ_μ-calculus is complete. More precisely, for every λ_μ-theory we construct a cartesian closed category such that the ensuing continuation-style interpretation of λ_μ, which maps terms to functions sending abstract continuations to responses, is full and faithful. Thus, any λ_μ-category in the sense of L. Ong (1996, in “Proceedings of LICS '96,” IEEE Press, New York) is isomorphic to a continuation model (Y. Lafont, B. Reus, and T. Streicher, “Continuous Semantics or Expressing Implication by Negation,” Technical Report 93-21, University of Munich) derived from a cartesian-closed category of continuations. We also extend this result to a later call-by-value version of λ_μ developed by C.-H. L. Ong and C. A. Stewart (1997, in “Proceedings of ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Paris, January 1997,” Assoc. Comput. Mach. Press, New York).     

Practical multiagent rendezvous through modified circumcenter algorithms

We present a class of modified circumcenter algorithms that allow a group of agents to achieve “practical rendezvous” when they are only able to take noisy measurements of their neighbors. Assuming a uniform detection probability in a disk of radius σ about each neighbor’s true position, we show how initially connected agents converge to a practical stability ball. More precisely, a deterministic analysis allows us to guarantee convergence to such a ball under r-disk graph connectivity in 1D under the condition that r/σ be sufficiently large. A stochastic analysis leads to a similar convergence result in probability, but for any r/σ>1, and under a sequence of switching graphs that contains a connected graph within bounded time intervals. We include several simulations to discuss the performance of the proposed algorithms.     

Fast algorithms for long transportation problems

The paper deals with large transportation problems in which the number of destinations (n) is much larger than the number of origins (m), and in which some or many of the paths from origins to destinations may be inadmissible. Using a new approach, with certain auxilary lists, it is proved that the ordinary simplex algorithm (“Most Negative Rule”) can be performed in O(m²+ m log n) computer operations per iteration, as against O(mn) in the usual approach. A search-in-a-row simplex algorithm (“Row Most Negative Rule”), for which the total number of iterations is probably only somewhat larger, is shown to require just O(m + σm log n) operations per iteration, where σ is the density of the cost matrix (i.e. the proportion of admissible paths). For these rigorous results one needs computer storage which is not considerably larger than that required for storing the cost matrix. For smaller memory an efficient algorithm is also proposed. A general tentative rule for die amount of scanning per iteration is introduced and applied. Computer experiments are reported, confirming theoretical estimates.     

An Overview of Cαml

Cαml is a tool that turns a so-called “binding specification” into an Objective Caml compilation unit. A binding specification resembles an algebraic data type declaration, but also includes information about names and binding. Cαml is meant to help writers of interpreters, compilers, or other programs-that-manipulate-programs deal with α-conversion in a safe and concise style. This paper presents an overview of Cαml's binding specification language and of the code that Cαml produces.     

Spines of random constraint satisfaction problems: definition and connection with computational complexity

We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobás et al. [10] to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a 2^Ω(n) resolution complexity (and thus a 2^Ω(n) complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of “large” (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition.We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are “qualitatively similar to random 2-SAT”. Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.AMS subject classification 68Q25, 82B27     

Term Collections in λ and ρ-calculi

The ρ-calculus generalises term rewriting and the λ-calculus by defining abstractions on arbitrary patterns and by using a pattern-matching algorithm which is a parameter of the calculus. In particular, equational theories that do not have unique principal solutions may be used. In the latter case, all the principal solutions of a matching problem are stored in a “structure” that can also be seen as a collection of terms.Motivated by the fact that there are various approaches to the definition of structures in the ρ-calculus, we study in this paper a version of the λ-calculus with term collections.The contributions of this work include a new syntax and operational semantics for a λ-calculus with term collections, which is related to the λ-calculi with strict parallel functions studied by Boudol and Dezani et al. and a proof of the confluence of the β-reduction relation defined for the calculus (which is a suitable extension of the standard rule of β-reduction in the λ-calculus).     

Modelling and solving the intrusion detection problem in computer networks

We introduce a novel anomaly intrusion detection method based on a Within-Class Dissimilarity, WCD. This approach functions by using an appropriate metric WCD to measure the distance between an unknown user and a known user defined respectively by their profile vectors. First of all, each user performs a set of commands (events) on a given system (Unix for example). The events vector of a given user profile is a binary vector, such that an element of this vector is equal to “1” if an event happens, and to “0” otherwise. In addition to this, each user's class k has a typical profile defined by the vector P_k, in order to test if a new user i defined by its profile vector P_i belongs to the same class k or not. The P_k vector is a weighted events vector E_k, such that each weight represents the number of occurrences of an event e_k. If the “distance” d_ki (measured by a dissimilarity parameter) between an unknown profile P_i and a known profile P_k is reasonable according to a given threshold and to some constraints, then there is no intrusion. Else, the user i is suspicious. A simple example illustrates the WCD procedure. A survey of intrusion detection methods is presented.Our proposed method based on clustering users and using simple statistical formulas is very easy for implementation.     

Extracting a polyhedron from a single-view sketch: Topological construction of a wireframe sketch with minimal hidden elements

An essential prerequisite to construct a manifold trihedral polyhedron from a given natural (or partial-view) sketch is solution of the “wireframe sketch from a single natural sketch (WSS)” problem, which is the subject of this paper. Published solutions view WSS as an “image-processing”/“computer vision” problem where emphasis is placed on analyzing the given input (natural sketch) using various heuristics. This paper proposes a new WSS method based on robust tools from graph theory, solid modeling and Euclidean geometry. Focus is placed on producing a minimal wireframe sketch that corresponds to a topologically correct polyhedron.     

A type theory which is complete for Kreisel's modified realizability

We define a type theory with a strong elimination rule for existential quantification. As in Martin-Löf's type theory, the “axiom of choice” is thus derivable. Proofs are also annotated by realizers which are simply typed λ-terms. A new rule called “type extraction” which extracts the type of a realizer allows us to derive the so-called “independance of premisses” schema. Consequently, any formula which is realizable in HA^ω, according to Kreisel's modified realizability, is derivable in this type theory.     

What is a “Good” Encoding of Guarded Choice?

The π-calculus with synchronous output and mixed-guarded choices is strictly more expressive than the π-calculus with asynchronous output and no choice. This result was recently proved by C. Palamidessi and, as a corollary, she showed that there is no fully compositional encoding from the former into the latter that preserves divergence-freedom and symmetries. This paper argues that there are nevertheless “good” encodings between these calculi. In detail, we present a series of encodings for languages with (1) input-guarded choice, (2) both input- and output-guarded choice, and (3) mixed-guarded choice, and investigate them with respect to compositionality and divergence-freedom. The first and second encoding satisfy all of the above criteria, but various “good” candidates for the third encoding—inspired by an existing distributed implementation—invalidate one or the other criterion. While essentially confirming Palamidessi's result, our study suggests that the combination of strong compositionality and divergence-freedom is too strong for more practical purposes.