Abstract: A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is d-collapsible is NP-complete for d greater or equal to 4 and polynomial time solvable for d at most 2. As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for d at most 2, but the greedy algorithm does not work for d greater or equal 3. A simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. The main motivation for studying d-collapsible complexes is that every d-representable complex is d-collapsible. We also observe that known results imply that analogical algorithmic question for d-representable complexes is NP-hard for d greater or equal to 2.

Abstract: Reversible finite automata with halting states (RFA) were first considered by Ambainis and Freivalds to facilitate the research of Kondacs-Watrous quantum finite automata. In this paper we consider some of the algebraic properties of RFA, namely the varieties these automata generate. Consequently, we obtain a characterization of the boolean closure of the classes of languages recognized by these models.

Abstract: We study the problem of identifying an n-bit string using a single quantum query to an oracle that computes the Hamming distance between the query and hidden strings. The standard action of the oracle on a response register of dimension r is by powers of the cycle (1 . . . r), all of which, of course, commute. We introduce a new model for the action of an oracle—by general permutations in Sr—and explore how the success probability depends on r and on the map from Hamming distances to permutations. In particular, we prove that when r = 2, for even n the success probability is 1 with the right choice of the map, while for odd n the success probability cannot be 1 for any choice. Furthermore, for small odd n and r = 3, we demonstrate numerically that the image of the optimal map generates a non-abelian group of permutations.

Abstract: Process complexity is one of the basic variants of Kolmogorov complexity. Unlike plain Kolmogorov complexity process complexity provides a simple characterization of randomness for real numbers in terms of initial segment complexity. Process complexity was first developed in (Schnorr 1973). Schnorr's definition of a process, while simple, can be difficult to work with. In many situations, a preferable definition of a process is that given by Levin in (Levin & Zvonkin 1970). In this paper we define a variant of process complexity based on Levin's definition of a process. We call this variant strict process complexity. Strict process complexity retains the main desirable properties of process complexity. Particularly, it provides simple characterizations of Martin-Lof random real numbers, and of computable real numbers. However, we will prove that strict process complexity does not agree within an additive constant with Schnorr's original process complexity. One of the basic properties of prefix-free complexity is that it is subadditive. Subadditive means that there is some constant d such that for all strings σ, τ the complexity of στ (σ and τ concatenated) is less than or equal to the sum of the complexities of σ and τ plus d. A fundamental question about any complexity measure is whether or not it is subadditive. In this paper we resolve this question for process complexity by proving that neither of these process complexities is subadditive.

Abstract: The Tutte polynomial of a graph, also known as the partition function of the q-state Potts model, is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants such as the number of spanning trees as complete evaluations. We have developed the most efficient algorithm to-date for computing the Tutte polynomial of a graph. An important component of the algorithm affecting efficiency is the choice of edge to work on at each stage in the computation. In this paper, we present and discuss two edge-selection heuristics which (respectively) give good performance on sparse and dense graphs. We also present experimental data comparing these heuristics against a range of others to demonstrate their effectiveness.

Abstract: In this paper we study the problem of deciding whether a given compressed string contains a square. A string x is called a square if x = zz and z = u^k implies k = 1 and u = z. A string w is said to be square-free if no substrings of w are squares. Many efficient algorithms to test if a given string is square-free, have been developed so far. However, very little is known for testing square-freeness of a given compressed string. In this paper, we give an O(max(n^2; n log^2 N))-time O(n^2)-space solution to test square-freeness of a given compressed string, where n and N are the size of a given compressed string and the corresponding decompressed string, respectively. Our input strings are compressed by balanced straight line program (BSLP). We remark that BSLP has exponential compression, that is, N = O(2^n). Hence no decompress-then-test approaches can be better than our method in the worst case.

Abstract: Z is a formal specification language combining typed set theory, predicate calculus, and a schema calculus. This paper describes an extension of Z that allows transformation and reasoning rules to be written in a Z-like notation. This gives a high-level, declarative, way of specifying transformations of Z terms, which makes it easier to build new Z manipulation tools. We describe the syntax and semantics of these rules, plus some example reasoning engines that use sets of rules to manipulate Z terms. The utility of these rules is demonstrated by discussing two sets of rules. One set defines expansion of Z schema expressions. The other set is used by the ZLive animator to preprocess Z expressions into a form more suitable for animation.; ;

Abstract: Many different random graph constructions are used to model large real life graphs, i.e., graphs that describe the structure of real systems. Often it is not clear, however, how the strength of the different models compare to each other, e.g., when does it hold that a certain model class contains another. We are particularly interested in random graph models that arise via abstract geometric constructions, motivated by the fact that these graphs can model wireless communication networks. We set up a general framework to compare the strength of random graph models, and present some results about the equality, inequality and proper containment of certain model classes, as well as some open problems.

Abstract: Data stream computations in domains such as internet applications are often performed in a highly distributed fashion in order to save time. An example is the class of applications that use the Google Mapreduce framework of scalable distributed processing as presented by (Dean & Ghemawat 2004). A basic question here is: what kind of data stream computations admit scalable and efficient distributed algorithms? We show that the class of data stream computations that approximate functions of the frequency vector of the stream can be computed efficiently in a distributed manner.

Abstract: A multiple quantifier is a quantifier having inference rules that introduce or eliminate arbitrary number of quantifiers by one inference. This paper introduces the lambda calculus with negation, conjunction, and multiple existential quantifiers, and the lambda calculus with implication and multiple universal quantifiers. Their type checking and type inference are proved to be undecidable. This paper also shows that the undecidability of type checking and type inference in the type-free-style lambda calculus with negation, conjunction, and existence is reduced to the undecidability of type checking and type inference in the type-free-style polymorphic lambda calculus.

Abstract: In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f 2 = I. We describe several generalisations of well-known results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich-Levin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of Bonami, Gross and Beckner.