Classification of Monogenic Ternary Semigroups

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

The present thesis studies some important random walk-based algorithms, which are randomized rumor spreading and balanced allocation protocols on networks. In the rst part of the thesis, we study the Push and the Push-Pull protocols introduced by [DGG+87], which are basic randomized protocols for information dissemination on networks. In Chapter 2, we propose a new model where the number of calls of each node in every round is chosen independently according to a probability distribution R with bounded mean determined at the beginning of the process. In addition to the model being a natural extension of the standard protocols, it also serves as a more realistic model for rumor spreading in a network whose entities are not completely uniform and may have di erent levels of power. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance. While it is well-known that the standard protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes, we show that, if R follows a power law distribution with exponent β ∈ (2, 3), then the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, we show a runtime of Θ ( logn log logn ) . In Chapter 3, we analyze the behavior of the standard Push-Pull protocol on a class of random graphs, called random k-trees for every integer k > 2, that are suitable to model poorly connected, small-world and scale free networks. Here, we show that the Push-Pull protocol propagates a rumor from a randomly chosen informed node to almost all nodes of a random k-tree with n nodes in O((log n)1+ck) rounds with high probability, where 0 3 can be any integer. Let l be a given positive integer. In each round t, 1 6 t 6 n, ball t picks a node of G uniformly at random and performs a nonbacktracking random walk of length l from the chosen node. Then it deterministically selects a subset of the visited nodes as the potential choices and allocates itself on one of the choices with minimum load (ties are broken uniformly at random). Provided G has a su ciently large girth, we establish an upper bound for the maximum number of balls at any bin after allocating n balls by the algorithm. We also show that the upper bound is tight up to a O(log log n) factor. In particular, we show that if we set l = b(log n) 1+ 2 c, for any constant ∈ (0, 1], and G has girth at least ω(l), then the maximum load is bounded by O(1/ ) with high probability.

Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and with ultimately periodic constants is exactly the class of hyper-arithmetical sets. Equations using addition only can represent every hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can also be represented by equations over sets of natural numbers equipped with union, addition and subtraction $S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}$. Testing whether a given system has a solution is $\Sigma^1_1$-complete for each model. These results, in particular, settle the expressive power of the most general types of language equations, as well as equations over subsets of free groups.

In this thesis, we introduce and examine four new temporal logic formalisms that can be used as specification languages for the automated verification of the reliability of hardware and software designs with respect to a desired behavior. The work is organized in two parts. In the first one, we reason about two logics for computations, GCTL* and MCTL*, which are useful to describe a correct execution of monolithic closed systems. In the second one, instead, we focus on two logics for strategies, SL and mATL*, which are useful to formalize several interesting properties about interactive plays in multi-entities systems modeled as multi-agent games. In the “Logics for Computations” part, we first study the immersion of the idea of graded quantifications into the temporal-logic framework. In first order logic, existential and universal quantifiers express the concepts of the existence of at least one individual object satisfying a formula, or that all individual objects satisfy a formula. In other logics, these quantifiers have been generalized to express that, for a given non-negative integer n, at least n or all but n individuals satisfy a particular formula. Here, we consider GCTL, a temporal logic with graded path quantifiers, which allows to describe properties like “there exist at least n different classes of computational fluxes in which a system reaches a predetermined state”, where the classes over paths are computed by means of a predetermined equivalence relation. More precisely, we uniformly extend the classic concept of graded quantifiers from states to paths, through the use of a concept of path equivalence with respect to a given path formula. About this logic, in particular, we study the expressiveness and succinctness relationships with respect to GµCalculus and the complexity of the satisfiability problem, which results to be ExpTime-Complete. This research is partially based on the works [BMM09] “Graded Computation Tree Logic” and [BMM10] “Graded Computation Tree Logic with Binary Coding” published, respectively, in the proceedings of the “IEEE Symposium on Logic in Computer Science, 2009” and “EACSL Annual Conference on Computer Science Logic, 2010”. Preliminary results can be also found in [Mog07]. Furthermore, we consider special quantifiers over substructures, which allow to select, using parametric criteria, small critical parts of a system to be successively verified. In literature, there are some attempts to define a logic that allows to modify the underlying structure under exam and then to verify on it some assigned property. However, as far as we know, none of them is able to select minimal submodels of a given property describing the criteria on which then execute the verification process. Here, we base our work on the search of a new operator that merges the concept of quantifiers on structures with that one derived by a generalization of the concept of pruning. The results of this work, is a class of three different extensions of CTL* with minimal model quantifiers, which we name MCTL*. Regarding these logics, we study several reductions among them, as well as the satisfiability problem that we prove to be highly undecidability, i.e., Σ_1^1-Hard, for two out of the three cases. This research is partially based on the work [MM09] “Branching-Time Temporal Logics with Minimal Model Quantifiers” published in the proceedings of the “International Conference on Developments in Language Theory, 2009”. In the “Logics for Strategies” part, we first study the problem of defining a new specification language through which it is possible to express several important properties of multi-entities systems that are neither expressible using classical monolithic temporal logics, such as CTL*, nor using two-agent-teams temporal logics, such as ATL*. In literature, we can found some proposal of logics that try to achieve this goal, but unfortunately, none of them succeeds completely on all the aspects. Among them, one of the most important attempts is CHP-SL, a logics in which one can use variables over strategies. However, this logic has a deep weakness, since it does not allow to describe games with more than two players and even two-players concurrent games. Here, we introduce SL, a logic with a syntax similar in some aspects to the first order logic, in which the strategies of the agent building the game are treated as first order objects on which we can quantify. This logic generalizes CHP-SL, by allowing the specification of the correct behavior of multi-agent concurrent games. In SL, for example, we are able to express very complex but useful Nash equilibria that are not expressible with CHP-SL. We enlighten that Nash equilibrium is one of the most important concepts in game theory. For the introduced logic, we solve two problems left open in the work on CHP-SL. Precisely, we show that the related model-checking problem is 2ExpTime-Complete, thus not harder of the same problem for several subsumed logics, while we prove that its satisfiability problem is highly undecidable, i.e., Σ_1^1-Hard. This research is partially based on the work [MMV10a] “Reasoning About Strategies” published in the proceedings of the “IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, 2010”. Finally, we consider the concept of relentful strategic reasoning, i.e., a formalism that expresses the ability of a strategy to be used not only to achieve a first given goal, but also to change its final goal in dependence of the history of the play. In the context of planning, memoryful quantification, i.e., quantification over computations that does not lose information about the past along the time, is one of the principal way to express the fact that a system is able to achieve a desired result, and shift to a different goal if some event happens. However, this kind of quantification was not considered before in the context of multi-agent planning. Here, we introduce mATL*, a fusion of the classic alternating temporal logic ATL* with memoryful quantification, with the aim of covering the previous idea. About this logics, we prove that, although it is equivalent to ATL*, it is exponentially more succinct. Nevertheless, we prove that both the model-checking and the satisfiability problems remain 2ExpTime-Complete, as for ATL*. This research is partially based on the work [MMV10b] “Relentful Strategic Reasoning in Alternating-Time Temporal Logic” published in the proceedings of the “International Conference on Logic for Programming Artificial Intelligence and Reasoning, 2010”.

Recent innovation in the design of type theories—foundational systems of mathematics with a focus on constructivity—has produced the concept of interval variable, which can be used to capture relations between objects that carry computational content. We examine two such relationships in type theory: equality, in particularquotients, and arbitrary relations as applied in parametricity interpretations. Cubical type theory, a system using an interval-based formulation of equality, enables a permissive kind of content-carrying equality that includes in particularisomorphism. Cubical type theory provides a constructive interpretation of homotopy type theory and the Univalent Foundations, formalisms that introduced the idea of isomorphism as equality but which lack intrinsic computational meaning. The cubical approach to equality also rectifies long-standing deficiencies in the behavior ofquotients in type theory. We realize a system of generalized quotients for cubical type theory originally conceived in homotopy type theory, called higher inductive types, that merges the concepts of inductive type and quotient. Such a mutual generalization is particularly essential in the contentful equality setting, but also has significant applications to ordinary mathematics. Parametricity is, among other things, a proof technique for deriving propertiesof constructions performed in type theories, based on the idea that constructions preserve all relations between objects. Traditionally a meta-theoretical tool, recentwork has shown that parametricity properties can be integrated into a type theory itself using an interval-based system. We develop internal parametricity on top ofcubical type theory, examining the similarities and distinctions between the two applications of intervals, finding that a background of cubical equality improves thebehavior of internal parametricity, and applying internal parametricity as a tool to solve difficult problems in cubical type theory. We introduce a system of cohesionmodalities to express the interaction between parametric and non-parametric type theory, enabling the use of parametricity results in a non-parametric setting.

Channel coding linear programming decoding (CC- LPD) and compressed sensing linear programming decoding (CS- LPD) are two setups that are formally tightly related. Recently, a connection between CC-LPD and CS-LPD was exhibited that goes beyond this formal relationship. The main ingredient was a lemma that allowed one to map vectors in the nullspace of some zero-one measurement matrix into vectors of the fundamental cone defined by that matrix. The aim of the present paper is to extend this connection along several directions. In particular, the above-mentioned lemma is extended from real measurement matrices where every entry is equal to either zero or one to complex measurement matrices where the absolute value of every entry is a non-negative integer. Moreover, this lemma and its generalizations are used to translate performance guarantees from CC-LPD to CS-LPD. In addition, the present paper extends the formal relationship between CC-LPD and CS-LPD with the help of graph covers. First, this graph-cover viewpoint is used to obtain new connec- tions between, on the one hand, CC-LPD for binary parity- check matrices, and, on the other hand, CS-LPD for complex measurement matrices. Secondly, this graph-cover viewpoint is used to see CS-LPD not only as a well-known relaxation of some zero-norm minimization problem but (at least in the case of real measurement matrices with only zeros, ones, and minus ones) also as a relaxation of a problem we call the zero-infinity operator minimization problem.

We define the reflection object of a simplicial object and the notion of a reflection simplicial map, which can be realized geometrically, too. Using the inversion in a group G, there is an interesting canonical reflection involution on the classifying space BG. We study its fixed point set and quotient space. As an application, we show that inversion invariant cohomology is isomorphic to the usual group cohomology if the coefficients are local away from 2. Then we construct reflection involutions on Eilenberg-MacLane spaces and simplicial pseudogroups which are local away from 2.

We prove that the localizations of the categories of dg categories, of cohomologically unital and strictly unital $A_\infty$ categories with respect to the corresponding classes of quasi-equivalences are all equivalent. Moreover we show that the last two localizations are equivalent to the corresponding quotients by the relation of being isomorphic in the cohomology of the $A_\infty$ category of $A_\infty$ functors. As an application we give a complete proof of a claim by Kontsevich stating that the category of internal Homs for two dg categories can be described as the category of strictly unital $A_\infty$ functors between them.

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.

Dλ-cycles in λ-claw-free graphs

In this paper we study the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N}),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $\mathscr{I}_{\infty}^{\,\Rsh\!\!\!\nearrow}(\mathbb{N})$.

Excluding any graph as a minor allows a low tree-width 2-coloring

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard'' on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.