Logics with Multiteam Semantics

Team semantics is the mathematical basis of modern logics of dependence and independence. In contrast to classical Tarski semantics, a formula is evaluated not for a single assignment of values to the free variables, but on a set of such assignments, called a team. Team semantics is appropriate for a purely logical understanding of dependency notions, where only the presence or absence of data matters, but being based on sets, it does not take into account multiple occurrences of data values. It is therefore insufficient in scenarios where such multiplicities matter, in particular for reasoning about probabilities and statistical independencies. Therefore, an extension from teams to multiteams (i.e. multisets of assignments) has been proposed by several authors. In this paper we aim at a systematic development of logics of dependence and independence based on multiteam semantics. We study atomic dependency properties of finite multiteams and discuss the appropriate meaning of logical operators to extend the atomic dependencies to full-fledged logics for reasoning about dependence properties in a multiteam setting. We explore properties and expressive power of a wide spectrum of different multiteam logics and compare them to second-order logic and to logics with team semantics. In many cases the results resemble what is known in team semantics, but there are also interesting differences. While in team semantics, the combination of inclusion and exclusion dependencies leads to a logic with the full power of both independence logic and existential second-order logic, independence properties of multiteams are not definable by any combination of properties that are downwards closed or union closed and thus are strictly more powerful than inclusion-exclusion logic. We also study the relationship of logics with multiteam semantics with existential second-order logic for a specific class of metafinite structures. It turns out that inclusion-exclusion logic can be characterised in a precise sense by the Presburger fragment of this logic, but for capturing independence, we need to go beyond it and add some form of multiplication. Finally, we also consider multiteams with weights in the reals and study the expressive power of formulae by means of topological properties.

Common complexes of decompositions and complex balanced equilibria of chemical reaction networks

A decomposition of a chemical reaction network (CRN) is produced by partitioning its set of reactions. The partition induces networks, called subnetworks, that are "smaller" than the given CRN which, at this point, can be called parent network. A complex is called a common complex if it occurs in at least two subnetworks in a decomposition. A decomposition is said to be incidence independent if the image of the incidence map of the parent network is the direct sum of the images of the subnetworks' incidence maps. It has been recently discovered that the complex balanced equilibria of the parent network and its subnetworks are fundamentally connected in an incidence independent decomposition. In this paper, we utilized the set of common complexes and a developed criterion to investigate decomposition’s incidence independence properties. A framework was also developed to analyze decomposition classes with similar structure and incidence independence properties. We identified decomposition classes that can be characterized by their sets of common complexes and studied their incidence independence. Some of these decomposition classes occur in some biological and chemical models. Finally, a sufficient condition was obtained for the complex balancing of some power law kinetic (PLK) systems with incidence independent and complex balanced decompositions. This condition led to a generalization of the Deficiency Zero Theorem for some PLK systems.

Chebyshev’s bias for analytic L-functions

We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set $ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.

On Monte-Carlo tree search for deterministic games with alternate moves and complete information

We consider a deterministic game with alternate moves and complete information, of which the issue is always the victory of one of the two opponents. We assume that this game is the realization of a random model enjoying some independence properties. We consider algorithms in the spirit of Monte-Carlo Tree Search, to estimate at best the minimax value of a given position: it consists in simulating, successively, n well-chosen matches, starting from this position. We build an algorithm, which is optimal, step by step, in some sense: once the n first matches are simulated, the algorithm decides from the statistics furnished by the n first matches (and the a priori we have on the game) how to simulate the (n + 1)th match in such a way that the increase of information concerning the minimax value of the position under study is maximal. This algorithm is remarkably quick. We prove that our step by step optimal algorithm is not globally optimal and that it always converges in a finite number of steps, even if the a priori we have on the game is completely irrelevant. We finally test our algorithm, against MCTS, on Pearl’s game [Pearl, Artif. Intell. 14 (1980) 113–138] and, with a very simple and universal a priori, on the game Connect Four and some variants. The numerical results are rather disappointing. We however exhibit some situations in which our algorithm seems efficient.

Outer products and frame coefficients

In this dissertation, we will examine two distinct areas of frame theory. The first will be the area of outer products. In particular, we will examine the spanning and independence properties of the collection [see abstract for equation] for a given set [see abstract for equation]. In the case that our collection of outer products is a Riesz sequence, we will examine the relation between the Riesz bounds of the outer products to those of the generating vectors. It is perhaps not surprising that an independent collection of vectors will produce an independent collection of outer products. What is surprising though, is that the outer products have the same or better Riesz bounds. However, linearly independent collections of vectors are by no means the only collections that produce independent outer products. We will see that almost all vector sequences produce independent outer product sequences, dimension and cardinality permitting. Next we will examine the distribution of frame coefficients. We will start this investigation by examining the number of indices for which the frame coefficient is non-zero. Then we will generalize and bound these coefficients away from zero. We will study products of frame coefficients and find that in cases of particular classes of frames, we can bound a particular sum away from zero. Finally, we will look at the distance from an arbitrary vector to the frame vectors for a given frame and find some surprising results for certain classes of frames.

Distribution of the smallest visited point in a greedy walk on the line

We consider a greedy walk on a Poisson process on the real line. It is known that the walk does not visit all points of the process. In this paper we first obtain some useful independence properties associated with this process which enable us to compute the distribution of the sequence of indices of visited points. Given that the walk tends to +∞, we find the distribution of the number of visited points in the negative half-line, as well as the distribution of the time at which the walk achieves its minimum.

Efficient Self-Stabilizing Algorithm for Independent Strong Dominating Sets in Arbitrary Graphs

In computer networks area, the minimal dominating sets (MDS) and maximal independent sets (MIS) structures are very useful for creating virtual network overlays. Often, these set structures are used for designing efficient protocols in wireless sensor and ad-hoc networks. In this paper, we give a particular interest to one kind of these sets, called Independent Strong Dominating Set (ISD-set). In addition to its domination and independence properties, the ISD-set considers also node’s degrees that make it very useful in practical applications where nodes with larger degrees play important role in the networks. For example, some network clustering protocols chose nodes with large degrees to be cluster-heads, which is exactly the result obtained by an ISD-set algorithm. Thence, we propose the first distributed self-stabilizing algorithm for computing an ISD-set of an arbitrary graph (called ISDS). Then, we prove that ISDS algorithm operates under the unfair distributed scheduler and converges after at most [Formula: see text] rounds requiring only [Formula: see text] space memory per node where Δ is the maximum node degree. The complexity of ISDS algorithm in rounds has the same order as the best known self-stabilizing algorithms for finding MDS and MIS. Moreover, performed simulations and comparisons with well-known self-stabilizing algorithms for MDS and MIS problems showed the efficiency of ISDS, especially for reducing the cardinality of dominating sets founded by the algorithms.