A Modal Proof Theory for Polynomial Coalgebras

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>

A Modal Proof Theory for Polynomial Coalgebras

<p>The abstract mathematical structures known as coalgebras are of increasing interest in computer science for their use in modelling certain types of data structures and programs. Traditional algebraic methods describe objects in terms of their construction, whilst coalgebraic methods describe objects in terms of their decomposition, or observational behaviour. The latter techniques are particularly useful for modelling infinite data structures and providing semantics for object-oriented programming languages, such as Java. There have been many different logics developed for reasoning about coalgebras of particular functors, most involving modal logic. We define a modal logic for coalgebras of polynomial functors, extending Rößiger’s logic [33], whose proof theory was limited to using finite constant sets, by adding an operator from Goldblatt [11]. From the semantics we define a canonical coalgebra that provides a natural construction of a final coalgebra for the relevant functor. We then give an infinitary axiomatization and syntactic proof relation that is sound and complete for functors constructed from countable constant sets.</p>

Computational Information Geometry For Binary Classification of High-Dimensional Random Tensors

The performance in terms of minimal Bayes&rsquo; error probability for detection of ahigh-dimensional random tensor is a fundamental under-studied difficult problem. In this work, weconsider two Signal to Noise Ratio (SNR)-based detection problems of interest. Under the alternativehypothesis, i.e., for a non-zero SNR, the observed signals are either a noisy rank-R tensor admitting aQ-order Canonical Polyadic Decomposition (CPD) with large factors of size Nq R, i.e, for 1 q Q,where R, Nq ! &yen; with R1/q/Nq converge towards a finite constant or a noisy tensor admittingTucKer Decomposition (TKD) of multilinear (M1, . . . ,MQ)-rank with large factors of size Nq Mq,i.e, for 1 q Q, where Nq,Mq ! &yen; with Mq/Nq converge towards a finite constant. The detectionof the random entries (coefficients) of the core tensor in the CPD/TKD is hard to study since theexact derivation of the error probability is mathematically intractable. To circumvent this technicaldifficulty, the Chernoff Upper Bound (CUB) for larger SNR and the Fisher information at low SNRare derived and studied, based on information geometry theory. The tightest CUB is reached forthe value minimizing the error exponent, denoted by s?. In general, due to the asymmetry of thes-divergence, the Bhattacharyya Upper Bound (BUB) (that is, the Chernoff Information calculated ats? = 1/2) can not solve this problem effectively. As a consequence, we rely on a costly numericaloptimization strategy to find s?. However, thanks to powerful random matrix theory tools, a simpleanalytical expression of s? is provided with respect to the Signal to Noise Ratio (SNR) in the twoschemes considered. A main conclusion of this work is that the BUB is the tightest bound at lowSNRs. This property is, however, no longer true for higher SNRs.

Decidability for some justification logics with negative introspection

Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.

Hardy-type inequalities for means

In this paper we consider inequalities of the form , Where M is a mean. The main results of the paper offer sufficient conditions on M so that the above inequality holds with a finite constant C. The results obtained extend Hardy's and Carleman's classical inequalities together with their various generalisations in a new dirction.

Constructions for Cubic Graphs with Large Girth

The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer $\mu_0(g)$, the smallest number of vertices for which a cubic graph with girth at least $g$ exists, and furthermore, the minimum value $\mu_0(g)$ is attained by a graph whose girth is exactly $g$. The values of $\mu_0(g)$ when $3 \le g \le 8$ have been known for over thirty years. For these values of $g$ each minimal graph is unique and, apart from the case $g=7$, a simple lower bound is attained. This paper is mainly concerned with what happens when $g \ge 9$, where the situation is quite different. Here it is known that the simple lower bound is attained if and only if $g=12$. A number of techniques are described, with emphasis on the construction of families of graphs $\{ G_i\}$ for which the number of vertices $n_i$ and the girth $g_i$ are such that $n_i\le 2^{cg_i}$ for some finite constant $c$. The optimum value of $c$ is known to lie between $0.5$ and $0.75$. At the end of the paper there is a selection of open questions, several of them containing suggestions which might lead to improvements in the known results. There are also some historical notes on the current-best graphs for girth up to 36.

Wave source potentials for two superposed fluids, each of finite depth

Problems dealing with the generation of internal waves at the surface separating two fluids involves the consideration of different types of singularities in one of the two fluids. In this paper the velocity potentials describing line sources are obtained for the case when each fluid is of finite constant depth, neglecting effects of surface tension at the surface of separation.