Probabilistic Interval Temporal Logic and Duration Calculus with Infinite Intervals: Complete Proof Systems

The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal logic with finite intervals and expanding modalities is established too.Comment: 43 page

Quasiclassical Dynamics in a Closed Quantum System

We consider Gell-Mann and Hartle's consistent histories formulation of quantum cosmology in the interpretation in which one history, chosen randomly according to the decoherence functional probabilities, is realised from each consistent set. We show that in this interpretation, if one assumes that an observed quasiclassical structure will continue to be quasiclassical, one cannot infer that it will obey the predictions of classical or Copenhagen quantum mechanics.Comment: Published version, to appear in Phys. Rev. A. Clarificatory remarks added on interpretations outside the scope of the paper. (TeX with harvmac, 13 pages.

Doob-Martin compactification of a Markov chain for growing random words sequentially

We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the $n^{\mathrm{th}}$ word is uniformly distributed over the set of words of length $2n$ in which $n$ letters are $a$ and $n$ letters are $b$: at each step an $a$ and a $b$ are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain. Writing $N(u)$ for the number of letters $a$ (equivalently, $b$) in the finite word $u$, we show that a sequence $(u_n)_{n \in \mathbb{N}}$ of finite words converges to a point in the boundary if, for an arbitrary word $v$, there is convergence as $n$ tends to infinity of the probability that the selection of $N(v)$ letters $a$ and $N(v)$ letters $b$ uniformly at random from $u_n$ and maintaining their relative order results in $v$. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set $\{a_1, b_1, a_2, b_2, \ldots \}$ that have distributions which are separately invariant under finite permutations of the indices of the $a'$s and those of the $b'$s. We establish a further bijective correspondence between the set of such random total orders and the set of pairs $(\mu,\nu)$ of diffuse probability measures on $[0,1]$ such that $\frac{1}{2}(\mu+\nu)$ is Lebesgue measure: the restriction of the random total order to $\{a_1, b_1, \ldots, a_n, b_n\}$ is obtained by taking $X_1, \ldots, X_n$ (resp. $Y_1, \ldots, Y_n$) i.i.d. with common distribution $\mu$ (resp. $\nu$), letting $(Z_1, \ldots, Z_{2n})$ be $\{X_1, Y_1, \ldots, X_n, Y_n\}$ in increasing order, and declaring that the $k^{\mathrm{th}}$ smallest element in the restricted total order is $a_i$ (resp. $b_j$) if $Z_k = X_i$ (resp. $Z_k = Y_j$).Comment: 24 pages, revised to deal with reviewer's comment

A Basic Compositional Model for Spiking Neural Networks

This paper is part of a project on developing an algorithmic theory of brain networks, based on stochastic Spiking Neural Network (SNN) models. Inspired by tasks that seem to be solved in actual brains, we are defining abstract problems to be solved by these networks. In our work so far, we have developed models and algorithms for the Winner-Take-All problem from computational neuroscience [LMP17a,Mus18], and problems of similarity detection and neural coding [LMP17b]. We plan to consider many other problems and networks, including both static networks and networks that learn. This paper is about basic theory for the stochastic SNN model. In particular, we define a simple version of the model. This version assumes that the neurons' only state is a Boolean, indicating whether the neuron is firing or not. In later work, we plan to develop variants of the model with more elaborate state. We also define an external behavior notion for SNNs, which can be used for stating requirements to be satisfied by the networks. We then define a composition operator for SNNs. We prove that our external behavior notion is "compositional", in the sense that the external behavior of a composed network depends only on the external behaviors of the component networks. We also define a hiding operator that reclassifies some output behavior of an SNN as internal. We give basic results for hiding. Finally, we give a formal definition of a problem to be solved by an SNN, and give basic results showing how composition and hiding of networks affect the problems that they solve. We illustrate our definitions with three examples: building a circuit out of gates, building an "Attention" network out of a "Winner-Take-All" network and a "Filter" network, and a toy example involving combining two networks in a cyclic fashion