String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes ("bang boxes"), on both sides of a rule can be copied any number of times or removed. After reviewing the string graph formalism, we show how string graphs can be extended to pattern graphs and how pattern graphs and pattern rewrite rules can be instantiated to concrete string graphs and rewrite rules. We then provide examples demonstrating the expressive power of pattern graphs and how they can be applied to study interacting algebraic structures that are central to categorical quantum mechanics.Comment: In Proceedings DCM 2012, arXiv:1403.757
We propose a coalgebraic model for constructing and reasoning about state-based protocols that implement efficient reductions among random processes. We provide basic tools that allow efficient protocols to be constructed in a compositional way and analyzed in terms of the tradeoff between latency and loss of entropy. We show how to use these tools to construct various entropy-conserving reductions between processes.
We present a new algorithm for multi-region segmentation of 2D images with objects that may partially occlude each other. Our algorithm is based on the observation that human performance on this task is based both on prior knowledge about plausible shapes and taking into account the presence of occluding objects whose shape is already known -once an occluded region is identified, the shape prior can be used to guess the shape of the missing part. We capture the former aspect using a deep learning model of shape; for the latter, we simultaneously minimize the energy of all regions and consider only unoccluded pixels for data agreement.Existing algorithms incorporating object shape priors consider every object separately in turn and can't distinguish genuine deviation from the expected shape from parts missing due to occlusion. We show that our method significantly improves on the performance of a representative algorithm, as evaluated on both preprocessed natural and synthetic images. Furthermore, on the synthetic images, we recover the ground truth segmentation with good accuracy.
We explore the impact of mutual altruism among the players belonging to the same set -their tribe -in a partition of all players in arbitrary strategic games upon the quality of equilibria attained. To this end, we introduce the notion of a τ -tribal extension of an arbitrary strategic game, in which players' subjective cost functions are updated to reflect this, and the associated Price of Tribalism, which is the ratio of the social welfare of the worst Nash equilibrium of the tribal extension to that of the optimum of social welfare. We show that in a well-known game of friendship cliques, network contribution games as well as atomic linear congestion games, the Price of Tribalism is higher than the Price of Anarchy of either the purely selfish players or fully altruistic players (i.e. ones who seek to maximise the social welfare). This phenomenon is observed under a variety of equilibrium concepts. In each instance, we present upper bounds on the Price of Tribalism that match the lower bounds established by our example.
We introduce a theoretical model of information acquisition under resource limitations in a noisy environment. An agent must guess the truth value of a given Boolean formula φ after performing a bounded number of noisy tests of the truth values of variables in the formula. We observe that, in general, the problem of finding an optimal testing strategy for φ is hard, but we suggest a useful heuristic. The techniques we use also give insight into two apparently unrelated, but well-studied problems: (1) rational inattention , that is, when it is rational to ignore pertinent information (the optimal strategy may involve hardly ever testing variables that are clearly relevant to φ), and (2) what makes a formula hard to learn/remember.
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