Approximating Holant problems by winding

We give an FPRAS for Holant problems with parity constraints and not-all-equal constraints, a generalisation of the problem of counting sink-free-orientations. The approach combines a sampler for near-assignments of "windable" functions -- using the cycle-unwinding canonical paths technique of Jerrum and Sinclair -- with a bound on the weight of near-assignments. The proof generalises to a larger class of Holant problems; we characterise this class and show that it cannot be extended by expressibility reductions. We then ask whether windability is equivalent to expressibility by matchings circuits (an analogue of matchgates), and give a positive answer for functions of arity three

Baumgarten on Sensible Perfection

One of the most important concepts Baumgarten introduces in his Reflections on Poetry is the concept of sensible perfection. It is surprising that Baumgarten does not elaborate upon this concept in his Metaphysics, since it plays such an important role in the new science of aesthetics that he proposes at the end of the Reflections on Poetry and then further develops in the Aesthetics. This article considers the significance of the absence of sensible perfection from the Metaphysics and its implications for Baumgarten’s aesthetics, before turning to the use Meier and Kant make of Baumgarten’s concept. In the end, this article shows that Baumgarten did not abandon his conception of sensible perfection in the Metaphysics, though its influence declined significantly after Kant rejected the idea that sensibility and the understanding could be distinguished by the perfections of their cognition.info:eu-repo/semantics/publishedVersio

The computational complexity of approximation of partition functions

This thesis studies the computational complexity of approximately evaluating partition functions. For various classes of partition functions, we investigate whether there is an FPRAS: a fully polynomial randomised approximation scheme. In many of these settings we also study “expressibility”, a simple notion of defining a constraint by combining other constraints, and we show that the results cannot be extended by expressibility reductions alone. The main contributions are: -� We show that there is no FPRAS for evaluating the partition function of the hard-core gas model on planar graphs at fugacity 312, unless RP = NP. -� We generalise an argument of Jerrum and Sinclair to give FPRASes for a large class of degree-two Boolean #CSPs. -� We initiate the classification of degree-two Boolean #CSPs where the constraint language consists of a single arity 3 relation. -� We show that the complexity of approximately counting downsets in directed acyclic graphs is not affected by restricting to graphs of maximum degree three. -� We classify the complexity of degree-two #CSPs with Boolean relations and weights on variables. -� We classify the complexity of the problem #CSP(F) for arbitrary finite domains when enough non-negative-valued arity 1 functions are in the constraint language. -� We show that not all log-supermodular functions can be expressed by binary logsupermodular functions in the context of #CSPs

Beyond the Analytic of Finitude: Kant, Heidegger, Foucault

The editors of the French edition of Michel Foucault's Introduction to Kant's Anthropology claim that Foucault started rereading Kant through Nietzsche in 1952 and then began rereading Kant and Nietzsche through Heidegger in 1953. This claim has not received much attention in the scholarly literature, but its significance should not be underestimated. In this article, I assess the likelihood that the editor’s claim is true and show how Foucault’s introduction to Kant’s Anthropology and his comments about Kant in The Order of Things echo the concerns about finitude and subjectivity in Heidegger’s Kant and the Problem of Metaphysics. I then argue that Foucault's later preoccupation with Kant's essay An Answer to the Question: What is Enlightenment? should be regarded as an attempt to develop an alternative to the Heideggerian interpretation of Kant, and the preoccupation with finitude, that had played such an important role in Foucault’s earlier works

Boolean approximate counting CSPs with weak conservativity, and implications for ferromagnetic two-spin

We analyse the complexity of approximate counting constraint satisfactions problems $\mathrm{\#CSP}(\mathcal{F})$, where $\mathcal{F}$ is a set of nonnegative rational-valued functions of Boolean variables. A complete classification is known in the conservative case, where $\mathcal{F}$ is assumed to contain arbitrary unary functions. We strengthen this result by fixing any permissive strictly increasing unary function and any permissive strictly decreasing unary function, and adding only those to $\mathcal{F}$: this is weak conservativity. The resulting classification is employed to characterise the complexity of a wide range of two-spin problems, fully classifying the ferromagnetic case. In a further weakening of conservativity, we also consider what happens if only the pinning functions are assumed to be in $\mathcal{F}$ (instead of the two permissive unaries). We show that any set of functions for which pinning is not sufficient to recover the two kinds of permissive unaries must either have a very simple range, or must satisfy a certain monotonicity condition. We exhibit a non-trivial example of a set of functions satisfying the monotonicity condition.Comment: 37 page

The complexity of approximating conservative counting CSPs

We study the complexity of approximately solving the weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, there is a classification known for the case in which the domain of functions in F is Boolean. In this paper, we give a classification for the more general problem where functions in F have an arbitrary finite domain. We define the notions of weak log-modularity and weak log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem of counting independent sets in bipartite graphs. #BIS is complete with respect to approximation-preserving reductions for a logically-defined complexity class #RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is NP-hard to approximate. Finally, we give a full trichotomy for the arity-2 case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in difficulty to #SAT, the problem of approximately counting the satisfying assignments of a Boolean formula in conjunctive normal form. We also discuss the algorithmic aspects of our classification.Comment: Minor revisio