Tropically convex constraint satisfaction

A semilinear relation S is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in the intersection of NP and co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinaer constraints in general into NP intersected co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

Differential Allocation of Constitutive and Induced Chemical Defenses in Pine Tree Juveniles: A Test of the Optimal Defense Theory

Optimal defense theory (ODT) predicts that the within-plant quantitative allocation of defenses is not random, but driven by the potential relative contribution of particular plant tissues to overall fitness. These predictions have been poorly tested on long-lived woody plants. We explored the allocation of constitutive and methyl-jasmonate (MJ) inducible chemical defenses in six half-sib families of Pinus radiata juveniles. Specifically, we studied the quantitative allocation of resin and polyphenolics (the two major secondary chemicals in pine trees) to tissues with contrasting fitness value (stem phloem, stem xylem and needles) across three parts of the plants (basal, middle and apical upper part), using nitrogen concentration as a proxy of tissue value. Concentration of nitrogen in the phloem, xylem and needles was found to be greater higher up the plant. As predicted by the ODT, the same pattern was found for the concentration of non-volatile resin in the stem. However, in leaf tissues the concentrations of both resin and total phenolics were greater towards the base of the plant. Two weeks after MJ application, the concentrations of nitrogen in the phloem, resin in the stem and total phenolics in the needles increased by roughly 25% compared with the control plants, inducibility was similar across all plant parts, and families differed in the inducibility of resin compounds in the stem. In contrast, no significant changes were observed either for phenolics in the stems, or for resin in the needles after MJ application. Concentration of resin in the phloem was double that in the xylem and MJ-inducible, with inducibility being greater towards the base of the stem. In contrast, resin in the xylem was not MJ-inducible and increased in concentration higher up the plant. The pattern of inducibility by MJ-signaling in juvenile P. radiata is tissue, chemical-defense and plant-part specific, and is genetically variable

Magnitude and Timing of Leaf Damage Affect Seed Production in a Natural Population of Arabidopsis thaliana (Brassicaceae)

Background: The effect of herbivory on plant fitness varies widely. Understanding the causes of this variation is of considerable interest because of its implications for plant population dynamics and trait evolution. We experimentally defoliated the annual herb Arabidopsis thaliana in a natural population in Sweden to test the hypotheses that (a) plant fitness decreases with increasing damage, (b) tolerance to defoliation is lower before flowering than during flowering, and (c) defoliation before flowering reduces number of seeds more strongly than defoliation during flowering, but the opposite is true for effects on seed size. Methodology/Principal Findings: In a first experiment, between 0 and 75% of the leaf area was removed in May from plants that flowered or were about to start flowering. In a second experiment, 0, 25%, or 50% of the leaf area was removed from plants on one of two occasions, in mid April when plants were either in the vegetative rosette or bolting stage, or in mid May when plants were flowering. In the first experiment, seed production was negatively related to leaf area removed, and at the highest damage level, also mean seed size was reduced. In the second experiment, removal of 50% of the leaf area reduced seed production by 60% among plants defoliated early in the season at the vegetative rosettes, and by 22% among plants defoliated early in the season at the bolting stage, but did not reduce seed output of plants defoliated one month later. No seasonal shift in the effect of defoliation on seed size was detected. Conclusions/Significance: The results show that leaf damage may reduce the fitness of A. thaliana, and suggest that in this population leaf herbivores feeding on plants before flowering should exert stronger selection on defence traits than those feeding on plants during flowering, given similar damage levels

An Imperfect Dopaminergic Error Signal Can Drive Temporal-Difference Learning

An open problem in the field of computational neuroscience is how to link synaptic plasticity to system-level learning. A promising framework in this context is temporal-difference (TD) learning. Experimental evidence that supports the hypothesis that the mammalian brain performs temporal-difference learning includes the resemblance of the phasic activity of the midbrain dopaminergic neurons to the TD error and the discovery that cortico-striatal synaptic plasticity is modulated by dopamine. However, as the phasic dopaminergic signal does not reproduce all the properties of the theoretical TD error, it is unclear whether it is capable of driving behavior adaptation in complex tasks. Here, we present a spiking temporal-difference learning model based on the actor-critic architecture. The model dynamically generates a dopaminergic signal with realistic firing rates and exploits this signal to modulate the plasticity of synapses as a third factor. The predictions of our proposed plasticity dynamics are in good agreement with experimental results with respect to dopamine, pre- and post-synaptic activity. An analytical mapping from the parameters of our proposed plasticity dynamics to those of the classical discrete-time TD algorithm reveals that the biological constraints of the dopaminergic signal entail a modified TD algorithm with self-adapting learning parameters and an adapting offset. We show that the neuronal network is able to learn a task with sparse positive rewards as fast as the corresponding classical discrete-time TD algorithm. However, the performance of the neuronal network is impaired with respect to the traditional algorithm on a task with both positive and negative rewards and breaks down entirely on a task with purely negative rewards. Our model demonstrates that the asymmetry of a realistic dopaminergic signal enables TD learning when learning is driven by positive rewards but not when driven by negative rewards

The Dichotomy for Conservative Constraint Satisfaction is Polynomially Decidable

International audienceGiven a fixed constraint language Γ , the conservative CSP over Γ (denoted by c-CSP(Γ)) is a variant of CSP(Γ) where the domain of each variable can be restricted arbitrarily. In [5] a dichotomy has been proven for conservative CSP: for every fixed language Γ , c-CSP(Γ) is either in P or NP-complete. However, the characterization of conservatively tractable languages is of algebraic nature and the recognition algorithm provided in [5] is super-exponential in the domain size. The main contribution of this paper is a polynomial-time algorithm that, given a constraint language Γ as input, decides if c-CSP(Γ) is tractable. In addition, if Γ is proven tractable the algorithm also outputs its coloured graph, which contains valuable information on the structure of Γ