Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix(TM) rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix(TM) rotation puzzle problem is as hard as the unique satisfiability problem, and so is DP-complete under polynomial-time randomized reductions, where DP is the second level of the boolean hierarchy over NP.Comment: 19 pages, 16 figures, appears in the Proceedings of "Machines, Computations and Universality" (MCU 2007

The Three-Color and Two-Color Tantrix(TM) Rotation Puzzle Problems are NP-Complete via Parsimonious Reductions

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix(TM) rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix(TM) tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.Comment: 30 pages, 25 figure

Acceptance in Incomplete Argumentation Frameworks

A Abstract argumentation frameworks (AFs), originally proposed by Dung, constitute a central formal model for the study of computational aspects of argumentation in AI. Credulous and skeptical acceptance of arguments in a given AF are well-studied problems both in terms of theoretical analysis-especially computational complexity-and the development of practical decision procedures for the problems. However, AFs make the assumption that all attacks between arguments are certain (i.e., present attacks are known to exist, and missing attacks are known to not exist), which can in various settings be a restrictive assumption. A generalization of AFs to incomplete AFs was recently proposed as a formalism that allows the representation of both uncertain attacks and uncertain arguments in AFs. In this article, we explore the impact of allowing for modeling such uncertainties in AFs on the computational complexity of natural generalizations of acceptance problems to incomplete AFs under various central AF semantics. Complementing the complexity-theoretic analysis, we also develop the first practical decision procedures for all of the NP-hard variants of acceptance in incomplete AFs. In terms of complexity analysis, we establish a full complexity landscape, showing that depending on the variant of acceptance and property/semantics, the complexity of acceptance in incomplete AFs ranges from polynomial-time decidable to completeness for Sigma(p)(3). In terms of algorithms, we show through an extensive empirical evaluation that an implementation of the proposed decision procedures, based on boolean satisfiability (SAT) solving, is effective in deciding variants of acceptance under uncertainties. We also establish conditions for what type of atomic changes are guaranteed to be redundant from the perspective of preserving extensions of completions of incomplete AFs, and show that the results allow for considerably improving the empirical efficiency of the proposed SAT-based counterexample-guided abstraction refinement algorithms for acceptance in incomplete AFs for problem variants with complexity beyond NP. (C) 2021 The Authors. Published by Elsevier B.V.Peer reviewe

Proposal for Quantum Simulation via All-Optically Generated Tensor Network States

We devise an all-optical scheme for the generation of entangled multimode photonic states encoded in temporal modes of light. The scheme employs a nonlinear down-conversion process in an optical loop to generate one- and higher-dimensional tensor network states of light. We illustrate the principle with the generation of two different classes of entangled tensor network states and report on a variational algorithm to simulate the ground-state physics of many-body systems. We demonstrate that state-of-the-art optical devices are capable of determining the ground-state properties of the spin-1/2 Heisenberg model. Finally, implementations of the scheme are demonstrated to be robust against realistic losses and mode mismatch.Comment: 6 pages main text plus 6 pages Supplementary Material and many figures. Updated to published version. Comments welcom

The Complexity of Computing Minimal Unidirectional Covering Sets

Given a binary dominance relation on a set of alternatives, a common thread in the social sciences is to identify subsets of alternatives that satisfy certain notions of stability. Examples can be found in areas as diverse as voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08] proved that it is NP-hard to decide whether an alternative is contained in some inclusion-minimal upward or downward covering set. For both problems, we raise this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other natural problems regarding minimal or minimum-size covering sets are hard or complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of our results is that neither minimal upward nor minimal downward covering sets (even when guaranteed to exist) can be computed in polynomial time unless P=NP. This sharply contrasts with Brandt and Fischer's result that minimal bidirectional covering sets (i.e., sets that are both minimal upward and minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure

Processing of social and monetary rewards in autism spectrum disorders

Background: Reward processing has been proposed to underpin the atypical social feature of autism spectrum disorder (ASD). However, previous neuroimaging studies have yielded inconsistent results regarding the specificity of atypicalities for social reward processing in ASD. Aims: Utilising a large sample, we aimed to assess reward processing in response to reward type (social, monetary) and reward phase (anticipation, delivery) in ASD. Method: Functional magnetic resonance imaging during social and monetary reward anticipation and delivery was performed in 212 individuals with ASD (7.6-30.6 years of age) and 181 typically developing participants (7.6-30.8 years of age). Results: Across social and monetary reward anticipation, whole-brain analyses showed hypoactivation of the right ventral striatum in participants with ASD compared with typically developing participants. Further, region of interest analysis across both reward types yielded ASD-related hypoactivation in both the left and right ventral striatum. Across delivery of social and monetary reward, hyperactivation of the ventral striatum in individuals with ASD did not survive correction for multiple comparisons. Dimensional analyses of autism and attention-deficit hyperactivity disorder (ADHD) scores were not significant. In categorical analyses, post hoc comparisons showed that ASD effects were most pronounced in participants with ASD without co-occurring ADHD. Conclusions: Our results do not support current theories linking atypical social interaction in ASD to specific alterations in social reward processing. Instead, they point towards a generalised hypoactivity of ventral striatum in ASD during anticipation of both social and monetary rewards. We suggest this indicates attenuated reward seeking in ASD independent of social content and that elevated ADHD symptoms may attenuate altered reward seeking in ASD