Evaluating Matrix Circuits

The circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups is studied. The best upper bound for this problem is $\mathsf{coRP}$, which is shown by a reduction to polynomial identity testing. Conversely, the compressed word problem for the linear group $\mathsf{SL}_3(\mathbb{Z})$ is equivalent to polynomial identity testing. In the paper, it is shown that the compressed word problem for every finitely generated nilpotent group is in $\mathsf{DET} \subseteq \mathsf{NC}^2$. Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits

Complexity of Equivalence and Learning for Multiplicity Tree Automata

We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to the size of the target automaton and the size of a largest counterexample, represented as a tree, that is returned by the Teacher. However, the smallest tree-counterexample may be exponential in the size of the target automaton. Thus the above algorithm does not run in time polynomial in the size of the target automaton, and has query complexity exponential in the lower bound. Assuming a Teacher that returns minimal DAG representations of counterexamples, we give a new exact learning algorithm whose query complexity is quadratic in the target automaton size, almost matching the lower bound, and improving the best previously-known algorithm by an exponential factor

The complexity of satisfaction problems in reverse mathematics

Satisfiability problems play a central role in computer science and engineering as a general framework for studying the complexity of various problems. Schaefer proved in 1978 that truth satisfaction of propositional formulas given a language of relations is either NP-complete or tractable. We classify the corresponding satisfying assignment construction problems in the framework of reverse mathematics and show that the principles are either provable over RCA or equivalent to WKL. We formulate also a Ramseyan version of the problems and state a different dichotomy theorem. However, the different classes arising from this classification are not known to be distinct.Comment: 19 page

Hardness of Sparse Sets and Minimal Circuit Size Problem

We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a $2^{n^{o(1)}}$-sparse set in $NTIME(2^{n^{o(1)}})$ that does not have any randomized streaming algorithm with $n^{o(1)}$ updating time, and $n^{o(1)}$ space, then $NEXP\not=BPP$, where a $f(n)$-sparse set is a language that has at most $f(n)$ strings of length $n$. We also show that if MCSP is $ZPP$-hard under polynomial time truth-table reductions, then $EXP\not=ZPP$

Polynomial Time Algorithms for Branching Markov Decision Processes and Probabilistic Min(Max) Polynomial Bellman Equations

We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic max(min) polynomial equations, referred to as maxPPSs (and minPPSs, respectively), in time polynomial in both the encoding size of the system of equations and in log(1/epsilon), where epsilon > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We establish this result using a generalization of Newton's method which applies to maxPPSs and minPPSs, even though the underlying functions are only piecewise-differentiable. This generalizes our recent work which provided a P-time algorithm for purely probabilistic PPSs. These equations form the Bellman optimality equations for several important classes of infinite-state Markov Decision Processes (MDPs). Thus, as a corollary, we obtain the first polynomial time algorithms for computing to within arbitrary desired precision the optimal value vector for several classes of infinite-state MDPs which arise as extensions of classic, and heavily studied, purely stochastic processes. These include both the problem of maximizing and mininizing the termination (extinction) probability of multi-type branching MDPs, stochastic context-free MDPs, and 1-exit Recursive MDPs. Furthermore, we also show that we can compute in P-time an epsilon-optimal policy for both maximizing and minimizing branching, context-free, and 1-exit-Recursive MDPs, for any given desired epsilon > 0. This is despite the fact that actually computing optimal strategies is Sqrt-Sum-hard and PosSLP-hard in this setting. We also derive, as an easy consequence of these results, an FNP upper bound on the complexity of computing the value (within arbitrary desired precision) of branching simple stochastic games (BSSGs)

The accuracy of breast volume measurement methods: a systematic review

Breast volume is a key metric in breast surgery and there are a number of different methods which measure it. However, a lack of knowledge regarding a method’s accuracy and comparability has made it difficult to establish a clinical standard. We have performed a systematic review of the literature to examine the various techniques for measurement of breast volume and to assess their accuracy and usefulness in clinical practice. Each of the fifteen studies we identified had more than ten live participants and assessed volume measurement accuracy using a gold-standard based on the volume, or mass, of a mastectomy specimen. Many of the studies from this review report large (> 200 ml) uncertainty in breast volume and many fail to assess measurement accuracy using appropriate statistical tools. Of the methods assessed, MRI scanning consistently demonstrated the highest accuracy with three studies reporting errors lower than 10% for small (250 ml), medium (500 ml) and large (1,000 ml) breasts. However, as a high-cost, non-routine assessment other methods may be more appropriate

Second harmonic light scattering induced by defects in the twist-bend nematic phase of liquid crystal dimers

The nematic twist-bend (NTB) phase, exhibited by certain thermotropic liquid crystalline (LC) dimers, represents a new orientationally ordered mesophase -- the first distinct nematic variant discovered in many years. The NTB phase is distinguished by a heliconical winding of the average molecular long axis (director) with a remarkably short (nanoscale) pitch and, in systems of achiral dimers, with an equal probability to form right- and left-handed domains. The NTB structure thus provides another fascinating example of spontaneous chiral symmetry breaking in nature. The order parameter driving the formation of the heliconical state has been theoretically conjectured to be a polarization field, deriving from the bent conformation of the dimers, that rotates helically with the same nanoscale pitch as the director field. It therefore presents a significant challenge for experimental detection. Here we report a second harmonic light scattering (SHLS) study on two achiral, NTB-forming LCs, which is sensitive to the polarization field due to micron-scale distortion of the helical structure associated with naturally-occurring textural defects. These defects are parabolic focal conics of smectic-like ``pseudo-layers", defined by planes of equivalent phase in a coarse-grained description of the NTB state. Our SHLS data are explained by a coarse-grained free energy density that combines a Landau-deGennes expansion of the polarization field, the elastic energy of a nematic, and a linear coupling between the two

Strong Enhancement of Superconducting Correlation in a Two-Component Fermion Gas

We study high-density electron-hole (e-h) systems with the electron density slightly larger than the hole density. We find a new superconducting phase, in which the excess electrons form Cooper pairs moving in an e-h BCS phase. The coexistence of the e-h and e-e orders is possible because e and h have opposite charges, whereas analogous phases are impossible in the case of two fermion species that have the same charge or are neutral. Most strikingly, the e-h order enhances the superconducting e-h order parameter by more than one order of magnitude as compared with that given by the BCS formula, for the same value of the effective e-e attractive potential \lambda^{ee}. This new phase should be observable in an e-h system created by photoexcitation in doped semiconductors at low temperatures.Comment: 5 pages including 5 PostScript figure