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Register a new userA non extensive approach to the entropy of symbolic sequences
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Symbolic sequences with long-range correlations are expected to result in a slow regression to a steady state of entropy increase. However, we prove that also in this case a fast transition to a constant rate of entropy increase can be obtained, provided that the extensive entropy of Tsallis with entropic index q is adopted, thereby resulting in a new form of entropy that we shall refer to as Kolmogorov-Sinai-Tsallis (KST) entropy. We assume that the same symbols, either 1 or -1, are repeated in strings of length l, with the probability distribution p(l) proportional to 1/(l^mu). The numerical evaluation of the KST entropy suggests that at the value mu = 2 a sort of abrupt transition might occur. For the values of mu in the range 1<mu<2 the entropic index q is expected to vanish, as a consequence of the fact that in this case the average length <l> diverges, thereby breaking the balance between determinism and randomness in favor of determinism. In the region mu > 2 the entropic index q seems to depend on mu through the power law expression q = (mu-2)^(alpha) with alpha approximately 0.13 (q = 1 with mu > 3). It is argued that this phase-transition like property signals the onset of the thermodynamical regime at mu = 2.
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Symbolic relative entropy, an efficient nonlinear complexity parameter measuring probabilistic divergences of symbolic sequences, is proposed in our nonlinear dynamics analysis of heart rates considering equal states. Equalities are not rare in discrete heartbeats because of the limits of resolution of signals collection, and more importantly equal states contain underlying important cardiac regulation information which is neglected by some chaotic deterministic parameters and temporal asymmetric measurements. The relative entropy of symbolization associated with equal states has satisfied nonlinear dynamics complexity detections in heartbeats and shows advantages to some nonlinear dynamics parameters without considering equalities. Researches on cardiac activities suggest the highest probabilistic divergence of the healthy young heart rates and highlight the facts that heart diseases and aging reduce the nonlinear dynamical complexity of heart rates.
Symbolic execution is a powerful technique for program analysis. However, it has many limitations in practical applicability: the path explosion problem encumbers scalability, the need for language-specific implementation, the inability to handle complex dependencies, and the limited expressiveness of theories supported by underlying satisfiability checkers. Often, relationships between variables of interest are not expressible directly as purely symbolic constraints. To this end, we present a new approach -- neuro-symbolic execution -- which learns an approximation of the relationship as a neural net. It features a constraint solver that can solve mixed constraints, involving both symbolic expressions and neural network representation. To do so, we envision such constraint solving as procedure combining SMT solving and gradient-based optimization. We demonstrate the utility of neuro-symbolic execution in constructing exploits for buffer overflows. We report success on 13/14 programs which have difficult constraints, known to require specialized extensions to symbolic execution. In addition, our technique solves $100$% of the given neuro-symbolic constraints in $73$ programs from standard verification and invariant synthesis benchmarks.
Statistical thermodynamics of small systems shows dramatic differences from normal systems. Parallel to the recently presented steady-state thermodynamic formalism for master equation and Fokker-Planck equation, we show that a ``thermodynamic theory can also be developed based on Tsallis generalized entropy $S^{(q)}=sum_{i=1}^N(p_i-p_i^q)/[q(q-1)]$ and Shiinos generalized free energy $F^{(q)}=[sum_{i=1}^Np_i(p_i/pi_i)^{q-1}-1]/[q(q-1)]$, where $pi_i$ is the stationary distribution. $dF^{(q)}/dt=-f_d^{(q)}le 0$ and it is zero iff the system is in its stationary state. $dS^{(q)}/dt-Q_{ex}^{(q)} = f_d^{(q)}$ where $Q_{ex}^{(q)}$ characterizes the heat exchange. For systems approaching equilibrium with detailed balance, $f_d^{(q)}$ is the product of Onsagers thermodynamic flux and force. However, it is discovered that the Onsagers force is non-local. This is a consequence of the particular transformation invariance for zero energy of Tsallis statistics.
A measure called Physical Complexity is established and calculated for a population of sequences, based on statistical physics, automata theory, and information theory. It is a measure of the quantity of information in an organisms genome. It is based on Shannons entropy, measuring the information in a population evolved in its environment, by using entropy to estimate the randomness in the genome. It is calculated from the difference between the maximal entropy of the population and the actual entropy of the population when in its environment, estimated by counting the number of fixed loci in the sequences of a population. Up to now, Physical Complexity has only been formulated for populations of sequences with the same length. Here, we investigate an extension to support variable length populations. We then build upon this to construct a measure for the efficiency of information storage, which we later use in understanding clustering within populations. Finally, we investigate our extended Physical Complexity through simulations, showing it to be consistent with the original.
In quantum many-body systems, a Hamiltonian is called an ``extensive entropy generator if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. Specializing to many-body localized systems, these results imply the observation stated in the title of Bardarson et al. [PRL 109, 017202 (2012)].
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