Probabilistic modal {\mu}-calculus with independent product

The probabilistic modal {\mu}-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic parity games. A shortcoming of the probabilistic modal {\mu}-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic modal {\mu}-calculus with independent product, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic {\mu}-calculus. This relies on the definition of a new class of games which generalize standard two-player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal

The Phillips Curve and Underlying Inflation

This paper examines methods of controlling the supply shock in the estimation of the Phillips curve and discusses the relationship between the supply shock and inflation inertia. The empirical results clearly show that controlling the supply shock effect, not only for current inflation but also for lagged inflation using the asymmetry of the price change distribution, substantially outperforms the traditional method in terms of the robustness to alternative lag specifications, predictive power, and parameter stability for changes in the estimation period, which are the essential properties for the practical use of the Phillips curve. These results suggest that (1) because supply shocks hit broad sectors, it is not appropriate to restrict the proxy for the supply shock to the relative price changes of a fixed commodity basket; and (2) the inflation inertia corresponds to the underlying inflation from which the supply shock effect has been eliminated.

Identifying Aggregate Demand and Aggregate Supply Components of Inflation Rate: A Structural Vector Autoregression Analysis for Japan

I estimate a bivariate output-price structural vector autoregression (VAR) model for Japan to decompose inflation rate time-series into two components explained by aggregate demand (AD) and aggregate supply (AS) shocks. For the modelfs identifying restriction, I assume that the long- run elasticity of output with respect to permanent changes in price due to AD shocks is zero; i.e., an AD shock has no long-run impact on the level of output. Dynamic properties of the estimated model are shown to be generally consistent with the predictions of the conventional AS-AD framework. The main features of the historical decomposition are the following: (1) the inflation rate explained by the AD shock shows a procyclical swing since 1970; (2) the inflation rate explained by the AS shock temporarily spikes during the two oil crises and experiences a large countercyclical swing in the 1990s; and (3) the coincidence of large and negative AS and AD shocks explains the combination of price stability and output stagnation during two recessions in the 1990s. These results are qualitatively robust to the sectoral shocks, alternative choices for the price variable, and assumptions for the lag length of VAR and the long-run elasticity of output with respect to permanent changes in price due to AD shocks. However, the bivariate approach does not allow the identification of more than three types of shocks with different dynaic effects on output and price. It might be necessary to expand the model to deal with this limitation.

On the Problem of Computing the Probability of Regular Sets of Trees

We consider the problem of computing the probability of regular languages of infinite trees with respect to the natural coin-flipping measure. We propose an algorithm which computes the probability of languages recognizable by \emph{game automata}. In particular this algorithm is applicable to all deterministic automata. We then use the algorithm to prove through examples three properties of measure: (1) there exist regular sets having irrational probability, (2) there exist comeager regular sets having probability $0$ and (3) the probability of \emph{game languages} $W_{i,k}$, from automata theory, is $0$ if $k$ is odd and is $1$ otherwise

Generic Entanglement Entropy for Quantum States with Symmetry

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.Comment: ver 1: 8 pages, 2 figures, ver 2: substantially updated, 19 pages, and 2 figure

Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation

We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.Comment: Section II and VIII adde