First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Transcendental number theory, by Alan Baker. Pp. x, 147. £4·90. 1975. SBN 0 521 20461 5 (Cambridge University Press)

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

We consider turn-based stochastic games on infinite graphs induced by game probabilistic lossy channel systems (GPLCS), the game version of probabilistic lossy channel systems (PLCS). We study games with Buchi (repeated reachability) objectives and almost-sure winning conditions. These games are pure memoryless determined and, under the assumption that the target set is regular, a symbolic representation of the set of winning states for each player can be effectively constructed. Thus, turn-based stochastic games on GPLCS are decidable. This generalizes the decidability result for PLCS-induced Markov decision processes in [10].

/pdf/stochastic-games-with-lossy-channels-4lfbmxienp.pdf

Stochastic games with lossy channels

https://www.georgejpappas.org/papers/TCS-FP09.pdf

Robustness of temporal logic specifications for continuous-time signals

This paper presents algorithms for the automatic synthesis of real-time controllers by finding a winning strategy for certain games defined by the timed-automata of Alur and Dill. In such games, the outcome depends on the players' actions as well as on their timing. We believe that these results will pave the way for the application of program synthesis techniques to the construction of real-time embedded systems from their specifications.

On the synthesis of discrete controllers for timed systems

Many protocols are designed to operate correctly even in the case where the underlying communication medium is faulty. To capture the behavior of such protocols, Lossy Channel Systems (LCS's) have been proposed. In an LCS the communication channels are modeled as unbounded FIFO buffers which are unreliable in the sense that they can nondeterministically lose messages. Recently, several attempts have been made to study Probabilistic Lossy Channel Systems (PLCS's) in which the probability of losing messages is taken into account. In this article, we consider a variant of PLCS's which is more realistic than those studied previously. More precisely, we assume that during each step in the execution of the system, each message may be lost with a certain predefined probability. We show that for such systems the following model-checking problem is decidable: to verify whether a linear-time property definable by a finite-state @w-automaton holds with probability one. We also consider other types of faulty behavior, such as corruption and duplication of messages, and insertion of new messages, and show that the decidability results extend to these models.

Verification of probabilistic systems with faulty communication

Many protocols are designed to operate correctly even in the case where the underlying communication medium is faulty. To capture the behaviour of such protocols, lossy channel systems (LCS) [AJ96b] have been proposed. In an LCS the communication channels are modelled as FIFO buffers which are unbounded, but also unreliable in the sense that they can nondeterministically lose messages.

Recently, several attempts [BE99, ABIJ00] have been made to study probabilistic Lossy Channel Systems (PLCS) in which the probability of losing messages is taken into account. In this paper, we consider a variant of PLCS which is more realistic than those studied in [BE99, ABIJ00]. More precisely, we assume that during each step in the execution of the system, each message may be lost with a certain predefined probability. We show that for such systems the following model checking problem is decidable: to verify whether a given property definable by finite state ω-automata holds with probability one. We also consider other types of faulty behavior, such as corruption and duplication of messages, and insertion of new messages, and show that the decidability results extend to these models.

/pdf/verification-of-probabilistic-systems-with-faulty-1cir3tzu41.pdf

Monadic second-order logic (MSOL) provides a general framework for expressing properties of reactive systems as modelled by trees. Monadic path logic (MPL) is obtained by restricting second-order quantification to paths reflecting computation sequences. In this paper we show that the expressive power of MPL over trees coincides with the usual branching time logic CTL* embellished with a simple form of counting. As a corollary, we derive an earlier result that CTL* coincides with the bisimulation-invariant properties of MPL. In order to prove the main result, we first prove a new Composition Theorem for trees.

/pdf/counting-on-ctl-on-the-expressive-power-of-monadic-path-2lgw5r9vbh.pdf

Counting on CTL: on the expressive power of monadic path logic

We study two variants of infinite games with imperfect information. In the first variant, in each round player-1 may decide to hide his move from player-2. This captures situations where the input signal is subject to fluctuations (noises), and every error in the input signal can be detected by the controller. In the second variant, all of player- 1 moves are visible to player-2; however, after the game ends, player-1 may change some of his moves. This captures situations where the input signal is subject to fluctuations; however, the controller cannot detect errors in the input signal.

We consider several cases, according to the amount of errors allowed in the input signal: a fixed number of errors, finitely many errors and the case where the rate of errors is bounded by a threshold. For each of these cases we consider games with regular and mean-payoff winning conditions. We investigate the decidability of these games.

There is a natural reduction for some of these games to (perfect information) multidimensional mean-payoff games recently considered in [7]. However, the decidability of the winner of multidimensional mean-payoff games was stated as an open question. We prove its decidability and provide tight complexity bounds.

/pdf/church-synthesis-problem-for-noisy-input-2tq5uenpi6.pdf

Church synthesis problem for noisy input

Many protocols are designed to operate correctly even in the case where the underlying communication medium is faulty. To capture the behaviour of such protocols, lossy channel systems (LCS) [3] have been proposed. In an LCS the communication channels are modelled as FIFO buffers which are unbounded, but also unreliable in the sense that they can nondeterministically lose messages.

Recently, several attempts [5,1,4,6] have been made to study Probabilistic Lossy Channel Systems (PLCS) in which the probability of losing messages is taken into account and the following qualitative model checking problem is investigated: to verify whether a given property holds with probability one.

Here we consider a more challenging problem, namely to calculate the probability by which a certain property is satisfied. Our main result is an algorithm for the following Quantitative model checking problem: Instance: A PLCS, its state s, a finite state ω-automaton A, and a rational θ > 0. Task: Find a rational r such that the probability of the set of computations that start at s and are accepted by A is between r and r + θ.

Alexander Rabinovich

Papers

Verification of probabilistic systems with faulty communication

Verification of probabilistic systems with faulty communication

Counting on CTL: on the expressive power of monadic path logic

Church synthesis problem for noisy input

Quantitative analysis of probabilistic lossy channel systems