Stochastic Sequential Machines Research Articles

On the complexity of minimizing probabilistic and quantum automata

Several types of automata, such as probabilistic and quantum automata, require to work with real and complex numbers. For such automata the acceptance of an input is quantified with a probability. There are plenty of results in the literature addressing the complexity of checking the equivalence of these automata, that is, checking whether two automata accept all inputs with the same probability. On the other hand, the critical problem of finding the minimal automata equivalent to a given one has been left open [C. Moore, J.P. Crutchfield, Quantum automata and quantum grammars, Theoret. Comput. Sci. 237 (2000) 275–306, see p. 304, Problem 5]. In this work, we reduce the minimization problem of probabilistic and quantum automata to finding a solution of a system of algebraic polynomial (in)equations. An EXPSPACE upper bound on the complexity of the minimization problem is derived by applying Renegarʼs algorithm. More specifically, we show that the state minimization of probabilistic automata, measure-once quantum automata, measure-many quantum automata, measure-once generalized quantum automata, and measure-many generalized quantum automata is decidable and in EXPSPACE. Finally, we also solve an open problem concerning minimal covering of stochastic sequential machines [A. Paz, Introduction to Probabilistic Automata, Academic Press, New York, 1971, p. 43].

Another approach to the equivalence of measure-many one-way quantum finite automata and its application

In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of measure many one-way quantum finite automata (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J.W. Carlyle, Reduced forms for stochastic sequential machines, J. Math. Anal. Appl. 7 (1963) 167–175]. Namely, we reduce the equivalence problem of MM-1QFAs to that of two (initial) vectors. As an application of the approach, we utilize it to address the equivalence problem of enhanced one-way quantum finite automata (E-1QFAs) introduced by Nayak [A. Nayak, Optimal lower bounds for quantum automata and random access codes, in: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), 1999, pp. 369–376]. We prove that two E-1QFAs A1 and A2 over Σ are equivalence if and only if they are n12+n22−1-equivalent where n1 and n2 are the numbers of states in A1 and A2, respectively. As an important consequence, we obtain that it is decidable whether or not L(A1)=L(A2) where L(A)⊆Σ⁎ denotes the set recognizable by MM-1QFA A (or by E-1QFA A) with cutpoint (or with non-strict cutpoint). This also extends a theorem of Eilenberg.

A note on quantum sequential machines

Quantum sequential machines (QSMs) are a quantum version of stochastic sequential machines (SSMs). Recently, we showed that two QSMs M 1 and M 2 with n 1 and n 2 states, respectively, are equivalent iff they are ( n 1 + n 2 ) 2 -equivalent [L.Z. Li, D.W. Qiu, Determination of equivalence between quantum sequential machines, Theoretical Computer Science 358 (2006) 65–74]. However, using this result to check the equivalence is likely to need exponential expected time. In this note, we consider the time complexity of deciding the equivalence between QSMs and related problems. The main results are as follows: (1) We present a polynomial-time algorithm for deciding the equivalence between QSMs, and, if two QSMs are not equivalent, this algorithm will produce an input–output pair with length not more than ( n 1 + n 2 ) 2 . (2) We improve the bound for the equivalence between QSMs from ( n 1 + n 2 ) 2 to n 1 2 + n 2 2 − 1 , by employing Moore and Crutchfield’s method [C. Moore, J.P. Crutchfield, Quantum automata and quantum grammars, Theoretical Computer Science 237 (2000) 275–306. Also quant-ph/9707031, 1997]. In addition, by viewing MO-1QFAs as a special case of QSMs, we briefly discuss the equivalence between MO-1QFAs, where the method used and the result obtained are slightly different from those given by Koshiba [T. Koshiba, Polynomial-time algorithms for the equivalence for one-way quantum finite automata, in: Proceedings of the 12th International Symposium on Algorithms and Computation, ISAAC’2001, Christchurch, New Zealand, in: Lecture Notes in Computer Science, vol. 2223, Springer, Berlin, 2001, pp. 268–278].

Determination of equivalence between quantum sequential machines

Quantum sequential machines (QSMs) that may be viewed as a quantum variant of stochastic sequential machines (SSMs) are one of the important quantum computing models. A very crucial result on SSMs is that two SSMs with n and n ′ states, respectively, and the same input and output alphabets are equivalent if and only if they are ( n + n ′ - 1 ) -equivalent. Therefore, Gudder asked whether or not it holds for QSMs, as an open problem in this direction, since the further study is closely related to this result. Qiu demonstrated that in QSMs this result does not hold, and therefore answered in part the problem. However, as Qiu indicated, the sufficient and necessary conditions for the equivalence between two QSMs are still not discovered. In this paper, we show that if the condition of ( n + n ′ - 1 ) -equivalence is appropriately relaxed, then we can give a sufficient and necessary condition for justifying the equivalence between two QSMs. More precisely, we show that any two QSMs M and M ′ are equivalent if and only if they are ( n + n ′ ) 2 -equivalent, where n and n ′ are, respectively, the numbers of states in M and M ′ . We therefore solve the open problem suggested by Gudder, and provide a basic result for further developing QSMs. As well, we discuss strongly factorizable QSMs and present the conditions for the equivalence between two strongly factorizable QSMs.

Stochastic sequential machine synthesis with application to constrained sequence generation

In power estimation, one is faced with two problems: (1) generating input vector sequences that satisfy a given statistical behavior (in terms of signal probabilities and correlations among bits); (2) making these sequences as short as possible so as to improve the efficiency of power simulators. Stochastic sequential machines (SSMs) can be used to solve both problems. In particular, this paper presents a general procedure for SSM synthesis and describes a new framework for sequence characterization to match designers' needs for sequence generation or compaction. Experimental results demonstrate that compaction ratios of 1–3 orders of magnitude can be obtained without much loss in accuracy of total power estimates.

Fitting probabilistic automata via the em algorithm

Probabilistic automata (PA) or stochastic sequential machines are very popular, because they are general enough to approximate an arbitrary process and at the same time they posses an attractive property of conditional independence which simplifies their applications. Therefore, it is important to develop an algorithm for approximating an arbitrary stochastic process with the PA. Standard fitting techniques are not robust and are difficult to implement, because of the large number of model parameters. The EM algorithm proved to be robust for maximum likelihood parameter estimation from experimental data of special cases of PA. We demonstrate that the EM algorithm can be used for fitting the PA to an arbitrary stochastic process if Kullback-Leibler discrimination information is used as a distance between the processes. We show that the EM algorithm can be used to calculate derivatives which allows us to combine it with other optimization algorithms. We develop also the modified forms of the algorithm for f...

Baum-Welch reestimation formula for coupled sequential machines

Extends the well known Baum-Welch algorithm for the parameter estimation of autonomous initialized stochastic sequential machines (ISSMs) to the case of feed-forward networks of ISSMs where the particular machines are communicating via hidden symbols. The approach should be appropriate if the stochastic process under consideration exhibits a modular structure. >

An Efficient Simplex Coverability Algorithm in E2 with Application to Stochastic Sequential Machines

The problem of determining the existence of a simplex which covers a given convex polytope inside another given convex polytope in two-dimensional Euclidean space is shown to be efficiently solvable, and an effective procedure is derived to find a suitable covering simplex or show that none exists. This solution provides a finite algorithm for satisfying a necessary condition in the search for a simplicial (fewest states) stochastic sequential machine (SSM) which either covers or is covered by a given SSM of rank three. Theorems are proved which establish necessary and sufficient conditions restricting the class of corresponding simplexes through which a search must proceed to those whose vertices lie in the boundary of the bounding polytope and whose facet-supporting flats contain certain specified vertices of the polytope to be covered. The problem is then reduced to testing roots in the finite solution tree of a set of second degree algebraic equations against a finite table of linear constraints. An algorithm to generate the constraint sets for the equations to be solved is also presented along with examples of its application.

Stochastic combinational networks

A new model is defined for combinational networks of probabilistic logic gates which differs from the earlier models of von Neumann and McCulloch in the way in which timing is handled. This differences makes the new model compatible with ordinary deterministic switching theory in that the outputs of any network containing feedforward connections, but not feedback loops, depend only on the network's inputs after a short, fixed time delay, not on any preceding inputs. A general method of analyzing arbitrary combinational networks within the new model is developed using a formalism based on the algebra of stochastic matrices. Methods of simplifying network analysis under certain special circumstances are also given. Probabilistic generalizations of several concepts from ordinary deterministic switching theory are developed in some detail. These include the complement of a function, the dual of a function, a fixed variable, a dummy variable, a symmetric function, and an associative function. De Morgan's law and the principle of duality are also shown to have probabilistic generalizations. Several methods are next developed for synthesizing one-output combinational networks within the new model. Included are generalizations of several basic methods from deterministic switching theory such as the and-or, or-and, and Shannon expansion methods. Several methods of multiple-output network synthesis are also developed, including a set of cascade networks as well as generalizations of the various one-output network synthesis methods. Finally, the concept of a sequential network is introduced, and the relationships between combinational networks, sequential networks, and stochastic sequential machines are developed in some detail.

On homomorphic images of rational stochastic languages

It is shown that a language is recursively enumerable if and only if it is a homomorphic image of a language belonging to a proper subfamily of all rational stochastic languages. Consequently, the homomorphic images of rational stochastic languages form the full principal AFL of all recursively enumerable languages. If only A-free homomorphisms are used, then the result is an intersection-closed AFL, the languages of which are accepted by deterministic linear bounded automata. Rational multistochastic automata and rational asynchronous stochastic sequential machines are machine models for these two AFLs.

Some remarks on multistochastic automata

We introduced in Turakainen (1973) several families of multistochastic languages which are closely related to the homomorphic images of stochastic languages and to asynchronous stochastic sequential machines. This paper has two goals. One is to establish some new results on these language families and stochastic sequential transducers, and the other is to restate the main results of Turakainen (1973) by using a different approach based on some ideas of Ginsburg et al. (1969) concerning the interdependence of AFL operations.

On the size of the set of all reducible stochastic sequential machines

Let us fix the number of states, input and output symbols of stochastic sequential machines (SSM) given over reals. Let us represent SSM naturally as points in the Euclidean space and consider 3 well-known reduction problems. Then, the set of all reducible SSM is of the Lebesgue measure zero (but not the set of all SSM) which also implies that this set is negligible for a large class of probability measures.

On mathematical models of adaptive systems

The aim of this paper is to present a mathematical model for a system to control an object favourably which is unknown for the system—an adaptive system. We make the concept of adaptiveness precise first and consider systems as stochastic sequential machines (SSM's) to answer the question : what kind of logical structure is necessary and sufficient for a SSM to be adaptive. Some necessary conditions for a SSM to be adaptive are shown, and for a specified class of SSM's, the necessary and sufficient conditions to be adaptive are given. Furthermore, some examples of adaptive SSM's are shown.

Cyclic decomposition of finite stochastic systems

Existence conditions and techniques for partitioning the states of finite stochastic systems into cyclic classes are given in this paper. For the special class of preserved finite stochastic systems, such cyclic partitions are shown to be evaluated, explicity, in terms of those cyclic partitions associated with the finite Markov chains embedded in the system. Generalizations and applications of above results to the periodic decomposition of stochastic sequential machines are investigated.

On multistochastic automata

The notion of a multistochastic automaton is defined. Several language families are obtained and investigated. It is proved that a language is multistochastic if and only if it is a homomorphic image of a stochastic language. The same holds also for λ -free homomorphisms and for those multistochastic automata in which spontaneous state transitions are not allowed. These two results are valid for regular substitutions and λ -free regular substitutions, too. Some closure properties of the considered language families are established. Finally, it is shown that multistochastic automata and stochastic sequential machines are closely related to each other.

State estimation algorithms for non-linear stochastic sequential machines

The problem of estimating the state sequence for a number of stochastic sequential machine models is considered. Basically, all models are assumed to have the structure commonly used in the stochastic control field, i.e. that of a deterministic system corrupted by random disturbances. A Bayesian sequential information processing approach is followed which is most convenient for dealing with finite-state systems. The resulting estimators are given in the form of sequential algorithms which are suitable for a digital computer. However, these estimators can be reduced in the form of finite machines with the aid of well-known state equivalence/reduction techniques. An example is presented which illustrates the effectiveness and usefulness of the theory. The results of the paper are applicable to a large variety of digital communication systems involving noisy channels and also to quantised stochastic control processes.

State-observer design for linear sequential machines

The problem of designing a state observer for a linear deterministic q-observable sequential machine is considered. It is shown that this observer has the structure of Kalman's linear estimator, i.e. it is a linear sequential machine driven by the measured outputs and inputs of the original machine. The main feature of this observer is that after q steps it gives the state of the original machine without error whatever the initial state estimate chosen. The theory is then applied for estimating the state of stochastic sequential machines. Observers are necessary when applying state feedback control to sequential machines, the state variables of which are not all accessible to direct measurement.

First and second covering problems of quasi stochastic systems

The concept of quasi stochastic systems, which is obtained from the concept of stochastic sequential machines by relaxing the positiveness condition, is introduced in the present paper. Reductions and the first and second covering problems of quasi stochastic systems, as well as stochastic sequential machines, are investigated.

Convex stochastic sequential machines

From the Convex addition of two independent stochastic vectors, one of them being the output of a Markov chain, the convex stochastic model is defined. Emphasis is on the Moore version of the model. Response equations are given. The relation between this model and the stochastic sequential machine is found. Paralleling the ideas used in systems approach a zero input machine is defined, which plays a very important role in determining the special characteristics of the convex stochastic machine. Equivalence, state minimization, and asymptotic behavior are among the problems discussed. Finally, the convex stochastic sequential machine is realized from stochastic sequential circuits. Its merits, from the economical point of view, as a sub-machine in a possible decomposition of the stochastic automata are discussed.

A Realizable Model for Stochastic Sequential Machines

A new model for stochastic sequential machines is introduced. This model consists of a deterministic Mealy-type synchronous sequential machine some of whose inputs are random number generators while the outputs of another set of random number generators are used to perturb the output function of the deterministic Mealy machine. Thus this model is physically realizable in terms of random number generators, logic and memory elements. It is shown that this model and the Shannon model of a stochastic sequential machine are coextensive and a procedure is given, through a proof of this result, for obtaining one from the other. The model given here is then compared with the realizable model introduced by Nieh and Carlyle [1] and it is shown that their model and ours may be realized with identical random number generators for the case of input-state calculable stochastic sequential machines.