Complete Proof Systems for Amortised Probabilistic Bisimulations

The notion of amortisation has been integrated in quantitative bisimulations to make long-term behavioral comparisons between nondeterministic systems. In this paper, we present sound and complete proof systems for amortised strong probabilistic bisimulation and its observational congruence on a process algebra with probability and nondeterminism, and prove their soundness and completeness. Our results make it possible to reason about long-term (observable) probabilistic behaviors by syntactic manipulations.     

Complexity of the Exact Domatic Number Problem and of the Exact Conveyor Flow Shop Problem

We prove that the exact versions of the domatic number problem are complete for the levels of the boolean hierarchy over NP. The domatic number problem, which arises in the area of computer networks, is the problem of partitioning a given graph into a maximum number of disjoint dominating sets. This number is called the domatic number of the graph. We prove that the problem of determining whether or not the domatic number of a given graph is exactly one of k given values is complete for BH_2k(NP), the 2kth level of the boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not the domatic number of a given graph equals exactly a given integer. Note that DP = BH₂(NP). We obtain similar results for the exact versions of generalized dominating set problems and of the conveyor flow shop problem. Our reductions apply Wagner's conditions sufficient to prove hardness for the levels of the boolean hierarchy over NP.     

Constraints and universal algebra

In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed by a constraint satisfaction problem with a given structure, or a given set of allowed constraint types. This revised version was published online in June 2006 with corrections to the Cover Date.     

Infinity problems and countability problems for ω-automata

The infinity problem for ω-automata is to decide if the ω-language recognized by a given automaton is infinite; the countability problem is to decide if a given automaton recognizes a countable ω-language. We prove that these problems are NLogspace-complete for (nondeterministic) Büchi, generalized Büchi, Muller, Rabin and parity automata.     

Efficient Update Strategies for Geometric Computing with Uncertainty

We consider the problems of computing maximal points and the convex hull of a set of points in two dimensions, when the points are “in motion.” We assume that the point locations (or trajectories) are not known precisely and determining these values exactly is feasible, but expensive. In our model the algorithm only knows areas within which each of the input points lie, and is required to identify the maximal points or points on the convex hull correctly by updating some points (i.e., determining their location exactly). We compare the number of points updated by the algorithm on a given instance to the minimum number of points that must be updated by a nondeterministic strategy in order to compute the answer provably correctly. We give algorithms for both of the above problems that always update at most three times as many points as the nondeterministic strategy, and show that this is the best possible. Our model is similar to that in [3] and [5].     

On finite automata with limited nondeterminism

We develop a new algorithm for determining if a given nondeterministic finite automaton is limited in nondeterminism. From this, we show that the number of nondeterministic moves of a finite automaton, if limited, is bounded by where is the number of states. If the finite automaton is over a one-letter alphabet, using Gohon's result the number of nondeterministic moves, if limited, is less than . In both cases, we present families of finite automata demonstrating that the upper bounds obtained are almost tight. We also show that the limitedness problem of the number of nondeterministic moves of finite automata is PSPACE-hard. Since the problem is already known to be in PSPACE, it is therefore PSPACE-complete. Received: 14 June 1994 / 3 December 1997     

Constrained Optimal Hybrid Control of a Flow Shop System

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.     

Observations on complete sets between linear time and polynomial time

There is a single set that is complete for a variety of nondeterministic time complexity classes with respect to related versions of m-reducibility. This observation immediately leads to transfer results for determinism versus nondeterminism solutions. Also, an upward transfer of collapses of certain oracle hierarchies, built analogously to the polynomial-time or the linear-time hierarchies, can be shown by means of uniformly constructed sets that are complete for related levels of all these hierarchies. A similar result holds for difference hierarchies over nondeterministic complexity classes. Finally, we give an oracle set relative to which the nondeterministic classes coincide with the deterministic ones, for several sets of time bounds, and we prove that the strictness of the tape-number hierarchy for deterministic linear-time Turing machines does not relativize.     

Robust and nonblocking supervisory control of nondeterministicdiscrete event systems using trajectory models

This note considers the robust supervisory control problems of uncertain nondeterministic discrete-event systems (DESs). The uncertain DES to be controlled is assumed to be modeled as a set of some possible nondeterministic automata. Then, the control objective is to achieve a given language specification and guarantee the nonblockingness of any nondeterministic automata of the set which are controlled by a robust nonblocking supervisor. Based on trajectory models, this note presents the necessary and sufficient conditions for the existence of a robust nonblocking supervisor for a given uncertain nondeterministic DES     

Compositional Asymmetric Cooperations for Process Algebras with Probabilities,Priorities, and Time

The modeling and analysis experience with process algebras has shown the necessity of extending them with priority, probabilistic internal/external choice, and time in order to be able to faithfully model the behavior of real systems and capture the properties of interest. An important open problem in this scenario is how to obtain semantic compositionality in the presence of all these mechanisms, to allow for an efficient analysis.In this paper we argue that, when abandoning the classical nondeterministic setting by considering the mechanisms above, a natural solution is to break the symmetry of the roles of the processes participating in a synchronization. We accomplish this by distinguishing between master actions – the choice among which is carried out generatively according to their priorities/probabilities or exponentially distributed durations – and slave actions – the choice among which is carried out reactively according to their priorities/probabilities – and by imposing that a master action can synchronize with slave actions only.Technically speaking, in this paper we define a process algebra called EMPA_gr including probabilities, priorities, exponentially distributed durations, and the generative master-reactive slaves synchronization mechanism. Then, we prove that the synchronization mechanism in EMPA_gr is correct w.r.t. the novel cooperation structure model, we show that the Markovian bisimulation equivalence is a congruence for EMPA_gr, and we present a sound and complete axiomatization for finite terms.     

Model matching for finite-state machines

The problem of model matching for finite state machines (FSMs) consists of finding a controller for a given open-loop system so that the resulting closed-loop system matches a desired input-output behavior. In this paper, a set of model matching problems is addressed: strong model matching (where the reference model and the plant are deterministic FSMs and the initial conditions are fixed), strong model matching with measurable disturbances (where disturbances are present in the plant), and strong model matching with nondeterministic reference model (where any behavior out of those in the reference model has to be matched by the closed-loop system). Necessary and sufficient conditions for the existence of controllers for all these problems are given. A characterization of all feasible control laws is derived and an efficient synthesis procedure is proposed. Further, the well-known supervisory control problem for discrete-event dynamical systems (DEDSs) formulated in its basic form is shown to be solvable as a strong model matching problem with measurable disturbances and nondeterministic reference model     

Polynomial synthesis of supervisor for partially observed discrete-event systems by allowing nondeterminism in control

We study the supervisory control of discrete-event systems (DESs) under partial observation using nondeterministic supervisors. We formally define a nondeterministic control policy and also a control & observation compatible nondeterministic state machine and prove their equivalence. The control action of a nondeterministic supervisor is chosen online, nondeterministically from among a set of choices determined offline. Also, the control action can be changed online nondeterministically (prior to any new observation) in accordance with choices determined offline. The online choices, once made, can be used to affect the set of control action choices in future. We show that when control is exercised using a nondeterministic supervisor, the specification language is required to satisfy a weaker notion of observability, which we define in terms of recognizability and achievability. Achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor, and it is weaker than controllability and observability combined. When all events are controllable, achievability reduces to recognizability. We show that both existence, and synthesis of nondeterministic supervisors can be determined polynomially. (For deterministic supervisors, only existence can be determined polynomially.) Both achievability and recognizability are preserved under union, and also under intersection (when restricted over prefix-closed languages). Using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix-closed case. Unlike the deterministic supervisory setting where the complexity of existence is exponential, our existence condition is polynomially verifiable, and also a supervisor can be polynomially synthesized.     

Intervals in generalized effect algebras

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Rie?anová and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.     

Congruences and ideals in pseudo effect algebras as total algebras

Congruences and ideals in pseudo-effect algebras and their total algebra versions are studied. It is shown that every congruence of the total algebra induces a Riesz congruence in the corresponding pseudo-effect algebra. Conversely, to every normal Riesz ideal in a pseudo-effect algebra there is a total algebra, in which the given ideal induces a congruence of the total algebra. Ideals of total algebras corresponding to lattice-ordered pseudo-effect algebras are characterized, and it is shown that they coincide with normal Riesz ideals in the pseudo-effect algebras.     

效应代数的T-模糊滤子

研究了效应代数上的[T-]模糊滤子的一些问题。利用水平截集及范数的方法获得了一些相关的结果,推广了效应代数模糊滤子的一些结果,得到存在一个映射使得效应代数的所有[T-]模糊滤子与这个效应代数的[T-]同余关系对应的结论。     

Dynamic effect algebras and their representations

For lattice effect algebras, the so-called tense operators were already introduced by Chajda and Kola?ík. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that in every effect algebra can be introduced tense operators which, for non-complete lattice effect algebras, can be only partial mappings. An effect algebra equipped with tense operators reflects changes of quantum events from past to future. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a frame such that each of these operators can be obtained by our construction. We solve this problem for (strict) dynamic effect algebras having a full set of homorphisms into a complete lattice effect algebra.     

State estimation and detectability of probabilistic discrete event systems

A probabilistic discrete event system (PDES) is a nondeterministic discrete event system where the probabilities of nondeterministic transitions are specified. State estimation problems of PDES are more difficult than those of non-probabilistic discrete event systems. In our previous papers, we investigated state estimation problems for non-probabilistic discrete event systems. We defined four types of detectabilities and derived necessary and sufficient conditions for checking these detectabilities. In this paper, we extend our study to state estimation problems for PDES by considering the probabilities. The first step in our approach is to convert a given PDES into a nondeterministic discrete event system and find sufficient conditions for checking probabilistic detectabilities. Next, to find necessary and sufficient conditions for checking probabilistic detectabilities, we investigate the “convergence” of event sequences in PDES. An event sequence is convergent if along this sequence, it is more and more certain that the system is in a particular state. We derive conditions for convergence and hence for detectabilities. We focus on systems with complete event observation and no state observation. For better presentation, the theoretical development is illustrated by a simplified example of nephritis diagnosis.     

Testing approximate symmetry in the plane is NP-hard

Let A be a set of n points in the Euclidean plane. We would like to know its symmetry group. If the input points are given by rational coordinates (which is the normal case for a computer), then this problem degenerates to a trivial case. This is why we introduce the notion of approximate symmetry. In this notion, we consider a fixed tolerance factor and ask for the maximal symmetry group that is possible for a set A' consisting of n points in the -neighborhoods of the points in A, each point of A' belonging to a different point of A. We prove that the corresponding decision problem is NP-hard even when and the symmetry group we ask for is fixed. This is a rather surprising result, because a similar concept has been developed for approximate congruence recently and several polynomial algorithms are known for that related problem.     

粗糙集代数与BR0代数

讨论粗糙集代数与BR₀代数的关系,以及由粗糙集代数构造BR₀代数的方法。借助近似代数的原子及同余关系,证明了在适当选取蕴涵算子及余运算之后,粗糙集代数就成为BR₀代数。