Łukasiewicz mu-Calculus

A behavioural pseudometric for probabilistic transition systems

We present a format for the specification of probabilistic transition systems that guarantees that bisimulation equivalence is a congruence for any operator defined in this format. In this sense, the format is somehow comparable to the ntyft/ntyxt format in a non-probabilistic setting. We also study the modular construction of probabilistic transition systems specifications and prove that some standard conservative extension theorems also hold in our setting. Finally, we show that the trace congruence for image-finite processes induced by our format is precisely bisimulation on probabilistic systems.

Larsen and Skou introduced a notion of bisimulation for probabilistic transition systems. They characterized probabilistic bisimilarity in terms of a probabilistic modal logic and also in terms of ‘button pressing’ tests. Desharnais et al. extended the notion of probabilistic bisimulation and the logical characterization of probabilistic bisimilarity to labelled Markov processes. These processes generalize probabilistic transition systems in that they also allow continuous state spaces. We extend the characterization of probabilistic bisimilarity in terms of testing to labelled Markov processes. One of our main technical contributions is the construction of a final object in a category of labelled Markov processes and the identification of a natural metric on the state space of the final labelled Markov process. This metric provides us with another characterization of probabilistic bisimilarity: states are probabilistic bisimilar if and only if they have distance 0.

We study the following problems for strong and weak bisimulation equivalence: given a labelled Petri net and a finite transition system, are they equivalent?; given a labelled Petri net, is it equivalent to some (unspecified) finite transition system? We show that both problems are decidable for strong bisimulation and undecidable for weak bisimulation.

Durational transition systems are finite transition systems where every transition is additionally equipped with a duration. We consider the problem of interpreting /spl mu/-formulas over durational transition systems. In case the formula contains only operations minimum, maximum, addition, and sequencing, we show that the interpretation ist not only computable but (up to a linear factor) as efficiently computable as the interpretation of /spl mu/-formulas over ordinary finite transition systems.

Internet of Things (IoT) and Cyber-Physical Systems (CPS) have become important topics in both theory and industry. In some application domains, such as when specifying the behaviour of precision mechanics, we need to include features of spatial-temporal consistency. How to model probabilistic real-time systems in such domains is a challenge. This paper presents a modelling formalism, called PSTeC, for describing the behaviour of probabilistic real-time systems focusing on spatial-temporal consistency with nondeterministic, probabilistic and real-time aspects. The consistency restricts a process to start and finish at the required location and time. Communications between agents is specified by interactive actions. The language we propose is an extension of STeC, which is a specification language for location-aware real-time systems, adding probabilistic operations so as to support the incorporation of probabilistic aspects. We first give a formal definition of the syntax for PSTeC, then focus on the details of its operational semantics, which maps a PSTeC term onto a Probabilistic Spatial-Temporal Transition System (PSTTS) following the structured operational semantics style. A simple example demonstrates the expressiveness of PSTeC.

A probabilistic Boolean network (PBN) is a discrete-time system composed of a collection of Boolean networks between which the PBN switches in a stochastic manner. This article focuses on the study of quotients of PBNs. Given a PBN and an equivalence relation on its state set, we consider a probabilistic transition system that is generated by the PBN; the resulting quotient transition system then automatically captures the quotient behavior of this PBN. We therefore describe a method for obtaining a probabilistic Boolean system that generates the transitions of the quotient transition system. Applications of this quotient description are discussed, and it is shown that for PBNs, controller synthesis can be performed easily by first controlling a quotient system and then lifting the control law back to the original network. A biological example is given to show the usefulness of the developed results.

In this paper, we investigate several equivalence relations for probabilistic labeled transition systems: bisimulation equivalence, readiness equivalence, failure equivalence, trace equivalence, maximal trace equivalence and finite trace equivalence. We formally prove the inclusions (equalities) among these equivalences. We also show that readiness, failure, trace, maximum trace and finite trace equivalences for finite probabilistic labeled transition systems are decidable in polynomial time. This should be contrasted with the PSPACE completeness of the same equivalences for classical labeled transition systems. Moreover, we derive an efficient polynomial time algorithm for deciding bisimulation equivalence for finite probabilistic labeled transition systems. The special case of initiated probabilistic transition systems will be considered. We show that the isomorphism problem for finite initiated labeled probabilistic transition systems is NC(1) equivalent to graph isomorphism.

Bisimulation for probabilistic transition systems: a coalgebraic approach

Linear reachability problems and minimal solutions to linear Diophantine equation systems

The linear reachability problem is to decide whether there is an execution path in a given finite state transition system such that the counts of labels on the path satisfy a given linear constraint. Using results on minimal solutions (in nonnegative integers) for linear Diophantine systems, we obtain new complexity results for the problem, as well as for other linear counting problems of finite state transition systems and timed automata. In contrast to previously known results, the complexity bounds obtained in this paper are polynomial in the size of the transition system in consideration, when the linear constraint is fixed.

Partition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed a partition refinement algorithm that is generic in the transition type of the given system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time analysis of our algorithm to cover additional instances, notably weighted automata and, more generally, weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular, and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors. Experiments show that even for complex system types, the tool is able to handle systems with millions of transitions.

Probabilistic models of computation have been studied for over three decades now, from foundational, logical, coalgebraic, categorical, as well as more practical verification-motivated point of view. In my work, and in this talk, we focus on the foundational, semantics side of probabilistic automata and transition systems, and in particular the relevant monads, and their algebras. The interplay of probability and non-determinism has been particularly challenging from a semantics point of view for some decades, as it does not just yield a monad. There are by now well-studied solutions to this and several monads for probability and non-determinism: some enriching the structure with convexity to obtain a monad, albeit a non-commutative one, others deliberately simplifying it, by imposing commutativity. It is well known that monads have two faces: a computational one - we think here of the powerset monad modelling non-determinism, or the probability distribution monad modelling probabilistic choices, and a universal-algebraic one - where we think of the algebraic presentation of the monads, like semilattices for the powerset monad and (variants of) convex algebras for (variants of) the probability distribution monad. Combining non-determinism and probability yields other combined monads, among which probably the most studied is the convex-subsets-of-distributions monad. This monad is presented by so-called convex semilattices, algebraic structures that are both a semilattice and a convex algebra, with suitable distributivity connecting the operations. From a semantics point of view, the algebraic presentations give us a nice way to define (and sometimes compute) language (aka trace) equivalence of the corresponding automata. Moreover, the presentations are useful for axiomatizations of language equivalence. From an algebraic point of view, these algebras are interesting and many questions about them are still open. We will discuss language semantics, its axiomatization, as well as some obtained and open algebraic problems for convex algebras and convex semilattices - and their computational consequences. In particular, we have a full characterization of congruences of (variants of) convex algebras, which for example yields decidability of distribution semantics; other results on congruences, e.g. a result on a congruence being finitely generated as a subalgebra, surprisingly accellerated the proof of completeness of infinite trace semanitcs, etc. We will review some existing results: all congruences are described and fp = fg for convex algebras, termination is a black hole, useful functors are proper on convex algebras, cancellativity of convex semi-lattices; as well as mention ongoing work on: mid-point-cancellativity, subalgebras, and homomorphisms for convex semi-lattices, as well as topological convex semilattices. This talk is based on previous joint works with Filippo Bonchi, Alexandra Silva, Valeria Vignudelli, and Harald Woracek, as well as on ongoing work and discussions with Matteo Mio, Alex Simpson, and Harald Woracek.

Probabilistic fuzzy logic controller for uncertain nonlinear systems

Effective inventory management in multi-echelon supply chains is challenged by stochastic demand and uncertain lead times, which amplify variability and increase operational costs. This study presents a Markov chain framework for modeling, analyzing, and optimizing multi-echelon inventory systems under stochastic lead-time conditions. The framework represents inventory levels and lead-time states as a probabilistic transition system, enabling computation of steady-state distributions, stockout probabilities, and transient recovery times. Analytical results are validated through Monte Carlo simulations, demonstrating high fidelity between theoretical and empirical distributions. Numerical experiments quantify the impact of lead-time variability on inventory performance, revealing nonlinear increases in stockout probability and total system cost as lead-time variance grows. Multi-echelon analyses demonstrate the emergence of the bullwhip effect and highlight the effectiveness of information sharing in mitigating variability propagation across echelons. Comparative benchmarking against deep reinforcement learning (DRL) policies shows that while DRL achieves marginally lower total costs, the Markov-based approach provides superior interpretability, robustness, and computational efficiency. The study offers theoretical contributions by unifying stochastic multi-echelon dynamics and transient analysis within a tractable Markov framework. Managerially, it provides actionable insights on lead-time variance reduction, cross-echelon visibility, and hybrid analytical–learning policy design. The framework establishes a foundation for resilient and cost-effective inventory control in complex, uncertain supply-chain networks.