Minimisation of Multiplicity Tree Automata

We consider the problem of minimising the number of states in a multiplicity tree automaton over the field of rational numbers. We give a minimisation algorithm that runs in polynomial time assuming unit-cost arithmetic. We also show that a polynomial bound in the standard Turing model would require a breakthrough in the complexity of polynomial identity testing by proving that the latter problem is logspace equivalent to the decision version of minimisation. The developed techniques also improve the state of the art in multiplicity word automata: we give an NC algorithm for minimising multiplicity word automata. Finally, we consider the minimal consistency problem: does there exist an automaton with $n$ states that is consistent with a given finite sample of weight-labelled words or trees? We show that this decision problem is complete for the existential theory of the rationals, both for words and for trees of a fixed alphabet rank.Comment: Paper to be published in Logical Methods in Computer Science. Minor editing changes from previous versio

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

Multiplayer Cost Games with Simple Nash Equilibria

Multiplayer games with selfish agents naturally occur in the design of distributed and embedded systems. As the goals of selfish agents are usually neither equivalent nor antagonistic to each other, such games are non zero-sum games. We study such games and show that a large class of these games, including games where the individual objectives are mean- or discounted-payoff, or quantitative reachability, and show that they do not only have a solution, but a simple solution. We establish the existence of Nash equilibria that are composed of k memoryless strategies for each agent in a setting with k agents, one main and k-1 minor strategies. The main strategy describes what happens when all agents comply, whereas the minor strategies ensure that all other agents immediately start to co-operate against the agent who first deviates from the plan. This simplicity is important, as rational agents are an idealisation. Realistically, agents have to decide on their moves with very limited resources, and complicated strategies that require exponential--or even non-elementary--implementations cannot realistically be implemented. The existence of simple strategies that we prove in this paper therefore holds a promise of implementability.Comment: 23 page

Quantitative multi-objective verification for probabilistic systems

We present a verification framework for analysing multiple quantitative objectives of systems that exhibit both nondeterministic and stochastic behaviour. These systems are modelled as probabilistic automata, enriched with cost or reward structures that capture, for example, energy usage or performance metrics. Quantitative properties of these models are expressed in a specification language that incorporates probabilistic safety and liveness properties, expected total cost or reward, and supports multiple objectives of these types. We propose and implement an efficient verification framework for such properties and then present two distinct applications of it: firstly, controller synthesis subject to multiple quantitative objectives; and, secondly, quantitative compositional verification. The practical applicability of both approaches is illustrated with experimental results from several large case studies

Determinising Parity Automata

Parity word automata and their determinisation play an important role in automata and game theory. We discuss a determinisation procedure for nondeterministic parity automata through deterministic Rabin to deterministic parity automata. We prove that the intermediate determinisation to Rabin automata is optimal. We show that the resulting determinisation to parity automata is optimal up to a small constant. Moreover, the lower bound refers to the more liberal Streett acceptance. We thus show that determinisation to Streett would not lead to better bounds than determinisation to parity. As a side-result, this optimality extends to the determinisation of B\"uchi automata

Learn with SAT to Minimize B\"uchi Automata

We describe a minimization procedure for nondeterministic B\"uchi automata (NBA). For an automaton A another automaton A_min with the minimal number of states is learned with the help of a SAT-solver. This is done by successively computing automata A' that approximate A in the sense that they accept a given finite set of positive examples and reject a given finite set of negative examples. In the course of the procedure these example sets are successively increased. Thus, our method can be seen as an instance of a generic learning algorithm based on a "minimally adequate teacher" in the sense of Angluin. We use a SAT solver to find an NBA for given sets of positive and negative examples. We use complementation via construction of deterministic parity automata to check candidates computed in this manner for equivalence with A. Failure of equivalence yields new positive or negative examples. Our method proved successful on complete samplings of small automata and of quite some examples of bigger automata. We successfully ran the minimization on over ten thousand automata with mostly up to ten states, including the complements of all possible automata with two states and alphabet size three and discuss results and runtimes; single examples had over 100 states.Comment: In Proceedings GandALF 2012, arXiv:1210.202