From LTL and Limit-Deterministic B\"uchi Automata to Deterministic Parity Automata

Controller synthesis for general linear temporal logic (LTL) objectives is a challenging task. The standard approach involves translating the LTL objective into a deterministic parity automaton (DPA) by means of the Safra-Piterman construction. One of the challenges is the size of the DPA, which often grows very fast in practice, and can reach double exponential size in the length of the LTL formula. In this paper we describe a single exponential translation from limit-deterministic B\"uchi automata (LDBA) to DPA, and show that it can be concatenated with a recent efficient translation from LTL to LDBA to yield a double exponential, \enquote{Safraless} LTL-to-DPA construction. We also report on an implementation, a comparison with the SPOT library, and performance on several sets of formulas, including instances from the 2016 SyntComp competition

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

Minimizing Expected Cost Under Hard Boolean Constraints, with Applications to Quantitative Synthesis

In Boolean synthesis, we are given an LTL specification, and the goal is to construct a transducer that realizes it against an adversarial environment. Often, a specification contains both Boolean requirements that should be satisfied against an adversarial environment, and multi-valued components that refer to the quality of the satisfaction and whose expected cost we would like to minimize with respect to a probabilistic environment. In this work we study, for the first time, mean-payoff games in which the system aims at minimizing the expected cost against a probabilistic environment, while surely satisfying an $\omega$-regular condition against an adversarial environment. We consider the case the $\omega$-regular condition is given as a parity objective or by an LTL formula. We show that in general, optimal strategies need not exist, and moreover, the limit value cannot be approximated by finite-memory strategies. We thus focus on computing the limit-value, and give tight complexity bounds for synthesizing $\epsilon$-optimal strategies for both finite-memory and infinite-memory strategies. We show that our game naturally arises in various contexts of synthesis with Boolean and multi-valued objectives. Beyond direct applications, in synthesis with costs and rewards to certain behaviors, it allows us to compute the minimal sensing cost of $\omega$-regular specifications -- a measure of quality in which we look for a transducer that minimizes the expected number of signals that are read from the input

Two Variable vs. Linear Temporal Logic in Model Checking and Games

Model checking linear-time properties expressed in first-order logic has non-elementary complexity, and thus various restricted logical languages are employed. In this paper we consider two such restricted specification logics, linear temporal logic (LTL) and two-variable first-order logic (FO2). LTL is more expressive but FO2 can be more succinct, and hence it is not clear which should be easier to verify. We take a comprehensive look at the issue, giving a comparison of verification problems for FO2, LTL, and various sublogics thereof across a wide range of models. In particular, we look at unary temporal logic (UTL), a subset of LTL that is expressively equivalent to FO2; we also consider the stutter-free fragment of FO2, obtained by omitting the successor relation, and the expressively equivalent fragment of UTL, obtained by omitting the next and previous connectives. We give three logic-to-automata translations which can be used to give upper bounds for FO2 and UTL and various sublogics. We apply these to get new bounds for both non-deterministic systems (hierarchical and recursive state machines, games) and for probabilistic systems (Markov chains, recursive Markov chains, and Markov decision processes). We couple these with matching lower-bound arguments. Next, we look at combining FO2 verification techniques with those for LTL. We present here a language that subsumes both FO2 and LTL, and inherits the model checking properties of both languages. Our results give both a unified approach to understanding the behaviour of FO2 and LTL, along with a nearly comprehensive picture of the complexity of verification for these logics and their sublogics.Comment: 37 pages, to be published in Logical Methods in Computer Science journal, includes material presented in Concur 2011 and QEST 2012 extended abstract

Controller synthesis & Ordinal Automata

with appendixOrdinal automata are used to model physical systems with Zeno behavior. Using automata and games techniques we solve a control problem formulated and left open by Demri and Nowak in 2005. It involves partial observability and a new synchronization between the controller and the environment

Lazy Probabilistic Model Checking without Determinisation

The bottleneck in the quantitative analysis of Markov chains and Markov decision processes against specifications given in LTL or as some form of nondeterministic B\"uchi automata is the inclusion of a determinisation step of the automaton under consideration. In this paper, we show that full determinisation can be avoided: subset and breakpoint constructions suffice. We have implemented our approach---both explicit and symbolic versions---in a prototype tool. Our experiments show that our prototype can compete with mature tools like PRISM.Comment: 38 pages. Updated version for introducing the following changes: - general improvement on paper presentation; - extension of the approach to avoid full determinisation; - added proofs for such an extension; - added case studies; - updated old case studies to reflect the added extensio