A Translation of Weighted LTL Formulas to Weighted Buchi Automata over omega-valuation Monoids

In this paper we introduce a weighted LTL over product omega-valuation monoids that satisfy specific properties. We also introduce weighted generalized Buchi automata with epsilon-transitions, as well as weighted Buchi automata with epsilon-transitions over product omega-valuation monoids and prove that these two models are expressively equivalent and also equivalent to weighted Buchi automata already introduced in the literature. We prove that every formula of a syntactic fragment of our logic can be effectively translated to a weighted generalized Buchi automaton with epsilon-transitions. For generalized product omega-valuation monoids that satisfy specific properties we define a weighted LTL, weighted generalized Buchi automata with epsilon-transitions, and weighted Buchi automata with epsilon-transitions, and we prove the aforementioned results for generalized product omega-valuation monoids as well. The translation of weighted LTL formulas to weighted generalized Buchi automata with epsilon-transitions is now obtained for a restricted syntactical fragment of the logic.

From LTL to unambiguous Büchi automata via disambiguation of alternating automata

Due to the high complexity of translating linear temporal logic (LTL) to deterministic automata, several forms of “restricted” nondeterminism have been considered with the aim of maintaining some of the benefits of deterministic automata, while at the same time allowing more efficient translations from LTL. One of them is the notion of unambiguity. This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike other approaches it is based on a known translation from very weak alternating automata (VWAA) to NBA. A notion of unambiguity for alternating automata is introduced and it is shown that the VWAA-to-NBA translation preserves unambiguity. Checking unambiguity of VWAA is determined to be PSPACE-complete, both for the explicit and symbolic encodings of alternating automata. The core of the LTL-to-UBA translation is an iterative disambiguation procedure for VWAA. Several heuristics are introduced for different stages of the procedure. We report on an implementation of our approach in the tool and compare it to an existing LTL-to-UBA implementation in the tool set. Our experiments cover model checking of Markov chains, which is an important application of UBA.

Synthesis with Mandatory Stop Actions

We study the impact of the need for the agent to obligatorily instruct the action stop in her strategies. More specifically we consider synthesis (i.e., planning) for LTLf goals under LTL environment specifications in the case the agent must mandatorily stop at a certain point. We show that this obligation makes it impossible to exploit the liveness part of the LTL environment specifications to achieve her goal, effectively reducing the environment specifications to their safety part only. This has a deep impact on the efficiency of solving the synthesis, which can sidestep handling Buchi determinization associated to LTL synthesis, in favor of finite-state automata manipulation as in LTLf synthesis. Next, we add to the agent goal, expressed in LTLf, a safety goal, expressed in LTL. Safety goals must hold forever, even when the agent stops, since the environment can still continue its evolution. Hence the agent, before stopping, must ensure that her safety goal will be maintained even after she stops. To do synthesis in this case, we devise an effective approach that mixes a synthesis technique based on finite-state automata (as in the case of LTLf goals) and model-checking of nondeterministic Buchi automata. In this way, again, we sidestep Buchi automata determinization, hence getting a synthesis technique that is intrinsically simpler than standard LTL synthesis.

Synthesizing Good-Enough Strategies for LTLf Specifications

We consider the problem of synthesizing good-enough (GE)-strategies for linear temporal logic (LTL) over finite traces or LTLf for short. The problem of synthesizing GE-strategies for an LTL formula φ over infinite traces reduces to the problem of synthesizing winning strategies for the formula (∃Oφ)⇒φ where O is the set of propositions controlled by the system. We first prove that this reduction does not work for LTLf formulas. Then we show how to synthesize GE-strategies for LTLf formulas via the Good-Enough (GE)-synthesis of LTL formulas. Unfortunately, this requires to construct deterministic parity automata on infinite words, which is computationally expensive. We then show how to synthesize GE-strategies for LTLf formulas by a reduction to solving games played on deterministic Büchi automata, based on an easier construction of deterministic automata on finite words. We show empirically that our specialized synthesis algorithm for GE-strategies outperforms the algorithms going through GE-synthesis of LTL formulas by orders of magnitude.

Coinductive Algorithms for Büchi Automata

We propose a new algorithm for checking language equivalence of non-deterministic Büchi automata. We start from a construction proposed by Calbrix, Nivat and Podelski, which makes it possible to reduce the problem to that of checking equivalence of automata on finite words. Although this construction generates large and highly non-deterministic automata, we show how to exploit their specific structure and apply state-of-the art techniques based on coinduction to reduce the state-space that has to be explored. Doing so, we obtain algorithms which do not require full determinisation or complementation.

UNITY and Büchi automata

UNITY is a model for concurrent specifications with a complete logic for proving progress properties of the form “P leads to Q”. UNITY is generalized to U-specifications by giving more freedom to specify the steps that are to be taken infinitely often. In particular, these steps can correspond to non-total relations. The generalization keeps the logic sound and complete. The paper exploits the generalization in two ways. Firstly, the logic remains sound when the specification is extended with hypotheses of the form “F leads to G”. As the paper shows, this can make the logic incomplete. The generalization is used to show that the logic remains complete, if the added hypotheses “F leads to G” satisfy “F unless G”. The main result extends the applicability and completeness of UNITY logic to proofs that a given concurrent program satisfies any given formula of LTL, linear temporal logic, without the next-operator which is omitted because it is sensitive to stuttering. For this purpose, the program, written as a UNITY program, is extended with a number of boolean variables. The proof method relies on implementing the LTL formula, i.e., restricting the specification in such a way that only those runs remain that satisfy the formula. This result is a variation of the classical construction of a Büchi automatonfor a given LTL formula that accepts precisely those runs that satisfy the formula.

Topological Characterisation of Multi-Buffer Simulation

Multi-buffer simulation is an extension of simulation preorder that can be used to approximate inclusion of languages recognised by Büchi automata up to their trace closures. DUPLICATOR can use some bounded or unbounded buffers to simulate SPOILER’s move. It has been shown that multi-buffer simulation can be characterised with the existence of a continuous function. In this paper, we show that such a characterisation can be refined to a more restricted case, that is, to the one where DUPLICATOR only uses bounded buffers, by requiring the function to be Lipschitz continuous instead of only continuous. This characterisation however only holds for some restricted classes of automata. One of the automata should only produce words where each letter cannot commute unboundedly. We show that this property can be syntactically characterised with cyclic-path-connectedness, a refinement of syntactic condition on automata that have regular trace closure. We further show that checking cyclic-path-connectedness is indeed co-NP-complete.

Certifying Inexpressibility

Different classes of automata on infinite words have different expressive power. Deciding whether a given language$$L \subseteq \varSigma ^\omega $$L⊆Σωcan be expressed by an automaton of a desired class can be reduced to deciding a game between Prover and Refuter: in each turn of the game, Refuter provides a letter in$$\varSigma $$Σ, and Prover responds with an annotation of the current state of the run (for example, in the case of Büchi automata, whether the state is accepting or rejecting, and in the case of parity automata, what the color of the state is). Prover wins if the sequence of annotations she generates is correct: it is an accepting run iff the word generated by Refuter is inL. We show how a winning strategy for Refuter can serve as a simple and easy-to-understand certificate to inexpressibility, and how it induces additional forms of certificates. Our framework handles all classes of deterministic automata, including ones with structural restrictions like weak automata. In addition, it can be used for refutingseparationof two languages by an automaton of the desired class, and for finding automata thatapproximateLand belong to the desired class.