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In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n \mathbf{G}\mathbf{F}\, \varphi_i \vee \mathbf{F}\mathbf{G}\, \psi_i $, where $\varphi_i$ and $\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata., Comment: Accepted in J. ACM. arXiv admin note: text overlap with arXiv:2304.08872, arXiv:2005.00472

In the mid 80s, Lichtenstein, Pnueli, and Zuck showed that every formula of Past LTL (the extension of Linear Temporal Logic with past operators) is equivalent to a conjunction of formulas of the form $\mathbf{G}\mathbf{F} \varphi \vee \mathbf{F}\mathbf{G} \psi$, where $\varphi$ and $\psi$ contain only past operators. Some years later, Chang, Manna, and Pnueli derived a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. In 2020, Sickert and Esparza presented a direct and purely syntactic normalization procedure for LTL yielding a normal form similar to the one by Chang, Manna, and Pnueli, with a single exponential blow-up, and applied it to the problem of constructing a succinct deterministic $\omega$-automaton for a given formula. However, their procedure had exponential time complexity in the best case. In particular, it does not perform better for formulas that are almost in normal form. In this paper we present an alternative normalization procedure based on a simple set of rewrite rules.

We report on the last four editions of the reactive synthesis competition (SYNTCOMP 2018-2021). We briefly describe the evaluation scheme and the experimental setup of SYNTCOMP. Then, we introduce new benchmark classes that have been added to the SYNTCOMP library and give an overview of the participants of SYNTCOMP. Finally, we present and analyze the results of our experimental evaluations, including a ranking of tools with respect to quantity and quality - that is, the total size in terms of logic and memory elements - of solutions., Comment: accepted for publication in STTT

While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not. In particular, there is no known elementary bound on the complexity of translating deterministic $\omega$-regular automata to LTL. Our first contribution consists of tight bounds for LTL over a unary alphabet: alternating, nondeterministic and deterministic automata can be exactly exponentially, quadratically and linearly more succinct, respectively, than any equivalent LTL formula. Our main contribution consists of a translation of general counter-free deterministic $\omega$-regular automata into LTL formulas of double exponential temporal-nesting depth and triple exponential length, using an intermediate Krohn-Rhodes cascade decomposition of the automaton. To our knowledge, this is the first elementary bound on this translation. Furthermore, our translation preserves the acceptance condition of the automaton in the sense that it turns a looping, weak, B\"uchi, coB\"uchi or Muller automaton into a formula that belongs to the matching class of the syntactic future hierarchy. In particular, it can be used to translate an LTL formula recognising a safety language to a formula belonging to the safety fragment of LTL (over both finite and infinite words)., Comment: Full version with appendix of a chapter with the same title that appears in the FoSSaCS 2022 conference proceedings

In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of LTL with past operators) is equivalent to a formula of the form $\bigwedge_{i=1}^n \mathbf{G}\mathbf{F} \varphi_i \vee \mathbf{F}\mathbf{G} \psi_i$, where $\varphi_i$ and $\psi_i$ contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalisation procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present a direct and purely syntactic normalisation procedure for LTL yielding a normal form, comparable to the one by Chang, Manna, and Pnueli, that has only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalises the formula, translates it into a special very weak alternating automaton, and applies a simple determinisation procedure, valid only for these special automata., Comment: This is the extended version of the referenced conference paper and contains an appendix with additional material

We report on the fifth reactive synthesis competition (SYNTCOMP 2018). We introduce four new benchmark classes that have been added to the SYNTCOMP library, and briefly describe the evaluation scheme and the experimental setup of SYNTCOMP 2018. We give an overview of the participants of SYNTCOMP 2018 and highlight changes compared to previous years. Finally, we present and analyze the results of our experimental evaluation, including a ranking of tools with respect to quantity and quality of solutions., Comment: arXiv admin note: substantial text overlap with arXiv:1711.11439, arXiv:1609.00507

The synthesis of reactive systems from linear temporal logic (LTL) specifications is an important aspect in the design of reliable software and hardware. We present our adaption of the classic automata-theoretic approach to LTL synthesis, implemented in the tool Strix which has won the two last synthesis competitions (Syntcomp2018/2019). The presented approach is (1) structured, meaning that the states used in the construction have a semantic structure that is exploited in several ways, it performs a (2) forward exploration such that it often constructs only a small subset of the reachable states, and it is (3) incremental in the sense that it reuses results from previous inconclusive solution attempts. Further, we present and study different guiding heuristics that determine where to expand the on-demand constructed arena. Moreover, we show several techniques for extracting an implementation (Mealy machine or circuit) from the witness of the tree-automaton emptiness check. Lastly, the chosen constructions use a symbolic representation of the transition functions to reduce runtime and memory consumption. We evaluate the proposed techniques on the Syntcomp2019 benchmark set and show in more detail how the proposed techniques compare to the techniques implemented in other leading LTL synthesis tools.

This continuously extended technical report collects and compares commonly used formulae from the literature and provides them in a machine readable way.

We present a unified translation of LTL formulas into deterministic Rabin automata, limit-deterministic B\"uchi automata, and nondeterministic B\"uchi automata. The translations yield automata of asymptotically optimal size (double or single exponential, respectively). All three translations are derived from one single Master Theorem of purely logical nature. The Master Theorem decomposes the language of a formula into a positive boolean combination of languages that can be translated into {\omega}-automata by elementary means. In particular, Safra's, ranking, and breakpoint constructions used in other translations are not needed.

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.

We consider the extensions of modal transition systems (MTS), namely Boolean MTS and parametric MTS and we investigate the refinement problems over both classes. Firstly, we reduce the problem of modal refinement over both classes to a problem solvable by a QBF solver and provide experimental results showing our technique scales well. Secondly, we extend the algorithm for thorough refinement of MTS providing better complexity then via reductions to previously studied problems. Finally, we investigate the relationship between modal and thorough refinement on the two classes and show how the thorough refinement can be approximated by the modal refinement.