In this report we study the problem of minimising deterministic automata over finite and infinite words. Deterministic finite automata are the simplest devices to recognise regular languages, and deterministic Buchi, Co-Buchi, and parity automata play a similar role in the recognition of ω-regular languages. While it is well known that the minimisation of deterministic finite and weak automata is cheap, the complexity of minimising deterministic Buchi and parity automata has remained an open challenge. We establish the NP-completeness of these problems. A second contribution of this report is the introduction of relaxed minimisation of deterministic finite automata. Like hyper-minimisation, relaxed minimisation allows for some changes in the language of the automaton: We seek a smallest automaton that, when used as a monitor, provides a wrong answer only a bounded number of times in any run of a system. We argue that minimisation of finite automata, hyper-minimisation, relaxed minimisation, and the minimisation of deterministic Buchi (or Co-Buchi) automata are operations of increasing reduction power, as the respective equivalence relations on automata become coarser from left to right. When we allow for minor changes in the language, relaxed minimisation can therefore be considered as a more powerful minimisation technique than hyper-minimisation from the perspective of finite automata. From the perspective of Buchi and Co-Buchi automata, we gain a cheap algorithm for state-space reduction that also turns out to be beneficial for further heuristic or exhaustive state-space reductions put on top of it.
This paper discusses the hardness of finding minimal good-for-games (GFG) Buchi, Co-Buchi, and parity automata with state based acceptance. The problem appears to sit between finding small deterministic and finding small nondeterministic automata, where minimality is NP-complete and PSPACE-complete, respectively. However, recent work of Radi and Kupferman has shown that minimising Co-Buchi automata with transition based acceptance is tractable, which suggests that the complexity of minimising GFG automata might be cheaper than minimising deterministic automata. We show for the standard state b…
Infinite words over infinite alphabets serve as models of the temporal development of the allocation and (re-)use of resources over linear time. We approach omega-languages over infinite alphabets in the setting of nominal sets, and study languages of infinite bar strings, i.e. infinite sequences of names that feature binding of fresh names; binding corresponds roughly to reading letters from input words in automata models with registers. We introduce regular nominal nondeterministic Büchi automata (Büchi RNNAs), an automata model for languages of infinite bar strings, repurposing the previous…
This work considers weak deterministic Büchi automata reading encodings of non-negative $d$-vectors of reals in a fixed base. A saturated language is a language which contains all encoding of elements belonging to a set of $d$-vectors of reals. A Real Vector Automaton is an automaton which recognizes a saturated language. It is explained how to decide in quasi-linear time whether a minimal weak deterministic Büchi automaton is a Real Vector Automaton. The problem is solved both for the two standard encodings of vectors of numbers: the sequential encoding and the parallel encoding. This algorit…
Probabilistic Büchi automata are a natural generalization of PFA to infinite words, but have been studied in-depth only rather recently and many interesting questions are still open. PBA are known to accept, in general, a class of languages that goes beyond the regular languages. In this work we extend the known classes of restricted PBA which are still regular, strongly relying on notions concerning ambiguity in classical omega-automata. Furthermore, we investigate the expressivity of the not yet considered but natural class of weak PBA, and we also show that the regularity problem for weak P…
Self-loop alternating automata (SLAA) with Büchi or co-Büchi acceptance are popular intermediate formalisms in translations of LTL to deterministic or nondeterministic automata. This paper considers SLAA with generic transition-based Emerson-Lei acceptance and presents translations of LTL to these automata and back. Importantly, the translation of LTL to SLAA with generic acceptance produces considerably smaller automata than previous translations of LTL to Büchi or co-Büchi SLAA. Our translation is already implemented in the tool LTL3TELA, where it helps to produce small deterministic or nond…
Aix-Marseille Université（艾克斯-马赛大学）；University of Kaiserslautern-Landau（凯泽斯劳滕-兰道工业大学）；CNRS（法国国家科学研究中心）；Université Libre de Bruxelles（布鲁塞尔自由大学）；Institute of Science and Technology Austria（奥地利科学技术研究院）
This work considers weak deterministic Büchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time whether a set of reals recognized by a given minimal weak deterministic RNA is ${FO}[\mathbb R;+,<,1]$-definable. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an existential (respectively, existential-universal) ${FO}[\mathbb R;+,<,1]$-formula which defines the set of reals recognized by th…
We study a categorical generalisation of tree automata, as $Σ$-algebras for a fixed endofunctor $Σ$ endowed with initial and final states. Under mild assumptions about the base category, we present a general minimisation algorithm for these automata. We then build upon and extend an existing generalisation of the Nerode equivalence to a categorical setting and relate it to the existence of minimal automata. Finally, we show that generalised types of side-effects, such as non-determinism, can be captured by this categorical framework, leading to a general determinisation procedure.
Büchi's theorem states that $ω$-regular languages are characterized as languages of the form $\bigcup_i U_i V_i^ω$, where $U_i$ and $V_i$ are regular languages. Parikh automata are automata on finite words whose transitions are equipped with vectors of positive integers, whose sum can be tested for membership in a given semi-linear set. We give an intuitive automata theoretic characterization of languages of the form $U_i V_i^ω$, where $U_i$ and $V_i$ are Parikh-recognizable. Furthermore, we show that the class of such languages, where $U_i$ is Parikh-recognizable and $V_i$ is regular is exact…
We introduce improvements in the algorithm by Gastin and Oddoux translating LTL formulae into Büchi automata via very weak alternating co-Büchi automata and generalized Büchi automata. Several improvements are based on specific properties of any formula where each branch of its syntax tree contains at least one eventually operator and at least one always operator. These changes usually result in faster translations and smaller automata. Other improvements reduce non-determinism in the produced automata. In fact, we modified all the steps of the original algorithm and its implementation known a…
In this paper, we propose a novel algorithm to learn a Büchi automaton from a teacher who knows an $ω$-regular language. The algorithm is based on learning a formalism named family of DFAs (FDFAs) recently proposed by Angluin and Fisman[10]. The main catch is that we use a classification tree structure instead of the standard observation table structure. The worst case storage space required by our algorithm is quadratically better than the table-based algorithm proposed in [10]. We implement the first publicly available library ROLL (Regular Omega Language Learning ), which consists of all $ω…
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.
Minimization of deterministic automata on finite words results in a {\em canonical\/} automaton. For deterministic automata on infinite words, no canonical minimal automaton exists, and a language may have different minimal deterministic Büchi (DBW) or co-Büchi (DCW) automata. In recent years, researchers have studied {\em good-for-games\/} (GFG) automata -- nondeterministic automata that can resolve their nondeterministic choices in a way that only depends on the past. Several applications of automata in formal methods, most notably synthesis, that are traditionally based on deterministic aut…
In this paper we revisit Safra's determinization constructions for automata on infinite words. We show how to construct deterministic automata with fewer states and, most importantly, parity acceptance conditions. Determinization is used in numerous applications, such as reasoning about tree automata, satisfiability of CTL*, and realizability and synthesis of logical specifications. The upper bounds for all these applications are reduced by using the smaller deterministic automata produced by our construction. In addition, the parity acceptance conditions allows to use more efficient algorithm…
In GFG automata, it is possible to resolve nondeterminism in a way that only depends on the past and still accepts all the words in the language. The motivation for GFG automata comes from their adequacy for games and synthesis, wherein general nondeterminism is inappropriate. We continue the ongoing effort of studying the power of nondeterminism in GFG automata. Initial indications have hinted that every GFG automaton embodies a deterministic one. Today we know that this is not the case, and in fact GFG automata may be exponentially more succinct than deterministic ones. We focus on the typen…
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 Büchi automata (LDBA) to DPA, and show that it can be concatenated with a recent efficient translation from LTL …
In this paper, we relate the problem of determining the chromatic memory requirements of Muller conditions with the minimisation of transition-based Rabin automata. Our first contribution is a proof of the NP-completeness of the minimisation of transition-based Rabin automata. Our second contribution concerns the memory requirements of games over graphs using Muller conditions. A memory structure is a finite state machine that implements a strategy and is updated after reading the edges of the game; the special case of chromatic memories being those structures whose update function only consid…
Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the n…
The determinization of a nondeterministic Büchi automaton (NBA) is a fundamental construction of automata theory, with applications to probabilistic verification and reactive synthesis. The standard determinization constructions, such as the ones based on the Safra-Piterman's approach, work on the whole NBA. In this work we propose a divide-and-conquer determinization approach. To this end, we first classify the strongly connected components (SCCs) of the given NBA as inherently weak, deterministic accepting, and nondeterministic accepting. We then present how to determinize each type of SCC i…
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 multipl…
This paper is concerned with the computational complexity of equivalence and minimisation for automata with transition weights in the field Q of rational numbers. We use polynomial identity testing and the Isolation Lemma to obtain complexity bounds, focussing on the class NC of problems within P solvable in polylogarithmic parallel time. For finite Q-weighted automata, we give a randomised NC procedure that either outputs that two automata are equivalent or returns a word on which they differ. We also give an NC procedure for deciding whether a given automaton is minimal, as well as a randomi…
When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a~monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property $L$ is a deterministic finite automaton $M_L$ that after each finite execution tells whether (1) every possible extension of the execution is in $L$, or (2) every possible extension is in the complement of $L$, or neither (1) nor (2) holds. Moreover, $L$ being monitorable means that it is always possible that in so…
The aim of this work is to thoroughly investigate Buchi automata augmented with spatial constraints. The input trees of such an automaton are infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest. The constraints, from an RCC8-like spatial Relation Algebra (RA) x, are used to impose spatial constraints on objects of the spatial scene, eventually at different nodes of the input trees. We show that a Buchi alternating automaton augm…
A word automaton recognizing a language $L$ is good for games (GFG) if its composition with any game with winning condition $L$ preserves the game's winner. While all deterministic automata are GFG, some nondeterministic automata are not. There are various other properties that are used in the literature for defining that a nondeterministic automaton is GFG, including "history-deterministic", "compliant with some letter game", "good for trees", and "good for composition with other automata". The equivalence of these properties has not been formally shown. We generalize all of these definitions…
Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially o…
By separating the principal acceptance mechanism from the concrete acceptance condition of a given Büchi automaton with $n$ states,Schewe presented the construction of an equivalent deterministic Rabin transition automaton with $o((1.65n)^n)$ states via \emph{history trees}, which can be simply translated to a standard Rabin automaton with $o((2.26n)^n)$ states. Apart from the inherent simplicity, Schewe's construction improved Safra's construction (which requires $12^nn^{2n}$ states). However, the price that is paid is the use of $2^{n-1}$ Rabin pairs (instead of $n$ in Safra's construction).…
Simulation relations are an important tool in automata theory because they provide efficiently computable approximations to language inclusion. In recent years, extensions of ordinary simulations have been studied, for instance multi-pebble and multi-letter simulations which yield better approximations and are still polynomial-time computable. In this paper we study the limitations of approximating language inclusion in this way: we introduce a natural extension of multi-letter simulations called buffered simulations. They are based on a simulation game in which the two players share a FIFO bu…
University of Twente（特文特大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Liverpool（利物浦大学）
When omega-regular objectives were first proposed in model-free reinforcement learning (RL) for controlling MDPs, deterministic Rabin automata were used in an attempt to provide a direct translation from their transitions to scalar values. While these translations failed, it has turned out that it is possible to repair them by using good-for-MDPs (GFM) Büchi automata instead. These are nondeterministic Büchi automata with a restricted type of nondeterminism, albeit not as restricted as in good-for-games automata. Indeed, deterministic Rabin automata have a pretty straightforward translation to…
We present a direct transformation of weak alternating $ω$-automata into equivalent backward deterministic $ω$-automata and show (1) how it can be used to obtain a transformation of non-deterministic Büchi automata into equivalent backward deterministic automata and (2) that it yields optimal equivalent backward deterministic automata when applied to linear-time temporal logic formulas. (1) uses the alternation-free fragment of the linear-time $μ$-calculus as an intermediate step; (2) is based on the straightforward translation of linear-time temporal logic into weak alternating $ω$-automata.
The determinization of Buchi automata is a celebrated problem, with applications in synthesis, probabilistic verification, and multi-agent systems. Since the 1960s, there has been a steady progress of constructions: by McNaughton, Safra, Piterman, Schewe, and others. Despite the proliferation of solutions, they are all essentially ad-hoc constructions, with little theory behind them other than proofs of correctness. Since Safra, all optimal constructions employ trees as states of the deterministic automaton, and transitions between states are defined operationally over these trees. The operati…
We follow a connection between tight determinisation and complementation and establish a complementation procedure from parity automata to nondeterministic Büchi automata and prove it to be tight up to an $O(n)$ factor, where $n$ is the size of the nondeterministic parity automaton. This factor does not depend on the number of priorities.
The languages of infinite timed words accepted by timed automata are traditionally defined using Buchi-like conditions. These acceptance conditions focus on the set of locations visited infinitely often along a run, but completely ignore quantitative timing aspects. In this paper we propose a natural quantitative semantics for timed automata based on the so-called frequency, which measures the proportion of time spent in the accepting locations. We study various properties of timed languages accepted with positive frequency, and in particular the emptiness and universality problems.
We present an efficient algorithm to reduce the size of nondeterministic Buchi word automata, while retaining their language. Additionally, we describe methods to solve PSPACE-complete automata problems like universality, equivalence and inclusion for much larger instances (1-3 orders of magnitude) than before. This can be used to scale up applications of automata in formal verification tools and decision procedures for logical theories. The algorithm is based on new transition pruning techniques. These use criteria based on combinations of backward and forward trace inclusions. Since these re…
Lehigh University（理海大学）；The University of Iowa（爱荷华大学）；University of Science and Technology of China（中国科学技术大学）
This paper studies the control synthesis of motion planning subject to uncertainties. The uncertainties are considered in robot motions and environment properties, giving rise to the probabilistic labeled Markov decision process (PL-MDP). A Model-Free Reinforcement The learning (RL) method is developed to generate a finite-memory control policy to satisfy high-level tasks expressed in linear temporal logic (LTL) formulas. Due to uncertainties and potentially conflicting tasks, this work focuses on infeasible LTL specifications, where a relaxed LTL constraint is developed to allow the agent to …
