30 research outputs found
Determinization of B\"uchi Automata: Unifying the Approaches of Safra and Muller-Schupp
Determinization of B\"uchi automata is a long-known difficult problem and
after the seminal result of Safra, who developed the first asymptotically
optimal construction from B\"uchi into Rabin automata, much work went into
improving, simplifying or avoiding Safra's construction. A different, less
known determinization construction was derived by Muller and Schupp and appears
to be unrelated to Safra's construction on the first sight. In this paper we
propose a new meta-construction from nondeterministic B\"uchi to deterministic
parity automata which strictly subsumes both the construction of Safra and the
construction of Muller and Schupp. It is based on a correspondence between
structures that are encoded in the macrostates of the determinization
procedures - Safra trees on one hand, and levels of the split-tree, which
underlies the Muller and Schupp construction, on the other. Our construction
allows for combining the mentioned constructions and opens up new directions
for the development of heuristics.Comment: Full version of ICALP 2019 pape
Simple and tight complexity lower bounds for solving Rabin games
We give a simple proof that assuming the Exponential Time Hypothesis (ETH),
determining the winner of a Rabin game cannot be done in time , where is the number of pairs of vertex subsets involved in
the winning condition and is the vertex count of the game graph. While this
result follows from the lower bounds provided by Calude et al [SIAM J. Comp.
2022], our reduction is simpler and arguably provides more insight into the
complexity of the problem. In fact, the analogous lower bounds discussed by
Calude et al, for solving Muller games and multidimensional parity games,
follow as simple corollaries of our approach. Our reduction also highlights the
usefulness of a certain pivot problem -- Permutation SAT -- which may be of
independent interest.Comment: 10 pages, 5 figures. To appear in SOSA 202
On finitely ambiguous B\"uchi automata
Unambiguous B\"uchi automata, i.e. B\"uchi automata allowing only one
accepting run per word, are a useful restriction of B\"uchi automata that is
well-suited for probabilistic model-checking. In this paper we propose a more
permissive variant, namely finitely ambiguous B\"uchi automata, a
generalisation where each word has at most accepting runs, for some fixed
. We adapt existing notions and results concerning finite and bounded
ambiguity of finite automata to the setting of -languages and present a
translation from arbitrary nondeterministic B\"uchi automata with states to
finitely ambiguous automata with at most states and at most accepting
runs per word
On the Minimisation of Transition-Based Rabin Automata and the Chromatic Memory Requirements of Muller Conditions
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 consider the colours of the edges. We prove that the minimal amount of chromatic memory required in games using a given Muller condition is exactly the size of a minimal Rabin automaton recognising this condition. Combining these two results, we deduce that finding the chromatic memory requirements of a Muller condition is NP-complete. This characterisation also allows us to prove that chromatic memories cannot be optimal in general, disproving a conjecture by Kopczy?ski
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
Constructing Deterministic ?-Automata from Examples by an Extension of the RPNI Algorithm
The RPNI algorithm (Oncina, Garcia 1992) constructs deterministic finite automata from finite sets of negative and positive example words. We propose and analyze an extension of this algorithm to deterministic ?-automata with different types of acceptance conditions. In order to obtain this generalization of RPNI, we develop algorithms for the standard acceptance conditions of ?-automata that check for a given set of example words and a deterministic transition system, whether these example words can be accepted in the transition system with a corresponding acceptance condition. Based on these algorithms, we can define the extension of RPNI to infinite words. We prove that it can learn all deterministic ?-automata with an informative right congruence in the limit with polynomial time and data. We also show that the algorithm, while it can learn some automata that do not have an informative right congruence, cannot learn deterministic ?-automata for all regular ?-languages in the limit. Finally, we also prove that active learning with membership and equivalence queries is not easier for automata with an informative right congruence than for general deterministic ?-automata