47 research outputs found
Can Nondeterminism Help Complementation?
Complementation and determinization are two fundamental notions in automata
theory. The close relationship between the two has been well observed in the
literature. In the case of nondeterministic finite automata on finite words
(NFA), complementation and determinization have the same state complexity,
namely Theta(2^n) where n is the state size. The same similarity between
determinization and complementation was found for Buchi automata, where both
operations were shown to have 2^\Theta(n lg n) state complexity. An intriguing
question is whether there exists a type of omega-automata whose determinization
is considerably harder than its complementation. In this paper, we show that
for all common types of omega-automata, the determinization problem has the
same state complexity as the corresponding complementation problem at the
granularity of 2^\Theta(.).Comment: In Proceedings GandALF 2012, arXiv:1210.202
IST Austria Technical Report
Gist is a tool that (a) solves the qualitative analysis problem of turn-based probabilistic games with ω-regular objectives; and (b) synthesizes reasonable environment assumptions for synthesis of unrealizable specifications. Our tool provides efficient implementations of several reduction based techniques to solve turn-based probabilistic games, and uses the analysis of turn-based probabilistic games for synthesizing environment assumptions for unrealizable specifications
Gist: A Solver for Probabilistic Games
GIST is a tool that (a) solves the qualitative analysis problem of turn-based probabilistic games with ω-regular objectives; and (b) synthesizes reasonable environment assumptions for synthesis of unrealizable specifications. Our tool provides the first and efficient implementations of several reduction-based techniques to solve turn-based probabilistic games, and uses the analysis of turn-based probabilistic games for synthesizing environment assumptions for unrealizable specifications
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
How unprovable is Rabin's decidability theorem?
We study the strength of set-theoretic axioms needed to prove Rabin's theorem
on the decidability of the MSO theory of the infinite binary tree. We first
show that the complementation theorem for tree automata, which forms the
technical core of typical proofs of Rabin's theorem, is equivalent over the
moderately strong second-order arithmetic theory to a
determinacy principle implied by the positional determinacy of all parity games
and implying the determinacy of all Gale-Stewart games given by boolean
combinations of sets. It follows that complementation for
tree automata is provable from - but not -comprehension.
We then use results due to MedSalem-Tanaka, M\"ollerfeld and
Heinatsch-M\"ollerfeld to prove that over -comprehension, the
complementation theorem for tree automata, decidability of the MSO theory of
the infinite binary tree, positional determinacy of parity games and
determinacy of Gale-Stewart games are all
equivalent. Moreover, these statements are equivalent to the
-reflection principle for -comprehension. It follows in
particular that Rabin's decidability theorem is not provable in
-comprehension.Comment: 21 page
One Theorem to Rule Them All: A Unified Translation of LTL into {\omega}-Automata
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
Satisfiability Games for Branching-Time Logics
The satisfiability problem for branching-time temporal logics like CTL*, CTL
and CTL+ has important applications in program specification and verification.
Their computational complexities are known: CTL* and CTL+ are complete for
doubly exponential time, CTL is complete for single exponential time. Some
decision procedures for these logics are known; they use tree automata,
tableaux or axiom systems. In this paper we present a uniform game-theoretic
framework for the satisfiability problem of these branching-time temporal
logics. We define satisfiability games for the full branching-time temporal
logic CTL* using a high-level definition of winning condition that captures the
essence of well-foundedness of least fixpoint unfoldings. These winning
conditions form formal languages of \omega-words. We analyse which kinds of
deterministic {\omega}-automata are needed in which case in order to recognise
these languages. We then obtain a reduction to the problem of solving parity or
B\"uchi games. The worst-case complexity of the obtained algorithms matches the
known lower bounds for these logics. This approach provides a uniform, yet
complexity-theoretically optimal treatment of satisfiability for branching-time
temporal logics. It separates the use of temporal logic machinery from the use
of automata thus preserving a syntactical relationship between the input
formula and the object that represents satisfiability, i.e. a winning strategy
in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner
closure of the input formula only. Last but not least, the games presented here
come with an attempt at providing tool support for the satisfiability problem
of complex branching-time logics like CTL* and CTL+
Qualitative Analysis of POMDPs with Temporal Logic Specifications for Robotics Applications
We consider partially observable Markov decision processes (POMDPs), that are
a standard framework for robotics applications to model uncertainties present
in the real world, with temporal logic specifications. All temporal logic
specifications in linear-time temporal logic (LTL) can be expressed as parity
objectives. We study the qualitative analysis problem for POMDPs with parity
objectives that asks whether there is a controller (policy) to ensure that the
objective holds with probability 1 (almost-surely). While the qualitative
analysis of POMDPs with parity objectives is undecidable, recent results show
that when restricted to finite-memory policies the problem is EXPTIME-complete.
While the problem is intractable in theory, we present a practical approach to
solve the qualitative analysis problem. We designed several heuristics to deal
with the exponential complexity, and have used our implementation on a number
of well-known POMDP examples for robotics applications. Our results provide the
first practical approach to solve the qualitative analysis of robot motion
planning with LTL properties in the presence of uncertainty