2,293 research outputs found
Residual Nominal Automata
We are motivated by the following question: which nominal languages admit an active learning algorithm? This question was left open in previous work, and is particularly challenging for languages recognised by nondeterministic automata. To answer it, we develop the theory of residual nominal automata, a subclass of nondeterministic nominal automata. We prove that this class has canonical representatives, which can always be constructed via a finite number of observations. This property enables active learning algorithms, and makes up for the fact that residuality - a semantic property - is undecidable for nominal automata. Our construction for canonical residual automata is based on a machine-independent characterisation of residual languages, for which we develop new results in nominal lattice theory. Studying residuality in the context of nominal languages is a step towards a better understanding of learnability of automata with some sort of nondeterminism
The Wadge Hierarchy of Deterministic Tree Languages
We provide a complete description of the Wadge hierarchy for
deterministically recognisable sets of infinite trees. In particular we give an
elementary procedure to decide if one deterministic tree language is
continuously reducible to another. This extends Wagner's results on the
hierarchy of omega-regular languages of words to the case of trees.Comment: 44 pages, 8 figures; extended abstract presented at ICALP 2006,
Venice, Italy; full version appears in LMCS special issu
Random Generation of Nondeterministic Finite-State Tree Automata
Algorithms for (nondeterministic) finite-state tree automata (FTAs) are often
tested on random FTAs, in which all internal transitions are equiprobable. The
run-time results obtained in this manner are usually overly optimistic as most
such generated random FTAs are trivial in the sense that the number of states
of an equivalent minimal deterministic FTA is extremely small. It is
demonstrated that nontrivial random FTAs are obtained only for a narrow band of
transition probabilities. Moreover, an analytic analysis yields a formula to
approximate the transition probability that yields the most complex random
FTAs, which should be used in experiments.Comment: In Proceedings TTATT 2013, arXiv:1311.5058. Andreas Maletti and
Daniel Quernheim were financially supported by the German Research Foundation
(DFG) grant MA/4959/1-
The Planning Spectrum - One, Two, Three, Infinity
Linear Temporal Logic (LTL) is widely used for defining conditions on the
execution paths of dynamic systems. In the case of dynamic systems that allow
for nondeterministic evolutions, one has to specify, along with an LTL formula
f, which are the paths that are required to satisfy the formula. Two extreme
cases are the universal interpretation A.f, which requires that the formula be
satisfied for all execution paths, and the existential interpretation E.f,
which requires that the formula be satisfied for some execution path.
When LTL is applied to the definition of goals in planning problems on
nondeterministic domains, these two extreme cases are too restrictive. It is
often impossible to develop plans that achieve the goal in all the
nondeterministic evolutions of a system, and it is too weak to require that the
goal is satisfied by some execution.
In this paper we explore alternative interpretations of an LTL formula that
are between these extreme cases. We define a new language that permits an
arbitrary combination of the A and E quantifiers, thus allowing, for instance,
to require that each finite execution can be extended to an execution
satisfying an LTL formula (AE.f), or that there is some finite execution whose
extensions all satisfy an LTL formula (EA.f). We show that only eight of these
combinations of path quantifiers are relevant, corresponding to an alternation
of the quantifiers of length one (A and E), two (AE and EA), three (AEA and
EAE), and infinity ((AE)* and (EA)*). We also present a planning algorithm for
the new language that is based on an automata-theoretic approach, and study its
complexity
Residual Nominal Automata
Nominal automata are models for accepting languages over infinite alphabets.
In this paper we refine the hierarchy of nondeterministic nominal automata, by
developing the theory of residual nominal automata. In particular, we show that
they admit canonical minimal representatives, and that the universality problem
becomes decidable. We also study exact learning of these automata, and settle
questions that were left open about their learnability via observations
Minimization and Canonization of GFG Transition-Based Automata
While many applications of automata in formal methods can use
nondeterministic automata, some applications, most notably synthesis, need
deterministic or good-for-games(GFG) automata. The latter are nondeterministic
automata that can resolve their nondeterministic choices in a way that only
depends on the past. The minimization problem for deterministic B\"uchi and
co-B\"uchi word automata is NP-complete. In particular, no canonical minimal
deterministic automaton exists, and a language may have different minimal
deterministic automata. We describe a polynomial minimization algorithm for GFG
co-B\"uchi word automata with transition-based acceptance. Thus, a run is
accepting if it traverses a set of designated transitions only
finitely often. Our algorithm is based on a sequence of transformations we
apply to the automaton, on top of which a minimal quotient automaton is
defined. We use our minimization algorithm to show canonicity for
transition-based GFG co-B\"uchi word automata: all minimal automata have
isomorphic safe components (namely components obtained by restricting the
transitions to these not in ) and once we saturate the automata with
-transitions, we get full isomorphism.Comment: 28 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:2009.1088
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