114 research outputs found
The complexity of linear-time temporal logic over the class of ordinals
We consider the temporal logic with since and until modalities. This temporal
logic is expressively equivalent over the class of ordinals to first-order
logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability
problem over the class of ordinals. Among the consequences of our proof, we
show that given the code of some countable ordinal alpha and a formula, we can
decide in PSPACE whether the formula has a model over alpha. In order to show
these results, we introduce a class of simple ordinal automata, as expressive
as B\"uchi ordinal automata. The PSPACE upper bound for the satisfiability
problem of the temporal logic is obtained through a reduction to the
nonemptiness problem for the simple ordinal automata.Comment: Accepted for publication in LMC
Complementation of Rational Sets on Scattered Linear Orderings of Finite Rank
International audienceIn a preceding paper, automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-finite and even transfinite words studied by Buchi Kleene's theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation
Weighted Automata and Logics for Infinite Nested Words
Nested words introduced by Alur and Madhusudan are used to capture structures
with both linear and hierarchical order, e.g. XML documents, without losing
valuable closure properties. Furthermore, Alur and Madhusudan introduced
automata and equivalent logics for both finite and infinite nested words, thus
extending B\"uchi's theorem to nested words. Recently, average and discounted
computations of weights in quantitative systems found much interest. Here, we
will introduce and investigate weighted automata models and weighted MSO logics
for infinite nested words. As weight structures we consider valuation monoids
which incorporate average and discounted computations of weights as well as the
classical semirings. We show that under suitable assumptions, two resp. three
fragments of our weighted logics can be transformed into each other. Moreover,
we show that the logic fragments have the same expressive power as weighted
nested word automata.Comment: LATA 2014, 12 page
Partially Ordered Two-way B\"uchi Automata
We introduce partially ordered two-way B\"uchi automata and characterize
their expressive power in terms of fragments of first-order logic FO[<].
Partially ordered two-way B\"uchi automata are B\"uchi automata which can
change the direction in which the input is processed with the constraint that
whenever a state is left, it is never re-entered again. Nondeterministic
partially ordered two-way B\"uchi automata coincide with the first-order
fragment Sigma2. Our main contribution is that deterministic partially ordered
two-way B\"uchi automata are expressively complete for the first-order fragment
Delta2. As an intermediate step, we show that deterministic partially ordered
two-way B\"uchi automata are effectively closed under Boolean operations.
A small model property yields coNP-completeness of the emptiness problem and
the inclusion problem for deterministic partially ordered two-way B\"uchi
automata.Comment: The results of this paper were presented at CIAA 2010; University of
Stuttgart, Computer Scienc
Randomness for Free
We consider two-player zero-sum games on graphs. These games can be
classified on the basis of the information of the players and on the mode of
interaction between them. On the basis of information the classification is as
follows: (a) partial-observation (both players have partial view of the game);
(b) one-sided complete-observation (one player has complete observation); and
(c) complete-observation (both players have complete view of the game). On the
basis of mode of interaction we have the following classification: (a)
concurrent (both players interact simultaneously); and (b) turn-based (both
players interact in turn). The two sources of randomness in these games are
randomness in transition function and randomness in strategies. In general,
randomized strategies are more powerful than deterministic strategies, and
randomness in transitions gives more general classes of games. In this work we
present a complete characterization for the classes of games where randomness
is not helpful in: (a) the transition function probabilistic transition can be
simulated by deterministic transition); and (b) strategies (pure strategies are
as powerful as randomized strategies). As consequence of our characterization
we obtain new undecidability results for these games
On the Sets of Real Numbers Recognized by Finite Automata in Multiple Bases
This article studies the expressive power of finite automata recognizing sets
of real numbers encoded in positional notation. We consider Muller automata as
well as the restricted class of weak deterministic automata, used as symbolic
set representations in actual applications. In previous work, it has been
established that the sets of numbers that are recognizable by weak
deterministic automata in two bases that do not share the same set of prime
factors are exactly those that are definable in the first order additive theory
of real and integer numbers. This result extends Cobham's theorem, which
characterizes the sets of integer numbers that are recognizable by finite
automata in multiple bases.
In this article, we first generalize this result to multiplicatively
independent bases, which brings it closer to the original statement of Cobham's
theorem. Then, we study the sets of reals recognizable by Muller automata in
two bases. We show with a counterexample that, in this setting, Cobham's
theorem does not generalize to multiplicatively independent bases. Finally, we
prove that the sets of reals that are recognizable by Muller automata in two
bases that do not share the same set of prime factors are exactly those
definable in the first order additive theory of real and integer numbers. These
sets are thus also recognizable by weak deterministic automata. This result
leads to a precise characterization of the sets of real numbers that are
recognizable in multiple bases, and provides a theoretical justification to the
use of weak automata as symbolic representations of sets.Comment: 17 page
What's Decidable About Sequences?
We present a first-order theory of sequences with integer elements,
Presburger arithmetic, and regular constraints, which can model significant
properties of data structures such as arrays and lists. We give a decision
procedure for the quantifier-free fragment, based on an encoding into the
first-order theory of concatenation; the procedure has PSPACE complexity. The
quantifier-free fragment of the theory of sequences can express properties such
as sortedness and injectivity, as well as Boolean combinations of periodic and
arithmetic facts relating the elements of the sequence and their positions
(e.g., "for all even i's, the element at position i has value i+3 or 2i"). The
resulting expressive power is orthogonal to that of the most expressive
decidable logics for arrays. Some examples demonstrate that the fragment is
also suitable to reason about sequence-manipulating programs within the
standard framework of axiomatic semantics.Comment: Fixed a few lapses in the Mergesort exampl
Journeying through Dementia: the story of a 14 year design-led research enquiry
Consider a linear ordering equipped with a finite sequence of monadic
predicates. If the ordering contains an interval of order type \omega or
-\omega, and the monadic second-order theory of the combined structure is
decidable, there exists a non-trivial expansion by a further monadic predicate
that is still decidable.Comment: 18 page
Trees over Infinite Structures and Path Logics with Synchronization
We provide decidability and undecidability results on the model-checking
problem for infinite tree structures. These tree structures are built from
sequences of elements of infinite relational structures. More precisely, we
deal with the tree iteration of a relational structure M in the sense of
Shelah-Stupp. In contrast to classical results where model-checking is shown
decidable for MSO-logic, we show decidability of the tree model-checking
problem for logics that allow only path quantifiers and chain quantifiers
(where chains are subsets of paths), as they appear in branching time logics;
however, at the same time the tree is enriched by the equal-level relation
(which holds between vertices u, v if they are on the same tree level). We
separate cleanly the tree logic from the logic used for expressing properties
of the underlying structure M. We illustrate the scope of the decidability
results by showing that two slight extensions of the framework lead to
undecidability. In particular, this applies to the (stronger) tree iteration in
the sense of Muchnik-Walukiewicz.Comment: In Proceedings INFINITY 2011, arXiv:1111.267
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