100 research outputs found
Order-Invariant MSO is Stronger than Counting MSO in the Finite
We compare the expressiveness of two extensions of monadic second-order logic
(MSO) over the class of finite structures. The first, counting monadic
second-order logic (CMSO), extends MSO with first-order modulo-counting
quantifiers, allowing the expression of queries like ``the number of elements
in the structure is even''. The second extension allows the use of an
additional binary predicate, not contained in the signature of the queried
structure, that must be interpreted as an arbitrary linear order on its
universe, obtaining order-invariant MSO.
While it is straightforward that every CMSO formula can be translated into an
equivalent order-invariant MSO formula, the converse had not yet been settled.
Courcelle showed that for restricted classes of structures both order-invariant
MSO and CMSO are equally expressive, but conjectured that, in general,
order-invariant MSO is stronger than CMSO.
We affirm this conjecture by presenting a class of structures that is
order-invariantly definable in MSO but not definable in CMSO.Comment: Revised version contributed to STACS 200
Parameterized Model Checking of Token-Passing Systems
We revisit the parameterized model checking problem for token-passing systems
and specifications in indexed .
Emerson and Namjoshi (1995, 2003) have shown that parameterized model checking
of indexed in uni-directional token
rings can be reduced to checking rings up to some \emph{cutoff} size. Clarke et
al. (2004) have shown a similar result for general topologies and indexed
, provided processes cannot choose the
directions for sending or receiving the token.
We unify and substantially extend these results by systematically exploring
fragments of indexed with respect to
general topologies. For each fragment we establish whether a cutoff exists, and
for some concrete topologies, such as rings, cliques and stars, we infer small
cutoffs. Finally, we show that the problem becomes undecidable, and thus no
cutoffs exist, if processes are allowed to choose the directions in which they
send or from which they receive the token.Comment: We had to remove an appendix until the proofs and notations there is
cleare
EPTCS
First cycle games (FCG) are played on a finite graph by two players who push a token along the edges until a vertex is repeated, and a simple cycle is formed. The winner is determined by some fixed property Y of the sequence of labels of the edges (or nodes) forming this cycle. These games are traditionally of interest because of their connection with infinite-duration games such as parity and mean-payoff games. We study the memory requirements for winning strategies of FCGs and certain associated infinite duration games. We exhibit a simple FCG that is not memoryless determined (this corrects a mistake in Memoryless determinacy of parity and mean payoff games: a simple proof by Bj⋯orklund, Sandberg, Vorobyov (2004) that claims that FCGs for which Y is closed under cyclic permutations are memoryless determined). We show that θ (n)! memory (where n is the number of nodes in the graph), which is always sufficient, may be necessary to win some FCGs. On the other hand, we identify easy to check conditions on Y (i.e., Y is closed under cyclic permutations, and both Y and its complement are closed under concatenation) that are sufficient to ensure that the corresponding FCGs and their associated infinite duration games are memoryless determined. We demonstrate that many games considered in the literature, such as mean-payoff, parity, energy, etc., satisfy these conditions. On the complexity side, we show (for efficiently computable Y) that while solving FCGs is in PSPACE, solving some families of FCGs is PSPACE-hard
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by
omega-automata, so called omega-automatic structures, and prove
that relations defined over such structures in first-order logic
expanded by the first-order quantifiers `there exist at most
many\u27, \u27there exist finitely many\u27 and \u27there exist
modulo many\u27 are omega-regular. The proof identifies certain
algebraic properties of omega-semigroups.
As a consequence an omega-regular equivalence relation of countable
index has an omega-regular set of representatives. This implies
Blumensath\u27s conjecture that a countable structure with an
-automatic presentation can be represented using automata
on finite words. This also complements a very recent result of
Hj"orth, Khoussainov, Montalban and Nies showing that there is an
omega-automatic structure which has no injective presentation
Generalized planning: Non-deterministic abstractions and trajectory constraints
We study the characterization and computation of general policies for families of problems that share a structure characterized by a common reduction into a single abstract problem. Policies mu that solve the abstract problem P have been shown to solve all problems Q that reduce to P provided that mu terminates in Q. In this work, we shed light on why this termination condition is needed and how it can be removed. The key observation is that the abstract problem P captures the common structure among the concrete problems Q that is local (Markovian) but misses common structure that is global. We show how such global structure can be captured by means of trajectory constraints that in many cases can be expressed as LTL formulas, thus reducing generalized planning to LTL synthesis. Moreover, for a broad class of problems that involve integer variables that can be increased or decreased, trajectory constraints can be compiled away, reducing generalized planning to fully observable nondeterministic planning
Strategy Logic with Imperfect Information
We introduce an extension of Strategy Logic for the imperfect-information
setting, called SLii, and study its model-checking problem. As this logic
naturally captures multi-player games with imperfect information, the problem
turns out to be undecidable. We introduce a syntactical class of "hierarchical
instances" for which, intuitively, as one goes down the syntactic tree of the
formula, strategy quantifications are concerned with finer observations of the
model. We prove that model-checking SLii restricted to hierarchical instances
is decidable. This result, because it allows for complex patterns of
existential and universal quantification on strategies, greatly generalises
previous ones, such as decidability of multi-player games with imperfect
information and hierarchical observations, and decidability of distributed
synthesis for hierarchical systems. To establish the decidability result, we
introduce and study QCTL*ii, an extension of QCTL* (itself an extension of CTL*
with second-order quantification over atomic propositions) by parameterising
its quantifiers with observations. The simple syntax of QCTL* ii allows us to
provide a conceptually neat reduction of SLii to QCTL*ii that separates
concerns, allowing one to forget about strategies and players and focus solely
on second-order quantification. While the model-checking problem of QCTL*ii is,
in general, undecidable, we identify a syntactic fragment of hierarchical
formulas and prove, using an automata-theoretic approach, that it is decidable.
The decidability result for SLii follows since the reduction maps hierarchical
instances of SLii to hierarchical formulas of QCTL*ii
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by omega-automata, so
called omega-automatic structures, and prove that relations defined over such
structures in first-order logic expanded by the first-order quantifiers `there
exist at most many', 'there exist finitely many' and 'there exist
modulo many' are omega-regular. The proof identifies certain algebraic
properties of omega-semigroups. As a consequence an omega-regular equivalence
relation of countable index has an omega-regular set of representatives. This
implies Blumensath's conjecture that a countable structure with an
-automatic presentation can be represented using automata on finite
words. This also complements a very recent result of Hj\"orth, Khoussainov,
Montalban and Nies showing that there is an omega-automatic structure which has
no injective presentation
Quantifying Bounds in Strategy Logic
Program synthesis constructs programs from specifications in an automated way. Strategy Logic (SL) is a powerful and versatile specification language whose goal is to give theoretical foundations for program synthesis in a multi-agent setting. One limitation of Strategy Logic is that it is purely qualitative. For instance it cannot specify quantitative properties of executions such as "every request is quickly granted", or quantitative properties of trees such as "most executions of the system terminate". In this work, we extend Strategy Logic to include quantitative aspects in a way that can express bounds on "how quickly" and "how many". We define Prompt Strategy Logic, which encompasses Prompt LTL (itself an extension of LTL with a prompt eventuality temporal operator), and we define Bounded-Outcome Strategy Logic which has a bounded quantifier on paths. We supply a general technique, based on the study of automata with counters, that solves the model-checking problems for both these logics
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