11 research outputs found
First-Order Model Checking on Generalisations of Pushdown Graphs
We study the first-order model checking problem on two generalisations of
pushdown graphs. The first class is the class of nested pushdown trees. The
other is the class of collapsible pushdown graphs. Our main results are the
following. First-order logic with reachability is uniformly decidable on nested
pushdown trees. Considering first-order logic without reachability, we prove
decidability in doubly exponential alternating time with linearly many
alternations. First-order logic with regular reachability predicates is
uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested
pushdown trees are first-order interpretable in collapsible pushdown graphs of
level 2. This interpretation can be extended to an interpretation of the class
of higher-order nested pushdown trees in the collapsible pushdown graph
hierarchy. We prove that the second level of this new hierarchy of nested trees
has decidable first-order model checking. Our decidability result for
collapsible pushdown graph relies on the fact that level 2 collapsible pushdown
graphs are uniform tree-automatic. Our last result concerns tree-automatic
structures in general. We prove that first-order logic extended by Ramsey
quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page
Collapsible Pushdown Graphs of Level 2 are Tree-Automatic
We show that graphs generated by collapsible pushdown systems of level 2 are
tree-automatic. Even when we allow -contractions and add a
reachability predicate (with regular constraints) for pairs of configurations,
the structures remain tree-automatic. Hence, their FO theories are decidable,
even when expanded by a reachability predicate. As a corollary, we obtain the
tree-automaticity of the second level of the Caucal-hierarchy.Comment: 12 pages Accepted for STACS 201
Collapsible Pushdown Graphs of Level 2 are Tree-Automatic
We show that graphs generated by collapsible pushdown systems of level 2 are
tree-automatic. Even if we allow epsilon-contractions and reachability
predicates (with regular constraints) for pairs of configurations, the
structures remain tree-automatic whence their first-order logic theories are
decidable. As a corollary we obtain the tree-automaticity of the second level
of the Caucal-hierarchy.Comment: Journal version of arXiv:0912.4110, accepted for publication in LMC
Reachability in Higher-Order-Counters
Higher-order counter automata (\HOCS) can be either seen as a restriction of
higher-order pushdown automata (\HOPS) to a unary stack alphabet, or as an
extension of counter automata to higher levels. We distinguish two principal
kinds of \HOCS: those that can test whether the topmost counter value is zero
and those which cannot.
We show that control-state reachability for level \HOCS with -test is
complete for \mbox{}-fold exponential space; leaving out the -test
leads to completeness for \mbox{}-fold exponential time. Restricting
\HOCS (without -test) to level , we prove that global (forward or
backward) reachability analysis is \PTIME-complete. This enhances the known
result for pushdown systems which are subsumed by level \HOCS without
-test.
We transfer our results to the formal language setting. Assuming that \PTIME
\subsetneq \PSPACE \subsetneq \mathbf{EXPTIME}, we apply proof ideas of
Engelfriet and conclude that the hierarchies of languages of \HOPS and of \HOCS
form strictly interleaving hierarchies. Interestingly, Engelfriet's
constructions also allow to conclude immediately that the hierarchy of
collapsible pushdown languages is strict level-by-level due to the existing
complexity results for reachability on collapsible pushdown graphs. This
answers an open question independently asked by Parys and by Kobayashi.Comment: Version with Full Proofs of a paper that appears at MFCS 201
An approach to computing downward closures
The downward closure of a word language is the set of all (not necessarily
contiguous) subwords of its members. It is well-known that the downward closure
of any language is regular. While the downward closure appears to be a powerful
abstraction, algorithms for computing a finite automaton for the downward
closure of a given language have been established only for few language
classes.
This work presents a simple general method for computing downward closures.
For language classes that are closed under rational transductions, it is shown
that the computation of downward closures can be reduced to checking a certain
unboundedness property.
This result is used to prove that downward closures are computable for (i)
every language class with effectively semilinear Parikh images that are closed
under rational transductions, (ii) matrix languages, and (iii) indexed
languages (equivalently, languages accepted by higher-order pushdown automata
of order 2).Comment: Full version of contribution to ICALP 2015. Comments welcom
Recursion Schemes and Logical Reflection
International audienceLet R be a class of generators of node-labelled infinite trees, and L be a logical language for describing correctness properties of these trees. Given r in R and phi in L, we say that r_phi is a phi-reflection of r just if (i) r and r_phi generate the same underlying tree, and (ii) suppose a node u of the tree t(r) generated by r has label f, then the label of the node u of t(r_phi) is f* if u in t(r) satisfies phi; it is f otherwise. Thus if t(r) is the computation tree of a program r, we may regard r_phi as a transform of r that can internally observe its behaviour against a specification phi. We say that R is (constructively) reflective w.r.t. L just if there is an algorithm that transforms a given pair (r,phi) to r_phi. In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal mu-calculus and monadic second order (MSO) logic. To obtain this result, we give the first characterisation of the winning regions of parity games over the transition graphs of collapsible pushdown automata (CPDA): they are regular sets defined by a new class of automata. (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.) As a corollary, we show that these schemes are closed under the operation of MSO-interpretation followed by tree unfolding a la Caucal
A pumping lemma for collapsible pushdown graphs of level 2
We present a pumping lemma for the class of collapsible pushdown graphs of level 2. This pumping lemma even applies to the ε-contractions of level 2 collapsible pushdown graphs. Our pumping lemma also improves the bounds of Hayashi’s pumping lemma for indexed languages
Collapsible Pushdown Graphs of Level 2 are Tree-Automatic
International audienceWe show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow -contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy