77 research outputs found
Recursion Schemes and the WMSO+U Logic
We study the weak MSO logic extended by the unbounding quantifier (WMSO+U), expressing the fact that there exist arbitrarily large finite sets satisfying a given property. We prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+U
Cost Automata, Safe Schemes, and Downward Closures
Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes
Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width
Marx (STOC 2010, J. ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class Phi of Boolean CQs of bounded submodular width, the model-checking problem for Phi on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marx\u27s result to non-Boolean queries as follows: For every class Phi of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marx\u27s splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently
The Safe Lambda Calculus
Safety is a syntactic condition of higher-order grammars that constrains
occurrences of variables in the production rules according to their
type-theoretic order. In this paper, we introduce the safe lambda calculus,
which is obtained by transposing (and generalizing) the safety condition to the
setting of the simply-typed lambda calculus. In contrast to the original
definition of safety, our calculus does not constrain types (to be
homogeneous). We show that in the safe lambda calculus, there is no need to
rename bound variables when performing substitution, as variable capture is
guaranteed not to happen. We also propose an adequate notion of beta-reduction
that preserves safety. In the same vein as Schwichtenberg's 1976
characterization of the simply-typed lambda calculus, we show that the numeric
functions representable in the safe lambda calculus are exactly the
multivariate polynomials; thus conditional is not definable. We also give a
characterization of representable word functions. We then study the complexity
of deciding beta-eta equality of two safe simply-typed terms and show that this
problem is PSPACE-hard. Finally we give a game-semantic analysis of safety: We
show that safe terms are denoted by `P-incrementally justified strategies'.
Consequently pointers in the game semantics of safe lambda-terms are only
necessary from order 4 onwards
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
Cost Automata, Safe Schemes, and Downward Closures
Higher-order recursion schemes are an expressive formalism used to define
languages of possibly infinite ranked trees. They extend regular and
context-free grammars, and are equivalent to simply typed -calculus
and collapsible pushdown automata. In this work we prove, under a syntactical
constraint called safety, decidability of the model-checking problem for
recursion schemes against properties defined by alternating B-automata, an
extension of alternating parity automata for infinite trees with a boundedness
acceptance condition. We then exploit this result to show how to compute
downward closures of languages of finite trees recognized by safe recursion
schemes.Comment: accepted at ICALP'2
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