34 research outputs found
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 Automata and Recursion Schemes
International audienceWe consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed {deterministic term} rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this paper we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it and contracts its silent transitions, which leads to an infinite tree which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving an effective transformations in both directions
Reachability analysis of first-order definable pushdown systems
We study pushdown systems where control states, stack alphabet, and
transition relation, instead of being finite, are first-order definable in a
fixed countably-infinite structure. We show that the reachability analysis can
be addressed with the well-known saturation technique for the wide class of
oligomorphic structures. Moreover, for the more restrictive homogeneous
structures, we are able to give concrete complexity upper bounds. We show ample
applicability of our technique by presenting several concrete examples of
homogeneous structures, subsuming, with optimal complexity, known results from
the literature. We show that infinitely many such examples of homogeneous
structures can be obtained with the classical wreath product construction.Comment: to appear in CSL'1
On Model-Checking Higher-Order Effectful Programs (Long Version)
Model-checking is one of the most powerful techniques for verifying systems
and programs, which since the pioneering results by Knapik et al., Ong, and
Kobayashi, is known to be applicable to functional programs with higher-order
types against properties expressed by formulas of monadic second-order logic.
What happens when the program in question, in addition to higher-order
functions, also exhibits algebraic effects such as probabilistic choice or
global store? The results in the literature range from those, mostly positive,
about nondeterministic effects, to those about probabilistic effects, in the
presence of which even mere reachability becomes undecidable. This work takes a
fresh and general look at the problem, first of all showing that there is an
elegant and natural way of viewing higher-order programs producing algebraic
effects as ordinary higher-order recursion schemes. We then move on to consider
effect handlers, showing that in their presence the model checking problem is
bound to be undecidable in the general case, while it stays decidable when
handlers have a simple syntactic form, still sufficient to capture so-called
generic effects. Along the way we hint at how a general specification language
could look like, this way justifying some of the results in the literature, and
deriving new ones