15 research outputs found
Model Checking Synchronized Products of Infinite Transition Systems
Formal verification using the model checking paradigm has to deal with two
aspects: The system models are structured, often as products of components, and
the specification logic has to be expressive enough to allow the formalization
of reachability properties. The present paper is a study on what can be
achieved for infinite transition systems under these premises. As models we
consider products of infinite transition systems with different synchronization
constraints. We introduce finitely synchronized transition systems, i.e.
product systems which contain only finitely many (parameterized) synchronized
transitions, and show that the decidability of FO(R), first-order logic
extended by reachability predicates, of the product system can be reduced to
the decidability of FO(R) of the components. This result is optimal in the
following sense: (1) If we allow semifinite synchronization, i.e. just in one
component infinitely many transitions are synchronized, the FO(R)-theory of the
product system is in general undecidable. (2) We cannot extend the expressive
power of the logic under consideration. Already a weak extension of first-order
logic with transitive closure, where we restrict the transitive closure
operators to arity one and nesting depth two, is undecidable for an
asynchronous (and hence finitely synchronized) product, namely for the infinite
grid.Comment: 18 page
The Complexity of Model Checking (Collapsible) Higher-Order Pushdown Systems
We study (collapsible) higher-order pushdown systems --- theoretically robust and well-studied models of higher-order programs --- along with their natural subclass called (collapsible) higher-order basic process algebras. We provide a comprehensive analysis of the model checking complexity of a range of both branching-time and linear-time temporal logics. We obtain tight bounds on data, expression, and combined-complexity for both (collapsible) higher-order pushdown systems and (collapsible) higher-order basic process algebra. At order-, results range from polynomial to -exponential time. Finally, we study (collapsible) higher-order basic process algebras as graph generators and show that they are almost as powerful as (collapsible) higher-order pushdown systems up to MSO interpretations
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
On the relationships between theories of time granularity and the monadic second-order theory of one successor
In this paper we explore the connections between the
monadic second-order theory of one successor MSO[<] for
short) and the theories of omega-layered structures for time
granularity. We first prove that the decision problem for
MSO[<] and that for a suitable first-order theory of the upward
unbounded layered structure are inter-reducible. Then, we show that a similar result holds for suitable chain variants of the MSO theory of the totally unbounded layered structure (this allows us to solve a decision problem about theories of time granularity left open by Franceschet et al. [FRA 06]))
Recurrent Reachability Analysis in Regular Model Checking
Abstract. We consider the problem of recurrent reachability over infinite systems given by regular relations on words and trees, i.e, whether a given regular set of states can be reached infinitely often from a given initial state in the given transition system. Under the condition that the transitive closure of the transition relation is regular, we show that the problem is decidable, and the set of all initial states satisfying the property is regular. Moreover, our algorithm constructs an automaton for this set in polynomial time, assuming that a transducer of the transitive closure can be computed in poly-time. We then demonstrate that transition systems generated by pushdown systems, regular ground tree rewrite systems, and the well-known process algebra PA satisfy our condition and transducers for their transitive closures can be computed in poly-time. Our result also implies that model checking EF-logic extended by recurrent reachability predicate (EGF) over such systems is decidable.
Transforming structures by set interpretations
We consider a new kind of interpretation over relational structures: finite
sets interpretations. Those interpretations are defined by weak monadic
second-order (WMSO) formulas with free set variables. They transform a given
structure into a structure with a domain consisting of finite sets of elements
of the orignal structure. The definition of these interpretations directly
implies that they send structures with a decidable WMSO theory to structures
with a decidable first-order theory. In this paper, we investigate the
expressive power of such interpretations applied to infinite deterministic
trees. The results can be used in the study of automatic and tree-automatic
structures.Comment: 36 page
Interpretations in Trees with Countably Many Branches
Abstract—We study the expressive power of logical interpreta-tions on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by ‘finitary’ set-interpretations. Index Terms—Composition method, finite-set interpretations, infinite scattered trees, monadic second order logic. I
Axiomatization of betweenness in order-theoretic trees
The ternary betweenness relation of a tree, B(x,y,z) expresses that y is on
the unique path between x and z. This notion can be extended to order-theoretic
trees defined as partial orders such that the set of nodes larger than any node
is linearly ordered. In such generalized trees, the unique "path" between two
nodes can have infinitely many nodes.
We generalize some results obtained in a previous article for the betweenness
of join-trees. Join-trees are order-theoretic trees such that any two nodes
have a least upper-bound. The motivation was to define conveniently the
rank-width of a countable graph. We called quasi-tree the structure based on
the betweenness relation of a join-tree. We proved that quasi-trees are
axiomatized by a first-order sentence.
Here, we obtain a monadic second-order axiomatization of betweenness in
order-theoretic trees. We also define and compare several induced betweenness
relations, i.e., restrictions to sets of nodes of the betweenness relations in
generalized trees of different kinds. We prove that induced betweenness in
quasi-trees is characterized by a first-order sentence. The proof uses
order-theoretic trees