754 research outputs found
Behavioural Equivalence for Infinite Systems—Partially Decidable!
For finite-state systems non-interleaving equivalences are computationallyat least as hard as interleaving equivalences. In this paper we showthat when moving to infinite-state systems, this situation may changedramatically.We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.We first study equivalences on Basic Parallel Processes, BPP, a processcalculus equivalent to communication free Petri nets. For this simpleprocess language our two notions of non-interleaving equivalences agree.More interestingly, we show that they are decidable, contrasting a result ofHirshfeld that standard interleaving language equivalence is undecidable.Our result is inspired by a recent result of Esparza and Kiehn, showingthe same phenomenon in the setting of model checking.We follow up investigating to which extent the result extends to largersubsets of CCS and TCSP. We discover a significant difference betweenour non-interleaving equivalences. We show that for a certain non-trivialsubclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets
Model Checking Dynamic-Epistemic Spatial Logic
In this paper we focus on Dynamic Spatial Logic, the extension of Hennessy-Milner logic with the parallel operator. We develop a sound complete Hilbert-style axiomatic system for it comprehending the behavior of spatial operators in relation with dynamic/temporal ones. Underpining on a new congruence we define over the class of processes - the structural bisimulation - we prove the finite model property for this logic that provides the decidability for satisfiability, validity and model checking against process semantics. Eventualy we propose algorithms for validity, satisfiability and model checking
Strong Turing Degrees for Additive BSS RAM's
For the additive real BSS machines using only constants 0 and 1 and order
tests we consider the corresponding Turing reducibility and characterize some
semi-decidable decision problems over the reals. In order to refine,
step-by-step, a linear hierarchy of Turing degrees with respect to this model,
we define several halting problems for classes of additive machines with
different abilities and construct further suitable decision problems. In the
construction we use methods of the classical recursion theory as well as
techniques for proving bounds resulting from algebraic properties. In this way
we extend a known hierarchy of problems below the halting problem for the
additive machines using only equality tests and we present a further
subhierarchy of semi-decidable problems between the halting problems for the
additive machines using only equality tests and using order tests,
respectively
Process Algebra, CCS, and Bisimulation Decidability
Over the past fifteen years, there has been intensive study of formal systems that can model concurrency and communication. Two such systems are the Calculus of Communicating Systems, and the Algebra of Communicating Processes. The objective of this paper has two aspects; (1) to study the characteristics and features of these two systems, and (2) to investigate two interesting formal proofs concerning issues of decidability of bisimulation equivalence in these systems. An examination of the processes that generate context-free languages as a trace set shows that their bisimulation equivalence is decidable, in contrast to the undecidability of their trace set equivalence. Recent results have also shown that the bisimulation equivalence problem for processes with a limited amount of concurrency is decidable
Further Results on Partial Order Equivalences on Infinite Systems
In [26], we investigated decidability issues for standard language equivalence for process description languages with two generalisations based on traditional approachesfor capturing non-interleaving behaviour: pomset equivalence reflecting global causal dependency, and location equivalence reflecting spatial distribution of events. In this paper, we continue by investigating the role played by TCSP-style renaming and hiding combinators with respect to decidability. One result of [26] was that in contrast to pomset equivalence, location equivalence remained decidable for a class of processes consisting of finite sets of BPP processes communicating in a TCSP manner. Here, we show that location equivalence becomes undecidable when either renaming or hiding is added to this class of processes. Furthermore, we investigate the weak versions of location and pomset equivalences.We show that for BPP with prefixing, both weak pomset and weak location equivalence are decidable. Moreover, we show that weak location equivalence is undecidable for BPP semantically extended with CCS communication
Expansions of MSO by cardinality relations
We study expansions of the Weak Monadic Second Order theory of (N,<) by
cardinality relations, which are predicates R(X1,...,Xn) whose truth value
depends only on the cardinality of the sets X1, ...,Xn. We first provide a
(definable) criterion for definability of a cardinality relation in (N,<), and
use it to prove that for every cardinality relation R which is not definable in
(N,<), there exists a unary cardinality relation which is definable in (N,<,R)
and not in (N,<). These results resemble Muchnik and Michaux-Villemaire
theorems for Presburger Arithmetic. We prove then that + and x are definable in
(N,<,R) for every cardinality relation R which is not definable in (N,<). This
implies undecidability of the WMSO theory of (N,<,R). We also consider the
related satisfiability problem for the class of finite orderings, namely the
question whether an MSO sentence in the language {<,R} admits a finite model M
where < is interpreted as a linear ordering, and R as the restriction of some
(fixed) cardinality relation to the domain of M. We prove that this problem is
undecidable for every cardinality relation R which is not definable in (N,<).Comment: to appear in LMC
A Decidable Characterization of a Graphical Pi-calculus with Iterators
This paper presents the Pi-graphs, a visual paradigm for the modelling and
verification of mobile systems. The language is a graphical variant of the
Pi-calculus with iterators to express non-terminating behaviors. The
operational semantics of Pi-graphs use ground notions of labelled transition
and bisimulation, which means standard verification techniques can be applied.
We show that bisimilarity is decidable for the proposed semantics, a result
obtained thanks to an original notion of causal clock as well as the automatic
garbage collection of unused names.Comment: In Proceedings INFINITY 2010, arXiv:1010.611
Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
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