9 research outputs found
Analysis of Boolean Equation Systems through Structure Graphs
We analyse the problem of solving Boolean equation systems through the use of
structure graphs. The latter are obtained through an elegant set of
Plotkin-style deduction rules. Our main contribution is that we show that
equation systems with bisimilar structure graphs have the same solution. We
show that our work conservatively extends earlier work, conducted by Keiren and
Willemse, in which dependency graphs were used to analyse a subclass of Boolean
equation systems, viz., equation systems in standard recursive form. We
illustrate our approach by a small example, demonstrating the effect of
simplifying an equation system through minimisation of its structure graph
Memory Reduction via Delayed Simulation
We address a central (and classical) issue in the theory of infinite games:
the reduction of the memory size that is needed to implement winning strategies
in regular infinite games (i.e., controllers that ensure correct behavior
against actions of the environment, when the specification is a regular
omega-language). We propose an approach which attacks this problem before the
construction of a strategy, by first reducing the game graph that is obtained
from the specification. For the cases of specifications represented by
"request-response"-requirements and general "fairness" conditions, we show that
an exponential gain in the size of memory is possible.Comment: In Proceedings iWIGP 2011, arXiv:1102.374
Structural Analysis of Boolean Equation Systems
We analyse the problem of solving Boolean equation systems through the use of
structure graphs. The latter are obtained through an elegant set of
Plotkin-style deduction rules. Our main contribution is that we show that
equation systems with bisimilar structure graphs have the same solution. We
show that our work conservatively extends earlier work, conducted by Keiren and
Willemse, in which dependency graphs were used to analyse a subclass of Boolean
equation systems, viz., equation systems in standard recursive form. We
illustrate our approach by a small example, demonstrating the effect of
simplifying an equation system through minimisation of its structure graph
Bisimulation minimisations for Boolean equation systems
Boolean equation systems (BESs) have been used to encode several complex verification problems, including model checking and equivalence checking. We introduce the concepts of strong bisimulation and oblivious bisimulation for BESs, and we prove that these can be used for minimising BESs prior to solving these. Our results show that large reductions of the BESs may be obtained efficiently. Minimisation is rewarding for BESs with non-trivial alternations: the time required for solving the original BES exceeds the time required for quotienting plus the time for solving the quotient. Furthermore, we provide a verification example that demonstrates that bisimulation minimisation of a process prior to encoding the verification problem on that process as a BES can be arbitrarily less effective than minimising the BES that encodes the verification problem
Fair Simulation for Nondeterministic and Probabilistic Buechi Automata: a Coalgebraic Perspective
Notions of simulation, among other uses, provide a computationally tractable
and sound (but not necessarily complete) proof method for language inclusion.
They have been comprehensively studied by Lynch and Vaandrager for
nondeterministic and timed systems; for B\"{u}chi automata the notion of fair
simulation has been introduced by Henzinger, Kupferman and Rajamani. We
contribute to a generalization of fair simulation in two different directions:
one for nondeterministic tree automata previously studied by Bomhard; and the
other for probabilistic word automata with finite state spaces, both under the
B\"{u}chi acceptance condition. The former nondeterministic definition is
formulated in terms of systems of fixed-point equations, hence is readily
translated to parity games and is then amenable to Jurdzi\'{n}ski's algorithm;
the latter probabilistic definition bears a strong ranking-function flavor.
These two different-looking definitions are derived from one source, namely our
coalgebraic modeling of B\"{u}chi automata. Based on these coalgebraic
observations, we also prove their soundness: a simulation indeed witnesses
language inclusion
Bisimulation minimisations for boolean equation systems
Abstract. Boolean equation systems (BESs) have been used to encode several complex verification problems, including model checking and equivalence checking. We introduce the concepts of strong bisimulation and idempotence-identifying bisimulation for BESs, and we prove that these can be used for minimising BESs prior to solving these. Our results show that large reductions of the BESs may be obtained efficiently. Minimisation is rewarding for BESs with non-trivial alternations: the time required for solving the original BES mostly exceeds the time required for quotienting plus the time for solving the quotient. Furthermore, we provide a verification example that demonstrates that bisimulation minimisation of a process prior to encoding the verification problem on that process as a BES can be arbitrarily less effective than minimising the BES that encodes the verification problem
Efficient reduction of nondeterministic automata with application to language inclusion testing
We present efficient algorithms to reduce the size of nondeterministic
B\"uchi word automata (NBA) and nondeterministic finite word automata (NFA),
while retaining their languages. Additionally, we describe methods to solve
PSPACE-complete automata problems like language universality, equivalence, and
inclusion for much larger instances than was previously possible (
states instead of 10-100). This can be used to scale up applications of
automata in formal verification tools and decision procedures for logical
theories. The algorithms are based on new techniques for removing transitions
(pruning) and adding transitions (saturation), as well as extensions of classic
quotienting of the state space. These techniques use criteria based on
combinations of backward and forward trace inclusions and simulation relations.
Since trace inclusion relations are themselves PSPACE-complete, we introduce
lookahead simulations as good polynomial time computable approximations
thereof. Extensive experiments show that the average-case time complexity of
our algorithms scales slightly above quadratically. (The space complexity is
worst-case quadratic.) The size reduction of the automata depends very much on
the class of instances, but our algorithm consistently reduces the size far
more than all previous techniques. We tested our algorithms on NBA derived from
LTL-formulae, NBA derived from mutual exclusion protocols and many classes of
random NBA and NFA, and compared their performance to the well-known automata
tool GOAL.Comment: 69 pages. arXiv admin note: text overlap with arXiv:1210.662