801 research outputs found
On P-transitive graphs and applications
We introduce a new class of graphs which we call P-transitive graphs, lying
between transitive and 3-transitive graphs. First we show that the analogue of
de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we
show that the mu-calculus fixpoint hierarchy is infinite for P-transitive
graphs. Both results contrast with the case of transitive graphs. We give also
an undecidability result for an enriched mu-calculus on P-transitive graphs.
Finally, we consider a polynomial time reduction from the model checking
problem on arbitrary graphs to the model checking problem on P-transitive
graphs. All these results carry over to 3-transitive graphs.Comment: In Proceedings GandALF 2011, arXiv:1106.081
The Arity Hierarchy in the Polyadic -Calculus
The polyadic mu-calculus is a modal fixpoint logic whose formulas define
relations of nodes rather than just sets in labelled transition systems. It can
express exactly the polynomial-time computable and bisimulation-invariant
queries on finite graphs. In this paper we show a hierarchy result with respect
to expressive power inside the polyadic mu-calculus: for every level of
fixpoint alternation, greater arity of relations gives rise to higher
expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282
On Verifying Complex Properties using Symbolic Shape Analysis
One of the main challenges in the verification of software systems is the
analysis of unbounded data structures with dynamic memory allocation, such as
linked data structures and arrays. We describe Bohne, a new analysis for
verifying data structures. Bohne verifies data structure operations and shows
that 1) the operations preserve data structure invariants and 2) the operations
satisfy their specifications expressed in terms of changes to the set of
objects stored in the data structure. During the analysis, Bohne infers loop
invariants in the form of disjunctions of universally quantified Boolean
combinations of formulas. To synthesize loop invariants of this form, Bohne
uses a combination of decision procedures for Monadic Second-Order Logic over
trees, SMT-LIB decision procedures (currently CVC Lite), and an automated
reasoner within the Isabelle interactive theorem prover. This architecture
shows that synthesized loop invariants can serve as a useful communication
mechanism between different decision procedures. Using Bohne, we have verified
operations on data structures such as linked lists with iterators and back
pointers, trees with and without parent pointers, two-level skip lists, array
data structures, and sorted lists. We have deployed Bohne in the Hob and Jahob
data structure analysis systems, enabling us to combine Bohne with analyses of
data structure clients and apply it in the context of larger programs. This
report describes the Bohne algorithm as well as techniques that Bohne uses to
reduce the ammount of annotations and the running time of the analysis
Complexity of Nested Circumscription and Nested Abnormality Theories
The need for a circumscriptive formalism that allows for simple yet elegant
modular problem representation has led Lifschitz (AIJ, 1995) to introduce
nested abnormality theories (NATs) as a tool for modular knowledge
representation, tailored for applying circumscription to minimize exceptional
circumstances. Abstracting from this particular objective, we propose L_{CIRC},
which is an extension of generic propositional circumscription by allowing
propositional combinations and nesting of circumscriptive theories. As shown,
NATs are naturally embedded into this language, and are in fact of equal
expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and
in particular the effect of nesting. The latter is found to be a source of
complexity, which climbs the Polynomial Hierarchy as the nesting depth
increases and reaches PSPACE-completeness in the general case. We also identify
meaningful syntactic fragments of NATs which have lower complexity. In
particular, we show that the generalization of Horn circumscription in the NAT
framework remains CONP-complete, and that Horn NATs without fixed letters can
be efficiently transformed into an equivalent Horn CNF, which implies
polynomial solvability of principal reasoning tasks. Finally, we also study
extensions of NATs and briefly address the complexity in the first-order case.
Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a
host language for expressing other formalisms such as action theories,
narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth
International Joint Conference on Artificial Intelligence (IJCAI-01), pages
169--174. Morgan Kaufmann, 200
Generalizing the Paige-Tarjan Algorithm by Abstract Interpretation
The Paige and Tarjan algorithm (PT) for computing the coarsest refinement of
a state partition which is a bisimulation on some Kripke structure is well
known. It is also well known in model checking that bisimulation is equivalent
to strong preservation of CTL, or, equivalently, of Hennessy-Milner logic.
Drawing on these observations, we analyze the basic steps of the PT algorithm
from an abstract interpretation perspective, which allows us to reason on
strong preservation in the context of generic inductively defined (temporal)
languages and of possibly non-partitioning abstract models specified by
abstract interpretation. This leads us to design a generalized Paige-Tarjan
algorithm, called GPT, for computing the minimal refinement of an abstract
interpretation-based model that strongly preserves some given language. It
turns out that PT is a straight instance of GPT on the domain of state
partitions for the case of strong preservation of Hennessy-Milner logic. We
provide a number of examples showing that GPT is of general use. We first show
how a well-known efficient algorithm for computing stuttering equivalence can
be viewed as a simple instance of GPT. We then instantiate GPT in order to
design a new efficient algorithm for computing simulation equivalence that is
competitive with the best available algorithms. Finally, we show how GPT allows
to compute new strongly preserving abstract models by providing an efficient
algorithm that computes the coarsest refinement of a given partition that
strongly preserves the language generated by the reachability operator.Comment: Keywords: Abstract interpretation, abstract model checking, strong
preservation, Paige-Tarjan algorithm, refinement algorith
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