344 research outputs found
Quantifier-Free Interpolation of a Theory of Arrays
The use of interpolants in model checking is becoming an enabling technology
to allow fast and robust verification of hardware and software. The application
of encodings based on the theory of arrays, however, is limited by the
impossibility of deriving quantifier- free interpolants in general. In this
paper, we show that it is possible to obtain quantifier-free interpolants for a
Skolemized version of the extensional theory of arrays. We prove this in two
ways: (1) non-constructively, by using the model theoretic notion of
amalgamation, which is known to be equivalent to admit quantifier-free
interpolation for universal theories; and (2) constructively, by designing an
interpolating procedure, based on solving equations between array updates.
(Interestingly, rewriting techniques are used in the key steps of the solver
and its proof of correctness.) To the best of our knowledge, this is the first
successful attempt of computing quantifier- free interpolants for a variant of
the theory of arrays with extensionality
Strict General Setting for Building Decision Procedures into Theorem Provers
The efficient and flexible incorporating of decision procedures into theorem provers is very important for their successful use. There are several approaches for combining and augmenting of decision procedures; some of them support handling uninterpreted functions, congruence closure, lemma invoking etc. In this paper we present a variant of one general setting for building decision procedures into theorem provers (gs framework [18]). That setting is based on macro inference rules motivated by techniques used in different approaches. The general setting enables a simple describing of different combination/augmentation schemes. In this paper, we further develop and extend this setting by an imposed ordering on the macro inference rules. That ordering leads to a âstrict settingâ. It makes implementing and using variants of well-known or new schemes within this framework a very easy task even for a non-expert user. Also, this setting enables easy comparison of different combination/augmentation schemes and combination of their ideas
Constraint LTL Satisfiability Checking without Automata
This paper introduces a novel technique to decide the satisfiability of
formulae written in the language of Linear Temporal Logic with Both future and
past operators and atomic formulae belonging to constraint system D (CLTLB(D)
for short). The technique is based on the concept of bounded satisfiability,
and hinges on an encoding of CLTLB(D) formulae into QF-EUD, the theory of
quantifier-free equality and uninterpreted functions combined with D. Similarly
to standard LTL, where bounded model-checking and SAT-solvers can be used as an
alternative to automata-theoretic approaches to model-checking, our approach
allows users to solve the satisfiability problem for CLTLB(D) formulae through
SMT-solving techniques, rather than by checking the emptiness of the language
of a suitable automaton A_{\phi}. The technique is effective, and it has been
implemented in our Zot formal verification tool.Comment: 39 page
On Algorithms and Complexity for Sets with Cardinality Constraints
Typestate systems ensure many desirable properties of imperative programs,
including initialization of object fields and correct use of stateful library
interfaces. Abstract sets with cardinality constraints naturally generalize
typestate properties: relationships between the typestates of objects can be
expressed as subset and disjointness relations on sets, and elements of sets
can be represented as sets of cardinality one. Motivated by these applications,
this paper presents new algorithms and new complexity results for constraints
on sets and their cardinalities. We study several classes of constraints and
demonstrate a trade-off between their expressive power and their complexity.
Our first result concerns a quantifier-free fragment of Boolean Algebra with
Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for
reducing the satisfiability of sets with symbolic cardinalities to constraints
on constant cardinalities, and give a polynomial-space algorithm for the
resulting problem.
In a quest for more efficient fragments, we identify several subclasses of
sets with cardinality constraints whose satisfiability is NP-hard. Finally, we
identify a class of constraints that has polynomial-time satisfiability and
entailment problems and can serve as a foundation for efficient program
analysis.Comment: 20 pages. 12 figure
Backward Reachability of Array-based Systems by SMT solving: Termination and Invariant Synthesis
The safety of infinite state systems can be checked by a backward
reachability procedure. For certain classes of systems, it is possible to prove
the termination of the procedure and hence conclude the decidability of the
safety problem. Although backward reachability is property-directed, it can
unnecessarily explore (large) portions of the state space of a system which are
not required to verify the safety property under consideration. To avoid this,
invariants can be used to dramatically prune the search space. Indeed, the
problem is to guess such appropriate invariants. In this paper, we present a
fully declarative and symbolic approach to the mechanization of backward
reachability of infinite state systems manipulating arrays by Satisfiability
Modulo Theories solving. Theories are used to specify the topology and the data
manipulated by the system. We identify sufficient conditions on the theories to
ensure the termination of backward reachability and we show the completeness of
a method for invariant synthesis (obtained as the dual of backward
reachability), again, under suitable hypotheses on the theories. We also
present a pragmatic approach to interleave invariant synthesis and backward
reachability so that a fix-point for the set of backward reachable states is
more easily obtained. Finally, we discuss heuristics that allow us to derive an
implementation of the techniques in the model checker MCMT, showing remarkable
speed-ups on a significant set of safety problems extracted from a variety of
sources.Comment: Accepted for publication in Logical Methods in Computer Scienc
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
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