83 research outputs found
New results on rewrite-based satisfiability procedures
Program analysis and verification require decision procedures to reason on
theories of data structures. Many problems can be reduced to the satisfiability
of sets of ground literals in theory T. If a sound and complete inference
system for first-order logic is guaranteed to terminate on T-satisfiability
problems, any theorem-proving strategy with that system and a fair search plan
is a T-satisfiability procedure. We prove termination of a rewrite-based
first-order engine on the theories of records, integer offsets, integer offsets
modulo and lists. We give a modularity theorem stating sufficient conditions
for termination on a combinations of theories, given termination on each. The
above theories, as well as others, satisfy these conditions. We introduce
several sets of benchmarks on these theories and their combinations, including
both parametric synthetic benchmarks to test scalability, and real-world
problems to test performances on huge sets of literals. We compare the
rewrite-based theorem prover E with the validity checkers CVC and CVC Lite.
Contrary to the folklore that a general-purpose prover cannot compete with
reasoners with built-in theories, the experiments are overall favorable to the
theorem prover, showing that not only the rewriting approach is elegant and
conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page
Automated deduction with built-in theories: completeness results and constraint solving techniques
Postprint (published version
A resolution principle for clauses with constraints
We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification
Ordering constraints on trees
We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature
Compositional Computational Reflection
Current work on computational reflection is single-minded; each reflective procedure is written with a specific application or scope in mind. Composition of these reflective procedures is done by a proof- generating tactic language such as Ltac. This composition, however, comes at the cost of both larger proof terms and redundant preprocessing. In this work, we propose a methodology for writing composable reflective procedures that solve many small tasks in a single invocation. The key technical insights are techniques for reasoning semantically about extensible syntax in intensional type theory. Our techniques make it possible to compose sound procedures and write generic procedures parametrized by lemmas mimicking Coq’s support for hint databases.Engineering and Applied Science
A theory of resolution
We review the fundamental resolution-based methods for first-order theorem proving and present them in a uniform framework. We show that these calculi can be viewed as specializations of non-clausal resolution with simplification. Simplification techniques are justified with the help of a rather general notion of redundancy for inferences. As simplification and other techniques for the elimination of redundancy are indispensable for an acceptable behaviour of any practical theorem prover this work is the first uniform treatment of resolution-like techniques in which the avoidance of redundant computations attains the attention it deserves. In many cases our presentation of a resolution method will indicate new ways of how to improve the method over what was known previously. We also give answers to several open problems in the area
Connectionist Inference Models
The performance of symbolic inference tasks has long been a challenge to connectionists. In this paper, we present an extended survey of this area. Existing connectionist inference systems are reviewed, with particular reference to how they perform variable binding and rule-based reasoning, and whether they involve distributed or localist representations. The benefits and disadvantages of different representations and systems are outlined, and conclusions drawn regarding the capabilities of connectionist inference systems when compared with symbolic inference systems or when used for cognitive modeling
Disproving in First-Order Logic with Definitions, Arithmetic and Finite Domains
This thesis explores several methods which enable a first-order
reasoner to conclude satisfiability of a formula modulo an
arithmetic theory. The most general method requires restricting
certain quantifiers to range over finite sets; such assumptions
are common in the software verification setting. In addition, the
use of first-order reasoning allows for an implicit
representation of those finite sets, which can avoid
scalability problems that affect other quantified reasoning
methods. These new techniques form a useful complement to
existing methods that are primarily aimed at proving validity.
The Superposition calculus for hierarchic theory combinations
provides a basis for reasoning modulo theories in a first-order
setting. The recent account of ‘weak abstraction’ and related
improvements make an mplementation of the calculus practical.
Also, for several logical theories of interest Superposition is
an effective decision procedure for the quantifier free fragment.
The first contribution is an implementation of that calculus
(Beagle), including an optimized implementation of Cooper’s
algorithm for quantifier elimination in the theory of linear
integer arithmetic. This includes a novel means of extracting
values
for quantified variables in satisfiable integer problems. Beagle
won an efficiency award at CADE Automated theorem prover System
Competition (CASC)-J7, and won the arithmetic non-theorem
category at CASC-25. This implementation is the start point for
solving the ‘disproving with theories’ problem.
Some hypotheses can be disproved by showing that, together with
axioms the hypothesis is unsatisfiable. Often this is relative to
other axioms that enrich a base theory by defining new functions.
In that case, the disproof is contingent on the satisfiability of
the enrichment.
Satisfiability in this context is undecidable. Instead, general
characterizations of definition formulas, which do not alter the
satisfiability status of the main axioms, are given. These
general criteria apply to recursive definitions, definitions over
lists, and to arrays. This allows proving some non-theorems which
are otherwise intractable, and justifies similar disproofs of
non-linear arithmetic formulas.
When the hypothesis is contingently true, disproof requires
proving existence of
a model. If the Superposition calculus saturates a clause set,
then a model exists,
but only when the clause set satisfies a completeness criterion.
This requires each
instance of an uninterpreted, theory-sorted term to have a
definition in terms of
theory symbols.
The second contribution is a procedure that creates such
definitions, given that a subset of quantifiers range over finite
sets. Definitions are produced in a counter-example driven way
via a sequence of over and under approximations to the clause
set. Two descriptions of the method are given: the first uses the
component solver modularly, but has an inefficient
counter-example heuristic. The second is more general, correcting
many of the inefficiencies of the first, yet it requires tracking
clauses through a proof. This latter method is shown to apply
also to lists and to problems with unbounded quantifiers.
Together, these tools give new ways for applying successful
first-order reasoning methods to problems involving interpreted
theories
Model generation style completeness proofs for constraint tableaux with superposition
We present several calculi that integrate equality handling
by superposition and ordered paramodulation into a free
variable tableau calculus. We prove completeness of this
calculus by an adaptation of the model generation technique
commonly used for completeness proofs of resolution calculi.
The calculi and the completeness proof are compared to earlier
results of Degtyarev and Voronkov
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