885 research outputs found
Abduction-Based Explanations for Machine Learning Models
The growing range of applications of Machine Learning (ML) in a multitude of
settings motivates the ability of computing small explanations for predictions
made. Small explanations are generally accepted as easier for human decision
makers to understand. Most earlier work on computing explanations is based on
heuristic approaches, providing no guarantees of quality, in terms of how close
such solutions are from cardinality- or subset-minimal explanations. This paper
develops a constraint-agnostic solution for computing explanations for any ML
model. The proposed solution exploits abductive reasoning, and imposes the
requirement that the ML model can be represented as sets of constraints using
some target constraint reasoning system for which the decision problem can be
answered with some oracle. The experimental results, obtained on well-known
datasets, validate the scalability of the proposed approach as well as the
quality of the computed solutions
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
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
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
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