288,948 research outputs found
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation
AC-completion efficiently handles equality modulo associative and commutative
function symbols. When the input is ground, the procedure terminates and
provides a decision algorithm for the word problem. In this paper, we present a
modular extension of ground AC-completion for deciding formulas in the
combination of the theory of equality with user-defined AC symbols,
uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our
algorithm, called AC(X), is obtained by augmenting in a modular way ground
AC-completion with the canonizer and solver present for the theory X. This
integration rests on canonized rewriting, a new relation reminiscent to
normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is
proved sound, complete and terminating, and is implemented to extend the core
of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized
Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for
publication by LMCS (Logical Methods in Computer Science
AC-KBO Revisited
Equational theories that contain axioms expressing associativity and
commutativity (AC) of certain operators are ubiquitous. Theorem proving methods
in such theories rely on well-founded orders that are compatible with the AC
axioms. In this paper we consider various definitions of AC-compatible
Knuth-Bendix orders. The orders of Steinbach and of Korovin and Voronkov are
revisited. The former is enhanced to a more powerful version, and we modify the
latter to amend its lack of monotonicity on non-ground terms. We further
present new complexity results. An extension reflecting the recent proposal of
subterm coefficients in standard Knuth-Bendix orders is also given. The various
orders are compared on problems in termination and completion.Comment: 31 pages, To appear in Theory and Practice of Logic Programming
(TPLP) special issue for the 12th International Symposium on Functional and
Logic Programming (FLOPS 2014
Modularity and Combination of Associative Commutative Congruence Closure Algorithms enriched with Semantic Properties
Algorithms for computing congruence closure of ground equations over
uninterpreted symbols and interpreted symbols satisfying associativity and
commutativity (AC) properties are proposed. The algorithms are based on a
framework for computing a congruence closure by abstracting nonflat terms by
constants as proposed first in Kapur's congruence closure algorithm (RTA97).
The framework is general, flexible, and has been extended also to develop
congruence closure algorithms for the cases when associative-commutative
function symbols can have additional properties including idempotency,
nilpotency, identities, cancellativity and group properties as well as their
various combinations. Algorithms are modular; their correctness and termination
proofs are simple, exploiting modularity. Unlike earlier algorithms, the
proposed algorithms neither rely on complex AC compatible well-founded
orderings on nonvariable terms nor need to use the associative-commutative
unification and extension rules in completion for generating canonical rewrite
systems for congruence closures. They are particularly suited for integrating
into the Satisfiability modulo Theories (SMT) solvers. A new way to view
Groebner basis algorithm for polynomial ideals with integer coefficients as a
combination of the congruence closures over the AC symbol * with the identity 1
and the congruence closure over an Abelian group with + is outlined
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
Learning Action Maps of Large Environments via First-Person Vision
When people observe and interact with physical spaces, they are able to
associate functionality to regions in the environment. Our goal is to automate
dense functional understanding of large spaces by leveraging sparse activity
demonstrations recorded from an ego-centric viewpoint. The method we describe
enables functionality estimation in large scenes where people have behaved, as
well as novel scenes where no behaviors are observed. Our method learns and
predicts "Action Maps", which encode the ability for a user to perform
activities at various locations. With the usage of an egocentric camera to
observe human activities, our method scales with the size of the scene without
the need for mounting multiple static surveillance cameras and is well-suited
to the task of observing activities up-close. We demonstrate that by capturing
appearance-based attributes of the environment and associating these attributes
with activity demonstrations, our proposed mathematical framework allows for
the prediction of Action Maps in new environments. Additionally, we offer a
preliminary glance of the applicability of Action Maps by demonstrating a
proof-of-concept application in which they are used in concert with activity
detections to perform localization.Comment: To appear at CVPR 201
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