5,737 research outputs found
A simple sequent calculus for nominal logic
Nominal logic is a variant of first-order logic that provides support for
reasoning about bound names in abstract syntax. A key feature of nominal logic
is the new-quantifier, which quantifies over fresh names (names not appearing
in any values considered so far). Previous attempts have been made to develop
convenient rules for reasoning with the new-quantifier, but we argue that none
of these attempts is completely satisfactory.
In this article we develop a new sequent calculus for nominal logic in which
the rules for the new- quantifier are much simpler than in previous attempts.
We also prove several structural and metatheoretic properties, including
cut-elimination, consistency, and equivalence to Pitts' axiomatization of
nominal logic
Formulating problems for real algebraic geometry
We discuss issues of problem formulation for algorithms in real algebraic
geometry, focussing on quantifier elimination by cylindrical algebraic
decomposition. We recall how the variable ordering used can have a profound
effect on both performance and output and summarise what may be done to assist
with this choice. We then survey other questions of problem formulation and
algorithm optimisation that have become pertinent following advances in CAD
theory, including both work that is already published and work that is
currently underway. With implementations now in reach of real world
applications and new theory meaning algorithms are far more sensitive to the
input, our thesis is that intelligently formulating problems for algorithms,
and indeed choosing the correct algorithm variant for a problem, is key to
improving the practical use of both quantifier elimination and symbolic real
algebraic geometry in general.Comment: To be presented at The "Encuentros de \'Algebra Computacional y
Aplicaciones, EACA 2014" (Meetings on Computer Algebra and Applications) in
Barcelon
Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition
Cylindrical algebraic decomposition (CAD) is a key tool for solving problems
in real algebraic geometry and beyond. In recent years a new approach has been
developed, where regular chains technology is used to first build a
decomposition in complex space. We consider the latest variant of this which
builds the complex decomposition incrementally by polynomial and produces CADs
on whose cells a sequence of formulae are truth-invariant. Like all CAD
algorithms the user must provide a variable ordering which can have a profound
impact on the tractability of a problem. We evaluate existing heuristics to
help with the choice for this algorithm, suggest improvements and then derive a
new heuristic more closely aligned with the mechanics of the new algorithm
DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL
We present the latest major release version 6.0 of the quantified Boolean
formula (QBF) solver DepQBF, which is based on QCDCL. QCDCL is an extension of
the conflict-driven clause learning (CDCL) paradigm implemented in state of the
art propositional satisfiability (SAT) solvers. The Q-resolution calculus
(QRES) is a QBF proof system which underlies QCDCL. QCDCL solvers can produce
QRES proofs of QBFs in prenex conjunctive normal form (PCNF) as a byproduct of
the solving process. In contrast to traditional QCDCL based on QRES, DepQBF 6.0
implements a variant of QCDCL which is based on a generalization of QRES. This
generalization is due to a set of additional axioms and leaves the original
Q-resolution rules unchanged. The generalization of QRES enables QCDCL to
potentially produce exponentially shorter proofs than the traditional variant.
We present an overview of the features implemented in DepQBF and report on
experimental results which demonstrate the effectiveness of generalized QRES in
QCDCL.Comment: 12 pages + appendix; to appear in the proceedings of CADE-26, LNCS,
Springer, 201
Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation
We apply multivariate Lagrange interpolation to synthesize polynomial
quantitative loop invariants for probabilistic programs. We reduce the
computation of an quantitative loop invariant to solving constraints over
program variables and unknown coefficients. Lagrange interpolation allows us to
find constraints with less unknown coefficients. Counterexample-guided
refinement furthermore generates linear constraints that pinpoint the desired
quantitative invariants. We evaluate our technique by several case studies with
polynomial quantitative loop invariants in the experiments
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