11,469 research outputs found
The NumericalCertification package in Macaulay2
The package \texttt{NumericalCertification} implements methods for certifying
numerical approximations of solutions for a given system of polynomial
equations. For certifying regular solutions, the package implements Smale's
-theory and Krawczyk method. For a singular solution, we implement soft
verification using the iterative deflation method. We demonstrate the
functionalities of the package focusing on interaction with current numerical
solvers in \texttt{Macaulay2}.Comment: 10 page
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Fatal Attractors in Parity Games: Building Blocks for Partial Solvers
Attractors in parity games are a technical device for solving "alternating"
reachability of given node sets. A well known solver of parity games -
Zielonka's algorithm - uses such attractor computations recursively. We here
propose new forms of attractors that are monotone in that they are aware of
specific static patterns of colors encountered in reaching a given node set in
alternating fashion. Then we demonstrate how these new forms of attractors can
be embedded within greatest fixed-point computations to design solvers of
parity games that run in polynomial time but are partial in that they may not
decide the winning status of all nodes in the input game.
Experimental results show that our partial solvers completely solve
benchmarks that were constructed to challenge existing full solvers. Our
partial solvers also have encouraging run times in practice. For one partial
solver we prove that its run-time is at most cubic in the number of nodes in
the parity game, that its output game is independent of the order in which
monotone attractors are computed, and that it solves all Buechi games and weak
games.
We then define and study a transformation that converts partial solvers into
more precise partial solvers, and we prove that this transformation is sound
under very reasonable conditions on the input partial solvers. Noting that one
of our partial solvers meets these conditions, we apply its transformation on
1.6 million randomly generated games and so experimentally validate that the
transformation can be very effective in increasing the precision of partial
solvers
Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation
We extend a template-based approach for synthesizing switching controllers
for semi-algebraic hybrid systems, in which all expressions are polynomials.
This is achieved by combining a QE (quantifier elimination)-based method for
generating continuous invariants with a qualitative approach for predefining
templates. Our synthesis method is relatively complete with regard to a given
family of predefined templates. Using qualitative analysis, we discuss
heuristics to reduce the numbers of parameters appearing in the templates. To
avoid too much human interaction in choosing templates as well as the high
computational complexity caused by QE, we further investigate applications of
the SOS (sum-of-squares) relaxation approach and the template polyhedra
approach in continuous invariant generation, which are both well supported by
efficient numerical solvers
Extending ACL2 with SMT Solvers
We present our extension of ACL2 with Satisfiability Modulo Theories (SMT)
solvers using ACL2's trusted clause processor mechanism. We are particularly
interested in the verification of physical systems including Analog and
Mixed-Signal (AMS) designs. ACL2 offers strong induction abilities for
reasoning about sequences and SMT complements deduction methods like ACL2 with
fast nonlinear arithmetic solving procedures. While SAT solvers have been
integrated into ACL2 in previous work, SMT methods raise new issues because of
their support for a broader range of domains including real numbers and
uninterpreted functions. This paper presents Smtlink, our clause processor for
integrating SMT solvers into ACL2. We describe key design and implementation
issues and describe our experience with its use.Comment: In Proceedings ACL2 2015, arXiv:1509.0552
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
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