59 research outputs found
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
On the Generation of Positivstellensatz Witnesses in Degenerate Cases
One can reduce the problem of proving that a polynomial is nonnegative, or
more generally of proving that a system of polynomial inequalities has no
solutions, to finding polynomials that are sums of squares of polynomials and
satisfy some linear equality (Positivstellensatz). This produces a witness for
the desired property, from which it is reasonably easy to obtain a formal proof
of the property suitable for a proof assistant such as Coq. The problem of
finding a witness reduces to a feasibility problem in semidefinite programming,
for which there exist numerical solvers. Unfortunately, this problem is in
general not strictly feasible, meaning the solution can be a convex set with
empty interior, in which case the numerical optimization method fails.
Previously published methods thus assumed strict feasibility; we propose a
workaround for this difficulty. We implemented our method and illustrate its
use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201
Sum of Squares Lower Bounds from Symmetry and a Good Story
In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of [1,n] and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes:
1) Are there natural pseudo-expectation values for the problem?
2) Are these pseudo-expectation values rational functions of the problem parameters?
3) Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution?
We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph K_n has no perfect matching when n is odd), and the following Turan type problem: Minimize the number of triangles in a graph G with a given edge density. For knapsack, we recover Grigoriev\u27s lower bound exactly. For the MOD 2 principle, we tighten Grigoriev\u27s linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities
AI Hilbert: A New Paradigm for Scientific Discovery by Unifying Data and Background Knowledge
The discovery of scientific formulae that parsimoniously explain natural
phenomena and align with existing background theory is a key goal in science.
Historically, scientists have derived natural laws by manipulating equations
based on existing knowledge, forming new equations, and verifying them
experimentally. In recent years, data-driven scientific discovery has emerged
as a viable competitor in settings with large amounts of experimental data.
Unfortunately, data-driven methods often fail to discover valid laws when data
is noisy or scarce. Accordingly, recent works combine regression and reasoning
to eliminate formulae inconsistent with background theory. However, the problem
of searching over the space of formulae consistent with background theory to
find one that fits the data best is not well-solved. We propose a solution to
this problem when all axioms and scientific laws are expressible via polynomial
equalities and inequalities and argue that our approach is widely applicable.
We further model notions of minimal complexity using binary variables and
logical constraints, solve polynomial optimization problems via mixed-integer
linear or semidefinite optimization, and prove the validity of our scientific
discoveries in a principled manner using Positivestellensatz certificates.
Remarkably, the optimization techniques leveraged in this paper allow our
approach to run in polynomial time with fully correct background theory, or
non-deterministic polynomial (NP) time with partially correct background
theory. We demonstrate that some famous scientific laws, including Kepler's
Third Law of Planetary Motion, the Hagen-Poiseuille Equation, and the Radiated
Gravitational Wave Power equation, can be derived in a principled manner from
background axioms and experimental data.Comment: Slightly revised from version 1, in particular polished the figure
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
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
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