19,905 research outputs found
LEARNING ARITHMETIC READ-ONCE FORMULAS*
Abstract. A formula is read-once if each variable appears at most once in it. An arithmetic read-once formula is one in which the operators are addition, subtraction, multiplication, and division. We present polynomial time algorithms for exact learning of arithmetic read-once formulas over a field. We present a membership and equivalence query algorithm that identifies arithmetic read-once formulas over an arbitrary field. We present a randomized membership query algorithm (i.e., a randomized black box interpolation algorithm) that identifies such formulas over finite fields with at least 2n + 5 elements (where n is the number of variables) and over infinite fields. We also show the existence of nonuniform deterministic membership query algorithms for arbitrary read-once formulas over fields of characteristic 0, and division-free read-once formulas over fields that have at least 2n + elements. For our algorithms, we assume we are able to perform efficiently arithmetic operations on field elements and compute square roots in the field. It is shown that the ability to compute square roots is necessary in the sense that the problem of computing n square roots in a field can be reduced to the problem of identifying an arithmetic formula over n variables in that field. Our equivalence queries are of a slightly nonstandard form, in which counterexamples are required not to be inputs on which the formula evaluates to 0/0. This assumption is shown to be necessary for fields of size o(n! log n) in the sense that we prove there exists no polynomial time identification algorithm that uses only membership and standard equivalence queries
Towards Verifying Nonlinear Integer Arithmetic
We eliminate a key roadblock to efficient verification of nonlinear integer
arithmetic using CDCL SAT solvers, by showing how to construct short resolution
proofs for many properties of the most widely used multiplier circuits. Such
short proofs were conjectured not to exist. More precisely, we give n^{O(1)}
size regular resolution proofs for arbitrary degree 2 identities on array,
diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs
for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result
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
Jacobian hits circuits: Hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
We present a single, common tool to strictly subsume all known cases of
polynomial time blackbox polynomial identity testing (PIT) that have been
hitherto solved using diverse tools and techniques. In particular, we show that
polynomial time hitting-set generators for identity testing of the two
seemingly different and well studied models - depth-3 circuits with bounded top
fanin, and constant-depth constant-read multilinear formulas - can be
constructed using one common algebraic-geometry theme: Jacobian captures
algebraic independence. By exploiting the Jacobian, we design the first
efficient hitting-set generators for broad generalizations of the
above-mentioned models, namely:
(1) depth-3 (Sigma-Pi-Sigma) circuits with constant transcendence degree of
the polynomials computed by the product gates (no bounded top fanin
restriction), and (2) constant-depth constant-occur formulas (no multilinear
restriction).
Constant-occur of a variable, as we define it, is a much more general concept
than constant-read. Also, earlier work on the latter model assumed that the
formula is multilinear. Thus, our work goes further beyond the results obtained
by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et
al. (CCC 2011), Beecken et al. (ICALP 2011) and Grenet et al. (FSTTCS 2011),
and brings them under one unifying technique.
In addition, using the same Jacobian based approach, we prove exponential
lower bounds for the immanant (which includes permanent and determinant) on the
same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our
results reinforce the intimate connection between identity testing and lower
bounds by exhibiting a concrete mathematical tool - the Jacobian - that is
equally effective in solving both the problems on certain interesting and
previously well-investigated (but not well understood) models of computation
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