Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to as goes to infinity, solving a conjecture of E.K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to and using exactly multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers , we compare the lengths of the arithmetic formulas for that each encoding produces with the length of the shortest formula for (which we estimate from below). We briefly discuss the time-space tradeoff offered by each

Counting arithmetic formulas

Anarithmeticformulaisanexpressioninvolvingonlytheconstant1, and the binary operations of addition and multiplication, withmultiplicationby1notallowed.Weobtainanasymptoticformulaforthenumberofarithmeticformulasevaluatingtonasngoestoinfinity, solving a conjecture of E.K. Gnang and D. Zeilberger. Wegivealsoanasymptoticformulaforthenumberofarithmeticfor-mulas evaluating tonand using exactlykmultiplications. Finallyweanalyzethreespecificencodingsforproducingarithmeticfor-mulas. For almost all integersn, we compare the lengths of thearithmetic formulas fornthat each encoding produces with thelength of the shortest formula forn(which we estimate from be-low).Webrieflydiscussthetime-spacetradeoffofferedbyeac

Counting arithmetic formulas

An arithmetic formula is an expression involving only the constant 1, and the binary operations of addition and multiplication, with multiplication by 1 not allowed. We obtain an asymptotic formula for the number of arithmetic formulas evaluating to as goes to infinity, solving a conjecture of E.K. Gnang and D. Zeilberger. We give also an asymptotic formula for the number of arithmetic formulas evaluating to and using exactly multiplications. Finally we analyze three specific encodings for producing arithmetic formulas. For almost all integers , we compare the lengths of the arithmetic formulas for that each encoding produces with the length of the shortest formula for (which we estimate from below). We briefly discuss the time-space tradeoff offered by each

Chebyshev's bias for products of irreducible polynomials

For any $k\geq 1$, this paper studies the number of polynomials having $k$ irreducible factors (counted with or without multiplicities) in $\mathbf{F}_q[t]$ among different arithmetic progressions. We obtain asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. In the generic case, the bias dissipates as the degree of the modulus or $k$ gets large, but there are cases when the bias is extreme. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. Several examples illustrate this new phenomenon.Comment: The exposition has been improved, we now present the case of the number of irreducible factors both counting and not counting multiplicities. We also add some results on the possible values of the bia

Arithmetic Circuits and the Hadamard Product of Polynomials

Motivated by the Hadamard product of matrices we define the Hadamard product of multivariate polynomials and study its arithmetic circuit and branching program complexity. We also give applications and connections to polynomial identity testing. Our main results are the following. 1. We show that noncommutative polynomial identity testing for algebraic branching programs over rationals is complete for the logspace counting class \ceql, and over fields of characteristic $p$ the problem is in \ModpL/\Poly. 2.We show an exponential lower bound for expressing the Raz-Yehudayoff polynomial as the Hadamard product of two monotone multilinear polynomials. In contrast the Permanent can be expressed as the Hadamard product of two monotone multilinear formulas of quadratic size.Comment: 20 page

Transforming numerical feature models into propositional formulas and the universal variability language

Real-world Software Product Lines (SPLs) need Numerical Feature Models (NFMs) whose features have not only boolean values that satisfy boolean constraints but also have numeric attributes that satisfy arithmetic constraints. An essential operation on NFMs finds near-optimal performing products, which requires counting the number of SPL products. Typical constraint satisfaction solvers perform poorly on counting and sampling. Nemo (Numbers, features, models) is a tool that supports NFMs by bit-blasting, the technique that encodes arithmetic expressions as boolean clauses. The newest version, Nemo2, translates NFMs to propositional formulas and the Universal Variability Language (UVL). By doing so, products can be counted efficiently by #SAT and Binary Decision Tree solvers, enabling finding near-optimal products. This article evaluates Nemo2 with a large set of synthetic and colossal real-world NFMs, including complex arithmetic constraints and counting and sampling experiments. We empirically demonstrate the viability of Nemo2 when counting and sampling large and complex SPLs.Munoz, Pinto and Fuentes work is supported by the European Union’s H2020 research and innovation programme under grant agreement DAEMON 101017109, by the projects co-financed by FEDER, Spain funds LEIA UMA18-FEDERJA-15, IRIS PID2021- 122812OB-I00 (MCI/AEI), and the PRE2019-087496 grant from the Ministerio de Ciencia e Innovación. Funding for open access charge: Universidad de Málaga / CBUA

Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$, $\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and a proof of $\tilde{O}(K \cdot d)$ length, and $\bullet$ the Verifier tosses $\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon))$ coins and can check the proof in about $\tilde{O}(K \cdot(n + d) + s)$ time, with probability of error less than $\varepsilon$. For small degree $d$, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in $c^{n}$ time (for various $c < 2$) for the Permanent, $\#$Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of $0$-$1$ linear programs. In general, the value of any polynomial in Valiant's class ${\sf VP}$ can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in $2^{2n/3+o(n)}$ time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in $n^{k/2+O(1)}$-time for counting $k$-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.Comment: 17 page