123 research outputs found
Formalization of the Lindemann-Weierstrass Theorem
International audienceThis article details a formalization in Coq of the Lindemann-Weierstrass theorem which gives a transcendence criterion for complex numbers: this theorem establishes a link between the linear independence of a set of algebraic numbers and the algebraic independence of the exponentials of these numbers. As we follow Baker's proof, we discuss the difficulties of its formalization and explain how we resolved them in Coq. Most of these difficulties revolve around multivariate polynomials and their relationship with the conjugates of a univariate polynomial. Their study ultimately leads to alternative forms of the fundamental theorem of symmetric polynomials. This formalization uses mainly the Mathcomp library for the part relying on algebra, and the Coquelicot library and the Coq standard library of real numbers for the calculus part
An Invitation to Formal Power Series
This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newtonâs binomial theorem, Jacobiâs triple product, the RogersâRamanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujanâs partition congruences, generating functions of Stirling numbers and Jacobiâs four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahonâs master theorem
A Formal Proof of the Expressiveness of Deep Learning
International audienceDeep learning has had a profound impact on computer science in recent years, with applications to image recognition, language processing, bioinformatics, and more. Recently , Cohen et al. provided theoretical evidence for the superiority of deep learning over shallow learning. We formalized their mathematical proof using Isabelle/HOL. The Isabelle development simplifies and generalizes the original proof, while working around the limitations of the HOL type system. To support the formalization, we developed reusable libraries of formalized mathematics, including results about the matrix rank, the Borel measure, and multivariate polynomials as well as a library for tensor analysis
Arithmetic Circuits with Locally Low Algebraic Rank
In recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply VP != VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits.
A depth-4 circuit is a representation of an N-variate, degree n polynomial P as P = sum_{i=1}^T Q_{i1} * Q_{i2} * ... * Q_{it} where the Q_{ij} are given by their monomial expansion. Homogeneity adds the constraint that for every i in [T], sum_{j} degree(Q_{ij}) = n. We study an extension where, for every i in [T], the algebraic rank of the set of polynomials {Q_{i1}, Q_{i2}, ... ,Q_{it}} is at most some parameter k. We call this the class of spnew circuits. Already for k=n, these circuits are a strong generalization of the class of homogeneous depth-4 circuits, where in particular t<=n (and hence k<=n).
We study lower bounds and polynomial identity tests for such circuits and prove the following results.
1. Lower bounds: We give an explicit family of polynomials {P_n} of degree n in N = n^{O(1)} variables in VNP, such that any spnewn circuit computing P_n has size at least exp{(Omega(sqrt(n)*log(N)))}. This strengthens and unifies two lines of work: it generalizes the recent exponential lower bounds for homogeneous depth-4 circuits [KLSS14, KS-full] as well as the Jacobian based lower bounds of Agrawal et al. which worked for spnew circuits in the restricted setting where T * k <= n.
2. Hitting sets: Let spnewbounded be the class of spnew circuits with bottom fan-in at most d. We show that if d and k are at most poly(log(N)), then there is an explicit hitting set for spnewbounded circuits of size quasipolynomial in N and the size of the circuit. This strengthens a result of Forbes which showed such quasipolynomial sized hitting sets in the setting where d and t are at most poly(log(N)).
A key technical ingredient of the proofs is a result which states that over any field of characteristic zero (or sufficiently large characteristic), upto a translation, every polynomial in a set of algebraically dependent polynomials can be written as a function of the polynomials in the transcendence basis. We believe this may be of independent interest. We combine this with shifted partial derivative based methods to obtain our final results
All Liouville Numbers are Transcendental
In this Mizar article, we complete the formalization of one of the items from Abad and Abadâs challenge list of âTop 100 Theoremsâ about Liouville numbers and the existence of transcendental numbers. It is item #18 from the âFormalizing 100 Theoremsâ list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated âquite closelyâ by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and
0 <
x â
p
q
<
1
q
n
.
It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and Ï [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouvilleâs theorem on Diophantine approximation.KorniĆowicz Artur - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandNaumowicz Adam - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandGrabowski Adam - Institute of Informatics, University of BiaĆystok, BiaĆystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and _ as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.CzesĆaw ByliĆski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.CzesĆaw ByliĆski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.CzesĆaw ByliĆski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.CzesĆaw ByliĆski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski and Artur KorniĆowicz. Introduction to Liouville numbers. Formalized Mathematics, 25(1):39-48, 2017.Adam Grabowski, Artur KorniĆowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.RafaĆ Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Joseph Liouville. Nouvelle dĂ©monstration dâun thĂ©orĂšme sur les irrationnelles algĂ©briques, insĂ©rĂ© dans le Compte Rendu de la derniĂšre sĂ©ance. Compte Rendu Acad. Sci. Paris, SĂ©r.A (18):910â911, 1844.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.MichaĆ Muzalewski and LesĆaw W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Andrzej Trybulec. Function domains and FrĂŠnkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291-299, 2016.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992
Nearly Optimal Algorithms for the Decomposition of Multivariate Rational Functions and the Extended L\"uroth's Theorem
The extended L\"uroth's Theorem says that if the transcendence degree of
\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK is 1 then there exists f \in
\KK(\underline{X}) such that \KK(\mathsf{f}_1,\dots,\mathsf{f}_m) is equal
to \KK(f). In this paper we show how to compute with a probabilistic
algorithm. We also describe a probabilistic and a deterministic algorithm for
the decomposition of multivariate rational functions. The probabilistic
algorithms proposed in this paper are softly optimal when is fixed and
tends to infinity. We also give an indecomposability test based on gcd
computations and Newton's polytope. In the last section, we show that we get a
polynomial time algorithm, with a minor modification in the exponential time
decomposition algorithm proposed by Gutierez-Rubio-Sevilla in 2001
Sums of products of polynomials in few variables : lower bounds and polynomial identity testing
We study the complexity of representing polynomials as a sum of products of
polynomials in few variables. More precisely, we study representations of the
form such that each is
an arbitrary polynomial that depends on at most variables. We prove the
following results.
1. Over fields of characteristic zero, for every constant such that , we give an explicit family of polynomials , where
is of degree in variables, such that any
representation of the above type for with requires . This strengthens a recent result of Kayal and Saha
[KS14a] which showed similar lower bounds for the model of sums of products of
linear forms in few variables. It is known that any asymptotic improvement in
the exponent of the lower bounds (even for ) would separate VP
and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity
testing (PIT) algorithm for circuits computed by the above model when and
the individual degree of each variable in are at most and
for any constant . We get quasipolynomial running
time when . The PIT algorithm is obtained by combining our
lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04].
To the best of our knowledge, this is the first nontrivial PIT algorithm for
this model (even for the case ), and the first nontrivial PIT algorithm
obtained from lower bounds for small depth circuits
Gr\"obner methods for representations of combinatorial categories
Given a category C of a combinatorial nature, we study the following
fundamental question: how does the combinatorial behavior of C affect the
algebraic behavior of representations of C? We prove two general results. The
first gives a combinatorial criterion for representations of C to admit a
theory of Gr\"obner bases. From this, we obtain a criterion for noetherianity
of representations. The second gives a combinatorial criterion for a general
"rationality" result for Hilbert series of representations of C. This criterion
connects to the theory of formal languages, and makes essential use of results
on the generating functions of languages, such as the transfer-matrix method
and the Chomsky-Sch\"utzenberger theorem.
Our work is motivated by recent work in the literature on representations of
various specific categories. Our general criteria recover many of the results
on these categories that had been proved by ad hoc means, and often yield
cleaner proofs and stronger statements. For example: we give a new, more
robust, proof that FI-modules (originally introduced by Church-Ellenberg-Farb),
and a family of natural generalizations, are noetherian; we give an easy proof
of a generalization of the Lannes-Schwartz artinian conjecture from the study
of generic representation theory of finite fields; we significantly improve the
theory of -modules, introduced by Snowden in connection to syzygies of
Segre embeddings; and we establish fundamental properties of twisted
commutative algebras in positive characteristic.Comment: 41 pages; v2: Moved old Sections 3.4, 10, 11, 13.2 and connected text
to arxiv:1410.6054v1, Section 13.1 removed and will appear elsewhere; v3:
substantial revision and reorganization of section
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