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Symmetries have been exploited successfully within the realms of SAT and QBF to improve solver performance in practical applications and to devise more powerful proof systems. As a first step towards extending these advancements to the class of dependency quantified Boolean formulas (DQBFs), which generalize QBF by allowing more nuanced variable dependencies, this work develops a comprehensive theory to characterize symmetries for DQBFs. We also introduce the notion of symmetry breakers of DQBFs, along with a concrete construction, and discuss how to detect DQBF symmetries algorithmically using a graph-based approach., Comment: 32 pages, 2 figures

One of the most important sequences in enumerative combinatorics is OEIS sequence A85, the number of involutions of length n. In the Art of Computer Programming, vol. 3, Don Knuth derived the O(1/n) asymptotic formula for these numbers. In this modest tribute to our two heroes, Neil Sloane who just turned 85, and Don Knuth who was 85 a year ago, we go all the way to an $O(1/n^{85})$ asymptotic formula., Comment: 4 pages. In honor of Neil Sloane and Don Knuth

For a given ideal I in K[x_1,...,x_n,y_1,...,y_m] in a polynomial ring with n+m variables, we want to find all elements that can be written as f-g for some f in K[x_1,...,x_n] and some g in K[y_1,...,y_m], i.e., all elements of I that contain no term involving at the same time one of the x_1,...,x_n and one of the y_1,...,y_m. For principal ideals and for ideals of dimension zero, we give a algorithms that compute all these polynomials in a finite number of steps.

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases., Comment: 20 pages

In 2014, R.H. Hardin contributed a family of sequences about king-moves on an array to the On-Line Encyclopedia of Integer Sequences (OEIS). The sequences were recently noticed in an automated search of the OEIS by Kauers and Koutschan, who conjectured a recurrence for one of them. We prove their conjecture as well as some older conjectures stated in the OEIS entries. We also have some new conjectures for the asymptotics of Hardin's sequences.

For various $2\leq n,m \leq 6$, we propose some new algorithms for multiplying an $n\times m$ matrix with an $m \times 6$ matrix over a possibly noncommutative coefficient ring., Comment: 9 pages

In an automatic search, we found conjectural recurrences for some sequences in the OEIS that were not previously recognized as being D-finite. In some cases, we are able to prove the conjectured recurrence. In some cases, we are not able to prove the conjectured recurrence, but we can prove that a recurrence exists. In some remaining cases, we do not know where the recurrence might come from.

Although in theory we can decide whether a given D-finite function is transcendental, transcendence proofs remain a challenge in practice. Typically, transcendence is certified by checking certain incomplete sufficient conditions. In this paper we propose an additional such condition which catches some cases on which other tests fail., Comment: 9 pages, 1 figure

Trager's Hermite reduction solves the integration problem for algebraic functions via integral bases. A generalization of this algorithm to D-finite functions has so far been limited to the Fuchsian case. In the present paper, we remove this restriction and propose a reduction algorithm based on integral bases that is applicable to arbitrary D-finite functions., Comment: 21 pages

A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.

We introduce a new method for discovering matrix multiplication schemes based on random walks in a certain graph, which we call the flip graph. Using this method, we were able to reduce the number of multiplications for the matrix formats (4, 4, 5) and (5, 5, 5), both in characteristic two and for arbitrary ground fields.

The orbit-sum method is an algebraic version of the reflection-principle that was introduced by Bousquet-M\'{e}lou and Mishna to solve functional equations that arise in the enumeration of lattice walks with small steps restricted to $\mathbb{N}^2$. Its extension to walks with large steps was started by Bostan, Bousquet-M\'{e}lou and Melczer. We continue it here, making use of the primitive element theorem, Gr\"{o}bner bases and the shape lemma, and the Newton-Puiseux algorithm., Comment: 8 pages

In response to a recent Nature article which announced an algorithm for multiplying $5\times5$-matrices over $\mathbb{Z}_2$ with only 96 multiplications, two fewer than the previous record, we present an algorithm that does the job with only 95 multiplications.

We compute the next few terms of the OEIS sequence A348456 and provide guessed equations for the generating functions of some sequences in its context.

Schemes for exact multiplication of small matrices have a large symmetry group. This group defines an equivalence relation on the set of multiplication schemes. There are algorithms to decide whether two schemes are equivalent. However, for a large number of schemes a pairwise equivalence check becomes cumbersome. In this paper we propose an algorithm to compute a normal form of matrix multiplication schemes. This allows us to decide pairwise equivalence of a larger number of schemes efficiently., Comment: 11 pages

It is known for linear operators with polynomial coefficients annihilating a given D-finite function that there is a trade-off between order and degree. Raising the order may give room for lowering the degree. The relationship between order and degree is typically described by a hyperbola known as the order-degree curve. In this paper, we add the height into the picture, i.e., a measure for the size of the coefficients in the polynomial coefficients. For certain situations, we derive relationships between order, degree, and height that can be viewed as order-degree-height surfaces.

Reconstructing a hypothetical recurrence equation from the first terms of an infinite sequence is a classical and well-known technique in experimental mathematics. We propose a variation of this technique which can succeed with fewer input terms.

Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis. We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions. The sharpened reduction is then used to design a reduction-based telescoping algorithm for algebraic functions in two variables., Comment: 20 pages

Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is always algebraic, and for the Kreweras step set, the generating function is always D-finite. However, apart from the particular case where the interactions are symmetric in $x$ and~$y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, we confirm their intuition that the generating function of Kreweras walks is not algebraic, apart from the particular case already identified., Comment: 13 pages, accepted at FPSAC'21

We investigate a functional equation which resembles the functional equation for the generating function of a lattice walk model for the quarter plane. The interesting feature of this equation is that its orbit sum is zero while its solution is not algebraic. The solution can be interpreted as the generating function of lattice walks in $\mathbb{Z}^2$ starting at $(-1,-1)$ and subject to the restriction that the coordinate axes can be crossed only in one direction. We also consider certain variants of the equation, all of which seem to have transcendental solutions. In one case, the solution is perhaps not even D-finite.

The number of Young Tableaux whose shape is a k by n rectangle is famously (nk)! 0! ... (k-1)!/((n+k-1)!(n+k-2)!... n!) implying that for each specific k, that sequence satisfies a linear recurrence equation with polynomial coefficients of the first order. But what about counting Young tableaux where certain "run lengths" are forbidden? Then things seem to get much more complicated. We conclude with four conjectures and pledge donations to the OEIS in honor of the first provers., Comment: 8 pages. Accompanied by Maple packages and C programs available from https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/cyt.html This version adds a footnote, and pledges donations to the OEIS in honor of the first provers

For sparse matrices up to size $8 \times 8$, we determine optimal choices for pivot selection in Gaussian elimination. It turns out that they are slightly better than the pivots chosen by a popular pivot selection strategy, so there is some room for improvement. We then create a pivot selection strategy using machine learning and find that it indeed leads to a small improvement compared to the classical strategy., Comment: 11 pages

We provide some first experimental data about generating functions of restricted lattice walks with small steps in NN^4.

In an earlier paper, the notion of integrality known from algebraic number fields and fields of algebraic functions has been extended to D-finite functions. The aim of the present paper is to extend the notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements for valued vector spaces and then show that this algorithm includes not only the algebraic and the D-finite cases but also covers the case of P-recursive sequences., Comment: 20 pages

We present an algorithm which for any given ideal $I\subseteq\mathbb{K} [x,y]$ finds all elements of $I$ that have the form $f(x) - g(y)$, i.e., all elements in which no monomial is a multiple of $xy$.

It is known since the 1970s that no more than 23 multiplications are required for computing the product of two 3 x 3-matrices. It is not known whether this can also be done with fewer multiplications. However, there are several mutually inequivalent ways of doing the job with 23 multiplications. In this article, we extend this list considerably by providing more than 13 000 new and mutually inequivalent schemes for multiplying 3 x 3-matrices using 23 multiplications. Moreover, we show that the set of all these schemes is a manifold of dimension at least 17.

Given a lattice L in Z^m and a subset A of R^m, we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.

Laderman discovered a scheme for computing the product of two 3x3 matrices using only 23 multiplications in 1976. Since then, some more such schemes were proposed, but it remains open how many there are and whether there exist schemes with fewer than 23 multiplications. In this paper we present two independent SAT-based methods for finding new schemes. Both methods allow computing a few hundred new schemes individually, and many thousands when combined. Local search SAT solvers outperform CDCL solvers consistently in this application.

Inspired by a recent article by Anthony Zaleski and Doron Zeilberger, we investigate the question of determining the largest k for which there exists boolean formulas in disjunctive normal form (DNF) with n variables, none of whose conjunctions are `parallel', and such that all of them have at least k literals. Using a SAT solver, we answer some of the questions they left open. We also determine the corresponding numbers for DNFs obeying certain symmetries.

We consider inhomogeneous lattice walk models in a half-space and in the quarter plane. For the models in a half-space, we show by a generalization of the kernel method to linear systems of functional equations that their generating functions are always algebraic. For the models in the quarter plane, we have carried out an experimental classification of all models with small steps. We discovered many (apparently) D-finite cases for most of which we have no explanation yet., Comment: 12 pages

In this case study, we illustrate the great potential of experimental mathematics and symbolic computation, by rederiving, ab initio, Onsager's celebrated solution of the twodimensional Ising model in zero magnetic field. Onsager's derivation is extremely complicated and ad hoc, as are all the subsequent proofs. Unlike Onsager's, our derivation is not rigorous, yet it is absolutely certain (even if Onsager did not do it before), and should have been acceptable to physicists who do not share mathematicians' fanatical (and often misplaced) insistence on rigor.

We exploit symmetries to give short proofs for two prominent formula families of QBF proof complexity. On the one hand, we employ symmetry breakers. On the other hand, we enrich the (relatively weak) QBF resolution calculus Q-Res with the symmetry rule and obtain separations to powerful QBF calculi.

While symmetries are well understood for Boolean formulas and successfully exploited in practical SAT solving, less is known about symmetries in quantified Boolean formulas (QBF). There are some works introducing adaptions of propositional symmetry breaking techniques, with a theory covering only very specific parts of QBF symmetries. We present a general framework that gives a concise characterization of symmetries of QBF. Our framework naturally incorporates the duality of universal and existential symmetries resulting in a general basis for QBF symmetry breaking.

We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems. Ordinary points of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.