Pitman Research Notes in Mathematics. Longman Sci and Tech. Krasnosel'skij A. Perov: On the existence of solutions of certain nonlinear operator equations. Nauk SSSR , Krasnosel'skij G. Vajnikko P. Zabrejko, Ja. Rutickij V. Stecenko: Approximate Solutions of Operator Equations. Krasnosel'skij P. Zabrejko: Geometric Methods of Nonlinear Analysis. Krasnosel'skij E. Lusnikov: Fixed points with special properties.
Nauk , Arouszajn's theorem on connectedness of the fixed point set of a compact mapping. Kuratowski: Topologie. Pelczar: Introduction to Theory of Differential Equations. Part 2. Elements of the Qualitative Theory of Differential Equations.
Warszawa In Polish. Pliss: Nonlocal Problems of Oscillation Theory. Rational Mech. Rouche P. This project focuses on deep connections between two rather complementary approaches to geometric integration methods by combining advanced differential geometry and modern algebraic combinatorics. Guided by real world applications, we develop algorithms and clarify mathematical theories. This will lead to the development of useful software packages. Under this assumption, the rule 2 reduces to a polynomial interpolation problem in each component, i. Indeed, any non-zero polynomial function that vanishes on the data inputs could be added to a function satisfying the conditions 3 and yield a different function that also satisfies 3.
Among all those possible solutions, the LS-algorithm chooses an interpolating polynomial function that does not contain any terms vanishing on the set X. Unfortunately, the LS-algorithm works within an algebraic framework that depends on the choice of a so called term order. For every different term order, the output of the algorithm might be a different one. In addition, term orders impose some quite arbitrary conditions on the set of possible candidates for the output of the LS-algorithm.
Furthermore, there is no clear criterion when it comes to actually choosing a term order. In the next subsection we will provide the definition of term order as well as a geometric framework in which the algebraic steps of the LS-algorithm can be visualized and better understood. In Section 1 of the Appendix S1 , we provide a concrete example in which the output of the algorithm is clearly presented. For the sake of completeness, we summarize here the technical steps of the LS-algorithm: To generate its output, the algorithm first takes as input the discretized time series and generates functions that satisfy 3 for each correspondingly.
Secondly, it takes a monomial order as input and generates the normal form of with respect to the vanishing ideal and the given order. For every , this normal form is the output of the algorithm. The mathematical framework presented here is based on a general algebraic result presented by the author in Section 4 of the Appendix S1. This result is known among algebraists, however, to the author's best knowledge, it has never been formulated within the context considered herein.
This framework will allow us to study the LS-algorithm as well as a generalized algorithm of it that is independent on the choice of term orders. Furthermore, within this framework, we will be able to provide answers to the two questions stated in the Introduction, see the Results section below. We use several well established linear algebraic results to construct the framework within which our investigations can be carried out. We start with the original problem: Given a time-discrete dynamical system over a finite field S in n variables and a data set generated by iterating the function F starting at one or more initial values, what are the chances of reconstructing the function F if the LS-algorithm or a similar algorithm is applied using X as input time series?
From an experimental point of view the following question arises: What is the function F in an experimental setting? Contrary to the situation when models with an infinite number of possible states are reverse engineered see 1. In this sense, even in an experimental setting, there is an underlying function F. The components of this function is what the author of  called. Since the algorithms studied here generate an output model by calculating every single coordinate function separately, we will focus on the reconstruction of a single coordinate function which we will simply call f.
We will use the notation for a finite field of cardinality. In what follows, we briefly review the main definitions and results stated and proved in Section 4 of the Appendix S1 :. We denote the vector space of functions with. A basis for is given by all the monomial functions where the exponents are non-negative integers satisfying. The basis of all those monomial functions is denoted with , where We call those monomial functions fundamental monomial functions.
This fact is basically telling us that all functions are polynomial functions of bounded degree 2.
When dealing with polynomial interpolation problems, it is convenient to establish the relationship between a polynomial function and the value it takes on a given point or set of points. A technique commonly used in algebra is to define an evaluation mapping that assigns to each polynomial function the list of the values it takes on each point of a given set of different points. Just to make sure this mapping is unique, we order this list of evaluations according to a fixed but arbitrary order.
This is equivalent to ordering the set in the first place see endnote 4 in the next page. Summarizing, consider a given finite field , natural numbers with and an ordered tuple of m different points with entries in the field. Then we can define the mapping where t denotes transpose. It can be shown see Theorem 21 in Section 4 of the Appendix S1 , that this mapping is a surjective linear operator 3.
We call this mapping the evaluation epimorphism of the tuple. For a given set of data points and a given vector , the interpolation problem of finding a function with the property can be expressed using the evaluation epimorphism as 4 follows: Find a function with the property 4 Since a basis of is given by the fundamental monomial functions , the matrix 5 representing the evaluation epimorphism of the tuple with respect to the basis of and the canonical basis of has always the full rank. That also means, that the dimension of the is 6 5 In the case where m is strictly smaller than we have and the solution of the interpolation problem is not unique.
There are exactly different solutions which constitute an affine subspace of see Fig. Only in the case , that means, when for all elements of the corresponding interpolation values are given, the solution is unique. Experimental data are typically sparse and therefore underdetermine the problem.
If the problem is underdetermined and no additional information about properties of the possible solutions is given, any algorithm attempting to solve the problem has to provide a selection criterion to pick a solution among the affine space of possible solutions. If we visualize the affine subspace of solutions of 4 in the space see Fig.
This solution does not contain any components pointing in the direction of the subspace, which, at least geometrically, seem redundant. A two-dimensional representation of the space of functions. Within this space, a one-dimensional representation of the affine subspace of solutions of. Three particular solutions are depicted; one red is the orthogonal solution. Interestingly, this simple geometric idea comprises the algebraic selection step in the LS-algorithm and at the same time generalizes the pool of possible candidates to be selected.
Of course we need to formalize this approach algebraically. The standard tool in this context is called orthogonality. For orthogonality to apply, a generalized inner product see  has to be defined on the space. We finish this subsection reviewing these concepts cf. Appendix S1. The space is endowed with a symmetric bilinear form 7 i. Two functions are called orthogonal if it holds. A family of functions is called orthonormal if it holds 8. For a given set of data points, consider the evaluation epimorphism of the tuple and its kernel. Now, let be a basis of.
By the basis extension theorem see  , we can extend the basis to a basis of the whole space , where.
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There are many possible ways this extension can be performed. See more details below.
As in Example 5 of Subsection 4. The way we extend the basis of to a basis of the whole space determines crucially the generalized inner product we get by setting 6. Consequently, the orthogonal solution of 4 may vary according to the extension chosen. In the Appendix S1 , a systematic way to extend the basis to a basis for the whole space is introduced. With the basis obtained, the process of defining a generalized inner product according to 6 is called the standard orthonormalization.
This is because the basis is orthonormal with respect to the generalized inner product defined by 6. A basis is by definition an ordered set. The basis of fundamental monomial functions is an ordered set arranged according to a fixed order relation defined on the set. Such order relations are called term orders. One of the key requirements for a term order is that it must be consistent with the algebraic operations performed with polynomials.
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In particular, the term order relation must be preserved after multiplication with an arbitrary term. Additionally, it has to be possible to always determine which is the smallest element among a set of arbitrary terms. Since every term in a polynomial in n indeterminates is uniquely determined by the exponents appearing in it, the order relation can as well be defined on the set of tuples of non negative integer exponents.
As stated above, in the context of polynomial functions in n variables over the finite field , the degrees are bounded above and therefore we only need to consider the order relation on the set. If it was a term order, then we could multiply both sides of the expression which holds by transitivity by to obtain. This result contradicts the order relation established above.
The precise definition of the standard orthonormalization procedure together with an example is provided in Subsection 4. The standard orthonormalization process depends on the way the elements of the basis of fundamental monomial functions are ordered. If they are ordered according to a term order, the calculation of the orthogonal solution of 4 11 yields precisely the same result as the LS-algorithm. If more general linear orders are allowed, a more general algorithm emerges that is not restricted to the use of term orders.
This algorithm can be seen as a generalization of the LS-algorithm. We call it the term-order-free reverse engineering method.
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In the next subsection we meticulously present the steps of the term-order-free reverse engineering method. It is pertinent to emphasize that although the term-order-free reverse engineering method generates the same solution as the LS-algorithm provided we use a term order to order the elements of the basis , the two algorithms differ significantly in their steps. This algebraic framework imposes restrictions on the type of order relations that can be used.
Our method is defined in a geometric and linear algebraic framework that is not subjected to those restrictions. Moreover, the fact that our method is capable of reproducing the input-output behavior of the LS-algorithm, allows us to study this behavior of the LS-algorithm within our, in our opinion, more tractable framework. In Section 1 of the Appendix S1 we present an illustrative example in which every step of the term-order-free reverse engineering method is carried out explicitly.
As we will show in the Results section, the monomial functions generated by the standard orthonormalization procedure to extend the basis of to a basis of the whole space constitute the pool of candidate monomials for the construction of the orthogonal solution.
In other words, the orthogonal solution is a linear combination of the. The use of term orders is a requirement imposed by the algebraic approach used in the LS-algorithm. However, it arbitrarily restricts the ways the basis of can be extended to a basis by virtue of the standard orthonormalization procedure. For instance, the constant function is always part of the extension when term orders are used. Furthermore, if an optimal data set to be defined below is used, some high degree monomials will never be among the candidates. Thus, a function f displaying such high degree terms could never be reverse engineered by the LS-algorithm, if fed with an optimal data set.
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It will also become apparent in the Results section, that the use of term orders makes it difficult to analyze the performance of the LS-algorithm. As a consequence, we tried to circumvent the issues related to the use of term orders by proposing the term order free reverse engineering method, a generalization of the LS-algorithm that does not depend on the choice of a term order.
The steps of the algorithm are as follows. The steps described above represent an intelligible description of the algorithm and are not optimized for an actual computational implementation. In Section 1 of the Appendix S1 we present an illustrative example in which every step of the method is carried out explicitly. Essentially, the steps of the term-order-free reverse engineering comprise standard matrix and linear algebra calculations.
However, the size or dimension of the matrices involved depends exponentially on the number n of variables and linearly on the number m of data points, as the reader can verify based on the dimensions of the matrices involved in the algorithm. The complexity of basic linear algebraic calculations such as Gaussian elimination and back substitution are well known, see, for instance, . With that in mind, we can briefly assess the complexity of our method: In step 1, matrix entries need to be calculated as the evaluation of the fundamental monomial functions on the data points.
In step 2, a basis of the nullspace of A is calculated.
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The number of data points m should be expressed as a proportion of the size of the entire space , thus, we write with a suitable factor. The basis of the nullspace is calculated using Gaussian elimination, which, neglecting the lower order terms in d , requires operations, and back substitution, which, given that , is. The standard orthonormalization procedure in step 3 is also accomplished via Gaussian elimination on an matrix.
Due to , we have , therefore, step 3 requires about operations. The calculation of the matrix S in step 4 requires the inversion of a matrix, whose columns are precisely the extended basis coordinate vectors. This inverted matrix is then multiplied by its transpose. The resulting product is the matrix S see Example 1 in the Appendix S1 for more details. Thus, step 4 requires operations. Finding the solution of the d -dimensional system of linear equations in step 5 requires again operations. According to  , the LS-algorithm is quadratic in the number n of variables and exponential in the number m of data points.
The exponential complexity of this type of algorithms should not be surprising, for it is an inherent property of even weaker reverse engineering problems see . Therefore, a computational implementation of these algorithms should take advantage of parallelization techniques and eventually of quantum computing. For what follows recall that. Let K be an arbitrary finite field, natural numbers and the polynomial ring in n indeterminates over K. It is a well known fact see, for instance,  ,  and  that the set of all polynomials of the form with coefficients is a vector space over K.
We denote this set with. It is not surprising see, for instance,  ,  and  that the vector space. Definition 1 Let K be a field , natural numbers and the polynomial ring in n indeterminates over K. Furthermore, let be polynomials. The set is called the ideal generated by. For a given set of data points and a given vector , consider the evaluation epimorphism of the tuple and its kernel. In what follows, will be a basis of. This basis will be extended to a basis of the whole space , according to the standard orthonormalization procedure.
The orthogonal solution of will be defined in terms of the generalized inner product defined by 6. In this subsection, by virtue of the mathematical framework developed in the Methods section, we will address the following two problems regarding the LS-algorithm and its generalization, the term-order-free reverse engineering method:. Problem 2 Given a function , what are the minimal requirements on a set , such that the LS-algorithm reverse engineers f based on the knowledge of the values that it takes on every point in the set?
Problem 3 Are there sets that make the LS-algorithm more likely to succeed in reverse engineering a function based only on the knowledge of the values that it takes on every point in the set? It is pertinent to emphasize that, contrary to the scenario studied in  , we do not necessarily assume that information about the number of variables actually affecting f is available. We will give further comments on this issue at the end of the Discussion. Definition 4 Let , be a polynomial function.
The following result tells us that if we are using the LS-algorithm to reverse engineer a nonzero function we necessarily have to use a data set containing points where the function does not vanish. Theorem 5 Let be a nonzero polynomial function. Furthermore let be a tuple of m different n-tuples with entries in the field , the vector defined by and the orthogonal solution of. Then if , it follows Proof: If , then by definition of , the vector would be equal to the zero vector. From Corollary 10 in Subsection 4. Theorem 6 Let and be as in the previous theorem. In addition, assume.
Then it holds. Proof: The claim follows directly from the definition of orthogonal solution and its uniqueness see Section 4 of the Appendix S1 for more details. Remark 7 From the necessary and sufficient condition 7 it becomes apparent, that if the function f is a linear combination of more than fundamental monomial functions, f can not be found as an orthogonal solution of where. In particular, if f is a linear combination containing all d fundamental monomial functions in , no proper subset of will allow us to find f as orthogonal solution of.
Remark 8 From the condition 7 it follows that in order to reverse engineer a monomial function appearing in f using the term-order-free reverse engineering method or the LS-algorithm, it is necessary that the monomial function is linearly independent of the basis vectors of. For this reason, the set X should be chosen in such a way that no fundamental monomial function is linearly dependent on the basis vectors of.
Otherwise, some of the terms appearing in f might vanish on the set X and would not be detectable by any reverse engineering method, as stated in [ 7 ]. Remark 10 Note that if U is in general position with respect to the basis , then, for any permutation of the elements of the basis , the general position of U remains unchanged. In other words, U is in general position with respect to the permuted basis.
Figure 2 shows two one-dimensional subspaces. The red subspace is not in general position since its basis cannot be extended to a basis of the entire space 2 dimensions by adjoining the first canonical unit vector horizontal black arrow to it. Within this space, two one-dimensional subspaces are depicted.