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Semantic Equivalence of Graph Polynomials Definable in Second Order Logic

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Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

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Abstract

We study semantic equivalence of multivariate graph polynomials via their distinctive power introduced in (Makowsky, Ravve, Blanchard 2014) under the name of d.p.-equivalence. There we studied only univariate graph polynomials. In this paper we extend our study to multivariate graph polynomials. We use the characterization from the previous paper of d.p.-equivalence of two graph polynomials in terms of computing their respective coefficients. To make our graph polynomials combinatorially meaningful we require them to be definable in Second Order Logic \(\ SOL\). The location of zeros in the multivariate case is captured by various versions of halfplane properties, also known as stability or Hurwitz stability. Our main application shows that every multivariate \(\mathrm {SOL}\)-definable graph polynomial \(P(G;X_1, X_2, ... X_k)\) is d.p.-equivalent to a substitution instance of a stable (Hurwitz stable) \(\mathrm {SOL}\)-definable graph polynomial \(Q(G;Y, X_1, X_2, ... X_k)\). In other words, two d.p.-equivalent \(\mathrm {SOL}\)-definable multivariate graph polynomials can also have very different behavior concerning their halfplane properties.

J.A. Makowsky—Partially supported by a grant of Technion Research Authority. Work done in part while the author was visiting the Simons Institute for the Theory of Computing in Spring 2016.

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Notes

  1. 1.

    Many are even definable in Monadic Second Order Logic \(\mathrm {MSOL}\), [35]. The exceptions are in [43]. The algorithmic advantages of \(\mathrm {MSOL}\)-definability, [14] are of no importance in this paper.

  2. 2.

    A univariate polynomial is monic if the leading coefficient equals 1.

  3. 3.

    A sequence of numbers \(a_i: i \le m\) is unimodal if there is \(k \le m\) such that \(a_i \le a_j\) for \(i<j < k\) and \(a_i \ge a_j\) for \( k \le i <j \le m\).

  4. 4.

    In engineering and stability theory, a square matrix A is called stable matrix (or sometimes Hurwitz matrix) if every eigenvalue of A has strictly negative real part. These matrices were first studied in the landmark paper [28] in 1895. The Hurwitz stability matrix plays a crucial part in control theory. A system is stable if its control matrix is a Hurwitz matrix. The negative real components of the eigenvalues of the matrix represent negative feedback. Similarly, a system is inherently unstable if any of the eigenvalues have positive real components, representing positive feedback. In the engineering literature, one also considers Schur-stable univariate polynomials, which are polynomials such that all their roots are in the open unit disk, see for example [55].

  5. 5.

    There is a polynomial time computable function \(F: {\mathbb Z}[\mathbf {X}] \rightarrow {\mathbb Z}[Y, \mathbf {X}]\) such that for all graphs G we have \(F(P(G;\mathbf {X})) = Q^s(G;Y,\mathbf {X})\).

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Acknowledgment

The authors would like to thank Petter Brändén for guiding us to the literature of stable polynomials, and Jason Brown and four anonymous readers of an earlier version of this paper for valuable comments. Thanks also to Jingcheng Lin for pointing out the references [2, 9]. We want to acknowledge that some of the definitions and examples were taken verbatim from our [40]. D.p-equivalence was first characterized in [37].

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Authors and Affiliations

  1. Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

    Johann A. Makowsky

  2. Department of Software Engineering, ORT-Braude College, Karmiel, Israel

    Elena V. Ravve

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Correspondence to Johann A. Makowsky .

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Makowsky, J.A., Ravve, E.V. (2016). Semantic Equivalence of Graph Polynomials Definable in Second Order Logic. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_18

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