First-Order Orbit Queries

Abstract

Orbit Problems are a class of fundamental reachability questions that arise in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. Instances of the problem comprise a dimension $d\in \mathbb {N}$, a square matrix $A\in \mathbb {Q}^{d\times d}$, and a query regarding the behaviour of some sets under repeated applications of A. For instance, in the Semialgebraic Orbit Problem, we are given semialgebraic source and target sets $S,T\subseteq \mathbb {R}^{d}$, and the query is whether there exists $n\in {\mathbb {N}}$ and x ∈ S such that Aⁿx ∈ T. The main contribution of this paper is to introduce a unifying formalism for a vast class of orbit problems, and show that this formalism is decidable for dimension d ≤ 3. Intuitively, our formalism allows one to reason about any first-order query whose atomic propositions are a membership queries of orbit elements in semialgebraic sets. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory—Baker’s theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of $\mathbb {R}^{d}$ for which membership is decidable. On the other hand, previous work has shown that in dimension d = 4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.

Appendices

Appendix A: The case of only real eigenvalues

In this section we consider the Semialgebraic Orbit Problem in the case where the matrix A has only real eigenvalues, denoted ρ₁,ρ₂,ρ₃. In this case, by converting A to Jordan normal form, there exists an invertible matrix $B\in (\AA \cap {\mathbb {R}})^{3\times 3}$ such that one of the following holds:

• 1. $A=B^{-1}\left (\begin {array}{ccc} \rho _{1} & 0 & 0\\ 0 & \rho _{2} & 0\\ 0& 0 & \rho _{3} \end {array}\right )B$, in which case $A^{n}=B^{-1}\left (\begin {array}{ccc} {\rho _{1}^{n}} & 0 & 0\\ 0 & {\rho _{2}^{n}} & 0\\ 0& 0 & {\rho _{3}^{n}} \end {array}\right )B$.

• 2. $A=B^{-1}\left (\begin {array}{ccc} \rho _{1} & 1 & 0\\ 0 & \rho _{2} & 0\\ 0& 0 & \rho _{3} \end {array}\right )B$ with ρ₁ = ρ₂, in which case $A^{n}=B^{-1}\left (\begin {array}{ccc} {\rho _{1}^{n}} & n\rho _{1}^{n-1} & 0\\ 0 & {\rho _{1}^{n}} & 0\\ 0& 0 & {\rho _{3}^{n}} \end {array}\right )B$.

• 3. $A=B^{-1}\left (\begin {array}{ccc} \rho _{1} & 1 & 0\\ 0 & \rho _{2} & 1\\ 0& 0 & \rho _{3} \end {array}\right )B$ with ρ₁ = ρ₂ = ρ₃, in which case Aⁿ = B^− 1 $\left (\begin {array}{ccc} {\rho _{1}^{n}} & n\rho _{1}^{n-1} & \frac 12n(n-1)\rho _{1}^{n-2}\\ 0 & {\rho _{1}^{n}} & n\rho _{1}^{n-1}\\ 0& 0 & {\rho _{1}^{n}} \end {array}\right )B$.

In any of the forms above, we can write

$$ A^{n} s=\left( \begin{array}{ccc} A_{1}(n){\rho_{1}^{n}}+B_{1}(n){\rho_{2}^{n}}+C_{1}(n){\rho_{3}^{n}}\\ A_{2}(n){\rho_{1}^{n}}+B_{2}(n){\rho_{2}^{n}}+C_{2}(n){\rho_{3}^{n}}\\ A_{3}(n){\rho_{1}^{n}}+B_{3}(n){\rho_{2}^{n}}+C_{3}(n){\rho_{3}^{n}} \end{array}\right) $$

where the A_i,B_i, and C_i are polynomials whose degree is less than the multiplicity of their corresponding eigenvalue.

In Sections 4 and 5, we reduce the problem to finding a solution to an almost self-conjugate system. In the case of real eigenvalues, the notion of almost self-conjugate is meaningless, as there are no complex numbers involved. Thus, following the analysis thereof, and plugging the entries of Aⁿs, we reduce the problem to solving a system of expressions of the form $\bigwedge _{J} R_{J}(A^{n}s)\sim _{J} 0$, where

$$ R_{J}(A^{n} s)= \sum\limits_{0\le p_{1},p_{2},p_{3}\le k} \alpha^{J}_{p_{1},p_{2},p_{3}}(n)\rho_{1}^{p_{1} n}\rho_{2}^{p_{2} n}\rho_{3}^{p_{3} n} $$

(4)

for some $k\in {\mathbb {N}}$, and $\alpha ^{J}_{p_{1},p_{2},p_{3}}(n)$ are polynomials.

Assuming ρ₁,ρ₂,ρ₃ > 0 (otherwise we can split according to odd and even n), for each such expression we can compute a bound $N\in {\mathbb {N}}$ based on the rate of growth of the summands, such that either for every n > N the equation holds, or for every n > N it does not hold.

Appendix B: The case where γ is a root of unity

We assume that $\gamma =\frac {\lambda }{|\lambda |}$ is a root of unity. That is, there exists $d\in {\mathbb {N}}$ such that γ^d = 1, so we have that $\left \{\gamma ^{n}:n\in {\mathbb {N}}\right \}=\left \{\gamma ^{0},\ldots ,\gamma ^{d-1}\right \}$.

Let $n\in {\mathbb {N}}$ and write m = (n mod d). We can now write

$$ A^{n}s=\left( \begin{array}{ccc} a_{1} |\lambda|^{n}\gamma^{m}+\overline{a_{1}}|\lambda|^{n}\overline{\gamma}^{m}+b_{1} \rho^{n}\\ a_{2} |\lambda|^{n}\gamma^{m}+\overline{a_{2}}|\lambda|^{n}\overline{\gamma}^{m}+b_{2} \rho^{n}\\ a_{3} |\lambda|^{n}\gamma^{m}+\overline{a_{3}}|\lambda|^{n}\overline{\gamma}^{m}+b_{3} \rho^{n} \end{array}\right) =\left( \begin{array}{ccc} 2\text{Re}(a_{1}\gamma^{m}) |\lambda|^{n}+b_{1} \rho^{n}\\ 2\text{Re}(a_{2}\gamma^{m}) |\lambda|^{n}+b_{2} \rho^{n}\\ 2\text{Re}(a_{3}\gamma^{m}) |\lambda|^{n}+b_{3} \rho^{n} \end{array}\right) $$

Observe that there exists $n\in {\mathbb {N}}$ such that Aⁿs ∈ T iff there exist 0 ≤ m ≤ d − 1 and $r\in {\mathbb {N}}\cup \left \{0\right \}$ such that A^rd+ms ∈ T. We can thus split our analysis according to $m\in \left \{0,\ldots ,d-1\right \}$. For every such m, we need to decide whether there exists $r\in {\mathbb {N}}\cup \left \{0\right \}$ such that $\left (\begin {array}{ccc} 2\text {Re}(a_{1}\gamma ^{m}) |\lambda |^{m}(|\lambda |^{d})^{r}+b_{1} \rho ^{m}(\rho ^{d})^{r}\\ 2\text {Re}(a_{2}\gamma ^{m}) |\lambda |^{m}(|\lambda |^{d})^{r}+b_{2} \rho ^{m}(\rho ^{d})^{r}\\ 2\text {Re}(a_{3}\gamma ^{m}) |\lambda |^{m}(|\lambda |^{d})^{r}+b_{3} \rho ^{m}(\rho ^{d})^{r} \end {array}\right )$ Note that γ^m, |λ|^m and ρ^m are constants. Therefore, these expressions contain only realalgebraic constants, the system can be viewed as a case handled in the setting of all real eigenvalues. We can thus proceed with the analysis in Section A.

Finally, we remark that $d\le \deg (\gamma )^{2}$. The proof appears in [13], and we bring it here for completeness. Since γ is a primitive root of unity of order d, then the defining polynomial p_γ of γ is the d-th Cyclotomic polynomial, so $\deg (\gamma )={\Phi }(d)$, where Φ is Euler’s totient function. Since ${\Phi }(d)\ge \sqrt {d}$, we get that $d\le \deg (\gamma )^{2}$. Therefore, the number of cases we consider is polynomial in the original input, and does not involve a blowup in the complexity.

Appendix C: Change of Basis Matrices in the 3 × 3 case

In this section we consider a diagonalisable matrix $A\in {\mathbb {Q}}^{3\times 3}$ with complex eigenvalues. Thus, we can write A = PDP^− 1 with $D=\text {diag}(\lambda ,\overline {\lambda },\rho )$ with λ ∈Å and $\rho \in \AA \cap {\mathbb {R}}$.

Note that the columns of the matrix P are eigenvectors of A, and moreover, conjugate eigenvalues have conjugate eigenvectors and real eigenvalues have real eigenvectors. We can therefore assume

$$ P=\left( \begin{array}{ccc} a & \overline{a} & d\\ b & \overline{b} & e\\ c & \overline{c} & f \end{array}\right) $$

for a,b,c ∈Å and $d,e,f\in {\mathbb {R}}\cap \AA$.

Lemma 4

Let E = diag(δ₁,δ₂,δ₃) be a diagonal matrix, then every coordinate of PEP^− 1 is of the form $\alpha \delta _{1}+\overline {\alpha }\delta _{2}+\upbeta \delta _{3}$, where α ∈Å and $\upbeta \in \AA \cap {\mathbb {R}}$.

Proof

The proof is straightforward: we compute the matrix P^− 1, and then the product PEP^− 1.

We leave it to the reader to verify the following: first, the determinant of P is pure-imaginary, i.e., $\det (P)=mi$ for $m\in {\mathbb {R}}\cap \AA$. Second, we have

$$ P^{-1}=\frac{1}{mi}\left( \begin{array}{ccc} f \overline{b}-e \overline{c} & d \overline{c}-f \overline{a} & e \overline{a}-d \overline{b} \\ c e-b f & a f-c d & b d-a e \\ b \overline{c}-c \overline{b} & c \overline{a}-a \overline{c} & a \overline{b}-b \overline{a} \end{array}\right) $$

Finally, it is very easy (yet tedious) to verify that PEP^− 1 satisfies the claim. We demonstrate by computing the coordinate (PEP^− 1)_1,2.

We have that the first row of PE is $(a\delta _{1} , \overline {a}\delta _{2} , d \delta _{3} )$, and hence

$$ \begin{array}{@{}rcl@{}} (PEP^{-1})_{1,2}&=&(PE)_{1,1}P^{-1}_{1,2}+(PE)_{1,2}P^{-1}_{2,2}+(PE)_{1,3}P^{-1}_{3,2}\\ &=&\frac{1}{mi}\left( a\delta_{1}(d \overline{c}-f \overline{a})+ \overline{a}\delta_{2}(a f-c d)+d \delta_{3} (c \overline{a}-a \overline{c})\right)\\ &=&\frac{1}{m}\left( -i \delta_{1}(ad \overline{c}-a f\overline{a})+i \delta_{2}(\overline{a}c d - a f \overline{a})- i \delta_{3} (d c \overline{a}-d a \overline{c})\right) \end{array} $$

It is now easy to see that the coefficients of δ₁ and δ₂ are conjugates, and the coefficient of δ₃ is real, as desired. □

Appendix D: Bounds on the Description Size of Points in Z _f

We complete the analysis of Remark 2.

Recall that $f(z)={\sum }_{m=0}^{k}{\upbeta }_{m} z^{m} +\overline {{\upbeta }_{m}}\overline {z}^{m}$, and $Z_{f}=\left \{z: f(z)=0 \wedge |z|=1\right \}$. Further recall that for every 0 ≤ m ≤ k, β_m is a polynomial in $a_{1},a_{2},a_{3},\overline {a_{1}},\overline {a_{2}},\overline {a_{3}},b_{1},b_{2},b_{3}$, where all the latter are linear combinations of roots of the characteristic polynomial of A, and are therefore algebraic numbers of degree at most 3 and description polynomial in ${\left \lVert A\right \rVert }+{\left \lVert s\right \rVert }$.

We can now express the condition f(z) = 0 using a quantified formula in the first-order theory of the reals by replacing each of the constants above (i.e. a₁, etc.) by their corresponding description, as per Section 2.2. It follows that in this description, there are at most 9 variables. We now employ the following result due to Renegar [21].

Theorem 5 (Renegar)

Let $M \in {\mathbb {N}}$ be fixed. Let τ(y) be a formula of the first-order theory of the reals. Assume that the number of (free and bound) variables in τ(y) is bounded by M. Denote the degree of τ(y) by d and the number of atomic predicates in τ(y) by n.

There is a polynomial time (polynomial in ${\left \lVert \tau (\mathbf {y})\right \rVert }$) procedure which computes an equivalent quantifier-free formula

$$ \begin{array}{@{}rcl@{}} \chi(\mathbf{y}) = \bigvee\limits_{i=1}^{I} \bigwedge\limits_{j=1}^{J_{i}} h_{i,j}(y) \sim_{i,j} 0 \end{array} $$

where each $\sim _{i,j}$ is either > or =, with the following properties:

• 1. Each of I and J_i (for 1 ≤ i ≤ I) is bounded by $(n+d)^{O(1)}$.

• 2. The degree of χ(y) is bounded by (n + d)^O(1).

• 3. The height of χ(y) is bounded by $2^{{\left \lVert \tau (\mathbf {y})\right \rVert }(n+d)^{O(1)}}$.

We apply this theorem to the description of Z_f given above, where we identify ${\mathbb {C}}$ with ${\mathbb {R}}^{2}$ so that f is indeed a polynomial. Then, we obtain in polynomial time a description of Z_f. Moreover, the degrees of the entries is bounded by ${\left \lVert f\right \rVert }^{{\mathcal {O}}(1)}$ and their height is bounded by $2^{{\left \lVert f\right \rVert }^{{{\mathcal {O}}(1)}}}$.