Probability estimates for reachability of linear systems defined over finite fields

This paper deals with the probability that random linear systems 
defined over a finite field are reachable. 
Explicit formulas are derived for the probabilities that a linear input-state 
system is reachable, that the reachability matrix has a 
prescribed rank, as well as for the number of cyclic vectors of a 
cyclic matrix. We also estimate the probability that the parallel 
 connection of finitely many single-input systems is reachable. These results 
 may be viewed as a first step to calculate the probability that a 
 network of linear systems is reachable.


Introduction
Linear systems over finite fields play an important role in many mathematical areas, including, e.g., convolutional codes [13,14], finite state automata and Boolean networks [9], network coding [7], and quantised information dissemination [15]. Since convolutional codes are dual to complete linear behaviors [13], they can be regarded as discrete linear systems over a finite field. Thus, fundamental concepts from control theory such as reachability and observability play a role in convolutional coding theory. In particular, reachability is necessary for the equivalence between non-catastrophicity and observability [14]. Moreover, observability is dual to reachability, so that the non-catastrophicity of a code can be guaranteed by two reachability conditions. Furthermore, it is stated in [14] that reachability and observability are necessary conditions if one tries to find a convolutional code with good distance properties.
In linear systems theory, the role of parallel, series and feedback interconnection has been recognized from early on since these operations constitute the building blocks for designing arbitrary networks of control systems. More recently, the importance of parallel and series interconnections for convolutional coding theory has been observed in, e.g., [1], where sufficient conditions for the minimality and observability of concatenations for two convolutional codes are derived. These concatenations consist of a specific combination of series and parallel connections of associated first order system realizations and thus allow for a system theoretic analysis. Moreover, in [8], so called woven convolutional codes, which are networks of convolutional codes consisting of series and parallel connections, are studied with the aim to achieve encoders with large free distances. This motivates us to investigate reachability problems for networks of linear systems over finite fields.
Further motivation for studying the interaction between systems theory and coding theory is provided from the fact that convolutional network codes as defined in [7] can be viewed as structured linear systems over finite fields. Typically, for network coding with acyclic networks, block codes are used. However, for networks containing cycles, convolutional codes suggest themselves as building blocks. Thus, using convolutional network coding, it becomes possible to extend solvability criteria for linear network coding problems to networks with cycles [7,Theorem 2.7].
In all such applications it becomes important to estimate the probability that randomly chosen interconnections in networks of linear systems entail reachability. A special case in point is the recent work [15], where probability estimates are derived for structural reachability of systems over finite fields. This covers the case where the node dynamics are first order one-dimensional systems. Similar investigations for general networks are unknown.
In this paper we consider parallel connections of N discrete-time linear systems defined over a finite field F. This interconnection structure could be interpreted as a network consisting of several independent blocks of nodes. Our goal is to establish a formula for the probability that such a system is reachable. Two cases of interest are considered, that of N = 1 and the parallel connection of single-input systems. Our first result is Theorem 1 that yields an explicit formula for the number of reachable pairs, i.e. covers the case N = 1. This extends earlier results by [10], derived for at most two inputs. The proof of Theorem 1 rests on the Hermite cell decomposition of the quotient space of reachable pairs, introduced in [5] and [6]. As a next step, Theorem 1 is extended by calculating the number of state space pairs with a reachability space of fixed dimension. Moreover, in section 2.4, we consider the closely related task of obtaining a formula for the probability that a vector is a cyclic vector of a matrix. Finally, Theorem 7 estimates the probability that the parallel connection of N ≥ 2 single-input systems is reachable. This is based on a formula for the number of N -tuples of pairwise coprime polynomials over F.

Probability of reachable properties
We begin with a brief summary of well-known properties of Grassmannians over a finite field F with cardinality |F|. Throughout this paper F is endowed with the uniform probability distribution that assigns to each field element the same probability t = 1 |F| .

2.1.
Counting points of the Grassmannian. The Grassmannian over a finite field F is the set G k (F n ) of all k-dimensional linear subspaces V ⊂ F n (more precisely, it is the set of F-rational points of the Grassmann variety but this distinction does not play a significant role in this paper). To count the number of elements in G k (F n ) one can proceed in at least two different ways. The first approach identifies the Grassmann manifold G k (F n ) of k-dimensional linear subspaces of F n with the homogeneous space GL n (F)/P by the parabolic subgroup of block upper triangular matrices. It is well-known and easily established that the general linear group GL n (F) of invertible n × n-matrices has exactly elements. Therefore the Grassmannian G k (F n ) has exactly points. In particular, one obtains the well-known formula 1 + t −1 + · · · + t 1−n for the cardinality of the projective space P n−1 (F). Alternatively, one constructs a cell decomposition of the Grassmannian G k (F n ) into finitely many disjoint Euclidean spaces S(a); see e.g. [11]. Recall that a Schubert-cell S(a) ⊂ G k (F n ) is defined for each sequence a = (a 1 , . . . , a k ) of strictly increasing integers 1 ≤ a 1 < . . . < a k ≤ n. Let A k,n denote the set of such sequences a. Thus S(a) can be identified with the set of all full row rank k × n matrices X that are in a-row echelon canonical form, i.e. X = (x ij ) ∈ F k×n has the standard basis vectors e 1 , . . . , e k at columns a 1 , . . . , a k and satisfies x ij = 0 for j < a i . For example, take k = 2, n = 5 and a = (2, 4). Then the 3-dimensional Schubert cell S(a) is described by all matrices of the form A simple counting argument shows that each Schubert cell S(a) is uniquely characterized by exactly free parameters and therefore consists of t −d(a) elements. This implies that the number of elements of the Grassmannian is given as In particular, we deduce from (2) the identity of power series 2.2. Probability of a system to be reachable. We consider linear control systems of the form (6) x(τ + 1) = Ax(τ ) + Bu(τ ), x(0) = 0, τ = 0, 1, 2, . . .
with system matrices A ∈ F n×n , B ∈ F n×m ; see [3] for a summary of linear systems theory developed over an arbitrary field.
Let Σ cr n,m (F) denote the set of all such reachable pairs and let |Σ cr n,m (F)| denote its cardinality. We are interested in calculating the number of reachable pairs (A, B) ∈ F n×n × F n×m , i.e., in calculating |Σ cr n,m (F)|. Equivalently, for the equidistribution on F n×(n+m) , we compute the probability P n,m (t) := |Σ cr n,m (F)| |F n×n × F n×m | of a pair (A, B) ∈ F n×n × F n×m to be reachable. Our first theorem generalizes an earlier result by [10], that has been restricted to the case m ≤ 2.
Theorem 1. The probability that a pair (A, B) ∈ F n×n × F n×m , n, m ≥ 1, is reachable is equal to In particular, one obtains for n ≥ 2: Proof. Clearly, reachability is invariant under the state space similarity transformations (A, B) → (T AT −1 , T B) with T ∈ GL n (F). Thus GL n (F) acts on Σ cr n,m (F) via similarity. Denote the corresponding orbit space by Σ n,m (F). Since (A, B) is reachable, the similarity action is a free action and therefore the map from the group to the group orbit is injective. This implies that the cardinalities of Σ cr n,m (F) and Σ n,m (F) are related as |Σ cr n,m (F)| = |GL n (F)| · |Σ n,m (F)|. Thus it amounts to determine the number of F-rational points in the quasi-affine algebraic variety Σ n,m (F). This we do following [5], using a cell decomposition of Σ n,m (F) that is obtained by fixing the so-called Hermite indices of reachable pairs. This is the main point where our analysis departs from [10], who use the more complicated Kronecker invariants rather the Hermite indices. We refer to [5] and [6] for further details and proofs of the subsequent statements on cell decompositions of Σ n,m (F). Specifically, Σ n,m (F) admits a disjoint decomposition into finitely many affine spaces that is parameterized by the combinations K = (K 1 , . . . , K m ) of n into m parts. Here the Hermite cells is bijective and therefore defines a bijection Her K → S(a K ) of Hermite cells and Schubert cells, respectively. Since for all K ∈ K n,m . In particular, both spaces Σ n,m (F) and G m−1 (F n+m−1 ) admit cell decompositions that are indexed by the combinations of n into m nonnegative parts. Moreover, the mutual dimensions of the Schubert and Hermite cells differ by n, respectively. Thus, although no direct relation between these very different spaces is known, the cardinalities of their F-rational points can be easily compared. Explicitly, one obtains (10) By (2), the Grassmannian G m−1 (F n+m−1 ) has exactly elements. Thus the cardinality of Σ cr n,m (F) is equal to which completes the proof of (8).
2.3. Systems with r-dimensional reachability subspace. One can extend Theorem 1 in a rather straightforward way to determine the number of pairs (A, B) with r-dimensional reachability subspace. Consider, for r = 0, . . . , n, the set In particular, Σ cr n,m (F) = S n n,m (F). To compute the cardinality of S r n,m (F), consider the set S r of all systems of the form where (A 1 , B 1 ) ∈ Σ cr r,m (F) and A 2 ∈ F (r×(n−r)) and A 3 ∈ F ((n−r)×(n−r)) are arbitrary. This space has cardinality Theorem 2. The cardinality of S r n,m (F) is equal to .
Proof. Let P ⊂ GL n (F) denote the parabolic subgroup of all block upper triangular matrices of the form Let GL n (F) × P S r denote the quotient space with respect to this free group action. This yields the well-defined map φ : GL n (F) × P S r −→ S r n,m (F) that sends each orbit [g, (A, B)] of (12) to the element (gAg −1 , gB). This map is a bijection and induces a G r (F n )-bundle on S r . Therefore one obtains the equality of cardinalities |GL n (F)× P S r | = |S r n,m (F)|. Moreover, the cardinality of the orbit space GL n (F)× P S r is equal to Using Theorem 1, this implies .
This completes the proof. Given an arbitrary cyclic matrix A ∈ F n×n , what is the probability that a vector b ∈ F n is cyclic?
and let q(z) = q 1 (z) ν1 · · · q r (z) νr denote the decomposition in irreducible factors q 1 (z), . . . , q r (z) of the characteristic polynomial q(z) = det(zI − A). Then the probability of cyclic vectors is given as the fraction The next result gives an explicit formula for the probability that a vector is cyclic.
Theorem 3. The probability of b ∈ F n to be a cyclic vector of A is equal to where n 1 , . . . , n r are the degrees of the distinct irreducible factors q 1 (z), . . . , q r (z) in the prime decomposition of det(zI − A).
Proof. Since A is cyclic, one can assume that A is in companion canonical form. Thus, without loss of generality, we can assume that A = S q is the shift operator on the polynomial model X q : and b can be identified with an arbitrary element p(z) with deg(p) < deg(q). Therefore the question is equivalent to that of determing the probability that a polynomial p(z) of degree < n is coprime with q(z). Now p(z) is coprime with q(z) if and only if p(z) is a unit of the ring X q . By the Chinese Remainder Theorem and using the primary decomposition of q(z), the ring X q is isomorphic to the direct product of rings Via this representation, the units of X q are seen to be in bijective correspondence with the r-tuples of units in X q ν 1 1 , . . . , X q νr r , respectively. On the other hand the number of units of X q ν i i is equal to |F niνi | − |F ni(νi−1) |, where n i := deg q i . Thus the number of units of X q is equal to The result follows.

Parallel connection of single-input systems
The aim of this section is to compute the probability that the parallel connected system with state vectors x i ∈ F ni for i = 1, . . . , N and input u ∈ F is reachable. In the first part we present explicit formulas for the probability of (14) to be reachable. For N > 2 the formula becomes very complex and difficult to evaluate. However, in part 2 of the section we derive asymptotic estimations.
3.1. Probability of reachability: explicit expressions. Throughout this section, let gcd and lcm denote the monic greatest common divisor and least common multiple, respectively.
be a (not necessarily coprime) factorization of the transfer function by polynomials p 1 , . . . , p n , d ∈ F[z], d ≡ 0. The next characterization of reachability is well-known; the proof is inserted for convenience of the reader.
, reachability is equivalent to the fact that c = 0 is the only solution of c(zI−A) −1 b ≡ 0 with c ∈ F n . This means that c(p 1 (z), . . . , p n (z)) ≡ 0 for some c ∈ F n implies c = 0, i.e. p 1 , . . . , p n are linearly independent over F.
Now we compute the probability that a parallel connection of two linear systems is reachable, i.e. we solve the problem for N = 2: Theorem 4. For i = 1, 2 and randomly chosen matrices A i ∈ F ni×ni and vectors b i ∈ F ni the probability that the parallel connected system with state vectors x i ∈ F ni and input u ∈ F is reachable is equal to the number of tuples (d 1 , d 2 ) of monic pairwise coprime polynomials of degrees n 1 and n 2 times |GL n1 (F)| · |GL n2 (F)|; in particular, this probability is equal to Proof. For i = 1, 2 assume that (A i , b i ) are reachable and consider factorizations From Lemma 1 it follows that the entries of P i are linearly independent, and consequently P i and d i are coprime (see Lemma 2). Since d i is scalar, these coprime factorizations are unique up to multiplying each factor with a nonzero constant. This constant must be one because d i is monic. Thus one can map each reachable pair (A i , b i ) to a unique element (P i , d i ), where d i monic, deg(P i ) < deg(d i ) = n i and the entries of P i are linearly independent. This map is denoted by f i and is injective because the realization of a transfer function is unique in the case of single input systems.
According to Theorem 1 (with m = 1) there are t −(n 2 On the other hand one has t −ni possibilities for d i and |GL ni (F)| = t −n 2 i ni j=1 (1 − t j ) possibilities for P i . Therefore the sets of pairs (A i , b i ) and (P i , d i ) have the same cardinality. Thus f i is a bijection and one can consider the pairs In [2] Fuhrmann has shown that reachability of (15) is equivalent to the fact that the scalar polynomials d 1 and d 2 are pairwise coprime if the single systems are reachable. The number of coprime pairs of monic polynomials (d 1 , d 2 ) is equal to t −n1−n2 (1 − t); see [4]. By Lemma 1, the pairs (A i , b i ) are reachable if and only if the entries of the polynomial vectors P i (z) are linearly independent over F. In particular, reachability of the systems depends only on P i . Thus the number of systems (A i , b i ) such that (15) is reachable coincides with the number of tuples (d 1 , d 2 ) of monic pairwise coprime polynomials of degrees n 1 and n 2 times |GL n1 (F)|· |GL n2 (F)|. Hence the total number of systems ( Therefore the considered probability is equal to which proves the theorem. We next attempt to extend these arguments to the parallel connection of N ≥ 2 single-input systems. As in the proof of Theorem 4, consider coprime factorizations of the corresponding transfer functions (zI − A i ) −1 b i = P i (z)d i (z) −1 for i = 1, . . . , N . Fuhrmann's characterization of reachability of two parallel connected systems is easily extended to more than two node systems. Explicitly, see [3], system (14) is reachable if and only if (A i , b i ) are reachable for i = 1, . . . , N and d 1 , . . . , d N are pairwise coprime. Thus one has to compute the number of N-tuples (d 1 , . . . , d N ) of monic pairwise coprime polynomials of given degrees n 1 , . . . , n N .
To this end we first prove the following theorem, which extends a result by [12] from the ring of integers to the ring of polynomials. Let n := (n 1 , . . . , n N ) ∈ N N and Γ be an undirected graph with set of vertices V = {1, . . . , N } and set of edges E, having cardinality E := |E|. The edges of Γ are denoted as ij, for suitable i, j ∈ V with i < j. For every vertex l ∈ V let E l := {ij ∈ E | i = l or j = l} denote the set of edges terminating at l. With each edge ij of Γ we associate a monic, square-free polynomial k ij (z) ∈ F[z]. We refer to this as a polynomial labeling of the graph and denote it by k. For each polynomial labeling and vertices l ∈ V let K l := lcm{k ij | ij ∈ E l }.
Then M (n) := {k ∈ F[z] E | k ij monic, square-free for ij ∈ E, deg(K l ) ≤ n l , l ∈ V} is the set of all polynomial labelings k of Γ satisfying the degree bounds deg(K l ) ≤ n l for all vertices l. For each monic square-free polynomial p let ω(p) denote the number of irreducible factors of p.
Proof. The sets Thus Γ(n) = X(n) \ r∈R D r . From the well-known inclusion-exclusion principle one obtains where D ∅ = X(n) and D S := r∈S D r for S = ∅. It remains to determine |S| and |D S |. For each edge ij define the monic and square-free polynomial while for each vertex l ∈ V we consider the monic and square-free polynomials From the definition of D S one obtains (d 1 , . . . , d N ) ∈ D S if and only if p | gcd(d i , d j ) is satisfied for all (p, ij) ∈ S. This implies the equivalence: holds for all ij ∈ E if and only if k S ij | d l for all ij ∈ E l and l ∈ V. Since k S ij are square-free, this in turn yields the characterization (d 1 , ..., d N ) ∈ D S ⇔ K S l | d l for l ∈ V. Thus one has to count the number of degree n l monic multiples of a monic polynomial K S l . This number coincides with the number of monic polynomials in F[z] with degree n l − deg(K S l ) if the last expression is non-negative; otherwise such a polynomial cannot exist. This shows that if deg(K S l ) ≤ n l holds for all l ∈ V and |D S | = 0 otherwise. To compute |S|, note that ω(k S ij ) coincides with the number of elements p ∈ P such that (p, ij) ∈ S. Thus Finally, for each non-empty subset S ⊂ R, equation (19) defines a unique polynomial labeling k S ∈ M (n). Conversely, for each k ∈ M (n) there exists S ⊂ R with k = k S . In fact, each polynomial labeling k = (k ij |ij ∈ E) ∈ M (n) admits a unique factorization into primes for subsets P ij ⊂ P . Defining S = ij∈E P ij × {ij} then yields k S ij = k ij for all edges ij ∈ E. Thus in (18) one can sum over polynomial labelings k instead of summing over S. Moreover, the restriction k ∈ M (n) in the sum of (17) allows us to use formula (21), i.e. we avoid summing up zeros. This completes the proof.
In the case that all pairs of vertices of Γ are connected by an edge, one obtains the probability that N monic polynomials are pairwise coprime. Going ahead like in the proof of Theorem 4 and using Theorem 5, one achieves: Corollary 1. For i = 1, . . . , N and randomly chosen A i ∈ F ni×ni and b i ∈ F n i the probability that system (14) is reachable is equal to Remark 1. For N = 2 and E = 1 formula (17) has the following form: Recall that the number of coprime pairs of monic polynomials is t −n1−n2 (1 − t), see [4]. Thus one obtains the combinatorical identity: |U (g)| := {(p 1 , . . . , p 2r−1 ) | r ∈ N, p i = p j monic, irreducible, are the numbers of monic, square-free polynomials with an even or odd number of irreducible factors, respectively. For n ≥ 2 it follows: i.e. |E(n)| = |U (n)| for every n ≥ 2.
For N > 2 the formula of Theorem 5 is very difficult to evaluate. If the degree of one of the polynomials is at least as large as the sum of the other degrees, the computation could be reduced to a computation with lower degrees. This fact is explicitely stated in the following corollary: Corollary 2. Let n 1 , ..., n N , h ∈ N. Then: Proof. For k ∈ M (n) it holds: The first and the third inequality follow because K l is the least common multiple of the corresponding k ij . The fourth inequality holds since k ∈ M (n). Finally, the last inequality holds because of the assumption n 1 = h + n 2 + ... + n N . From (23) it follows that in the given situation increasing n 1 does not increase the number of elements in M (n), because deg(K l ) are restricted more strongly by n 2 , ..., n N than by n 1 . Thus the only expression in (17) that changes when increasing n 1 is t −n1 , which causes the factor t −h .
3.2. Probability of reachability: asymptotic expansions. Now we calculate the asymptotic behaviour when 1/t -the size of the field -becomes large. Theorem 6. Let n 1 , ..., n N ∈ N and let E be the number of edges in a graph Γ. Then: Proof. To prove this result, first sort the elements of M (n) with respect to the degrees of the entries of the vector k = (k 1 , ..., k E ). To this end for each vector of non-negative integers g := (g 1 , ..., g E ) define M (n, g) := {k ∈ M (n) | deg(k m ) = g m f or 1 ≤ m ≤ E}. Let A be the set of all g with M (n, g) = ∅. Note that the degree bounds for M (n) ensure that A is finite. One achieves: Starting with small values for the entries of g the first summands are computed.
For g = (0, ..., 0), i.e. k = (1, ..., 1), one gets the summand 1 because of ω(1) = 0 and K l = 1 for l = 1, . . . , N . If g m0 = 1 for exactly one 1 ≤ m 0 ≤ E and g m = 0 for m = m 0 , there are |F| = 1/t possibilities for the linear polynomial k m0 and E possibilities for the choice of m 0 . Moreover, ω(k m0 ) = 1, so that these summands have negative sign. As k m0 is relevant for exactly those K l for which its associated edge is terminating at l, there are exactly two K l which are of degree 1. Hence the resulting sum of these terms is equal to −E · 1 t · t 2 = −E · t.
Thus one only has to show that each of the remaining summands behaves asymptotically as O(t 2 ), which is done by showing for every fixed g for which the sum of the entries of g is at least two. This will be done by induction with respect to E. For E = 1 note that g and k = k 12 are scalar. Moreover K 1 = K 2 = k 12 . Therefore R(g) = 0 if g > min(n 1 , n 2 ) and otherwise This computation starts with an inequality since the condition that k has to be square-free is dropped. The first equality follows from the fact that there are (1/t) g monic polynomials of degree g.
Next we take the step from E − 1 to E.
To this end choose one of the smallest entries of g and denote it without loss of generality by g E . Then the edge with which k E is associated -in the following denoted by ij -is taken away form the original graph and thus a graph with E − 1 edges is achieved. In the following the index (E − 1) above an expression means that it belongs to a graph with E − 1 edges; in the same way we use the index (E). Similarly, k (E−1) and g (E−1) should denote the vectors consisting of the first E − 1 entries of k and g, respectively.
The degrees of the K l can never increase, when taking an edge away. Therefore k ∈ M (n, g) implies k (E−1) ∈ M (E−1) (n, g (E−1) ). Next we set

Moreover let
The number of summands in the first sum is finite and thus one only has to show that for any fixed v i , v j , w i , w j the following is true: To do this one computes analogous. For l / ∈ {i, j} it holds K because nothing changes at the associated vertices. It follows: Here the product vi,vj the number of polynomials k E such that k ∈ B (E) vi,vj ,wi,wj should be determined. k (E−1) uniquely determines K (E−1) i and since W i is a divisor of K (E−1) i of degree w i , there are only finitely many possibilities for W i . Define C as this number of possibilities for W i . One knows that k E has to be a multiple of W i of degree g E . Thus for every W i there are at most t wi−g E possibilities for k E .
Using this and the fact that the product in (24) is independent of k E , it follows for the expression in (24): because w j ≤ g E since W j | k E . Now we distinguish three cases: 1. The sum of the entries of g (E−1) is at least two. Then R(g (E−1) ) is O(t 2 ) per induction and we are done. 2. g (E−1) has a component that is equal to zero. Here g E must be zero since it was choosen to be one of the smallest entries. But then the sum of the entries of g (E−1) is equal to the sum of the entries of g (E) and thus in particular at least two. Consequently, we are done, too. 3. E = 2 and g (E−1) = g 1 = 1. Then g E = g 2 ≤ 1. If g 2 = 0, we argue as before.
If g 2 = 1 and the two edges of the graph meet at a vertex, one gets R(g) = R(1, 1) ≤ k1, k2 monic deg(km)=1 t 2+deg(lcm(k1,k2)) = k1 = k2 monic deg(km)=1 If g 2 = 1 and the two edges of the graph are isolated, one gets R(g) = R(1, 1) ≤ k1, k2 monic deg(km)=1 t 4 = t 2 since there are two K l that coincide with k 1 and k 2 , respectively. Moreover, there are t −2 pairs of monic polynomials of degree one. Thus this case is done as well and our proof is complete. Now we come to the situation of parallel connected systems again. Recall that here all pairs of vertices of Γ(n) are connected by an edge, i.e. it holds E = N (N −1) 2 .
The following result is an easy consequence of Theorem 6: Corollary 3. For n := (n 1 , . . . , n N ) ∈ N N the set G(n) of N-tuples (d 1 , . . . , d N ) of monic pairwise coprime polynomials d i ∈ F[z] with deg(d i ) = n i for i = 1, . . . , N has the following cardinality: .
Therefore the probability that d 1 , . . . , d N are pairwise coprime is equal to Using the preceding corollary, it is possible to estimate the probabilty that the parallel connection of N linear single-input systems is reachable.
Theorem 7. For i = 1, . . . , N and randomly chosen A i ∈ F ni×ni and b i ∈ F n i the probability that system (14) is reachable is equal to Proof. Going ahead like in the proof of Theorem 4 and using Corollary 3, one achieves the expression for the considered probability.

Conclusions
We compare cell decompositions of the moduli space of reachable linear systems with the Grassmannian to derive an explicit formula for the probability that a linear system is reachable. The formula is extended to count the number of reachable linear systems with r-dimensional reachability subspace. We calculate the probability that finitely many monic polynomials of given degrees are pairwise coprime. This allows us to estimate the probability that the parallel connection of finitely many linear single-input systems is reachable. Future research will concern the extension to the parallel connection of finitely many multivariable systems and to general networks of systems.

Acknowledgments
We would like to thank the referees very much for their valuable comments and suggestions. This research has been partially supported by the grant DFG 1858/13-1 from the German Research Foundation.