**2. Semigroups**

A *semigroup S* ≡ (*<sup>S</sup>*, ◦) is a set *S* and an associative binary operation " ◦ " called multiplication defined upon the set (contrast this with the above definition of a group and note that a group is a semigroup endowed with the additional special properties given by items (*iii*) and (*iv*)). The one-sided right (one-sided left) multiplication of *x* ∈ *S* by *y* ∈ *S* is the product *x* ◦ *y* ∈ *S* (*y* ◦ *x* ∈ *<sup>S</sup>*). An element *e* ∈ *S* is an *identity* if *x* ◦ *e* = *e* ◦ *x* = *x* for *x* ∈ *S*. An identity can be adjoined to *S* by setting *S*1 = *S* ∪ {*e*} and defining *x* ◦ *e* = *e* ◦ *x* = *x* for *x* ∈ *S*1. Semigroup *S* ≡ (*<sup>S</sup>*, ◦) and the semigroup *T* ≡ (*<sup>T</sup>*, ∗) on set *T* with associative binary operation " ∗ " are *isomorphic* (denoted *S* ≈ *T*) when there is a bijective map (i.e., an isomorphism) *θ* : *S* → *T* such that *<sup>θ</sup>*(*x* ◦ *y*) = *<sup>θ</sup>*(*x*) ∗ *<sup>θ</sup>*(*y*) for all *x*, *y* ∈ *S*.

The well-known *L*, *R*, *H*, and *D* Green's equivalence relations on a semigroup *S* partition *S* into a highly organized "egg box" structure using their relatively simple algebraic properties. In particular, the equivalence relation *L*(*R*) on *S* is defined by the rule that *xLy* (*xRy*) if and only if *S*1*x* = *<sup>S</sup>*<sup>1</sup>*y xS*<sup>1</sup> = *yS*<sup>1</sup> for *x*, *y* ∈ *S* and the equivalence relation *H* = *L* ∩ *R* is similarly defined so that *xHy* if and only if *xLy* and *<sup>x</sup>Ry*. The relations *L* and *R* commute under the composition "•" of binary relations and *D*≡ *L*•*R* = *R*•*L* is the smallest equivalence relation containing *L* and *R*.

For *x* ∈ *S* and *X* <sup>∈</sup>{*L*, *R*, *H*, *D*} denote the *X* class containing *x* by *<sup>X</sup>*(*x*) where *X* = *L*, *R*, *H*, or *D* when *X* = *L*, *R*, *H*, or *D*, respectively. Thus, *xXy* if and only if *<sup>X</sup>*(*x*) = *<sup>X</sup>*(*y*). If *x*, *y* ∈ *S* and *<sup>R</sup>*(*x*) = *<sup>R</sup>*(*y*)(*L*(*x*) = *<sup>L</sup>*(*y*)), then there exist elements *s* (*t*) in *S*1 such that *xs* = *y* (*tx* = *y*) (hereafter the juxtaposition *xy* will *also* be used for the multiplication *x* ◦ *y*).

#### **3. The Semigroups** *Bn* **and** *BX*

The semigroup *Bn* of Boolean matrices is the set of all *n* × *n* matrices over {0, 1} with Boolean composition *γ* = *α* ◦ *β* defined by

$$
\gamma\_{ij} = \vee\_{k \in f} \left( \alpha\_{ik} \wedge \beta\_{kj} \right),
\tag{1}
$$

as the semigroup multiplication operation. Here *J* = {1, 2, ··· , *<sup>n</sup>*}, where *n* ≥ 1 is a counting number, ∧ denotes Boolean multiplication (i.e., 0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0, 1 ∧ 1 = 1), and ∨ denotes Boolean addition (i.e., 0 ∨ 0 = 0, 0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1).

The rows (columns) of any *α* ∈ *Bn* are Boolean row (column) *n*, vectors, i.e., row (column) *n*, tuples over {0, <sup>1</sup>}, and come from the set *Vn*(*Wn*) of all Boolean row (column) *n*-vectors. These vectors can be added coordinate-wise using Boolean addition. If *u*, *v* ∈ *Vn*(*Wn*), then *u* # *v* when the *i*th coordinate *ui* = 1 implies the *i*th coordinate *vi* = 1, 1 ≤ *i* ≤ *n* (# is a partial order).

Let **0**(**1**) be either the zero (unit) row or zero (unit) column vector (the context in which **0**(**1**) is used defines whether it is a row or column vector). The matrix with **0** in every row, i.e., the zero matrix, is denoted by "Z" and the matrix with **1** in every row is denoted by "*ω*". For *α* ∈ *Bn*, the row space <sup>Γ</sup>(*α*) of *α* is the subset of *Vn* consisting of **0** and all possible Boolean sums of (one or more) nonzero rows of *α*. <sup>Γ</sup>(*α*) is a lattice (Γ(*α*), #) under the partial order #. The row (column) *basis r*(*α*) (*c*(*α*)) of *α* is the set of all row (column) vectors in *α* that are not Boolean sums of other row (column) vectors in *α*. Please note that each vector in *r*(*α*) (*c*(*α*)) must be a row (column) vector of *α*. The vector **0** is never a basis vector and the empty set ∅ is the basis for the Z matrix [6,7].

The semigroup *BX* of binary relations on a set *X* of cardinality *n* (denoted |*X*| = *n*) is the power set of *X* × *X* with multiplication *a* = *bc* being the "composition of binary relations" defined by

$$a = \{(\mathbf{x}, \mathbf{y}) \in X \times X : (\mathbf{x}, \mathbf{z}) \in b, \ (\mathbf{z}, \mathbf{y}) \in c, \ \text{when } \mathbf{z} \in X\}.\tag{2}$$

It is easy to see that a bijective index map *f* : *X* → *J* induces an isomorphism *λ* : *BX* → *Bn* defined by *<sup>λ</sup>*(*a*) = *α*, where *<sup>α</sup>ij* = 1 if *f* <sup>−</sup><sup>1</sup>(*i*), *f* −<sup>1</sup>(*j*) ∈ *a* and is 0 if *f* <sup>−</sup><sup>1</sup>(*i*), *f* −<sup>1</sup>(*j*) ∈/ *a*. *Bn* is therefore the Boolean matrix representation of *BX* [8].

#### **4. The Semigroup** *NV*

A network *E* of order *n* is the pair *E* = (*<sup>V</sup>*, *<sup>C</sup>*), where *V* is a nonempty set of nodes with |*V*| = *n*, and the binary relation *C* ⊆ *V* × *V* is the set of directed links connecting the nodes of the network. Thus, *E* is both a digraph and a binary relation. If (*<sup>x</sup>*, *y*) ∈ *C*, then node *x*(*y*) is an in(out)-neighbor of node *y*(*x*). The *in-neighborhood of x* ∈ *V* is the set *<sup>I</sup>*(*<sup>E</sup>*; *x*) of all in-neighbors of *x* and the *out-neighborhood of x* ∈ *V* is the set *<sup>O</sup>*(*<sup>E</sup>*; *x*) of all out-neighbors of *x*.

Let *NV* be the set of networks on *V* and define "multiplication of networks" by *EF* = *G* ≡ *<sup>V</sup>*, *<sup>C</sup>*#, where *E* = (*<sup>V</sup>*, *<sup>C</sup>*), *F* = (*<sup>V</sup>*, *<sup>C</sup>*), and

$$\mathbb{C}^{\#} = \{ (\mathbf{x}, \mathbf{y}) \in V \times V : (\mathbf{x}, \mathbf{z}) \in \mathbb{C}, (\mathbf{z}, \mathbf{y}) \in \mathbb{C}', \text{ when } \mathbf{z} \in V \}. \tag{3}$$

**Lemma 1.** *NV is a semigroup that is isomorphic to BV.*

**Proof.** The operation "multiplication of networks" is the same as the operation "composition of binary relations". Since it is clearly an associative binary operation on *NV*, then *NV* is a semigroup under the operation "multiplication of networks". Also, the bijective map *ϕ* : *NV* → *BV* defined by *ϕ*(*E*) = *C* preserves multiplication. Thus, *ϕ* is a semigroup isomorphism and *NV* ≈ *BV*.

**Lemma 2.** *If* |*V*| = *n, then NV* ≈ *Bn.*

**Proof.** This follows from the facts that *NV* ≈ *BV* (Lemma 1) and *BV* ≈ *Bn* [8].

Thus, *Bn* is also a Boolean matrix representation of *NV*.

#### **5. Green's Equivalence Classifications of** *NV*

Let *θ* : *NV* → *Bn* be the isomorphism of Lemma 2 and *f* : *V* → *J* be an associated index bijection. If *αi*∗ is the *i*th Boolean row vector and *<sup>α</sup>*∗*j* is the *j*th Boolean column vector in the matrix *α* = *θ*(*E*) corresponding to network *E*, then *αi*∗ encodes the out-neighbors of node *f* −<sup>1</sup>(*i*) in *E* as the set

$$O\left(E; f^{-1}(i)\right) = \left\{f^{-1}(k) : a\_{ik} = 1, k \in J\right\}\tag{4}$$

and *<sup>α</sup>*∗*j* encodes the in-neighbors of node *f* −<sup>1</sup>(*j*) in *E* as the set

$$I\left(E; f^{-1}(j)\right) = \{f^{-1}(j) : \mathfrak{a}\_{kj} = 1, k \in I\}.\tag{5}$$

When *αi*∗ ∈ *r*(*α*) and *<sup>α</sup>*∗*j* ∈ *<sup>c</sup>*(*α*), then *Or<sup>E</sup>*; *f* −<sup>1</sup>(*i*) ≡ *<sup>O</sup><sup>E</sup>*; *f* −<sup>1</sup>(*i*) is a basis out-neighborhood and *Ic<sup>E</sup>*; *f* −<sup>1</sup>(*j*) ≡ *<sup>I</sup><sup>E</sup>*; *f* −<sup>1</sup>(*j*) is a basis in-neighborhood for network *E*. Thus, a basis neighborhood in *E* is a nonempty neighborhood in *E* which is not the set union of other neighborhoods in *E*.

Let *Or*(*E*) be the set of basis out-neighborhoods and *Ic*(*E*) be the set of basis in-neighborhoods in network *E*. Also, define *P*(*E*) as the set whose elements are ∅ and the sets generated by the closure under set union of the out-neighborhoods in *E* and let (*P*(*E*), ⊆) be the poset ordered by the set inclusion relation " ⊆ ". Thus, when *θ*(*E*) = *α*, it may be formally stated that:

**Lemma 3.** (*P*(*E*), ⊆) *is a lattice that is isomorphic to* (Γ(*α*), #)*.*

**Proof.** The proof for this Lemma is the same as that given as the proof of Lemma 3.3 in [3]. In what follows, (*P*(*E*), ⊆) will be referred to as the *Π lattice* for *E*.

The following major theorem provides complete *L*, *R*, *H*, and *D* equivalence classifications of all fixed-order networks:

**Theorem 1.** *Let E*, *F* ∈ *NV. Then*

*i. L*(*E*) = *L*(*F*) *if and only if Or*(*E*) = *Or*(*F*);


**Proof.** The proof of this result is the same as the proof of Theorem 3.4 in [3].

Thus, the Green's *L*, *R*, and *H* equivalence classifications of the networks in *NV* depend entirely upon their having (generally distinct) nodes with identical out-neighborhoods, identical in-neighborhoods, and both identical out-neighborhoods and in-neighborhoods, respectively, whereas the *D* equivalence classification of networks in *NV* depends entirely upon their having isomorphic *Π* lattices which are generated by their out-neighborhoods. As an illustration of this theorem the reader is invited to consult the simple example given in [3] which corresponds to the complete Green's equivalence classification of (and the associated "egg box" structure for) all order two networks.

#### **6. Green's Evolutions of Fixed-Order Networks**

For *E*, *F* ∈ *NV*, let *E* → *F* denote the evolution of a network during a time interval [*<sup>t</sup>*1, *<sup>t</sup>*2], where *E* is the initial network at *t*1 and *F* is the final network at *t*2 > *t*1. If *L*(*E*) = *L*(*F*)(*R*(*E*) = *R*(*F*))[*H*(*E*) = *H*(*F*)] {*D*(*E*) = *<sup>D</sup>*(*F*)}, then the evolution *E* → *F* is a *Green's L*(*R*)[*H*]{*D*} *evolution*. It is important to note that since *D* = *L*•*R* = *R*•*L* and *H* = *L* ∩ *R*, then *L* and *R* evolutions are also *D* evolutions, whereas *H* evolutions are both *L* and *R* evolutions, as well as *D* evolutions.

**Theorem 2.** *The following statements are true for network evolutions in NV:*


**Proof.** This is a direct and obvious consequence of the definitions of Green's evolutions and Theorem 1.

To illustrate this theorem, consider the order two networks *ψ* ≡ *<sup>V</sup>*, *<sup>C</sup>ψ* and *μ* ≡ *<sup>V</sup>*, *<sup>C</sup>μ* in the example in [3], where *V* = {*<sup>a</sup>*, *b*}, *<sup>C</sup>ψ* = {(*<sup>a</sup>*, *<sup>a</sup>*)}, and *Cμ* = {(*<sup>a</sup>*, *<sup>a</sup>*),(*b*, *<sup>a</sup>*)}. As can be seen from the associated Green's equivalence classification performed there, since *L*(*ψ*) = *<sup>L</sup>*(*μ*) and *D*(*ψ*) = *<sup>D</sup>*(*μ*), the evolution *ψ* → *μ* is both a Green's *L* evolution and a Green's *D* evolution. Theorem 2 (*i*) is satisfied, since, from Table 1 and the discussion in [3], it is also seen that *Or*(*ψ*) = {{*a*}} = *Or*(*μ*) and that the *Π* lattices are isomorphic undirected paths of length 1.

#### **7. The Green's Symmetry Problem**

In general, a symmetry associated with a "situation" is defined as an "immunity to change" for some aspect of the "situation". For a "situation" to have a symmetry: (a) the aspect of the "situation" remains unchanged when a change is performed; and (b) it must be possible to perform the change, although the change does not actually have to be performed [9].

Recall from Section 2 that for an *R*(*L*) evolution *E* → *F* in *NV*, there exists at least one *A* ∈ *NV* (*T* ∈ *NV*) such that *EA* = *F* (*TE* = *F*). Although *A*(*T*) does not have to be applied to *E*, it can produce the desired evolution when applied as a right (left) multiplication of *E*. In so doing, this multiplication not only preserves *Ic*(*E*) (*Or*(*E*)), but also *E*'s *Π* lattice structure. Thus, (a) and (b) above are satisfied and both *Ic*(*E*)(*Or*(*E*)) and the associated *Π* lattice structure can be considered as the invariant properties associated with the *symmetries A* (*T*) which produce the evolution. Symmetries such as *A* (*T*) are *Green's R*(*L*) *symmetries*.

The "*Green's symmetry problem*" is defined here as the determination of all symmetries that produce an evolution from an initial to a final network within an *R* or an *L* class such that each symmetry preserves the structural invariants required by Theorem 2. As will be discussed below, such symmetries encode information about which node neighborhoods in the initial network can be joined to form neighborhoods in the final network such that the structural invariants required by the evolution are preserved.

#### **8. Satisfiability and Computational Complexity of the Green's Symmetry Problem**

The Green's symmetry problem for an evolution is *m*− *satisfiable* if there are *m* symmetries which can produce the evolution.

#### **Theorem 3.** *The Green's symmetry problem is at least* 1− *satisfiable for both Green's R and L evolutions.*

**Proof.** Semigroup theory guarantees the existence of at least one Green's symmetry in *NV* that can produce a Green's *R* evolution and at least one Green's symmetry in *NV* that can produce a Green's *L* evolution.

#### *8.1. Green's R Evolutions*

The isomorphism established in Lemma 2 provides for computational solutions to the Green's symmetry problem. In particular, if *E* → *F* is a Green's *R* evolution, then, since *E* and *F* are known, the equation *EA* = *F* can be solved for *A* for each *i*, *j* ∈ *J* using the disjunctive normal form logical expression

$$\vee\_{k \in \mathcal{J}} \left( E\_{i\mathbf{k}} \wedge A\_{k\mathbf{j}} \right) = F\_{i\mathbf{j}\prime} \tag{6}$$

where use is now made of the Boolean matrix representations of *E*, *F*, and *A*. This expression for fixed *j* and all *i* ∈ *J* defines a system of |*J*| equations for node *j*.

This system of equations is *column-j satisfied* if there exists a column vector *<sup>A</sup>*∗*<sup>j</sup>* ∈ *Wn* for which (1) is a true statement for each *i* ∈ *J*. For each *j* ∈ *J*, let *<sup>G</sup>*∗*<sup>j</sup>* be the set of all *<sup>A</sup>*∗*<sup>j</sup>* for which the associated system of equations is satisfied and define *γ* ≡ <sup>∏</sup>*j*∈*J*--*<sup>G</sup>*∗*j*--. Clearly, if *γ* > 0, then *EA* = *F* is column-*j* satisfied for each *j* ∈ *J* and *the evolution E* → *F is γ-satisfiable*. Each instantiation of *A* is represented by a Boolean matrix in *Bn* which has an *x* ∈ *<sup>G</sup>*∗*<sup>j</sup>* as its *j*th column.

Let *Mi* = {*k* ∈ *J* : *Eik* = 1} index the unit valued entries in the row vector *Ei*∗ ∈ *Vn*.

**Lemma 4.** *Let Fij* = 0 *for some i*, *j* ∈ *J and Mi* = ∅*. If <sup>A</sup>*∗*<sup>j</sup>* ∈ *Wn column- j satisfies EA* = *F, then <sup>A</sup>*∗*<sup>j</sup> has Akj* = 0 *when k* ∈ *Mi.*

**Proof.** Assume for some *j* ∈ *J* that *<sup>A</sup>*∗*<sup>j</sup>* ∈ *Wn* column-*j* satisfies *EA* = *F*. If *Fij* = 0 and *Mi* = ∅ for some *i* ∈ *J*, then (1) is true and zero valued for *<sup>A</sup>*∗*<sup>j</sup>* and that *i* value, and the following implication chain is valid: <sup>∨</sup>*k*∈*<sup>J</sup> Eik* ∧ *Akj*! = 0 ⇒ <sup>∨</sup>*l*∈*J*−*Mi* <sup>0</sup> ∧ *Alj*! <sup>∨</sup>*k*∈*Mi* <sup>1</sup> ∧ *Akj*! = 0 ⇒ <sup>∨</sup>*k*∈*Mi* <sup>1</sup> ∧ *Akj*! = 0 ⇒ *Akj* = 0, *k* ∈ *Mi*. However, since *<sup>A</sup>*∗*<sup>j</sup>* ∈ *Wn* column-*j* satisfies *EA* = *F*, it must also satisfy (1) for all *k* ∈ *J* ⇒ *<sup>A</sup>*∗*<sup>j</sup>* has *Akj* = 0 when *k* ∈ *Mi*.

**Corollary 1.** *If E* = *ω, then <sup>A</sup>*∗*<sup>j</sup>* = **0**.

$$\mathbf{Proof}. \text{ } E = \omega \Rightarrow M\_{\mathbf{i}} = J \Rightarrow \vee\_{k \in \mathcal{J}} \left( 1 \land A\_{k\mathbf{j}} \right) = 0 \Rightarrow A\_{\mathbf{k}\mathbf{j}} = 0, k \in \mathcal{J} \Rightarrow A\_{\ast \mathbf{j}} = \mathbf{0}. \quad \square$$

The computational complexity CR of the Green's symmetry problem for Green's *R* evolutions is the number of remaining combinations of *Akj* ∈ {0, 1} values which must be checked for *EA* = *F* satisfiability after the *Akj* = 0 assignments specified by Lemma 4 have been made. Assume that *E* = *ω*, *z* and for each *j* ∈ *J* let *Q*(*j*) = *i* ∈ *J* : *Fij* = 0 index the zero valued Boolean equations of form (1).

$$\text{Theorem 4. } \mathcal{C}\_{\mathcal{R}} = \sum\_{j \in J} \left[ 2^{n - \left\lfloor \mathbb{J}\_{i \in Q(j)} \mid \mathcal{M}\_i \right\rfloor} \right].$$

**Proof.** For each *j* ∈ *J*, the set <sup>∪</sup>*i*∈*Q*(*j*)*Mi* (which can possibly be empty) indexes all row locations *k* ∈ *J* in *<sup>A</sup>*∗*<sup>j</sup>* for which *Akj* = 0 in every *<sup>A</sup>*∗*<sup>j</sup>* that column-*j* satisfies *EA* = *F*. The set *J* − <sup>∪</sup>*i*∈*Q*(*j*)*Mi* indexes all *k* ∈ *J* for which *Akj* must be evaluated to determine the column-*j* satisfiability of an associated *<sup>A</sup>*∗*j*. Since there are *Zj* = <sup>2</sup>*<sup>n</sup>*−|∪*<sup>i</sup>*∈*Q*(*j*) *Mi*| such evaluations for each *j* ∈ *J*, then for all *j* ∈ *J* there are a total of CR = ∑*j*∈*<sup>J</sup> Zj* evaluations required to determine all *<sup>A</sup>*∗*<sup>j</sup>* ∈ *Wn* which column-*j* satisfy *EA* = *F*.

#### *8.2. Green's L Evolutions*

If *E* → *F* is a Green's *L* evolution, then, since *TE* = *F*, it can be solved for *T* for each *i*, *j* ∈ *J* using the disjunctive normal form logical expression

$$\vee\_{k \in \mathcal{J}} \left( T\_{ik} \wedge E\_{kj} \right) = F\_{ij\prime} \tag{7}$$

which, for fixed *i* and all *j* ∈ *J*, defines a system of |*J*| equations for node *i*. This system is *row-i satisfied* if there exists a row vector *Ti*∗ ∈ *Vn* for which (2) is a true statement for each *j* ∈ *J*. For each *i* ∈ *J*, let *Hi*∗ be the set of all *Ti*∗ for which the associated system of equations is row-*i* satisfied and define *δ* ≡ ∏*<sup>i</sup>*∈*<sup>J</sup>*|*Hi*∗|. If *δ* > 0, then *TE* = *F* is row-*i* satisfied for each *i* ∈ *J* and *the evolution E* → *F is δ*− *satisfiable*. Each instantiation of *T* is represented by a Boolean matrix in *Bn* which has a *y* ∈ *Hi*∗ as its *i*th row.

Let *Kj* = *k* ∈ *J* : *Ekj* = 1 index the unit valued entrees in the column vector *<sup>E</sup>*∗*<sup>j</sup>* ∈ *Wn*.

**Lemma 5.** *Let Fij* = 0 *for some i*, *j* ∈ *J and Kj* = ∅*. If Ti*∗ ∈ *Vn row-i satisfies TE* = *F, then Ti*∗ *has Tik* = 0 *when k* ∈ *Kj.*

**Proof.** Assume for some *i* ∈ *J* that *Ti*∗ ∈ *Vn* row, *i* satisfies *TE* = *F*. If *Fij* = 0 for some *j* ∈ *J* and *Kj* = ∅, then (2) is true and zero valued for *Ti*∗ and that *j* value, and the following implication chain is valid: <sup>∨</sup>*k*∈*<sup>J</sup> Tik* ∧ *Ekj*! = 0 ⇒ <sup>∨</sup>*l*∈*J*−*Kj*(*Til* ∧ 0) <sup>∨</sup>*k*∈*Kj* (*Tik* ∧ 1) = 0 ⇒ <sup>∨</sup>*k*∈*Kj*(*Tik* ∧ 1) = 0 ⇒ *Tik* = 0, *k* ∈ *Kj*. However, since *Ti*∗ row-*i* satisfies *TE* = *F*, it must also satisfy (2) for all *j* ∈ *J* ⇒ *Ti*∗ has *Tik* = 0 when *k* ∈ *Kj*.

**Corollary 2.** *If E* = *ω, then Ti*∗ = 0*.*

$$\mathbf{Proof}. \quad E = \omega \Rightarrow K\_{\mathbf{j}} = J \Rightarrow \lor\_{k \in I} (T\_{i\mathbf{k}} \land 1) = 0 \Rightarrow T\_{i\mathbf{k}} = 0, \; k \in I \Rightarrow T\_{i\bullet} = 0. \quad \mathbf{D}$$

The computational complexity CL of the Green's symmetry problem for Green's *L* evolutions is the number of remaining combinations of *Tik* ∈ {0, 1} values which must be checked for *TE* = *F* satisfiability after the *Tik* = 0 assignments specified by Lemma 5 have been made. Assume that *E* = *ω*, *z* and for each *i* ∈ *J* let *Y*(*i*) = *j* ∈ *J* : *Fij* = 0 index the zero valued Boolean equations of form (2).

**Theorem 5.** CL = <sup>∑</sup>*<sup>i</sup>*∈*<sup>J</sup>*2*<sup>n</sup>*−|∪*j*∈*<sup>Y</sup>*(*i*) *Kj*| .

**Proof.** For each *i* ∈ *J*, the set <sup>∪</sup>*j*∈*<sup>Y</sup>*(*i*)*Kj* (which can possibly be empty) indexes all column locations *k* ∈ *J* for which *Tik* = 0 in every *Ti*∗ that row-*i* satisfies *TE* = *F*. The set *J* − <sup>∪</sup>*j*∈*<sup>Y</sup>*(*i*)*Kj* indexes all *k* ∈ *J* for which *Tik* must be evaluated to determine the row-*i* satisfiability of an associated *Ti*<sup>∗</sup>. Since there are *Zi* = <sup>2</sup>*<sup>n</sup>*−|∪*j*∈*<sup>Y</sup>*(*i*) *Kj*| such evaluations for each *i* ∈ *J*, then for all *i* ∈ *J* there are a total of CL = ∑*<sup>i</sup>*∈*<sup>J</sup> Zi* evaluations required in order to determine all *Ti*∗ ∈ *Vn* which row-*i* satisfy *TE* = *F*.

#### **9. Symmetries: Instantiations of Internal Dynamics**

Since Green's symmetries are themselves effectively elements of *Bn*, they correspond to special binary relations between network nodes that encode aspects of the internal dynamics of a Green's evolution *E* → *F* . In particular, they generally identify many-to-one correspondences between neighborhood sets in *E* that are joined by set union to produce a neighborhood in *F*. Each of these correspondences occurs in such a way as to preserve the structural invariants required by Theorem 2. These correspondences are the *internal dynamics* of the evolution.

Consider a Green's *R* evolution *E* → *F* where each symmetry *A* satisfies *EA* = *F* and is one instantiation of a possible set of symmetries which produce the evolution and preserve the required invariants. If *j* ∈ *J* is a column in *A* with a 1 in each of the rows in the set *Ψj* = {*<sup>i</sup>*1, *i*2, ··· , *ik*} and zeros in every other row location (i.e., there are --*Ψj*-- = *k* 1's and *n* − *k* 0's), then this column encodes an internal dynamic of the evolution where the in-neighborhoods of nodes *i*1, *i*2, ··· , *ik* in *E* are joined together as <sup>∪</sup>*i*∈*Ψj<sup>I</sup>*(*<sup>E</sup>*∗*<sup>i</sup>*) and associated with the in-neighborhood *<sup>I</sup><sup>F</sup>*∗*<sup>j</sup>* in *F* according to

$$\cup\_{i \in \Psi\_j^\circ} I(E\_{\*i}) \subseteq I(F\_{\*j}). \tag{8}$$

This expression is called a *Ψj internal R dynamic* of the evolution and the set *Ψj* is *the associated motion of the dynamic*. Clearly, for the special case where *Ψj* = {*i*},

$$I(E\_{\ast i}) = I(F\_{\ast i}).$$

If *E* → *F* is a Green's *L* evolution, a symmetry *T* which produces the invariant preserving evolution satisfies *TE* = *F*. If *i* is a row in *T* with a 1 in each of the column locations in *Φi* = {*j*1, *j*2, ··· , *jl*}, then this row encodes an internal dynamic of the evolution where the out-neighborhoods of nodes *j*1, *j*2, ··· , *jl* in network *E* are joined by set union and associated with the out-neighborhood *<sup>O</sup>*(*Fi*∗) in network *F* according to

$$\cup\_{j\in\Phi\_l} O(E\_{j\ast}) \subseteq O(F\_{l\ast}).\tag{9}$$

This expression is a *Φi* internal *L* dynamic of the evolution and the set *Φi* is the associated motion of the dynamic. When *Φi* = {*j*}, then

$$O(E\_{j\*}) = O(F\_{i\*})\,.$$

These notions will be clarified below using a simple example.

#### **10. Symmetry Ensembles, Propensities, and Energies**

Since the symmetry which produces a Green's evolution is not necessarily unique, it can be unclear as to how to assign a specific symmetry to an evolution. However, the collection of symmetries obtained from Green's symmetry problem, i.e., the *symmetry ensembles*, can be used to construct *propensities*. Propensities can be viewed as weighted symmetries which, in some sense, represent their respective ensembles.

Let *I*R (*<sup>I</sup>*L) = ∅ index the symmetries which are solutions to the Green's symmetry problem for some Green's *R*(*L*) evolution *E* → *F* . The sets

$$\mathcal{E}\_{\mathbb{R}} = \left\{ A^{(i)} : i \in I\_{\mathbb{R}\_{\prime}} \, \middle| \, EA^{(i)} = F \right\}$$

and

$$\mathcal{E}\_{\mathcal{L}} = \left\{ T^{(i)} : i \in I\_{\mathcal{L}}, T^{(i)}E = F \right\}$$

are the associated symmetry ensembles. The propensities associated with each ensemble are defined as

$$\overline{A} \equiv \left| I\_{\mathcal{R}} \right|^{-1} \sum\_{i \in I\_{\mathcal{R}}} A^{(i)}$$

and

$$\overline{T} = \left| I\_{\mathcal{L}} \right|^{-1} \sum\_{i \in I\_{\mathcal{L}}} T^{(i)}.$$

Thus, *<sup>A</sup>*∗*<sup>j</sup>* is a measure of the tendency of the nodes in column *j* in network *E* to form motions *Ψj* that associate in-neighborhoods in *E* with in-neighborhoods in network *F* according to the internal dynamic (3). Similarly, *Ti*∗ is a measure of the tendency of nodes in row *i* in *E* to form motions *Φi* that associate out-neighborhoods in *E* with out-neighborhoods in *F* according to the internal dynamic (4).

Propensities can be used to associate energies with both ensembles and specific symmetries. These energies quantify in a directly proportional manner the complexity level of the internal dynamical activity that is associated with an evolution. The *propensity energies* provide a representative measure of the "overall" complexity of internal dynamical activity for an evolution based upon ensemble propensity. The propensity energies for ensembles ER and EL are defined as

$$\mathfrak{e}\_{\mathbb{R}} \equiv \sum\_{i,j \in J} \overline{A}\_{i\bar{j}}$$

and

$$\mathfrak{e}\_{\mathcal{L}} \equiv \sum\_{\vec{a}, \vec{j} \in J} \overline{T}\_{\vec{i}\vec{j}\prime}$$

respectively.

The *energies of evolution* for the specific symmetries in an ensemble measure the complexity of internal dynamical activity for an evolution produced by a specific symmetry in an ensemble. In particular, if *A*(*k*) ∈ ER and *B*(*k*) ∈ EL, then the associated energies of evolution are defined as

$$\mathfrak{E}\_{\mathcal{R}}\left[\boldsymbol{A}^{(k)}\right] \equiv \sum\_{i,j \in \mathcal{J}} \mathcal{A}\_{ij}^{(k)} \overline{\mathcal{A}}\_{ij}$$

and

$$\mathfrak{E}\_{\mathcal{L}}\left[T^{(k)}\right] \equiv \sum\_{i,j \in J} T^{(k)}\_{ij} T\_{ij}.$$

The following Lemma guarantees that the energy of evolution for a symmetry never exceeds the propensity energy for the associated ensemble.

**Lemma 6.** *For any Green's R or L evolution,* E*x* ≥ <sup>E</sup>*x*[*y*]*, where y* = *A*(*k*) *or T*(*k*) *when x* = R *or* L .

**Proof.** A(k) ij , T(k) ij ∈ {0, 1} ⇒ Aij ≥ A(k) ij Aij, Tij ≥ T(k) ij Tij ⇒ ∑i,j∈<sup>J</sup> Aij ≥ ∑i,j∈<sup>J</sup> A(k) ij Aij, ∑i,j∈<sup>J</sup> Tij ≥ ∑i,j∈<sup>J</sup> T(k) ij Tij ⇒ ER ≥ <sup>E</sup>RA(k), EL ≥ <sup>E</sup>LT(k).

Recall that internal *R* and *L* dynamics are strictly defined by their motions. These motions also have energies that provide a measure of the level of internal dynamical activity induced by the motion. Since the symmetries *A* and *T* encode *R* and *L* internal dynamics with motions *Ψj* and *Φi*, respectively, then the associated *energies of motion* are the quantities

$$\mathfrak{E}\_{\mathcal{R}}\left[A;\Psi\_{\vec{j}}\right] \equiv \sum\_{i \in \Psi\_{\vec{j}}} A\_{ij} \overline{A}\_{i\vec{j}}$$

and

$$\mathfrak{E}\_{\mathcal{L}}[T; \Phi\_i] \equiv \sum\_{j \in \Phi\_i} T\_{ij} T\_{ij}.$$

The energies of motion are related to their energies of evolution by the following theorem: **Theorem 6 (Conservation of Energy of Evolution).** *The energy of evolution of a Green's symmetry is conserved by the energies of motion of its internal dynamics.*

**Proof.** Let *A* ∈ ER and set *M* index all the *Ψj* internal *R* dynamics encoded by *A*. Then ∑*j*∈*<sup>M</sup>* <sup>E</sup>R*<sup>A</sup>*; *Ψj* = ∑*j*∈*<sup>M</sup>* ∑*<sup>i</sup>*∈*Ψj AijAij* = ∑*<sup>i</sup>*,*j*∈*<sup>J</sup> AijAij* = <sup>E</sup>R[*A*], where use has been made of the fact that ∑*j*∈*<sup>M</sup>* ∑*<sup>i</sup>*∈*Ψj* is equivalent to ∑*<sup>i</sup>*,*j*∈*<sup>J</sup>* because *Aij* = 0 when *i* ∈ *J* − *Ψj* and *j* ∈ *J* − *M*. It is similar for the *L* dynamics.
