Prerequisite Mathematical Definitions

We consider functions $f:S_1\times \mathrm{\cdots }\times S_n\to T$, where the factor sets $S_i$ and the codomain $T$ carry no assumed structure unless otherwise noted. We write ${𝒂|}_{a_i=a_i^{\prime }}$ for the tuple identical to $𝒂$ except that the $i$-th component is replaced by $a_i^{\prime }$, and write

for the joint range of $(f,{\mathrm{pr}}_i)$ — the set of $(v,a_i)$ pairs realized by some $𝒂$. The auxiliary functions introduced in the conditions below are defined on ${𝒟}_i$ (or ${𝒟}_i\times S_i$).

We introduce three formulations of transition independence forming a hierarchy of decreasing generality: the transition-map condition requires no structure on $T$; the finite-difference condition requires that $(T,+)$ be a group, so that differences are defined, writing $x-y$ for $x+(-y)$; and the derivative condition requires that $S_i$ be an open interval of $ℝ$, that $T$ be a normed vector space, and that $f$ be differentiable in $s_i$, with the regularity supporting its equivalences stated in the summary below. Each formulation is stated at its natural level of generality.

The variable $s_i$ is transition-independent of the remaining variables ${𝒔}_{j\ne i}$ with respect to $f$ iff there exists a transition map $H_i:{𝒟}_i\times S_i\to T$ (necessarily unique on its declared domain) such that

The non-trivial content of the condition is that the remaining coordinates ${𝒂}_{j\ne i}$ are dropped in favor of the output $f(𝒂)$. No structure on $T$ is required: the condition is meaningful for functions valued in discrete spaces, ordered sets, manifolds, or any other codomain. For $H_i$ to be well-defined as a function on ${𝒟}_i\times S_i$, the following consistency condition is necessary and sufficient.

When $(T,+)$ is a group, the transition-map condition has an equivalent algebraic formulation: there exists $G_i:{𝒟}_i\times S_i\to T$ such that

The two formulations are related by $G_i(v,a_i,a_i^{\prime })=H_i(v,a_i,a_i^{\prime })-v$ and $H_i(v,a_i,a_i^{\prime })=G_i(v,a_i,a_i^{\prime })+v$.

Suppose $S_i$ is an open interval of $ℝ$ and $T$ is a normed vector space. When $f$ is differentiable in $s_i$, the derivative condition is: there exists $F_i:{𝒟}_i\to T$ such that

When (3) holds, the map $t\mapsto f({𝒂|}_{a_i=t})$ satisfies the initial value problem

whose solution determines $f({𝒂|}_{a_i=a_i^{\prime }})$ entirely from $(f(𝒂),a_i)$ provided the solution is unique.¹¹1Solution and uniqueness are taken, here and below, in the class specified in Appendix A. The ODE (4) is the bridge between the derivative condition and the transition map.

All integrals below are norm-Riemann integrals on compact intervals. Integrability of $g:[u,v]\to T$, for , means that all tagged Riemann sums converge in the norm of $T$, as their mesh tends to zero, to one element of $T$. Reversing the limits changes the sign, and equal limits give zero. If a coordinate trajectory is continuously differentiable, its derivative is integrable in this sense: embed $T$ isometrically in its Banach completion and apply the fundamental theorem of calculus there. The resulting integral is the difference of two trajectory values and hence lies in $T$, so the same Riemann sums converge in $T$.

The relationships among (1), (2), and (3) can be summarized as follows:

1. (a)

   Condition (1) is equivalent to output-determinacy (Remark 1). No structure on $T$ is required.

2. (b)

   If $(T,+)$ is a group, conditions (1) and (2) are equivalent via $H_i(v,a_i,a_i^{\prime })=G_i(v,a_i,a_i^{\prime })+v$ and $G_i(v,a_i,a_i^{\prime })=H_i(v,a_i,a_i^{\prime })-v$.

3. (c)

   Suppose $S_i$ is an open interval of $ℝ$, $T$ is a normed vector space, and $f$ is differentiable in $s_i$ throughout. Then (1) $\Rightarrow$ (3) holds, via $F_i(v,a_i):={{\partial }_{a_i^{\prime }}{H}_i(v,a_i,a_i^{\prime })|}_{a_i^{\prime }=a_i}$. The reverse direction (3) $\Rightarrow$ (1) holds provided the IVP (4) admits a unique solution from each initial datum $(v,a_i)\in {𝒟}_i$. Under these hypotheses the first two formulations are equivalent and imply the third; when in addition the IVPs (4) arising from (3) are uniquely solvable in the sense of Appendix A, the third returns the first, and the three formulations coincide. The trajectory $t\mapsto f({𝒂|}_{a_i=t})$ solves the initial value problem

   $$\frac{dy}{dt}=F_i(y,t),y(a_i)=f(𝒂).$$

   The identity below additionally assumes the transition-map condition (1) and integrability of the trajectory's derivative; continuous differentiability of the trajectory suffices for this integrability. The transition-map condition is ensured, in particular, by (3) together with the IVP uniqueness hypothesis above. The corresponding finite-difference map then satisfies

   $$G_i(f(𝒂),a_i,a_i^{\prime })={\int }_{a_i}^{a_i^{\prime }}{F}_i(f({𝒂|}_{a_i=t}),t)𝑑t.$$

   (5)

   Examples: the role of uniqueness. The derivative condition alone does not imply transition independence, even when every coordinate trajectory is smooth. Let $f(t,q)=q{t}^3$ on $\mathbb{R}\times \{0,1\}$, with $t$ the distinguished coordinate. Then ${\partial }_{t}f(t,q)=F(f(t,q),t)$ for

   $$F(v,t)=3|v{|}^{2/3}.$$

   But $f(0,0)=f(0,1)=0$, whereas $f(1,0)=0$ and $f(1,1)=1$. The two tuples share an output and an initial distinguished-coordinate value, but not their output after the same substitution.

   Conversely, uniqueness in Appendix A's solution class is sufficient but not necessary for transition independence. Let $f(t,c)=(t+c{)}^3$ on ${\mathbb{R}}^2$. A transition map is

   $$H(v,s,u)=(\sqrt[3]{v}+u-s{)}^3,$$

   where $s$ and $u$ are the initial and target values of the first coordinate. The induced ODE is $y^{\prime }=3|y{|}^{2/3}$. Here ${\mathcal{D}}_1={\mathbb{R}}^2$, and both $y(t)\equiv 0$ and $y(t)=t^3$ are solutions on $\mathbb{R}$ through $(v,t)=(0,0)$. Thus transition independence holds although uniqueness in the specified class fails.

   By default we will mean (1) when referring to ‘transition independence.’

Transition independence may be partial: $s_i$ might be transition-independent of some variables, with the others retained as its context, without being fully independent. We formalize this by partitioning $\{1,\mathrm{\dots },n\}$ as $\{i\}\cup J\cup K$, where $J$ is the context and $K=\{1,\mathrm{\dots },n\}\setminus (\{i\}\cup J)$ contains the remaining variables. Each of the three formulations of Section 1 extends to this conditional setting, the auxiliary maps $H_{i,J}$, $G_{i,J}$, $F_{i,J}$ now allowed to depend on the context values ${𝒂}_J$.

The variable $s_i$ is transition-independent of ${𝐬}_K$ given context ${𝐬}_J$ iff there exists a map $H_{i,J}:{𝒟}_i^J\times S_i\to T$ (denoted $H_{i,J}(f(𝒂),a_i,a_i^{\prime },{𝒂}_J)$, with the target $a_i^{\prime }$ written before the context ${𝒂}_J$) such that

where ${𝒟}_i^J≔\{(f(𝒂),a_i,{𝒂}_J):𝒂\in S_1\times \mathrm{\cdots }\times S_n\}$. Setting $J=\mathrm{\varnothing }$ recovers (1); setting $J=\{1,\mathrm{\dots },n\}\setminus \{i\}$ holds trivially.

When $(T,+)$ is a group, the condition has the equivalent algebraic form: there exists $G_{i,J}:{𝒟}_i^J\times S_i\to T$ such that

The two formulations are related by $G_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)=H_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)-v$ and $H_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)=G_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)+v$.

Suppose $S_i$ is an open interval of $ℝ$ and $T$ is a normed vector space. When $f$ is differentiable in $s_i$, a weaker condition is: there exists $F_{i,J}:{𝒟}_i^J\to T$ such that

When (8) holds, the map $t\mapsto f({𝒂|}_{a_i=t})$ satisfies the initial value problem

whose solution determines $f({𝒂|}_{a_i=a_i^{\prime }})$ entirely from $(f(𝒂),a_i,{𝒂}_J)$ provided the solution is unique.³³3In the class of Appendix A, with $(y(t),t,{𝒂}_J)\in {𝒟}_i^J$ in place of $(y(t),t)\in {𝒟}_i$. The ODE (9) is the bridge between the derivative condition and the transition map. By default we will mean (6) when referring to ‘transition context’, ’partial transition independence’, or ’transition independence given …’.

The relationships among (6), (7), and (8) parallel those of Section 1, with the context tuple ${𝒂}_J$ entering each auxiliary map as a passenger argument:

1. (a)

   Condition (6) is equivalent to output-determinacy on context $J$ (Remark 2). No structure on $T$ is required.

2. (b)

   If $(T,+)$ is a group, conditions (6) and (7) are equivalent via $H_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)=G_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)+v$ and $G_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)=H_{i,J}(v,a_i,a_i^{\prime },{𝒂}_J)-v$.

3. (c)

   Suppose $S_i$ is an open interval of $ℝ$, $T$ is a normed vector space, and $f$ is differentiable in $s_i$ throughout. Then (6) $\Rightarrow$ (8) holds. The reverse direction (8) $\Rightarrow$ (6) holds provided the IVP (9) admits a unique solution from each initial datum. Under these hypotheses the first two formulations are equivalent and imply the third; when in addition the IVPs (9) arising from (8) are uniquely solvable in the sense of Appendix A, the third returns the first, and the three formulations coincide. The trajectory $t\mapsto f({𝒂|}_{a_i=t})$ solves the initial value problem

   $$\frac{dy}{dt}=F_{i,J}(y,t,{𝒂}_J),y(a_i)=f(𝒂).$$

   The identity below additionally assumes the context transition-map condition (6) and integrability of the trajectory's derivative; continuous differentiability of the trajectory suffices for this integrability. The context transition-map condition is ensured, in particular, by (8) together with the IVP uniqueness hypothesis above for the fixed context. The corresponding finite-difference map then satisfies

   $$G_{i,J}(f(𝒂),a_i,a_i^{\prime },{𝒂}_J)={\int }_{a_i}^{a_i^{\prime }}{F}_{i,J}(f({𝒂|}_{a_i=t}),t,{𝒂}_J)𝑑t.$$

   (10)

From now on we work with the transition-map version (6).

Sections 1 and 2 describe the change in a variable by its initial and final values. This section describes it by a transform carrying the initial value to the final one. Fix a coordinate $i$, and let ${\mathrm{\Delta }}_i$ be a family of transforms of $S_i$: each ${\delta }_i\in {\mathrm{\Delta }}_i$ is a map ${\delta }_i:S_i\to S_i$, and we write ${\delta }_i\ast a_i$ for ${\delta }_i(a_i)$, so that the substitution of $a_i^{\prime }$ for $a_i$ is represented by a transform ${\delta }_i$ if $a_i^{\prime }={\delta }_i\ast a_i$. No structure on the family is assumed; in particular ${\mathrm{\Delta }}_i$ need not be closed under composition. We write

for the product of the range of $f$ with the family — the value and the transform do not constrain each other, since ${\delta }_i$ is chosen independently of $𝒂$. The auxiliary functions introduced in the conditions below are defined on ${ℰ}_i$: they consult the transform applied, not the initial value $a_i$ or the final value $a_i^{\prime }$.

The variable $s_i$ is transform-independent of the remaining variables ${𝒔}_{j\ne i}$ and of its own initial value with respect to $f$ and ${\mathrm{\Delta }}_i$ iff there exists a transform map $H_i^{\mathrm{\Delta }}:{ℰ}_i\to T$ (necessarily unique on its declared domain) such that

The non-trivial content of the condition is that the remaining coordinates ${𝒂}_{j\ne i}$ and the initial value $a_i$ are both dropped in favor of the output $f(𝒂)$ and the transform ${\delta }_i$ alone. For $H_i^{\mathrm{\Delta }}$ to be well-defined as a function on ${ℰ}_i$, the following consistency condition is necessary and sufficient.

When $(T,+)$ is a group, the transform-map condition has an equivalent algebraic formulation: there exists $G_i^{\mathrm{\Delta }}:{ℰ}_i\to T$ such that

The two formulations are related by $G_i^{\mathrm{\Delta }}(v,{\delta }_i)=H_i^{\mathrm{\Delta }}(v,{\delta }_i)-v$ and $H_i^{\mathrm{\Delta }}(v,{\delta }_i)=G_i^{\mathrm{\Delta }}(v,{\delta }_i)+v$. No derivative (instantaneous) formulation is included; the two formulations above are the only ones we use.

The relationships between (11) and (12) parallel items (a) and (b) of Section 1, with the pair $(f(𝒂),{\delta }_i)$ replacing $(f(𝒂),a_i,a_i^{\prime })$ throughout:

1. (a)

   Condition (11) is equivalent to transform output-determinacy (Remark 4). No structure on $T$ is required.

2. (b)

   If $(T,+)$ is a group, conditions (11) and (12) are equivalent via $H_i^{\mathrm{\Delta }}(v,{\delta }_i)=G_i^{\mathrm{\Delta }}(v,{\delta }_i)+v$ and $G_i^{\mathrm{\Delta }}(v,{\delta }_i)=H_i^{\mathrm{\Delta }}(v,{\delta }_i)-v$.