Behavioral Mereology: A Modal Logic for Passing Constraints

Luckily, we won’t need to settle on what precisely a behavior type is, so long as we can reason about it logically. For this, we need the behavior type of our system and its parts to be objects in a topos; then, we can use the internal logic of the topos to reason about our behavior types. A topos can be understood as a system of variable sets; in our case, this allows us the freedom to have sets varying in time, in space, or according to different observers. We will just work in the topos of sets and functions, leaving it to the experts to extend the theory to arbitrary toposes.

So, a behavior type is simply a set whose elements are regarded as possible behaviors of a system. We can think of it as the “phase space” of our system, in a general sense. This terminology is inspired by Willem’s behavioral approach to control theory [WP13], which describes a dynamical system as a subset $B\subseteq W^{T}$ of lawful trajectories parameterized by time $T$ in some value space $W$. The set $B$ is the behavior type of this system, where a behavior of the system is simply a lawful trajectory. Many of our examples follow this general form.

If we know a whole system $S$ (say, my body) is behaving like $b$, then we also know how any part $P$ of $S$ (say, my hand) is behaving: we just look at what $P$ is doing while $S$ does $b$. In other words, there should be a restriction function, which we denote $\big{|}_{P}\colon B_{S}\to B_{P}$, from the behavior type of the whole system to the behavior type of the part.

Moreover, every behavior of a part $P$ will arise from some behavior of the whole system: how could a part of the system do something if the system as a whole had no behaviors in which $P$ was doing that thing? Remember, we are considering the part $P$ as a part of the system $S$, not on its own: my hand, not a severed hand. If we sever $P$ from the system $S$, it may be able to behave in ways that have no extension to $S$. But as a part of the system $S$, every behavior of the $P$ must be restricted from a behavior of $S$. We will give examples below, but first a definition.

For example, suppose that a system $S$ is divided into a plant $P$ and a controller $C$. For example, $B_{S}$ might be $\{\alpha:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}^{p+c}\mid\operatorname{\mathcal{L}}(\alpha)\}$ a set of $p+c$ real variables satisfying a dynamical law $f$, the first $p$ of which concern the plant $P$ and the last $c$ of which concern the controller $C$. Then $B_{P}$ would be $\{\rho:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}^{p}\mid\exists\gamma:\operatorname{\mathbb{R}}\to\operatorname{\mathbb{R}}^{c}.\,\operatorname{\mathcal{L}}(\rho,\gamma)\}$ of $p$ real variables for which there is some extension of $c$ variables (the behavior of the controller) which is valid according to the dynamical law. The projection map $\operatorname{\mathbb{R}}^{p+c}\to\operatorname{\mathbb{R}}^{p}$ gives a surjection from $B_{S}\twoheadleftarrow B_{P}$, witnessing that the plant is a part of the whole system.

In practice, we may have certain parts of $S$ in mind, and so we may consider a sublattice of that defined in Definition 2.

A surjection $\big{|}_{P}\colon B_{S}\to B_{P}$ out of a set may equally be presented by its kernel pair, the equivalence relation on $B_{S}$ where $b\sim_{P}b^{\prime}$ iff $b\big{|}_{P}=b^{\prime}\big{|}_{P}$. We may call this relation observational equivalence; the behaviors $b,b^{\prime}$ with $b\sim_{P}b^{\prime}$ are observationally equivalent relative to $P$. This is clearest when thinking of $P$ as some measuring device in a larger system; two behaviors of the whole system are observationally equivalent relative to our measuring device when it measures them to be the same. Two behaviors of my body are hand-equivalent if they are indistinguishable by looking at my hand; and two times are clock-equivalent when they read the same on the clock face.

There is essentially (i.e. up to isomorphism) no distinciton between a quotient of $B_{S}$ and an equivalence relation on $B_{S}$. Thus we are defining parts of $S$ to be equivalence relations on $S$-behaviors. This seems to be a novel approach to mereology, though we cannot claim to know the literature well enough to be sure.

Now we turn our attention to how behaviors of various parts of the system relate to one another. The most basic relation between behaviors of two parts is that of being simultaneously realizable by a behavior of the whole system. We call this relation compatibility.

In other words, two behaviors (one on each of two parts) are compatible if there is a behavior of the whole system that restricts to both of them.

We can express some of the parts-lattice operations in terms of the compatibility relation.

In general, we will be more interested in joins than in meets because joins are easier to work with (being subsets of a product, rather than quotients of a disjoint union by a non-transitive relation).

This is a much stronger notion than compatibility, and we shall show in Section 11 that it can be used to recover the original order relation $\geq$ on parts.

We will identify a constraint $\phi$ on a part $P$ with the predicate “satisfies $\phi$” on behaviors $B_{P}$ of $P$. In other words, we have the following definition.

Let $\operatorname{\textbf{Prop}}$ be the two element set $\{\mathtt{true},\mathtt{false}\}$ of truth values. We think of functions $\phi:X\to\operatorname{\textbf{Prop}}$ as predicates concerning the elements of $X$ — applied to $x\in X$, $\phi$ gives a truth value $\phi(x)$ which says whether or not $x$ satisfies the predicate $\phi$.

For parts $P\geq Q$, we get an adjoint triple that allows us to transform constraints on $P$ to those on $Q$, and vice versa, given by the logical quantifiers:

These functors are defined logically as follows:

	$\displaystyle\exists^{P}_{Q}\phi(q)$	$\displaystyle:=\exists p\in B_{P}.\,\big{(}(p\big{|}_{Q}=q)\operatorname{\wedge}\phi(p)\big{)}$
	$\displaystyle\Delta^{Q}_{P}\psi(p)$	$\displaystyle:=\psi(p\big{|}_{Q})$
	$\displaystyle\forall^{P}_{Q}\phi(q)$	$\displaystyle:=\forall p\in B_{P}.\,\big{(}(p\big{|}_{Q}=q)\Rightarrow\phi(p)\big{)}$

The fact that they are adjoint means that

	$\displaystyle\exists^{P}_{Q}\phi\vdash\psi\quad$	$\displaystyle\iff\quad\phi\vdash\Delta^{Q}_{P}\psi$
	$\displaystyle\Delta^{Q}_{P}\psi\vdash\xi\quad$	$\displaystyle\iff\quad\psi\vdash\forall^{P}_{Q}\xi$

We will write $\exists_{P}$, $\Delta^{P}$, and $\forall_{P}$ for $\exists^{S}_{P}$, $\Delta^{P}_{S}$ and $\forall^{S}_{P}$ respectively. As mentioned, these operations are functorial, meaning that $\exists^{P}_{P}(\phi)=\Delta^{P}_{P}(\phi)=\forall^{P}_{P}(\phi)=\phi$ and for $R\leq Q\leq P$,

	$\displaystyle\exists^{Q}_{R}\exists^{P}_{Q}$	$\displaystyle=\exists^{P}_{R},$
	$\displaystyle\Delta^{R}_{Q}\Delta^{Q}_{P}$	$\displaystyle=\Delta^{R}_{P},$
	$\displaystyle\forall^{Q}_{R}\forall^{P}_{Q}$	$\displaystyle=\forall^{P}_{R}.$

Thinking again of $P$ as a way to observe behaviors, $\Delta^{P}\exists_{P}\phi$ is the set of system behaviors $s$ that our observer says plausibly satisfy $\phi$: there is something $P$-equivalent to $s$ that satisfies $\phi$. And $\Delta^{P}\forall_{P}\phi$ is the set of system behaviors that our observer can guarantee satisfy $\phi$.

Now we turn to the question of how constraints on the behavior of some part of the system constrain the behavior of other parts. We discuss two ways to pass constraints between parts.

A behavior $q$ of $Q$ allows a constraint $\phi$ on $P$ if $Q$ can be doing $q$ while $P$ is satisfying $\phi$; we write this as $\Diamond_{Q}^{P}\phi(q)$. A behavior $q$ of $Q$ ensures $\phi$ on $P$ if whenever $Q$ does $q$, $P$ must satisfy $\phi$; we write this as $\Box_{Q}^{P}\phi(q)$.

These symbols are chosen due to their relation to the usual modalities of possibility ($\Diamond$) and necessity ($\Box$) [Kri63]; a behavior $q$ allows $\phi$ if it is possible that $P$ satisfies $\phi$ while $Q$ does $q$, and a behavior $q$ ensures $\phi$ if it is necessary that $P$ satisfies $\phi$ while $Q$ does $q$. Indeed, in the case that the accessibility relation in the Kripke frame is an equivalence relation, we will be able to recover the usual possibility and necessity modalities from our allowance and ensurance operators (see Section 3.3).

Note that compatibility and determination appear as particular, pointwise cases of the allowance and ensurance operators. For any $p\in B_{P}$ and $q\in B_{Q}$, we have

	$\displaystyle\mathfrak{c}(p,q)$	$\displaystyle=\Diamond_{Q}^{P}(=p)(q)=\Diamond_{P}^{Q}(=q)(p)$
	$\displaystyle\mathfrak{d}(p,q)$	$\displaystyle=\Box_{P}^{Q}(=q)(p)$

We write $(=p)$ for the map $B_{P}\to\operatorname{\textbf{Prop}}$ that sends $p^{\prime}$ to $\mathtt{true}$ if and only if $p=p^{\prime}$.

Assuming the law of excluded middle, our operators are inter-definable by conjugating with negation.

Note that Proposition 17 does not generalize to arbitrary toposes, where the variation of the sets (in time or in space) means that one must reason constructively in general.

Finally, we describe the manner in which our intermodalities generalize the classical alethic modalities of possibility and necessity.

Fix a part $P$. Then for any part $Q$, we obtain two modalities on $P$ by composing our intermodalities from $P$ to $Q$ with their corresponding intermodality from $Q$ to $P$.

These modalities describe constraints on $P$ as seen through the part $Q$. To obtain a description of possibility and necessity, assume that $B_{S}$ is inhabted — that there is some behavior of the system. We let $Q=\bot$ be the system whose behavior type $B_{Q}=\ast$ consists of just a single element. Then the adjoint modalities

$$\Diamond_{P}^{\bot}\Diamond_{\bot}^{P}\qquad\textrm{left adjoint to}\qquad\Box_{P}^{\bot}\Box_{\bot}^{P}$$

describe possibility and necessity on $B_{P}$.

For example, for any constraint $\varphi$ on $B_{P}$, the constraint $\Diamond_{P}^{\bot}\Diamond_{\bot}^{P}(\varphi)$ maps all elements of $B_{P}$ to $\mathtt{true}$ if there is some behavior $p$ that satisfies $\varphi$, and maps all elements to $\mathtt{false}$ otherwise. Thus the modality detects whether $\varphi$ is possible: that is, whether there is some behavior that satisfies $\varphi$.

On the other hand, $\Box_{P}^{\bot}\Box_{\bot}^{P}(\varphi)$ is the constraint that maps all elements of $B_{P}$ to $\mathtt{true}$ if all behaviors $p\in B_{P}$ satisfy $\varphi$, and maps all elements to $\mathtt{false}$ otherwise; thus this modality detects whether $\varphi$ is always satisfied, and hence necessary.

More generally, the usual semantics of the “it is possible that” and “it is necessary that” modalities $\Diamond$ and $\Box$ take place in a Kripke frame $(W,A)$, where $W$ is a set, known as the set of worlds, and $A$ is a binary relation on $W$ known as the accessibility relation. The predicate then $\Diamond(\varphi)(w)$ holds if $\varphi(w^{\prime})$ for some $w^{\prime}$ such that $wAw^{\prime}$, and $\Box(\varphi)(w)$ holds if $\varphi(w^{\prime})$ for all $w^{\prime}$ such that $wAw^{\prime}$ [Kri63]. If $A$ is an equivalence relation, then we may equivalently describe $A$ by an epi $W\twoheadrightarrow W/A$. In this case, we have $\Diamond=\Diamond_{W}^{W/A}\Diamond_{W/A}^{W}$ and $\Box=\Box_{W}^{W/A}\Box_{W/A}^{W}$ as modalities on $\operatorname{\textbf{Prop}}^{W}$.