Partially Specified Model (PS-model)

• PS-model is a partially specified Boolean network where unknown logical sub-expressions are modeled by uninterpreted functions, preserving all admissible completions.
• It represents the union of asynchronous dynamics by encoding completions as input valuations, allowing analysis of attractors and phenotype stabilization.
• The model supports perturbation-based control strategies with robustness metrics to evaluate intervention efficacy under biological uncertainty.

In "Robust Control of Partially Specified Boolean Networks," the Partially Specified Model (PS-model) denotes a partially specified Boolean network for regulatory networks in which some parts of the Boolean update logic are left unknown, but constrained to be fixed Boolean functions (Brim et al., 2022). The formalism is motivated by biological uncertainty: the wiring of a regulatory network may be known, while exact logical update functions are only partially established from data or literature. Rather than selecting one guessed completion, the PS-model represents all admissible fully specified networks at once, and interprets dynamics, attractors, and control over that entire uncertainty class (Brim et al., 2022).

1. Definition and formal setting

The underlying object is a Boolean network with inputs. Let

$\mathbb{B}=\{0,1\}, \qquad \mathbb{B}_*=\{0,1,*\},$

where $*$ denotes a free value. A Boolean network has $n$ Boolean variables and $m$ Boolean inputs, with states $s\in\mathbb{B}^n$, input valuations $c\in\mathbb{B}^m$, and update functions

$$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$

The semantics are asynchronous: each transition updates exactly one variable (Brim et al., 2022).

A PS-model is obtained by allowing update functions to contain uninterpreted function symbols. The paper assumes symbols $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$, where $a_i$ is the arity, and each update function $b'_i$ is a Boolean expression that may contain such symbols. What remains unspecified is therefore not the whole network structure, but selected sub-expressions inside the Boolean update rules. Each uninterpreted symbol denotes an unknown but fixed truth table, so one completion corresponds to choosing concrete Boolean functions for all such symbols (Brim et al., 2022).

This distinguishes the PS-model from ad hoc model completion. The model does not collapse uncertainty into a single guessed network. Instead, it preserves the full admissible family of Boolean regulatory mechanisms consistent with current knowledge. This suggests a precise uncertainty formalism for systems biology in which structural knowledge and logical uncertainty are represented separately.

2. Completions, colours, and asynchronous dynamics

The paper operationalizes completions by translating the partially specified Boolean network into a Boolean network with inputs. Each uninterpreted function $*$0 is encoded by $*$1 fresh Boolean inputs, one per truth-table row. The key expansion rule is

$*$2

where $*$3 and $*$4 are fresh uninterpreted functions of arity $*$5. Repeated application yields a standard Boolean network with inputs (Brim et al., 2022).

A completion or instantiation is then equivalently an input valuation $*$6, that is, one assignment to all fresh input bits encoding all unknown truth-table entries. The number of such completions is exponential in the arity of the uninterpreted functions and, in the worst case, doubly exponential in the number of inputs of update functions (Brim et al., 2022).

The induced coloured state-transition graph is

$*$7

with colours $*$8, vertices $*$9, and edges

$n$0

If a state has no outgoing edge under a colour, a self-loop $n$1 is added so the graph is total. The notation $n$2 denotes $n$3, and $n$4 denotes membership in the reflexive-transitive closure $n$5 (Brim et al., 2022).

The dynamics of a PS-model are interpreted as the union of the dynamics of all completions. Concretely, $n$6 represents a family of ordinary asynchronous transition graphs over the same vertex set, one graph per colour $n$7. The model therefore stores every transition relation arising from every admissible completion, rather than a single transition relation. This suggests that uncertainty is treated as an explicit semantic dimension rather than as external noise.

3. Fair runs, attractors, and phenotype interpretation

For any coloured graph, a run under colour $n$8 is a maximal sequence $n$9 with $m$0. Because of totality, runs are infinite. The set $m$1 contains states appearing infinitely often in run $m$2. As in asynchronous biological models, the paper considers only fair runs, meaning enabled transitions cannot be postponed indefinitely. $m$3 denotes the set of all fair runs under colour $m$4 starting from $m$5 (Brim et al., 2022).

Attractors are defined per completion. For $m$6, an attractor $m$7 is “the smallest set such that for every $m$8, if $m$9 for some $s\in\mathbb{B}^n$0, then $s\in\mathbb{B}^n$1.” Equivalently, attractors are minimal closed sets of the asynchronous transition graph for that completion. Every fair run $s\in\mathbb{B}^n$2 satisfies that $s\in\mathbb{B}^n$3 is some attractor (Brim et al., 2022).

Biologically, different attractors correspond to phenotypes. The paper studies source-target attractor control in the partially specified setting, where the target is represented by a target state $s\in\mathbb{B}^n$4 assumed to belong to a relevant attractor. Stabilization means that after perturbation, the long-run behavior of every fair run is confined to an attractor containing the target state, captured by the condition

$s\in\mathbb{B}^n$5

The PS-model is therefore not merely a representational device for uncertainty; it is a control-theoretic model in which phenotype stabilization is evaluated completion by completion.

4. Perturbation-based control in the partially specified setting

Control is defined by perturbing variable values. A perturbation is

$s\in\mathbb{B}^n$6

where $s\in\mathbb{B}^n$7 means variable $s\in\mathbb{B}^n$8 is unperturbed, and $s\in\mathbb{B}^n$9 means it is fixed to that value. Applying perturbation $c\in\mathbb{B}^m$0 to state $c\in\mathbb{B}^m$1 gives $c\in\mathbb{B}^m$2, defined by

$c\in\mathbb{B}^m$3

The notation $c\in\mathbb{B}^m$4 denotes fair runs under colour $c\in\mathbb{B}^m$5 starting in $c\in\mathbb{B}^m$6, but restricted to edges that update only unperturbed variables in $c\in\mathbb{B}^m$7; if no such transition exists, a self-loop is assumed. During such runs, the perturbed variables are frozen (Brim et al., 2022).

The paper studies three source-target control modes for fixed $c\in\mathbb{B}^m$8, $c\in\mathbb{B}^m$9, and completion $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$0:

Control type	Defining condition	Operational meaning
One-step control	$$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$1	Perturb once, then release all variables
Permanent control	$$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$2	Keep perturbed variables fixed forever
Temporary control	$$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$3	Hold variables fixed for a finite time, then release

In the partially specified setting, a control is not required to succeed for all completions. Instead, for each control type $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$4, the paper computes a relation

$$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$5

where $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$6 means perturbation $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$7 controls the network under completion $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$8 (Brim et al., 2022).

This is a defining feature of the PS-model. Success is judged per completion, and only then aggregated quantitatively. A plausible implication is that the PS-model treats control synthesis as a family-of-models problem rather than as a single-model verification problem.

5. Robustness, perturbation size, and quantitative assessment

The main quantitative notion is robustness, defined as the fraction of admissible completions for which a perturbation succeeds: $$b_i:\mathbb{B}^n\times \mathbb{B}^m \to \mathbb{B}.$$9 The denominator excludes completions in which the target is not part of any attractor (Brim et al., 2022).

Perturbations are also evaluated by size, namely the number of perturbed variables. The paper explicitly aims for controls that are minimal in size because biological interventions are costly and difficult. For sets of perturbations $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$0, two aggregate measures are introduced: $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$1 and

$f_1^{(a_1)}, f_2^{(a_2)}, \ldots$2

Here $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$3 is the best single perturbation’s robustness, while $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$4 measures coverage of the uncertain model space by the whole set (Brim et al., 2022).

Experimentally, the paper reports a consistent pattern: one-step perturbations tend to be substantially less robust than temporary or permanent ones, while temporary and permanent are often very similar. Temporary control sometimes achieves better union robustness because some perturbations work only when eventually released, and permanent fixation can destroy the target attractor (Brim et al., 2022).

These metrics give the PS-model a distinct practical interpretation. A perturbation with high robustness means that, if the real biological system is one of the admissible completions, the perturbation is likely to remain valid across that uncertainty class. High $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$5 and high $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$6 therefore quantify different intervention strategies under model uncertainty.

6. Symbolic representation and control algorithms

The paper addresses the combined explosion over states, perturbations, and completions using a fully symbolic BDD-based representation. The unspecified parts of the model are encoded as input valuations, and perturbations are encoded separately. Instead of representing perturbations directly as elements of $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$7, the symbolic domain uses

$f_1^{(a_1)}, f_2^{(a_2)}, \ldots$8

where $f_1^{(a_1)}, f_2^{(a_2)}, \ldots$9 means variable $a_i$0 is perturbed, while the actual perturbed value is stored in the state component $a_i$1. This yields the extended symbolic domain

$a_i$2

A pair $a_i$3 is equivalent to a perturbation $a_i$4, written $a_i$5, if $a_i$6 when $a_i$7, and $a_i$8 when $a_i$9 (Brim et al., 2022).

Symbolic predecessor and successor operators are defined over $b'_i$0: $b'_i$1

$b'_i$2

These mirror asynchronous transitions while forbidding updates of perturbed variables (Brim et al., 2022).

Using BDD operations, the paper performs symbolic fixpoint computations with:

• Bwd for backward reachability,
• Trap for trap-set computation,
• CanPerturb(source, targets) for filtering symbolic states to those reachable from the source by a single perturbation step.

The control algorithms are then built bottom-up. Permanent control computes the symbolic basin around the target under perturbation-aware dynamics and filters it by $b'_i$3. One-step control computes the basin with perturbations disabled and enables perturbations only when checking perturbability from the source. Temporary control combines both ideas: it computes the unperturbed basin of the target, then expands symbolically under perturbation-enabled dynamics to states that can be held temporarily and eventually released into the target basin (Brim et al., 2022).

The methodological point is explicit: the paper does not build a separate perturbed graph for each perturbation or each completion. Instead, it represents all perturbations and all completions in one transition-labelled symbolic graph. This suggests that the PS-model is inseparable from the symbolic representation used to make it computationally tractable.

7. Empirical behavior and significance in systems biology

The paper evaluates biological models including Myeloid, Cardiac, ERBB, Tumour, and MAPK, where partially specified versions were created by replacing some update functions with uninterpreted functions of the same arity (Brim et al., 2022). Reported model sizes reach up to $b'_i$4 states and very large colour spaces. Runtime grows much slower than the number of colours: for the tumour model, increasing relevant colours from $b'_i$5 to $b'_i$6, a $b'_i$7 increase, caused only about a $b'_i$8–$b'_i$9 runtime increase (Brim et al., 2022).

The paper also compares the fully symbolic approach with the authors’ earlier semi-symbolic method, reporting that the fully symbolic approach especially improves tractability for larger state spaces, even when semi-symbolic methods fail for memory reasons (Brim et al., 2022). This is significant because the PS-model deliberately retains all admissible completions, which would otherwise induce severe state explosion.

Practically, the PS-model is useful because it captures a common biological situation: some regulatory influences are known, but the precise Boolean rule combining them is not. Unknown logical subrules are treated as arbitrary but fixed functions, so each completion is a plausible fully specified regulatory mechanism consistent with current knowledge (Brim et al., 2022). The paper’s overall formulation can therefore be summarized as follows: unknown update-function fragments are represented by uninterpreted functions; all completions are encoded as input valuations or colours; asynchronous fair dynamics and attractors are defined per completion; control is solved over the joint space of states, completions, and perturbations; and robustness measures how much of the completion space a given intervention covers (Brim et al., 2022).

In that sense, the PS-model is a mathematically precise uncertainty model for Boolean regulatory networks, designed not only to represent incomplete knowledge but also to support source-target attractor control under biological uncertainty.