Complexity Bounds for the Controllability of Temporal Networks with Conditions, Disjunctions, and Uncertainty

To show WC-TCSPU is $\Pi_{2}^{P}$-hard, we will provide a reduction from $\forall\exists$3SAT. In other words, we want to construct a TCSPU $T$ such that a formula $\forall\vec{y},\exists\vec{x}:\phi(\vec{x},\vec{y})$ is weakly controllable if and only if $T$ is weakly controllable, where $\vec{x},\vec{y}$ are vectors of boolean values, and $\phi$ is a 3-CNF formula.

We start by defining our timepoints, starting with a reference timepoint $Z$. For each $x_{i}$, we construct timepoint $t_{x_{i}}$ with disjunctive constraint $t_{x_{i}}-Z\in[0,0]\cup[1,1]$. For each $y_{j}$, we also construct timepoint $t_{y_{j}}$ with contingent constraint $t_{y_{j}}-Z\in[0,0]\cup[1,1]$. These timepoints will represent the initial values chosen against which we will evaluate $\phi$ with 0 corresponding to an assignment of false and 1 corresponding to true.

For convenience, we also add timepoints corresponding to the negations of each variable. $t_{\overline{x}_{i}}$ has two corresponding constraints, $t_{\overline{x}_{i}}-Z\in[0,0]\cup[1,1]$ and $t_{\overline{x}_{i}}-t_{x_{i}}\in[-1,-1]\cup[1,1]$. This ensures that $t_{\overline{x}_{i}}$ takes on a different value than $t_{x_{i}}$. Similarly, we add new timepoints $t_{\overline{y}_{j}}$ with requirement links $t_{\overline{y}_{j}}-Z\in[0,0]\cup[1,1]$ and $t_{\overline{y}_{j}}-t_{y_{j}}\in[-1,-1]\cup[1,1]$. We will rely on the fact that we are evaluating weak controllability to ensure that we set the timepoints for the negated variables in response to the values assigned by nature.

We now move on to encoding each individual clause of $\phi$ into our TCSPU $T$. Our approach is going to be highly inspired by the reduction from 3SAT to the 3-coloring problem on graphs and the reduction from 3-coloring to computing feasibility of a TCSP [6]. We will emulate the three colors by requiring all timepoints to occur at time 0, 1, or 2 and enforce that two nodes $t_{i},t_{j}$ differ in value by requiring that $t_{i}-t_{j}\in\{-2,-1,1,2\}$.

For each clause $c_{k}$ of $\psi$, we create a new gadget whose output represents the truth value of $c_{k}$ (see Figure 3). Each timepoint $G_{k,l}$ represents the truth value of literal $l$ of clause $c_{k}$. We require that the value matches the initially assigned value of literal $q$ by adding the constraint $G_{k,l}-t_{q}=0$. The layout of timepoints $A_{k}$ weakly emulate an or-gate, where $A_{k,6}$ is the output and constrained to have a value of 1. For any values of the timepoints $G_{k}$, it is possible to assign all of the timepoints $A_{k}$ such that $A_{k,6}=1$ except for when $G_{k,1}=G_{k,2}=G_{k,3}=0$. As a result, it is possible to choose a set of values for the timepoints to satisfy the constraints of the gadget so long as at least one literal of the original clause $c_{k}$ is true.

Taken together, if there exists an assignment of values to timepoints such that each gadget’s constraints are satisfied, then for whichever particular $\vec{y}$ we start with, then $\exists\vec{x}:\phi(\vec{x},\vec{y})$. When checking weak controllability, all executable timepoints are assigned values after the contingent timepoints, so as we have constructed it, $T$ is weakly controllable if and only if $\forall\vec{y},\exists\vec{x}:\phi(\vec{x},\vec{y})$. Thus, WC-TCSPU is $\Pi_{2}^{P}$-hard. ∎

To show that DC-TCSPU is PSPACE-hard, we provide a reduction from TQBF, which is known to be PSPACE-complete, to DC-TCSPU. In particular, for a problem of the form $\exists x_{1}\forall y_{1}...\exists x_{n}\forall y_{n}:\phi(\vec{x},\vec{y})$, we can construct a TCSPU $T$ such that $T$ is dynamically controllable if and only if $\exists x_{1}\forall y_{1}...\exists x_{n}\forall y_{n}:\phi(\vec{x},\vec{y})$, where $\phi$ is a 3-CNF formula.

Ideally, we would employ a strategy similar to our transformation for WC-TCSPU in Lemma 1, but in that construction, many of the clausal gadget timepoints can occur before the contingent timepoints they relate to are assigned by nature. Because dynamic controllability requires timepoints to be assigned reactively in a just-in-time manner, we must make sure that all values of $\vec{y}$ are encoded and specified by the network before we do any subsequent computation.

We start by encoding the alternating choice of $x_{i}$ and $y_{i}$ as represented by the values decided by the scheduler and nature. We start with an anchor point $O$ and for each $x_{i}$ and $y_{i}$, we create timepoints $\tau_{x_{i},s}$, $\tau_{x_{i},e}$, $\tau_{y_{i},s}$, and $\tau_{y_{i},e}$. For each $x_{i}$, we create a requirement constraint of $\tau_{x_{i},e}-\tau_{x_{i},s}\in[0,0]\cup[1,1]$, and for each $y_{i}$, we create a contingent constraint of $\tau_{y_{i},e}-\tau_{y_{i},s}\in[0,0]\cup[1,1]$. This enforces that the difference between the start and end values is either 0 or 1, corresponding to an assignment of false or true in the original formula. To ensure that the values are chosen in order when evaluated in a dynamic controllability setting, we require that $\tau_{x_{i},s}-O=2i-2$ and that $\tau_{y_{i},s}-O=2i-1$. This gives us the exact alternating pattern as described by the original formula, and what remains is to evaluate the truth condition.

Our strategy for evaluating the truth of the formula is to replicate the same structures used by the constructed TCSPU in Lemma 1. We create a secondary anchor point $Z$ with $Z-O=2n+2$ to ensure that $Z$ happens after all boolean values have been assigned, and then create new timepoints corresponding to the values of $\vec{x}$ and $\vec{y}$ that are anchored at $Z$ instead of at different times during the execution. For each $x_{i}$, we create $t_{x_{i}}$ with the constraint $t_{x_{i}}-\tau_{x_{i},e}=2(n-i)+4$, and for each $y_{i}$, we create $t_{y_{i}}$ with the constraint $t_{y_{i}}-\tau_{y_{i},e}=2(n-i)+3$. The rest of the construction, namely the construction of the negated literal values and the clausal gadgets, remains the same, and by the same reasoning, we see that it is possible for a given assignment, it is possible for all constraints to be respected if and only if $\phi$ is satisfied by that assignment of values. Since the initial timepoints are set up such that when the entire network is dynamically controllable the values of timepoints are chosen in the same order as the quantification of the original TQBF formula, we know that $T$ is dynamically controllable if and only if $\exists x_{1}\forall y_{1}...\exists x_{n}\forall y_{n}:\phi(\vec{x},\vec{y})$. Because the new network can be constructed in polynomial time, we have a polynomial time reduction from TQBF to DC-TCSPU, so DC-TCSPU is PSPACE-hard.

∎

To prove that SC-DTNU is $\Sigma_{2}^{P}$-hard, we will reduce the complement of $\forall\exists$3SAT, a $\Pi_{2}^{P}$-complete problem, to SC-DTNU.

An example problem of $\forall\exists$3SAT is of the form $\forall\vec{x},\exists\vec{y}:\phi(\vec{x},\vec{y})$, where $\vec{x},\vec{y}$ are vectors of boolean values and $\phi$ is a 3-CNF formula. The complementary problem is $\exists\vec{x},\forall\vec{y}:\psi(\vec{x},\vec{y})$, where $\psi$ is a 3-DNF formula representing the negation of $\phi$. Given the input problem, we construct a corresponding DTNU $D$ that is strongly controllable if and only if the complementary formula $\psi$ is true (if the original formula $\phi$ is false).

First we define the timepoints of $D$. We start with a reference timepoint $Z$, which represents the first point to be executed. For each $x_{i}\in\vec{x}$, we add points $t_{x_{i}}$ and $t_{\overline{x}_{i}}$ to represent the value of $x_{i}$ and its negation during some candidate assignment to our formula. We do the same thing for $\vec{y}$ adding $t_{y_{j}}$ and $t_{\overline{y}_{j}}$ for each $y_{j}\in\vec{y}$. We also introduce a new gadget per clause of $\psi$ (see Figures 4 and 5) and in each gadget, we introduce ten new timepoints. Timepoints $G_{k,1}$, $G_{k,2}$, and $G_{k,3}$ represent the values of each literal of clause $k$ and timepoint $G_{k,and}$ represents the value of the conjunction of those literals. For each clause, we also add $A_{k,1}$, $A_{k,2}$, $A_{k,3}$, $A_{k,4}$, $A_{k,5}$, and $A_{k,6}$ which are used collectively to simulate an and clause. By appropriately adding contingent and requirement links between these timepoints, we will get a DTNU that is controllable if and only if the original formula $\psi$ is true.

We start by adding constraints to encode the initial assignment of values. For each $t_{x_{i}}$ we add a simple disjunctive constraint requiring that $t_{x_{i}}-Z\in[0,0]\cup[1,1]$. Similarly, for each $t_{y_{j}}$, we add a disjunctive contingent constraint enforcing $t_{y_{j}}-Z\in[0,0]\cup[1,1]$. The choice of values for these initial timepoints maps directly back to an assignment of values in the 3-DNF formula $\psi$ with 0 representing false and 1 representing true.

We also enforce the values of the negations of these variables for convenience, with the same simple disjunctive constraint requiring $t_{\overline{x}_{i}}-Z\in[0,0]\cup[1,1]$ and the disjunctive contingent constraint enforcing $t_{\overline{y}_{j}}-Z\in[0,0]\cup[1,1]$. To ensure that $x_{i}$ and its negation take on values we also add the requirement that $t_{x_{i}}-t_{\overline{x}_{i}}\in\cup[-1,-1]\cup[1,1]$. We will discuss our strategy for ensuring that the values of $t_{y_{j}}$ and $t_{\overline{y}_{j}}$ differ below.

We now move on to the constraints associated with the clausal gadgets. $G_{k,l}$ represents the truth value of the $l^{th}$ element of clause $k$, and $G_{k,and}$ represents the truth value of the entire clause; each timepoint, $G_{k,*}$, that is newly created for the gadget is initialized using a contingent constraint enforcing $G-Z\in[0,0]\cup[1,1]$. We also create a disjunctive constraint across all gadgets, such that if for any $k$, $G_{k,and}-Z=1$, then the constraint is satisfied. We call this disjunctive constraint the goal constraint. This has an immediate correspondence to the notion that the entire formula $\psi$ is satisfied if any of its constituent clauses is satisfied.

Our current construction makes heavy use of contingent constraints, and while we may want the timepoints in our gadgets to represent certain values, their values are chosen by nature, meaning we have no way to directly control their values.

However, we do have control over the constraints of $D$ and, in particular, the disjunctive constraint that spans the gadgets. We can think of checking strong controllability as a two-player game, where the scheduler goes first and nature goes second. Nature’s goal is to construct an assignment such that some constraint is violated. Upon closer examination, we see that in our construction, the only constraint that can be affected by the contingent link durations chosen by nature is the goal constraint. If there exist certain combinations of contingent link durations that we want to preclude from our evaluation, we can do so by adding additional disjunct to the goal constraint that are satisfied when those contingent links take on those durations. In this way, any contingent link values that do not conform to our desired structure make $D$ trivially controllable, and controllability then reduces to controllability under our desired set of constraints across contingent links.

First, we need to make sure that the timepoints $t_{y_{j}}$ and $t_{\overline{y}_{j}}$ take on different values. We can ensure this by adding $t_{y_{j}}-t_{\overline{y}_{j}}=0$ to our goal constraint; if nature gives $t_{y_{j}}$ and $t_{\overline{y}_{j}}$ the same value, then we trivially ignore this case. Similarly, since we want $G_{k,l}$ to take on the same value as the literal $q$ it represents, we augment our disjunctive goal constraint with $G_{k,l}-t_{q}\in[-1,-1]$ and $G_{k,l}-t_{q}\in[1,1]$ where $t_{q}$ is the timepoint associated with literal $q$. As a result, if the clausal representation of the variable differs from our assignment, our network is trivially controllable.

We enforce the conjunction of the elements of each clause by augmenting our goal constraint with $G_{k,and}-G_{k,l}\geq 1$ for each $G_{k,l}$ of our clause gadget. Since each timepoint of our gadget can take on a value of 0 or 1, this constraint will only be satisfied if some literal value is 0 while $G_{k,and}$ has a value of 1. In these situations, $G_{k,and}$ does not represent the conjunction of the literals of clause $k$, and our network then becomes trivially controllable.

Unfortunately, our network still does not perfectly encode the conjunction seen in a DNF clause. It is possible for each $G_{k,l}$ to take on a value of 1 while $G_{k,and}$ is assigned a value of 0. As a result, it may be the case that the original problem, $\exists\vec{x}\forall\vec{y}\psi(\vec{x},\vec{y})$ is true but each $G_{k,and}$ is set to 0, meaning that the network is not strongly controllable.

To fix this, we must augment our gadget to enforce that identical inputs have the same output. This is the reason for introducing timepoints $A_{k,m}$, and these timepoints’ values are set by new contingent links that enforce $A_{k,m}-Z\in[0,0]\cup[1,1]\cup[2,2]$. Through an exhaustive enumeration of possible values, we can confirm that whenever $G_{k,1},G_{k,2},G_{k,3}$ are all 1, either $A_{k,6}$ will be 1 or one of the disjuncts of the goal constraints (see Figure 4) will be satisfied. In this case, when we add $A_{k,6}-G_{k,and}\geq 1$ to the goal constraint, we know that when $G_{k,1},G_{k,2},G_{k,3}$ are all equal to 1, $D$ is controllable, as either $G_{k,and}=1$, meaning $G_{k,and}-Z=1$, which satisfies the goal constraint, or $G_{k,and}=0$, implying $A_{k,6}-G_{k,and}=1$, which also satisfies the goal constraint.

Before continuing, we need to confirm that the addition of the new sub-gadget does not introduce any new problems. For all other values of $G_{k,1}$, $G_{k,2}$, and $G_{k,3}$, we know it is possible for $A_{k,6}$ to take on a value of 0. Since $A_{k,6}$ is the only timepoint of the or-gate that is related to other values by the goal constraint and setting it to 0 does not satisfy the goal constraint, we know that if $G_{k,1},G_{k,2},G_{k,3}$ are not all 1, then there exists a choice of values by nature such that the goal constraint is not satisfied by gadget $k$.

Our transformation is complete and because there is one gadget per clause in $\psi$ and each gadget is of constant size, we see that the transformation takes polynomial time. What remains is to show that $D$ is strongly controllable if and only if $\exists\vec{x}\forall\vec{y}\psi(\vec{x},\vec{y})$ is true. This is evident from our construction.

If $D$ is strongly controllable, there must be some set of assignment to values $t_{x_{i}}$ such that no possible assignment of values to the other timepoints violates any of the constraints. We can prove this by contradiction, assuming that although our choice of $t_{x_{i}}$ guarantees the satisfaction of all other constraints in $D$, there is no choice of $\vec{x}$ that guarantees satisfaction of $\forall\vec{y}\psi(\vec{x},\vec{y})$. Let $\vec{x}$ be specified such that $x_{i}$ is true if and only if $t_{x_{i}}=1$. If $\psi$ is not guaranteed to be satisfied, there must be some $\vec{y}$ such that $\psi(\vec{x},\vec{y})$ is false. Returning to $D$, assume that nature specifies $t_{y_{j}}$ such that $t_{y_{j}}=1$ if and only if $y_{j}$ is true. Since $D$ is strongly controllable, we know that some disjunctive goal constraint is satisfied no matter the assignment of contingent timepoint variables. Let’s assume that all $t_{\overline{y}_{j}}$ are chosen such that they represent the negation of their corresponding $t_{y_{j}}$, that all $A_{k,m}$ of the gadgets are chosen such that the disjunctive constraints involved between all $G_{k,l}$ and $A_{k,m}$ are not satisfied, and that all $G_{k,and}$ are chosen to be 0. The only remaining disjunctive constraints are those involving each $G_{k,and}$. For any particular $k$, setting $G_{k,and}$ to 0 only satisfies a constraint if $A_{k,6}$ is 1, so given all these assumptions, at least one $A_{k,6}$ must be set to 1 (otherwise the system would be uncontrollable). As we demonstrated earlier, $A_{k,6}$ is only constrained to be 1 when all of $G_{k,1}$, $G_{k,2}$, and $G_{k,3}$ are also 1. But those three values correspond exactly to literals in a clause of $\psi(\vec{x},\vec{y})$. If all three are 1, then we have a true clause and because $\psi$ is a 3-DNF formula, this means that $\psi$ is true. We have a contradiction. Therefore if $D$ is controllable, $\exists\vec{x}\forall\vec{y}\psi(\vec{x},\vec{y})$.

To conclude we show the reverse direction, that if $\exists\vec{x}\forall\vec{y}\psi(\vec{x},\vec{y})$ is true, then $D$ is strongly controllable. Let $\vec{x}$ be the assignment of variables that guarantees $\forall\vec{y}\psi(\vec{x},\vec{y})$; we show how we can use $\vec{x}$ to show that $D$ is strongly controllable. We will pick our $t_{x_{i}}$ such that $t_{x_{i}}=1$ if and only if $x_{i}$ is true and will pick our $t_{\overline{x}_{i}}$ such that $t_{\overline{x}_{i}}\neq t_{x_{i}}$. Again we will proceed with proof by contradiction, assuming that $D$ is not strongly controllable. Our choice of $t_{x_{i}}$ and $t_{\overline{x}_{i}}$ satisfy all constraints except the disjunctive goal constraint, so there must be a choice of contingent timepoints that violate the disjunctive goal constraint. We know setting $G_{k,and}=1$ satisfies the goal constraint, so all $G_{k,and}=0$. By proxy, for all $k$, $A_{k,6}=0$ to ensure that the goal constraint is not satisfied because of the link between $A_{k,6}$ and $G_{k,and}$. Because $A_{k,6}=0$, it must be the case that for each gadget, at least one of $G_{k,1}$, $G_{k,2}$, or $G_{k,3}$ must equal zero. In order for the goal disjunctive constraint to remain unsatisfied, each $G_{k,l}$ must maintain the same value as some $t_{x_{i}}$, $t_{\overline{x}_{i}}$, $t_{y_{j}}$, or $t_{\overline{y}_{j}}$ based on the values of the clauses of $\phi$. This forces a particular assignment of values to $t_{y_{j}}$ which we can map back to some $\vec{y}$. For that particular assignment, we know that $\psi(\vec{x},\vec{y})$ is true, or that there is some clause $k^{\prime}$ with all literals set to true. This contradicts the fact that for all $k$, at least one of $G_{k,1}$, $G_{k,2}$, or $G_{k,3}$ must be zero. Thus, $D$ must be strongly controllable, and we have proven that SC-DTNU is $\Sigma_{2}^{P}$-hard.

∎