Precise parameter synthesis for stochastic biochemical systems

Abstract

We consider the problem of synthesising rate parameters for stochastic biochemical networks so that a given time-bounded CSL property is guaranteed to hold, or, in the case of quantitative properties, the probability of satisfying the property is maximised or minimised. Our method is based on extending CSL model checking and standard uniformisation to parametric models, in order to compute safe bounds on the satisfaction probability of the property. We develop synthesis algorithms that yield answers that are precise to within an arbitrarily small tolerance value. The algorithms combine the computation of probability bounds with the refinement and sampling of the parameter space. Our methods are precise and efficient, and improve on existing approximate techniques that employ discretisation and refinement. We evaluate the usefulness of the methods by synthesising rates for three biologically motivated case studies: infection control for a SIR epidemic model; reliability analysis of molecular computation by a DNA walker; and bistability in the gene regulation of the mammalian cell cycle.

Appendix: Proofs

1.1 Proposition 2

Consider a parameter region ${\mathcal {R}}= [x_1^\bot , x_1^\top ] \times \cdots \times [x_n^\bot , x_n^\top ] \subseteq {\mathcal {P}}$. Fix an arbitrary state $s$ and let the maximizing argument of the transient probability in $s$ be (cf. Eq. 8):

$$\begin{aligned} {p^*}&= {{\mathrm{arg\,max}}}_{p \in {\mathcal {R}}} \pi _{t}(s) \end{aligned}$$

(44)

and let ${\tau ^*_{i}} = \pi _{0} \mathbf {P}_{p^*}^{i}$. The local error introduced during a single discrete time step for state s, in the multi-affine case, is given by

$$\begin{aligned} e_{i} (s) = \frac{1}{q} \left| \text {flux}\left( \tau ^*_{i-1},s\right) (p^*) - \max _{p\in V_{{\mathcal {R}}}} \text {flux}\left( \tau ^*_{i-1},s\right) (p) \right| . \end{aligned}$$

(45)

Note that in this analysis the local error is expressed for the discrete distribution $\tau _{i}^{\max }$, from which the solution $\pi _{i}^{\max }$ is derived as a linear combination. We now analyse the global error, which for $i>1$ is given as:

$$\begin{aligned} g_{i}(s)&= \left| \tau _{i}^{*}(s) - \tau _{i}^{\max }(s) \right| \end{aligned}$$

(46)

$$\begin{aligned}&= \left| \tau _{i-1}^{*}(s) + \frac{1}{q} \text{ flux }\left( \tau _{i-1}^{*},s\right) (p^{*}) - \tau _{i-1}^{\max }(s) - \frac{1}{q} \max _{p \in V_{{\mathcal {R}}}} \text{ flux }\left( \tau _{i-1}^{\max },s\right) (p) \right| \end{aligned}$$

(47)

where we now use the definition of $g_{i}(s)$ and employ triangle inequality:

$$\begin{aligned}&\le g_{i-1}(s) + \frac{1}{q} \left| \text{ flux }\left( \tau _{i-1}^{*},s\right) (p^{*}) - \max _{p \in V_{{\mathcal {R}}}} \text{ flux }\left( \tau _{i-1}^{\max },s\right) (p) \right| \end{aligned}$$

(48)

$$\begin{aligned}&\le g_{i-1}(s) + \frac{1}{q} \left| \text{ flux }\left( \tau _{i-1}^{*},s\right) (p^{*}) - \max _{p \in V_{{\mathcal {R}}}} \text{ flux }\left( \tau _{i-1}^{*} + g_{i-1},s\right) (p) \right| \end{aligned}$$

(49)

where the overall global error $g_{i}$ is the vector-wise equivalent of $g_{i}(s)$ and we continue

$$\begin{aligned} g_{i}(s)&\le g_{i-1}(s) + \frac{1}{q} \max _{p \in V_{{\mathcal {R}}}} \text{ flux }(g_{i-1},s)(p) \end{aligned}$$

(50)

$$\begin{aligned}&\quad + \frac{1}{q} \left| \text{ flux }\left( \tau _{i-1}^{*},s\right) (p^{*}) - \max _{p \in V_{{\mathcal {R}}}} \text{ flux }\left( \tau _{i-1}^{*},s\right) (p) \right| \end{aligned}$$

(51)

$$\begin{aligned}&\le g_{i-1}(s) + \frac{1}{q} \max _{p \in V_{{\mathcal {R}}}} \text{ flux }(g_{i-1},s)(p) + e_{i}(s) \end{aligned}$$

(52)

The form of $g_{i}(s)$ is understood as follows. The error in the current step is less than the error in the previous step plus the maximal local error plus the worst-case additional flux resulting from the approximation error in the previous step. Let the width of the parameter space ${\mathcal {R}}$ be given as $w_{{\mathcal {R}}} = \max _{j} (x_{j}^{\top } - x_{j}^{\bot })$.

We now show how to derive bounds $M_{1},M_{2} < \infty$ such that

$$\begin{aligned} e_{i}&\le \frac{M_{1}}{q} w_{{\mathcal {R}}} \end{aligned}$$

(53)

$$\begin{aligned} \max _{p \in V_{{\mathcal {R}}}} \text{ flux }(g_{i}, s)(p)&\le M_{2} \cdot \max _{s\in S} g_{i-1}(s) \end{aligned}$$

(54)

Observe that, if the two inequalities above hold, we get the same over-approximation as in Eq. 42, which would prove the proposition true. For the local error we find

$$\begin{aligned} e_{i}(s) \le \frac{1}{q} \max _{p \in V_{{\mathcal {R}}}} | \text {flux}\left( \tau ^*_{i-1},s\right) (p^*) - \text {flux}\left( \tau ^*_{i-1},s\right) (p) | \end{aligned}$$

(55)

and in this case a Lipschitz constant $M_{1,s}$ exists such that

$$\begin{aligned} e_{i}(s) \le \frac{M_{1,s}}{q} w_{{\mathcal {R}}} \end{aligned}$$

(56)

Thus, $M_{1} = \max _{s\in S} M_{1,s}$ is such that Eq. 53 holds. The maximum flux that propagates due to the approximation error in the previous step is given as

$$\begin{aligned} \max _{p \in V_{{\mathcal {R}}}} \text{ flux }(g_{i-1}, s)(p). \end{aligned}$$

(57)

Now regard $\text{ flux }(g_{i-1}, s)(p)$ as a function of $g_{i-1}$, with $p,s$ fixed. Note that the domain of the function is given by ${\mathcal {R'}} = \times _{s \in S} [0, g_{i-1}(s)]$, and thus $w_{{\mathcal {R}}'}=\max _{s\in S} g_{i-1}(s)$. Also, $\text{ flux }(\mathbf {0}, s)(p) = 0$ for $\mathbf {0} : S\rightarrow 0$. Thus, another Lipschitz constant $M_{2,p,s}$ exists such that

$$\begin{aligned} \text{ flux }(g_{i-1}, s)(p) = \text{ flux }(g_{i-1}, s)(p) - \text{ flux }(\mathbf {0}, s)(p) \le M_{2,p,s} \cdot \max _{s\in S} g_{i-1}(s) \end{aligned}$$

(58)

By taking $M_{2} = \max _{p \in {\mathcal {R}}, s \in S} M_{2,p,s}$, Eq. 54 holds. Summarizing, the global error on $\tau _{i}^{\max }(s)$ is bounded by $\overline{g}_{i}$, under the assumption of multi-affine rate functions, as

$$\begin{aligned} \overline{g}_i = {\left\{ \begin{array}{ll} 0 &{}\quad \text {if } i = 0 \\ \overline{g}_{i-1} \cdot \left( 1 + \frac{M_{2}}{q} \right) + \frac{M_{1}}{q} w_{{\mathcal {R}}} &{}\quad \text {if } i > 0, \end{array}\right. } \end{aligned}$$

(59)

1.2 Theorem 1

The probability of satisfying an unnested and time-bounded CSL property given a CTMC ${\mathcal {C}}_{p}=(S,s_0, \mathbf {R}_{p}, L)$ reduces to the computation of transient-state probabilities over a CTMC ${\mathcal {C}}_{p}'$ for which the transition relation $\mathbf {R}_{p}'$ is easily derived from the original CTMC [4, 31]. Recalling Definition 4, the transient-state probability is given by standard uniformisation:

$$\begin{aligned} \hat{\pi }_{t,p} = \pi _{0} \sum _{i=0}^{k_{\epsilon }} \gamma _{i,qt} \mathbf {P}_{p}^{i}. \end{aligned}$$

(60)

Provided the entries in $\mathbf {P}$ are polynomials of finite degree, the expression $\hat{\pi }_t(s)$ is also a finite-degree polynomial over domain ${\mathcal {P}}$. We now prove that the approximate satisfaction function $\hat{{\varLambda }}_{\phi }$ of any time-bounded finitely-nested path formula can be expressed as a piecewise polynomial function with a finite number of subdomains, using the above as the base case in our induction. Note that only the path operators until ($\text{ U }$) and next ($\mathsf {X}$) allow nesting. We prove the induction step only for the until operator, since the proof for the next operator can be obtained in a similar way.

Let $\phi _1, \phi _2$ be time-bounded CSL path formulas such that $\textit{P}_{=?}[\phi _1]$, $\textit{P}_{=?}[\phi _2]$ are piecewise polynomial with a finite number of subdomains. Consider the nested formula:

$$\begin{aligned} \textit{P}_{=?} \left[ \textit{P}_{\sim _1 r_1} [\phi _1] \ \text{ U }^I\ \textit{P}_{\sim _2 r_2} [ \phi _2] \right] \end{aligned}$$

(61)

for a subdomain $I\in {\mathbb {R}}_{\ge 0}$, bounds $r_1,r_2$ and $\sim _i \ \in \left\{ <, \le , \ge , > \right\}$.

Observe that, if the satisfaction sets $v_{i}=\{ s\in S\mid s \models \textit{P}_{\sim _{i} r_i} [ \phi _i ] \}$ for $i=1,2$ are constant over a subspace ${\mathcal {R}}\subseteq {\mathcal {P}}$, then the expression in Eq. 61 is given by a polynomial function over ${\mathcal {R}}$, cf. Eq. 60. We will demonstrate that $\hat{{\varLambda }}_{ \tiny \textit{P}_{\sim _1 r_1} [\phi _1] \ \text{ U }^I\ \textit{P}_{\sim _2 r_2} [ \phi _2]}$ is piecewise polynomial over finitely many subdomains by constructing a partition of ${\mathcal {P}}$ that is conditioned on the truth assignment of each state. Given a state $s$, allow the partition ${\mathcal {T}}_{1}(s) \cup {\mathcal {F}}_{1}(s) = {\mathcal {P}}$ where

$$\begin{aligned} \forall p \in {\mathcal {T}}_{1}(s)&: s \models \textit{P}_{\sim _{1} r_1} [ \phi _1 ] \end{aligned}$$

(62)

$$\begin{aligned} \forall p \in {\mathcal {F}}_{1}(s)&: s \not \models \textit{P}_{\sim _{1} r_1} [ \phi _1 ]. \end{aligned}$$

(63)

By the induction hypothesis $\textit{P}_{=?} [ \phi _1 ] - r_1$ is piecewise polynomial over finite many subspaces, so that ${\mathcal {T}}_{1}(s),{\mathcal {F}}_{1}(s)$ are unions of finitely many subspaces of ${\mathcal {P}}$. Assume a similar partition ${\mathcal {T}}_{2}(s) \cup {\mathcal {F}}_{2}(s) = {\mathcal {P}}$. We now wish to construct a partition $\bigcup _{v_{1},v_{2} \in 2^{S}} {\mathcal {R}}(v_{1},v_{2}) = {\mathcal {P}}$ that is conditioned on the truth assignment of all states, so that $\forall p \in {\mathcal {R}}(v_{1},v_{2})$:

$$\begin{aligned} s\in v_{1} \Leftrightarrow s\models \textit{P}_{\sim _{1} r_1} [ \phi _1 ] \quad \wedge \quad s\in v_{2} \Leftrightarrow s\models \textit{P}_{\sim _{2} r_2} [ \phi _2 ] \end{aligned}$$

(64)

in which case the expression of Eq. 61 is a polynomial function over ${\mathcal {R}}(v_{1},v_{2})$ because the truth-assignment of the nested formulas is fixed. We provide a constructive definition for ${\mathcal {R}}(v_{1},v_{2})$ by finite intersection of the sets ${\mathcal {T}}_{i}(s)$ and ${\mathcal {F}}_{i}(s)$, that is:

$$\begin{aligned} {\mathcal {R}}(v_{1},v_{2}) = \left[ \cap _{s\in v_{1}} {\mathcal {T}}_{1}(s) \right] \cap \left[ \cap _{s\not \in v_{1}} {\mathcal {F}}_{1}(s) \right] \cap \left[ \cap _{s\in v_{2}} {\mathcal {T}}_{2}(s) \right] \cap \left[ \cap _{s\not \in v_{2}} {\mathcal {F}}_{2}(s) \right] . \end{aligned}$$

(65)

Then ${\mathcal {R}}(v_{1},v_{2})$ is a union of a finite number of subdomains of ${\mathcal {P}}$.

1.3 Proposition 3

We first show termination of the algorithm for an unnested property $P_{\ge r} [ \phi ]$. Define

$$\begin{aligned} f = \hat{{\varLambda }}_\phi (\cdot )(s_0) - r \end{aligned}$$

and let the zero-set of f be given as $Z(f)=\{ p \in {\mathcal {P}}\mid f(p) = 0 \}$. In other words, Z(f) is the set of parameters p yielding satisfaction probability equal to r, i.e. $\hat{{\varLambda }}_\phi (p)(s_0) = r$. Now note that, at a generic step of the algorithm, the undecided space is composed by those regions ${\mathcal {R}}_{i}$ such that ${\varLambda }_{\min }^{{\mathcal {R}}_{i}} < r$ and ${\varLambda }_{\max }^{{\mathcal {R}}_{i}} \ge r$. Assuming infinite precision, ${\varLambda }_{\min }^{{\mathcal {R}}_{i}}$ and ${\varLambda }_{\max }^{{\mathcal {R}}_{i}}$ can be made arbitrarily tight, i.e. the approximation error made arbitrarily small (cf Proposition 2). Therefore, any such ${\mathcal {R}}_{i}$ intersects the zero-set of f and the undecided space covers the zero-set. Formally, given a decomposition $\cup _{i} {\mathcal {R}}_{i} = {\mathcal {P}}$, the undecided region is given as:

$$\begin{aligned} {\mathcal {U}}= \cup _{i} {\mathcal {R}}_{i} \text { s.t. } {\mathcal {R}}_{i} \cap Z(f) \ne \emptyset . \end{aligned}$$

(66)

Excluding the trivial case that f is identically zero ($Z(f) = {\mathcal {P}}$), we prove termination in two steps. First, we show that Z(f) can be covered by finitely many rectangular regions whose total volume can be made arbitrarily small. Call such cover C. Second, we show that our algorithm can reach in a finite number of steps a decomposition where ${\mathcal {U}}$ covers C and has volume no larger than the tolerance $\varepsilon$. Finally, we extend the termination proof to nested CSL properties.

• 1. We state now two useful properties of zero-sets: (i) the zero-set of a multi-variate polynomial is negligible, i.e. it has Lebesgue measure (volume) 0 (a proof is available in [12]). (ii) the zero-set $Z(f')$ of any continuous function $f'$ is closed. This follows from the fact that $Z(f')$ is the pre-image of $f'$ on the closed set $\{0\}$. Now note that the parameter space ${\mathcal {P}}$ is compact (bounded and closed). It follows that $Z(f) \subset {\mathcal {P}}$ is compact too. Z(f) meeting condition i) corresponds to saying that for each $\epsilon ' > 0$ there exists a finite or countable collection $R_1, R_2, \ldots$ of (possibly overlapping) open rectangles such that

  $$\begin{aligned} Z(f) \subseteq \bigcup _{k \in K} R_k \text { and } \sum _{k \in K} \text{ vol }(R_k) < \epsilon ' \text { (see e.g. [8], Section 1).} \end{aligned}$$

  Finally, by compactness every open cover of Z(f) has finite a subcover, i.e. there exists a finite index set $K' \subseteq K$ such that:

  $$\begin{aligned} Z(f) \subseteq \bigcup _{k' \in K'} R_{k'} \subseteq \bigcup _{k \in K} R_k. \end{aligned}$$

  It follows that:

  $$\begin{aligned} \text{ vol }\left( \bigcup _{k' \in K'} R_{k'} \right) \le \sum _{k' \in K'} \text{ vol }(R_{k'}) \le \sum _{k \in K} \text{ vol }(R_k) < \epsilon '. \end{aligned}$$

  The first inequality holds since rectangles in the cover can overlap. Thus, $C \overset{\text {def}}{=} \bigcup _{k' \in K'} R_{k'}$ is the required finite rectangular cover of Z(f) with arbitrarily small volume. Note that these properties hold also if we replace in C each rectangle $R_{k'}$ with its closure.

• 2. Since any finite union of overlapping rectangles can be rewritten as a finite union of almost disjoint rectangles (i.e. intersecting only at their extrema) [15], we rewrite the cover C as

  $$\begin{aligned} C = \bigcup _{j \in J} R_{j}, \text { such that } \forall i,j \in J. \ \mathsf {int}(R_{i}) \cap \mathsf {int}(R_{j}) = \emptyset \end{aligned}$$

  where $\mathsf {int}(R)$ is the interior of rectangle R. In particular, this transformation can be done in a such way that each $R_{j}$ is a box of width $\delta$, for some $\delta > 0$. We can hence derive the following:

  $$\begin{aligned} |J|\cdot \delta ^n = \sum _{j \in J} \text{ vol }(R_{j}) = \text{ vol }(C) < \epsilon ' \end{aligned}$$

  (67)

  where n is the number of dimensions/parameters. Without loss of generality assume that the parameter space ${\mathcal {P}}$ is the unit cube, meaning that the algorithm terminates for $\text{ vol }({\mathcal {U}}) \le \varepsilon$. Assume also that at each step undecided parameter regions are decomposed by bisection. Consider a number of refinement steps i such that each undecided region at the i-th step has width $w \in [\delta , 2\delta ]$, yielding $i = \lfloor -\log _2 \delta \rfloor$. From Eq. 66 and C being a cover of Z(f), we derive that:

  $$\begin{aligned} {\mathcal {U}}\subseteq \cup _{i} {\mathcal {R}}_{i} \text { s.t. } {\mathcal {R}}_{i} \cap C \ne \emptyset . \end{aligned}$$

  Observe that each rectangle in the cover C is intersected by at most $2^n$ undecided regions of width w. Let $N = |J|$ be the number of boxes in the cover C. Then,

  $$\begin{aligned} \text{ vol }({\mathcal {U}}) \le 2^n \cdot N \cdot w^n \le 2^n \cdot N \cdot \left( 2\delta \right) ^n < \epsilon ' \cdot 4^n \end{aligned}$$

  (68)

  where the last inequality holds by Eq. 67. Since the bound $\epsilon '$ on the volume of C is arbitrary, termination follows. Indeed, to satisfy the termination condition ($\text{ vol }({\mathcal {U}}) \le \varepsilon$), we can chose $\epsilon ' = \varepsilon \cdot 4^{-n}$ which implies the existence of a suitable $\delta$ and, in turn, of a finite number of steps $i = \lfloor -\log _2 \delta \rfloor$.

• 3. By Theorem 1, we know that the satisfaction function of a nested CSL property is piecewise polynomial, with a finite number of (bounded) subdomains. We proceed by borrowing from steps (1) and (2) of this proof. Let D be an index set identifying the subdomains and $f_d$ be the satisfaction function at the d-th subdomain. Then, there exist $\delta > 0$, arbitrary positive constants $\{\epsilon '_d\}_{d\in D}$ and a set of finite covers $\{C_d\}_{d\in D}$ such that for all $d \in D$, $\text{ vol }(C_d) < \epsilon '_d$ and $C_d$ is composed of pairwise-disjoint boxes of width $\delta$. We can now prove termination by showing that Eq. 68 holds also for the nested case if we set $N = \sum _{d \in D} |C_d|$ to the total number of boxes and $\epsilon ' = \sum _{d\in D} \epsilon '_d$.

However, there is an important caveat to discuss. In this case, the satisfaction function might exhibit jump discontinuities characterized by coarse probability bounds that cannot be “mitigated” by the iterative refinement procedure. It follows that we need to take into account all the additional undecided regions that contain jump discontinuity points. Fortunately, since piecewise continuous functions are continuous almost everywhere, the set of such discontinuities has measure 0, and hence, by a similar argument to the above proof, the total volume of the undecided regions containing discontinuities can be made arbitrarily small in a finite number of steps. $\square$

1.4 Proposition 4

We prove the proposition by first showing that the safe bounds can be made arbitrarily small in a finite number of steps. Second, we derive the number of steps for which the termination condition is met.

• 1. Define $f = \hat{{\varLambda }}_\phi (\cdot )(s_0)$ and let $D^k$ be the set of regions at the k-th decomposition step. Fix a region ${\mathcal {R}}^k_j \in D^k$. Let $[f({\mathcal {R}}^k_j)]$ be the interval describing the range of f over ${\mathcal {R}}^k_j$. Since f is a polynomial function over a bounded domain, then it is also Lipschitz continuous, implying that there exists a constant $m_j^k$ s.t. $\mathbf {w}([f({\mathcal {R}}^k_j)]) \le m_j^k \cdot \mathbf {w}({\mathcal {R}}^k_j)$, where $\mathbf {w}$ denotes the width of the rectangle. The safe approximation we compute is such that $[f({\mathcal {R}}^k_j)] \subset [{\varLambda }_{\min }^{{\mathcal {R}}^k_j}, {\varLambda }_{\max }^{{\mathcal {R}}^k_j}]$ and, in particular, ${\varLambda }_{\max }^{{\mathcal {R}}^k_j} - {\varLambda }_{\min }^{{\mathcal {R}}^k_j} = \mathbf {w}([f({\mathcal {R}}^k_j)]) + e^\top + e^\bot$, where $e^\top$ and $e^\bot$ are the global approximation errors for the upper and lower bound of $\hat{{\varLambda }}_\phi$. We now derive a closed-form expression for $e^\top$, based on Proposition 2. Let $e^\top = \overline{g}_{k_\epsilon }$, where $k_\epsilon$ is the number of uniformisation steps and, by Eq. 42, there exist positive constants $M_1$ and $M_2$ such that

  $$\begin{aligned} \overline{g}_{k_\epsilon } = \overline{g}_{k_{\epsilon }-1} \cdot \left( 1 + \frac{M_{2}}{q} \right) + \frac{M_{1}}{q} \left( \mathbf {w}\left( {\mathcal {R}}^k_j\right) \right) . \end{aligned}$$

  (69)

  Let $c_2 = 1 + \frac{M_{2}}{q}$ and $c_1 = \frac{M_{1}}{q}$. Solving the recurrence at Eq. 69, we get the following expression:

  $$\begin{aligned} \overline{g}_{k_\epsilon } = c_j^{k,\top } \cdot \mathbf {w}({\mathcal {R}}^k_j)) \end{aligned}$$

  (70)

  where $c_j^{k,\top } = \dfrac{c_1 \cdot \left( c_2^{k_\epsilon } -1 \right) }{c_2 -1}$. Following the same derivation for $e^\bot$, we conclude that there exist positive constants $c_j^{k,\top }$ and $c_j^{k,\bot }$ such that $e^\top = c_j^{k,\top } \cdot \mathbf {w}({\mathcal {R}}^k_j)$ and $e^\bot = c_j^{k,\bot } \cdot \mathbf {w}({\mathcal {R}}^k_j)$. Then, for all ${\mathcal {R}}^k_j \in D^k$:

  $$\begin{aligned} {\varLambda }_{\max }^{{\mathcal {R}}^k_j} - {\varLambda }_{\min }^{{\mathcal {R}}^k_j} \le \left( m+c^\top +c^\bot \right) \cdot \mathbf {w}\left( {\mathcal {R}}^k_j \right) \end{aligned}$$

  (71)

  where $m = \max \{m_j^k \mid {\mathcal {R}}^k_j \in D_k \}$, $c^\top = \max \{c_j^{k,\top } \mid {\mathcal {R}}^k_j \in D_k \}$ and $c^\bot = \max \{c_j^{k,\bot } \mid {\mathcal {R}}^k_j \in D_k \}$. Since regions are decomposed by sectioning at the mid-point of each parameter interval, the width of a region is halved at every step, so the equation is expressed as:

  $$\begin{aligned} {\varLambda }_{\max }^{{\mathcal {R}}^k_j} - {\varLambda }_{\min }^{{\mathcal {R}}^k_j} \le \left( m+c^\top +c^\bot \right) \cdot \dfrac{\mathbf {w}({\mathcal {P}})}{2^k}. \end{aligned}$$

  (72)

  Then, for an arbitrary precision $\epsilon '$, there exists a finite number of decomposition steps k such that ${\varLambda }_{\max }^{{\mathcal {R}}^k_j} - {\varLambda }_{\min }^{{\mathcal {R}}^k_j} \le \epsilon '$ for all ${\mathcal {R}}^k_j \in D^k$.

• 2. Now we show how to determine $\epsilon '$ s.t. the termination condition ${\varLambda }_\phi ^\top -{\varLambda }_\phi ^\bot \le \epsilon$ is met. Let ${\mathcal {R}}^k_\top$ be one of the regions with the highest upper probability bound, thus ${\varLambda }_\phi ^\top = {\varLambda }_{\max }^{{\mathcal {R}}^k_\top }$. Note that the highest lower bound M is at least ${\varLambda }_{\min }^{{\mathcal {R}}^k_\top }$, and thus every region ${\mathcal {R}}^k_j$ in ${\mathcal {T}}$ is such that

  $$\begin{aligned} {\varLambda }_{\max }^{{\mathcal {R}}^k_j} \ge M \ge {\varLambda }_{\min }^{{\mathcal {R}}^k_\top } \ge {\varLambda }_{\max }^{{\mathcal {R}}^k_\top } - \epsilon ' = {\varLambda }_\phi ^\top - \epsilon '. \end{aligned}$$

  Then, the smallest lower bound in the true region, ${\varLambda }_\phi ^\bot$, is at least $M - \epsilon '$ and so, ${\varLambda }_\phi ^\bot \ge {\varLambda }_\phi ^\top - 2\cdot \epsilon '$. This implies that termination is achieved with $\epsilon ' = \frac{\epsilon }{2}$ and, according to Eq. 72, in a number of steps equal to:

  $$\begin{aligned} k = \left\lceil \log _2 \left( \dfrac{2 \cdot (m + c^\top +c^\bot ) \cdot \mathbf {w}({\mathcal {P}}) }{\epsilon } \right) \right\rceil . \end{aligned}$$