Correction to: From Emerson-Lei automata to deterministic, limit-deterministic or good-for-MDP automata

1 Correction to: Innovations in Systems and Software Engineering https://doi.org/10.1007/s11334-022-00445-7

In the original publication of the article, the following corrections have been updated.

2 Introduction

Is:

Dedicated translations from LTL directly to (limit) deterministic [21] and unambiguous [25] automata have been studied.

Should be:

Dedicated translations from LTL directly to (limit-) deterministic [21] and unambiguous [25] automata have been studied.

3 In 3.1.1, Definition 3.3

Is:

Figure 3 is inside Definition 3.3, and breaks a list of items.

Should be:

Figure 3 should be outside of Definition 3.3., e.g., directly following it.

4 In 5.2: Proof of Proposition 5.3

Is:

In NP for limit-det. TELA. We now show that the problem of computing $\textbf{Pr}_{\mathcal {M}}^{\max }(\mathcal {L}(\mathcal {A})) > 0$ is in NP if TELA $\mathcal {A} = (Q,\Sigma ,\delta ,I,\alpha )$ is limit-deterministic.

[$\ldots$]

Claim. $\textbf{Pr}^{\max }_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) > 0$ holds iff there exists a reachable end-component $\mathcal E$ of $\mathcal {M}\times \mathcal {A}$ such that ${{\,\textrm{atrans}\,}}(\mathcal{E}) \models \alpha$.

[$\ldots$]

In P for fin-less limit-det. TELA. If $\alpha$ is fin-less, then the existence of an end-component $\mathcal E$ with ${{\,\textrm{atrans}\,}}(\mathcal{E}) \models \alpha$ is equivalent to the existence of a maximal end-component satisfying the same property.

Should be:

In NP for limit-det. TELA. We now show that the problem of computing $\textbf{Pr}_{\mathcal {M}}^{\max }(\mathcal {L}(\mathcal {A})) > 0$ is in NP if TELA $\mathcal {A} = (Q,\Sigma ,\delta ,I,\alpha )$ is limit-deterministic.

[$\ldots$]

Claim. $\textbf{Pr}^{\max }_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) > 0$ holds iff there exists a reachable end-component $\mathcal E$ of $\mathcal {M}\times \mathcal {A}$ such that ${{\,\textrm{atrans}\,}}(\mathcal{E}) \models \alpha$.

[$\ldots$]

In P for fin-less limit-det. TELA. If $\alpha$ is fin-less, then the existence of an end-component $\mathcal E$ with ${{\,\textrm{atrans}\,}}(\mathcal{E}) \models \alpha$ is equivalent to the existence of a maximal end-component satisfying the same property.

5 Sec 5.2: between Lemma 5.5 and Lemma 5.6

Is:

[$\ldots$] and show that words labeling a loop in the deterministic automaton $\mathscr {D}_i$ induce a path through the powerset automaton of [$\ldots$]

Should be:

[$\ldots$] and show that words labeling a loop in the deterministic automaton $\mathcal {D}_i$ induce a path through the powerset automaton of [$\ldots$]

Is:

Recall that we are given a finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}$, and to prove the GFM property, we have to provide a scheduler $\mathfrak {S}'$ such that $\textrm{Pr}^{\mathfrak {S}}_{\mathcal {M}}(\mathcal {L}(\mathscr {A})) \le \textrm{Pr}^{\mathfrak {S}'}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathscr {A}}}(\Pi _{acc})$. [$\ldots$] It stays inside the initial component of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathscr {A}}$ and mimics the action chosen by $\mathfrak {S}$ until the corresponding path in $\mathcal {M}_{\mathfrak {S}} \times \mathcal {D}$ reaches an accepting bottom strongly connected component (BSCC) B.

Should be:

Recall that we are given a finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}$, and to prove the GFM property, we have to provide a scheduler $\mathfrak {S}'$ such that $\textrm{Pr}^{\mathfrak {S}}_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) \le \textrm{Pr}^{\mathfrak {S}'}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}(\Pi _{acc})$. [$\ldots$] It stays inside the initial component of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}$ and mimics the action chosen by $\mathfrak {S}$ until the corresponding path in $\mathcal {M}_{\mathfrak {S}} \times \mathcal {D}$ reaches an accepting bottom strongly connected component (BSCC) B.

6 Sec 5.2: Lemma 5.6

Is:

$$\begin{aligned} \textrm{Pr}_{s}(\{\pi \mid L(\pi ) \text { is accepted from } (Q',\varnothing ,0) \text { in } \mathfrak {B}_i\bigl \}) = 1. \end{aligned}$$

Should be:

$$\begin{aligned} \textrm{Pr}_{\mathfrak {s}}(\{\pi \mid L(\pi ) \text { is accepted from } (Q',\varnothing ,0) \text { in } \mathfrak {B}_i\bigl \}) = 1. \end{aligned}$$

7 Sec 5.2: Lemma 5.7

Is:

$$\begin{aligned} \mathop {\textrm{Pr}}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}^{\mathfrak {S}'}(\Pi _{acc}) \ge \mathop {\textrm{Pr}}_{\mathcal {M}}^{\mathfrak {S}}(\mathcal {L}(\mathcal {A})) \end{aligned}$$

Should be:

$$\begin{aligned} \textrm{Pr}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}^{\mathfrak {S}'}(\Pi _{acc}) \ge \textrm{Pr}_{\mathcal {M}}^{\mathfrak {S}}(\mathcal {L}(\mathcal {A})) \end{aligned}$$

8 Sec 5.2: Theorem 5.2

Is:

• 1. For every finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}$ there exists a scheduler $\mathfrak {S}'$ of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}$ such that

  $$\begin{aligned} \mathop {\textrm{Pr}}^{\mathfrak {S}}_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) \le \mathop {\textrm{Pr}}^{\mathfrak {S}'}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}(\Pi _{acc}) \end{aligned}$$

• 2. For every finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}$ there exists a scheduler $\mathfrak {S}'$ of $\mathcal {M}$ such that

  $$\begin{aligned} \mathop {\textrm{Pr}}^{\mathfrak {S}}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}(\Pi _{acc}) \le \mathop {\textrm{Pr}}^{\mathfrak {S}'}_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) \end{aligned}$$

Should be:

• 1. For every finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}$ there exists a scheduler $\mathfrak {S}'$ of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}$ such that

  $$\begin{aligned} \textrm{Pr}^{\mathfrak {S}}_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) \le \textrm{Pr}^{\mathfrak {S}'}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}(\Pi _{acc}) \end{aligned}$$

• 2. For every finite memory scheduler $\mathfrak {S}$ of $\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}$ there exists a scheduler $\mathfrak {S}'$ of $\mathcal {M}$ such that

  $$\begin{aligned} \textrm{Pr}^{\mathfrak {S}}_{\mathcal {M}\times \mathcal {G}^{{\text {GFM}}}_{\mathcal {A}}}(\Pi _{acc}) \le \textrm{Pr}^{\mathfrak {S}'}_{\mathcal {M}}(\mathcal {L}(\mathcal {A})) \end{aligned}$$

9 Sec 6: Table 1, first column (“Algorithm”)

Is:

• Spot

• remFin$\rightarrow$split$\alpha$

• split$\alpha$ $\rightarrow$remFin

• remFin$\rightarrow$rewrite$\alpha$

• product

• product (no langcover)

• limit-det.

• limit-det. via GBA

• good-for-MDP

• good-for-MDP via GBA

• Spot

• product

Should be:

• Spot

• remFin $\rightarrow$ split $\alpha$

• split $\alpha \rightarrow$ remFin

• remFin $\rightarrow$ rewrite

• product

• product (no langcover)

• limit-det.

• limit-det. via GBA

• good-for-MDP

• good-for-MDP via GBA

• Spot

• product

10 Sec 6: Table 1

Should be:

• the horizontal line after the 6th line shold not cross through the first column

• random in the first column should be vertically aligned in the middle of the first 10 lines