Fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices

In computation theory, nondeterminism has played important roles([10, 13]). Viewing nondeterministic computations as words, systems and its specifications can be seen as languages, then we can translate problems about model checking, satisfiability and synthesis to ones about languages of automata. These transforms provide a new automata-theoretic approach to study system specification, verification and synthesis, and meanwhile such method is proven to be effective ([22]). Nondeterministic computation has only existential quantifier, but as a generalization of nondeterminism, “alternation”, it has existential and universal quantifiers ([3]). In [3], A.K.Chandra studied the properties about alternating Turing machines and their languages. Moreover, some information about alternating finite state automata and alternating pushdown automata were also introduced. Alternating automata is a useful model to study formal verification, and more information about it can be referred to [12, 20].

In the study of linear temporal logic ([22]), Vardi translated the problems about programs and specifications to the ones about languages of automata: he illustrated that alternating ($\mathrm{B\ddot{u}chi}$) automata have same expressive power as nondeterministic ($\mathrm{B\ddot{u}chi}$) ones, and the former ones are exponentially more succinct than the latters; The result automaton obtained after taking dual operation and exchanging final and non-final states is the complement to the original alternating ($\mathrm{B\ddot{u}chi}$) automaton, which reflects the great advantage of “alternation”. Then he use these conclusions to build an alternating $B\ddot{u}chi$ automaton for an LTL formula and such that the language of such automaton is exactly the set of computations satisfying that LTL formula.

Are these conclusions about alternating $\mathrm{B\ddot{u}chi}$ automata all suitable for weighted cases? i.e., (from the perspective of automata) Are there automata with weighted existential and universal quantifiers? In [1, 4], O.Kupferman et al. had already introduced the definition of weighted alternating $\mathrm{B\ddot{u}chi}$ automata, which answers the above question. O.Kupferman et al. studied the expressive powers of weighted alternating $\mathrm{B\ddot{u}chi}$ automata for special semantics such as $Max$, $Sum$, $Sup$, $LimSup$ and so on over real number set, and discussed the relationship between them simultaneously. But these specific semantics make the conclusions restricted, and the discussion about the relation between weighted alternating $\mathrm{B\ddot{u}chi}$ automata and nondeterministic $\mathrm{B\ddot{u}chi}$ ones is not involved. Furthermore, their automata have no final state, which is not comprehensive and general: the influences exerted by final states are not taken into the consideration, and thus, the Boolean cases cannot be seen as the special case of theirs. It shows the drawbacks of the version of weighted alternating $\mathrm{B\ddot{u}chi}$ automata in [1, 4]. So we want to give another one to avoid above shortcomings.

Derived from these ideas, we will introduce a new version of weighted alternating $\mathrm{B\ddot{u}chi}$ automata with weights in distributive lattices, of which the properties such as: the equivalence relation between weighted nondeterministic $\mathrm{B\ddot{u}chi}$ automata and weighted alternating $\mathrm{B\ddot{u}chi}$ ones, the closure properties can be established. In ours, the factor about final states are considered, and our version are more convenient to calculate the weights of their languages: for a word, to describe how likely it can be accepted depends on all successful runs on it, and the weight of each run is obtained just by taking conjunction of the weights of all branches, to be specific, if the branch is finite, its weight is equal to the label of its leaf node, otherwise, it is equal to $\bigwedge\limits_{i\geq 0}\bigvee\limits_{j\geq i}F(q_{j})$, where $q_{0},q_{1},\cdots$ is the label sequence of such branch and $F$ is the $L$-valued fuzzy sets of final states. Such advantage is due to our weights’ and transitions’ settings: the image set of the transition function “$\delta$” is a subset of Boolean formulas over $L\cup Q$ ($L$ is a distributive lattice and $Q$ is the states set) rather than that in [1, 4], a subset of Boolean formulas over $L\times Q$. Then in the runs of our version, weights and states are the labels of nodes (weights can and only can label the leaf nodes), which is much clearer and simpler than the case of [1, 4]: weights label the edges between nodes and each node is labeled by states.

In section 2, some pre-knowledge about alternating $\mathrm{B\ddot{u}chi}$ automata are introduced. In section 3, with the notion of fuzzy Boolean formulas, we give the definitions of fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices, show how to calculate the weights of run trees of our version (leaf nodes labeled by weights), and illustrate the equivalence relation between $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata and $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ ones. The closure properties about $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata are introduced in section 4. A construction showing the languages recognized by $L$-fuzzy alternating co-$\mathrm{B\ddot{u}chi}$ automata are also $L$-fuzzy $\omega$-regular without using the equivalence relation between $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata and $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ ones and closure properties of $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ ones is provided. In section 5, we discuss the decision problems (emptiness-value, universality-value, implication-value problems) for $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata: these problems can be decidable in exponential time and are PSPACE-complete. Some specific examples are given in the last section, which can be evidences to testify our theorems’ correctness. Similarly to classical case, the above conclusions could also be seen as an effective approach to study fuzzy temporal logic, which can be leaving as one future study. For example, how to build a fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton for a fuzzy LTL formula such that the language of this automaton is exactly the fuzzy set of computations satisfying that fuzzy LTL formula.

For a set $X$, let $\mathbf{\mathcal{B}}^{+}(X)$ denote the set of all positive Boolean formulas over it (i.e., Boolean formulas built by elements of $X$ using $\wedge$ and $\vee$). Besides, $\mathbf{\mathcal{B}}^{+}(X)$ includes two special formulas, $\mathbf{true}$ and $\mathbf{false}$. For $Y\subseteq X$ and $\theta\in\mathbf{\mathcal{B}}^{+}(X)$, we say that $Y$ satisfies $\theta$, if the truth value is true after assigning $true$ to the members of $Y$ and assigning $false$ to the members of $X-Y$; furthermore, if there is no proper subset of $Y$ satisfying $\theta$, then we say $Y$ satisfies $\theta$ in a minimal manner. Obviously, $\{x_{1},x_{2},x_{3}\}$ satisfies the formula $(x_{1}\vee x_{2})\wedge x_{3}$, and $\{x_{1},x_{2}\}$, $\{x_{1},x_{3}\}$ satisfy it in a minimal manner, while the set $\{x_{2},x_{3}\}$ does not.

For any nondeterministic $\mathrm{B\ddot{u}chi}$ automaton $\mathcal{A}=(Q,\Sigma,\delta,q_{0},F)$, some formulas from $\mathbf{\mathcal{B}}^{+}(Q)$ can be used to represent its $\delta$. For example, for a transition $\delta(q,a)=\{q_{1},q_{2},q_{3}\}$, it can be described by formula $q_{1}\vee q_{2}\vee q_{3}$. Based on such representation, there is a new notion: alternating $B\ddot{u}chi$ automata. The only distinctions between nondeterministic and alternating ones are transitions “$\delta$”.

In an alternating $\mathrm{B\ddot{u}chi}$ automaton, the transitions can be any formula of $\mathbf{\mathcal{B}}^{+}(Q)$. The language recognized by an alternating $\mathrm{B\ddot{u}chi}$ automaton is characterized by induction, for instance, if $\delta(q,a)=(q_{1}\wedge q_{2})\vee q_{3}$ is a transition of some alternating $\mathrm{B\ddot{u}chi}$ automaton, which means this automaton accepts $aw$ from $q$, if it accepts $w$ from both $q_{1}$ and $q_{2}$ or from $q_{3}$, where $w$ is a word of $\Sigma^{\omega}$. It is clear that such transition includes both the features of existential choice (the disjunction in the formula) and universal choice (the conjunction in the formula).

Because of the universal choice, a run of an alternating $\mathrm{B\ddot{u}chi}$ automaton is a tree rather than a sequence. $|x|$ denotes the level which the node $x$ occurring at; in particular, for root $\varepsilon$, $|\varepsilon|=0$ ($x$ and $\varepsilon$ are symbols rather than specific states). A branch $\beta=x_{0},x_{1},\cdots$ of a tree is a nodes sequence, where $x_{0}$ is $\varepsilon$ and $x_{i}$ is the parent of $x_{i+1}$ for all $i\geq 0$. In fact, a run $r$ of an alternating $\mathrm{B\ddot{u}chi}$ automaton is a $Q$-labeled tree, in which nodes are labeled by states. $r(x)=q$ means that the node $x$ of $r$ labeled by $q$ ($x$ is a symbol and $q$ is a specific state).

For example, if $\delta(q,a_{i})=(q_{1}\vee q_{2})\wedge q_{3}$, then the labels of $q$’s children include one element of $\{q_{1},q_{2}\}$ and also include state $q_{3}$ after putting $a_{i}$. Notice that if $\delta(r(x),a_{i})=\mathbf{true}$, then $x$ does not have any children, i.e., $x$ is a leaf node. In addition, there is no run taking a transition with $\theta=\mathbf{false}$. The run tree $r$ is accepting if every infinite branch in $r$ infinitely passes $F$.

The relationships between alternating $\mathrm{B\ddot{u}chi}$ automata and nondeterministic $\mathrm{B\ddot{u}chi}$ automata have been studied ([22]): they have the same expressive power, furthermore, the former ones are more succinct than latters, and the blow-ups of states during the transforms from alternating to nondeterministic ones are unavoidable ([22]).

One advantage of alternating $\mathrm{B\ddot{u}chi}$ automata is that they are easy to be complemented. For equivalent alternating and nondeterministic $\mathrm{B\ddot{u}chi}$ automata, it is more easy to complement the former ones, cf.[3]: just interchanging the conjunctions and disjunctions in every transition, as well as final and non-finial states.

If not illustrate especially, the lattice $L$ we used below is distributive. In addition, we require that $L$ have the largest element $1$ and the least element $0$. In the following, we firstly introduce some preparation works. Our version of $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata is distinct from [1, 4]: weights belong to $L$ rather than the real set, and in ours, weights labels every leaf node of run tree instead of along with every edge from each node to its child. In order to overcome shortcomings of [1, 4], i.e., Boolean case cannot be seen as its special case, we put factor about final states in consideration.

For any $Y\subseteq X$ and a formula $\theta\in\mathbf{\mathcal{F_{L}B}}^{+}(X)$, we define a value $v(\theta,Y)$ in $L$, which is obtained by substituting any element of $Y$ occurring in $\theta$ by $1$, and that of $X-Y$ by $0$. Let $\theta_{1},\theta_{2}\in\mathbf{\mathcal{F_{L}B}}^{+}(X)$, if for any $Y\subseteq X$, $v(\theta_{1},Y)=v(\theta_{2},Y)$ holds, then we call them equivalent, denoted by $\theta_{1}\equiv\theta_{2}$. For example, for $\theta_{1}=0.5\vee(x_{2}\wedge 0.2\wedge x_{3})\vee(0.8\wedge x_{2})$ and $\theta_{2}=0.5\vee(((0.3\wedge x_{3})\vee 0.8))\wedge x_{2})$, we can verify that $\theta_{1}\equiv\theta_{2}$.

For any $\theta\in\mathbf{\mathcal{F_{L}B}}^{+}(X)$, it is easy to find its equivalent formula $\theta^{\prime}$, called standard form: in it each term between every two “$\vee$” is in the form: $l\wedge\bigwedge_{i\in I}x_{i}$ for some index set $I$ (if $l=1$, we always omit it and just write $\bigwedge_{i\in I}x_{i}$), $l$ is a element in $L-\{0\}$, and we call it “coefficient” of such term.

In fact, the factor impacting on the equivalence relation between formulas are their simplest final expansions: for above $\theta_{1}$ and $\theta_{2}$, they are equivalent because they have the identical simplest final expansions $0.5\vee(0.8\wedge x_{2})$. To be specific, we divide the procedures of obtaining the simplest final expansion for a given formula into the following steps:

$\mathbf{Step\ 1}$: Expand the formula;

$\mathbf{Step\ 2}$: Write above expansion in the standard form. In particular, there maybe exists a term $l$ ($l\in L$), called constant term, where its index set $I=\emptyset$;

$\mathbf{Step\ 3}$: If there exist two term $l_{1}\wedge\bigwedge_{i\in I}x_{i}$ and $l_{2}\wedge\bigwedge_{j\in J}x_{j}$ such that $l_{1}\leq l_{2}$ and $J\subseteq I$, then remove the former one (indeed the former one is absorbed in the latters in the calculations of runs’ weights).

Further on, we let $v(\mathbf{true},Y)=1$ for any set $Y$ (include empty set) and correspondingly, we let no set satisfy formula $\mathbf{false}$ (these settings are compatible with classic logic). Obviously, $\mathbf{true}\equiv 1$.

For $\theta$, we define its satisfaction sets: if there exists a term $l\wedge\bigwedge_{i\in I}x_{i}$ in the standard form of $\theta$, we call $\{x_{i}|i\in I\}$ satisfies $\theta$ with weight $l$. Moreover, if it is also in the simplest final expansion of $\theta$, we say $\{x_{i}|i\in I\}$ satisfies $\theta$ in a minimal manner with weight $l$. In particular, for the constant term $l^{\prime}$ in the simplest final expansion, we call $\emptyset$ satisfies $\theta$ in a minimal manner with weight $l^{\prime}$. Also for formulas $\theta_{1}$ and $\theta_{2}$ mentioned above, we know $\{x_{2},x_{3}\}$ satisfies $\theta_{1}$ and $\theta_{2}$ with weights $0.2$ and $0.3$ respectively; $\emptyset$ and $x_{2}$ satisfies $\theta_{1}$ and $\theta_{2}$ in a minimal manner with weights $0.5$ and $0.8$ respectively.

Considering an $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automaton $\mathcal{A}=(Q,\Sigma,\delta,\\ I,F)$, its transition function $\delta$ maps each state $q\in Q$ to an $L$-fuzzy set by inputting a symbol of $\Sigma$. We can represent $\delta$ by some formulas of $\mathbf{\mathcal{F_{L}B}}^{+}(Q)$: for example, $\delta(q,a)=\frac{l_{1}}{q_{1}}+\frac{l_{2}}{q_{2}}+\frac{l_{3}}{q_{3}}$ (sometimes, we also use $\delta(q,a)(q_{i})=l_{i}$ or $\delta(q,a,q_{i})=l_{i}$ ($i=1,2,3$) to characterize such transition) can be described as $\delta(q,a)=(l_{1}\wedge q_{1})\vee(l_{2}\wedge q_{2})\vee(l_{3}\wedge q_{3})$ of $\mathbf{\mathcal{F_{L}B}}^{+}(Q)$. Generally, in an $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton, the transitions can be any formula of $\mathbf{\mathcal{F_{L}B}}^{+}(Q)$.

For example, if $\delta(q,a)=(l_{1}\vee q_{2})\wedge q_{1}$, then $q$’s children are $l_{1}$ and $q_{1}$ or $q_{2}$ and $q_{1}$ after inputting $a$.

If the total weight of $r$ is not $0$, i.e., $weight(r)=I(r(\varepsilon))\wedge wt(r)\neq 0$, then we call $r$ an accepting run of $\mathcal{A}$, where $wt(r)$ is equal to the conjunction of all branches’ weights in $r$. The weight of a branch $\beta$ is defined by:

If it is finite, its weight equals to $l$ ($\in L$), the label of the leaf node;

If it is infinite, $\beta=x_{0},x_{1},\cdots$, and $r(x_{i})=q_{i}$, then $wt(\beta)$ equals to $\bigwedge\limits_{i\geq 0}\bigvee\limits_{j\geq i}F(q_{j})$.

Then for any $w\in\Sigma^{\omega}$, $L_{\omega}(\mathcal{A})(w)=\bigvee\limits_{r\in R_{\mathcal{A}}(w)}I(r(\varepsilon))\wedge wt(r)$, where $R_{\mathcal{A}}(w)$ denotes the set of all runs on $w$ of $\mathcal{A}$.

Here we give the corresponding construction (similar to [18, 19]): let $\mathcal{A}=(Q,\Sigma,\delta,I,F)$ be an $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton, define an automaton with a crisp initial state $\mathcal{A}^{\prime}$ as $(Q\cup\{q_{0}\},\Sigma,\delta^{\prime},q_{0},F)$, where $q_{0}\notin Q$, $\delta^{\prime}(q_{0},a)=\bigvee\limits_{I(q)\neq 0}I(q)\wedge\delta(q,a)$ and otherwise, $\delta^{\prime}(q,a)=\delta(q,a)$.

After introducing the basic definitions, we are ready to study the equivalence relation between fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automata and fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata over distributive lattices. Firstly, we show that $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automata are at least as expressive and as succinct as $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automata.

Proof. Let $\mathcal{A}=(Q,\Sigma,\delta,I,F)$ be the given $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automaton. Define an $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton $\mathcal{A}_{a}=(Q,\Sigma,\delta_{a},I,F)$: where $\delta_{a}(q,b)=\bigvee\limits_{\delta(q,b)(q^{\prime})=l_{q^{\prime}}\neq 0}l_{q^{\prime}}\wedge q^{\prime}$, $b\in\Sigma$, and otherwise, if $\delta(q,b)(q^{\prime})=0$ for any $q^{\prime}\in Q$, we set $\delta_{a}(q,b)=\mathbf{false}$.

Let $w$ be an arbitrary word of $\Sigma^{\omega}$ (denoted by $w=a_{1}a_{2}\cdots$) such that $L(\mathcal{A})(w)\neq 0$. Assume that $P$ is a run on $w$ of $\mathcal{A}$ such that $weight(P)\neq 0$, i.e., there a sequence of states $q,q_{1},q_{2},\cdots$ such that $I(q)\neq 0$, $\delta(q,a_{1},q_{1})\neq 0$ and $\delta(q_{i},a_{i+1},q_{i+1})\neq 0$ ($i\geq 1$), $\bigwedge\limits_{i\geq 0}\bigvee\limits_{j\geq i}F(q_{j})\neq 0$, then there exists a corresponding successful run tree $r$ on $w$ of $\mathcal{A}_{a}$ satisfying:

At $0$-th level of $r$, there is only one element $q$, and $I(r(\varepsilon))=I(q)\neq 0$;

At $1$-th level of $r$, there are two elements: a leaf node labeled by $\delta(q,a_{1},q_{1})$ and a non-leaf node labeled by $q_{1}$;

$\cdots$

At $i$-th level of $r$, there are two elements: a leaf node labeled by $\delta(q_{i-1},a_{i},q_{i})$ and a non-leaf node labeled by $q_{i}$;

$\cdots$.

Then we have,

Conversely, we can also show that $L(\mathcal{A})(w)=L(\mathcal{A}_{a})(w)$, for any $w\in\Sigma^{\omega}$ such that $L(\mathcal{A}_{a})(w)\neq 0$.  $\Box$

This part is easy to be obtained, and afterwards, we will turn to the other one. We divides it into two steps: firstly, we shall prove that any $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton with crisp final states can be transformed to an equivalent $L$-fuzzy nondeterministic $\mathrm{B\ddot{u}chi}$ automaton; Secondly, we will show that every $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton can be converted to another one with crisp final states. The next proposition shows the first step:

Proof. Let $\mathcal{A}=(Q,\Sigma,\delta,I,F)$ be an $L$-fuzzy alternating $\mathrm{B\ddot{u}chi}$ automaton, where $F$ is a crisp set of final states. Define $\mathcal{A}_{n}=(Q_{n},\Sigma,\delta_{n},I_{n},F_{n})$ as follows: $Q_{n}=2^{Q}\times 2^{Q}$; for any $q\in Q$, we let $I_{n}((\{q\},\emptyset))=I(q)$, and otherwise, $I_{n}((A,B))=0$, where $A,B\in 2^{Q}$; $F_{n}=2^{Q}\times\{\emptyset\}$;

For any $(U,V)\in Q_{n}$, $V\neq\emptyset$, and if $U=\{q_{1},\cdots,q_{s}\}$, $V=\{q^{\prime}_{1},\cdots,q^{\prime}_{m}\}(\subseteq U)$, we define $\delta_{n}$ by:

For any $(U,\emptyset)\in Q_{n}\times Q_{n}$, if $U=\{p_{1},\cdots,p_{k}\}$, then $\delta_{n}$ is defined as:

We take an empty conjunction in the definition of $\delta_{n}$ to be $1$, i.e., $\delta_{n}((\emptyset,\emptyset),a,\\ (\emptyset,\emptyset))=1$. In addition, the others not mentioned are defined to $0$.

On one hand, we need to prove that for any $w\in\Sigma^{\omega}$, if it satisfies $L_{\omega}(\mathcal{A}_{n})(w)\neq 0$, then $L_{\omega}(\mathcal{A}_{n})(w)=L_{\omega}(\mathcal{A})(w)$.

In fact, for any successful run $P:(\{q\},\emptyset)\stackrel{{\scriptstyle a_{1}}}{{\rightarrow}}(A_{1},B_{1})\stackrel{{\scriptstyle a_{2}}}{{\rightarrow}}(A_{2},B_{2})\rightarrow\cdots$, we can construct a run $r$ of $\mathcal{A}$:

Put $r(\varepsilon)=q$ firstly;

Let all states of $A_{1}$ ($A_{1}\triangleq\{q_{11},\cdots,q_{1s}\}$) be the children of $q$ occurring at $1$-th level of $r$;

If $B_{1}\triangleq\{\widehat{q}_{11},\cdots,\widehat{q}_{1m}\}\neq\emptyset$, we follow the steps below:

Let $\delta(q_{1i},a_{2})=\theta_{1i}$, $\delta(\widehat{q}_{1j},a_{2})=\widehat{\theta}_{1j}$, $i=1,\cdots,s$, $j=1,\cdots,m$, we choose sets $B_{2j}=\{\widehat{q}_{j1}^{\prime},\cdots,\widehat{q}_{jl_{j}}^{\prime}\}\subseteq A_{2}$ such that $B_{2j}$ satisfies $\widehat{\theta}_{1j}$ in a minimal manner with weight $\mu_{l_{j}}(a_{2})_{\widehat{q}_{j1}^{\prime}\cdots\widehat{q}_{jl_{j}}^{\prime},\widehat{q}_{1j}}$, $j=1,\cdots,m$, and $\bigcup\limits_{j=1}^{m}B_{2j}-F=B_{2}$. Meanwhile, we choose sets $A_{2i}=\{q_{i1}^{\prime},\cdots,q_{il_{i}}^{\prime}\}\subseteq A_{2}$ such that $A_{2i}$ satisfies $\theta_{1i}$ in a minimal manner with weight $\mu_{l_{i}}(a_{2})_{q_{i1}^{\prime}\cdots q_{il_{i}}^{\prime},q_{1i}}$, $i=1,\cdots,s$, $\bigcup\limits_{i=1}^{s}A_{2i}=A_{2}$ and if there are $t_{1},t_{2}$ such that $q_{1t_{1}}=\widehat{q}_{1t_{2}}$, then $A_{2t_{1}}=B_{2t_{2}}$. Then we let $\mu_{l_{i}}(a_{2})_{q_{i1}^{\prime}\cdots q_{il_{i}}^{\prime},q_{1i}},q_{i1}^{\prime},\cdots,q_{il_{i}}^{\prime}$ be the children of $q_{1i}$ occurring at $2$-th level of $r$.

If $B_{1}=\emptyset$, we just choosing sets $A_{2i}=\{q_{i1}^{\prime},\cdots,q_{il_{i}}^{\prime}\}\subseteq A_{2}$ such that $A_{2i}$ satisfies $\theta_{1i}$ in a minimal manner with weight $\mu_{l_{i}}(a_{2})_{q_{i1}^{\prime}\cdots q_{il_{i}}^{\prime},q_{1i}}$, $i=1,\cdots,s$, and $\bigcup\limits_{i=1}^{s}A_{2i}=A_{2}$, $\bigcup\limits_{i=1}^{s}A_{2i}-F=B_{2}$. Then we let $\mu_{l_{i}}(a_{2})_{q_{i1}^{\prime}\cdots q_{il_{i}}^{\prime},q_{1i}},q_{i1}^{\prime},\cdots,q_{il_{i}}^{\prime}$ be the children of $q_{1i}$ occurring at $2$-th level of $r$.

Similarly, the choices of other levels are considered.

We observe that even though the run tree constructed is not unique (under isomorphism), the disjunction of all these probabilities’ total weights is equal to $weight(P)=I_{n}((\{q\},\emptyset))\wedge\delta((\{q\},\emptyset),a_{1},(A_{1},\cdots,B_{1}))\wedge\bigwedge\limits_{i\geq 1}\delta((A_{i},B_{i}),a_{i+1},\\ (A_{i+1},B_{i+1}))$ (because $L$ is distributive). Then we have:

On the other hand, for any successful run $r$ of $\mathcal{A}$ on a infinite word $w=a_{1}a_{2}\cdots$, we can construct a run $P^{\prime}$ of $\mathcal{A}_{n}$:

$\widehat{A_{0}}=(r(\varepsilon),\emptyset)$;

$\widehat{A_{1}}=(A_{1},B_{1})$ (where $A_{1}=\{q|q\ is\ the\ child\ of\ r(\varepsilon)\}$, $B_{1}=\{q|q\ is\ the\\ \ child\ of\ r(\varepsilon)\}-F$);