Preservation of normality by transducers
aa r X i v : . [ c s . F L ] A p r Preservation of normality by transducers
Olivier Carton Elisa OrdunaOctober 31, 2018
Abstract
We consider input-deterministic finite state transducers with infiniteinputs and infinite outputs, and we consider the property of Borel nor-mality on infinite words. When these transducers are given by a stronglyconnected set of states, and when the input is a Borel normal sequence,the output is an infinite word such that every word has a frequency givenby a weighted automaton over the rationals. We prove that there isan algorithm that decides in cubic time whether an input-deterministictransducer preserves normality.
Keywords: transducers, weighted automata, normal sequences
We start with the definition of normality for real numbers, given by ´EmileBorel [6] more than one hundred years ago. A real number is normal to aninteger base if, in its infinite expansion expressed in that base, all blocks ofdigits of the same length have the same limiting frequency. Borel proved thatalmost all real numbers are normal to all integer bases. However, very little isknown on how to prove that a given number has the property.The definition of normality was the first step towards a definition of random-ness. Normality formalizes the least requirements about a random sequence. Itis indeed expected that in a random sequence, all blocks of symbols with thesame length occur with the same limiting frequency. Normality, however, isa much weaker notion than the one of purely random sequences defined byMartin-L¨of.The motivation of this work is the study of transformations preservingrandomness, hence preserving normality. The paper is focused on very simpletransformations, namely those that can be realized by finite-state machines. Weconsider input-deterministic automata with outputs, also known as sequentialtransducers mapping infinite sequences of symbols to infinite sequences of sym-bols. The main result is that it can be decided in cubic time whether sucha machine preserves or not normality. Preserving normality means that theoutput sequence is normal whenever the input sequence is.The main result is obtained through a second result involving weightedautomata. This second result states that if a sequential transducer is stronglyconnected, then the frequency of each block in the output of a run with anormal input is given by a weighted automaton on rational numbers. It implies,in particular, that the frequency of each block in the output sequence does notdepend on the input as long as this input sequence is normal.1his is not the first result linking normality and automata. A fundamentaltheorem relates normality and finite automata: an infinite word is normalto a given alphabet if and only if it cannot be compressed by lossless finitetransducers. These are deterministic finite automata with injective input-outputbehaviour. This result was first obtained by joining a theorem by Schnorr andStimm [14] with a theorem by Dai, Lathrop, Lutz and Mayordomo [12]. Becherand Heiber gave a direct proof [3].Agafonov’s Theorem [1] is another striking result relating normality andautomata. It establishes that oblivious selection of symbols with a regular setpreserves normality. Oblivious selection with a regular set L means that asymbol a i is selected whenever the prefix a · · · a i − belongs to L . This obliviousselections can actually be realized by sequential transducers considered in thiswork.The paper is organized as follows. Notions of normal sequences and trans-ducers are introduced in Section 2. Main results are stated in Section 3. Proofsof the results and algorithms are given in Section 4. Before giving the formal definition of normality, let us introduce some simpledefinitions and notation. Let A be a finite set of symbols that we refer to asthe alphabet . We write A ω for the set of all infinite words on the alphabet A and A ∗ for the set of all finite words. The length of a finite word w is denotedby | w | . The positions of finite and infinite words are numbered starting from 1.To denote the symbol at position i of a word w we write w [ i ], and to denotethe substring of w from position i to j inclusive we write w [ i . . . j ]. The emptyword is denoted by λ . The cardinality of a finite set E is denoted by E .Given two words w and v in A ∗ , the number | w | v of occurrences of v in w isdefined by: | w | v = { i : w [ i . . . i + | v | −
1] = v } . For example, | abbab | ab = 2.Given a finite word w ∈ A + and an infinite word x ∈ A ω , we refer to the frequency of w in x as freq( x, w ) = lim n →∞ | x [1 . . . n ] | w n when this limit is well-defined.An infinite word x ∈ A ω is normal on the alphabet A if for every word w ∈ A ∗ : freq( x, w ) = 1( A ) | w | An alternative definition of normality can be given by counting aligned occurrences, and it is well-known that they are equivalent (see for example [2]).We refer the reader to [8, Chap.4] for a complete introduction to normality.The most famous example of a normal word is due to Champernowne [10],who showed in 1933 that the infinite word obtained from concatenating all the2atural numbers (in their usual order):0123456789101112131415161718192021222324252627282930 . . . is normal on the alphabet { , , . . . , } . In this paper we consider automata with outputs, also known as transduc-ers. Such finite-state machines are used to realize functions mapping wordsto words and especially infinite words to infinite words. We only considerinput-deterministic transducers, also known as sequential in the literature (See[13, Sec. V.1.2] and [4, Chap. IV]). Each transition of these transducers consumesexactly one symbol of their input and outputs a finite word which might beempty. Furthermore, ignoring the output label of each transition must yield adeterministic automaton.More formally a transducer T is a tuple h Q, A, B, δ, q i , where Q is a finiteset of states, A and B are the input and output alphabets respectively, δ ⊆ Q × A × B ∗ × Q is a finite transition relation and q ∈ Q is the initial state. Atransition is a tuple h p, a, v, q i in Q × A × B ∗ × Q and it is written p a | v −−→ q .1 23 a | a b | λ a | λb | bb a | λb | ba Figure 1: A deterministic complete transducerA transducer T is input-deterministic , or sequential for short, if whenever p a | v −−→ q and p a | v ′ −−→ q ′ are two of its transitions, then q = q ′ and v = v ′ . Atransducer T is complete if for each symbol a ∈ A and each state p ∈ Q there isa transition from p and consuming a , that is, there exists a transition p a | v −−→ q .A finite (respectively infinite) run in T is a finite (respectively infinite)sequence of consecutive transitions, q a | v −−−→ q a | v −−−→ · · · q n − a n | v n −−−−→ q n Its input and output labels are the words a a . . . a n and v v · · · v n respectively.Note that there is no accepting condition and note also that the output label ofan infinite run might be finite since the output label of some transitions mightbe empty. An infinite run is accepting if its first state is initial and its outputlabel is infinite. If T is an input-deterministic transducer, each infinite word x is the input label of at most one accepting run in T . When this run does exist,its output is denoted by T ( x ).Each transducer T can be seen as a graph by ignoring the labels of itstransitions. For this reason, we may consider strongly connected components T . Using the terminology of Markov chains, a strongly connectedcomponent is called recurrent if no transition leaves it.We say that a input-deterministic transducer T preserves normality if foreach normal word x , T ( x ) is also normal. We now introduce weighted automata. In this paper we only consider weightedautomata whose weights are rational numbers with the usual addition andmultiplication (See [13, Chap. III] for a complete introduction).A weighted automaton A is a tuple h Q, B, ∆ , I, F i , where Q is the state set, B is the alphabet, I : Q → Q and F : Q → Q are the functions that assign toeach state an initial and a final weight and ∆ : Q × B × Q → Q is a functionthat assigns to each transition a weight.As usual, the weight of a run is the product of the weights of its transitionstimes the initial weight of its first state and times the final weight of its laststate. Furthermore, the weight of a word w ∈ B ∗ is the sum of the weights ofall runs with label w . q q p a −→ q withweight x is pictured p a : x −−→ q . Non-zero initial and final weights are given oversmall incoming and outgoing arrows. The weight of the run q −→ q −→ q −→ q −→ q is 1 · · · · · w = 1010 is 8 + 2 = 10.More generally the weight of a word w = a · · · a k is the integer n = P ki =1 a i k − i ( w is a binary expansion of n with possibly some leading zeros). We now state the main results of the paper. The first one states that when atransducer is strongly connected, deterministic and complete, the frequency ofeach finite word w in the output of a run with a normal input label is given bya weighted automaton over Q . The second one states that it can be checked incubic time whether an input-deterministic transducer preserves normality. Theorem 1.
Given a transducer T = h Q, A, B, δ, q i which is strongly con-nected, deterministic and complete, there exists a weighted automaton A suchthat for any normal word x and for any finite word w , freq( T ( x ) , w ) = weight A ( w ) .Furthermore, the weighted automaton A can be computed in cubic time withrespect to the size of the transducer T . x is the input label of an infinite run. Nevertheless, this run may not beaccepting since its output label is not necessarily infinite. The really restrictivehypothesis is that the transducer must be input-deterministic.To illustrate Theorem 1 we give in Figure 3 a weighted automaton A whichcomputes the frequency of each finite word w in T ( x ) for a normal input x andthe transducer T pictured in Figure 1. States 2 and 3 are useless and could beremoved since their initial weight is zero and they have no incoming transitions.They have been kept because the automaton pictured in Figure 3 is the onegiven by the procedure described in the next section.1 234 52 / / /
611 1 11 a : 1 / b : 1 / b : 1 / b : 1 / b : 1 / b : 1 b : 1 a : 1Figure 3: A weighted automaton for the transducer pictured in Fig. 1 Theorem 2.
Given a transducer T = h Q, A, B, δ, q i which is complete anddeterministic, it can be decided in cubic time with respect to the size of T whether T preserves normality. In this section we provide the proofs for Theorems 1 and 2. The next propositionshows that it suffices to independently analyze each recurrent strongly connectedcomponent of the transducer.
Proposition 3.
A deterministic and complete transducer preserves normalityif and only if each of its recurrent strongly connected components preservesnormality.
The previous proposition follows directly from the next lemma which isSatz 2.5 in [14].
Lemma 4.
A run labeled with a normal word in a deterministic and completeautomaton always reaches a recurrent strongly connected component.
By Proposition 3 it suffices to analyze preservation of normality in eachrecurrent strongly connected component. In what follows we mainly considerstrongly connected transducers. If all transitions have an empty output label,the output of any run is empty and the transducer does not preserve normality.5herefore, we assume that transducers have at least one transition with a nonempty output label. By Lemmas 5 and 6, this transition is visited infinitelyoften if the input is normal because all entries of the stationary distribution arepositive [15, Thm 1.1(b)]. This guarantees that each normal word x is the inputlabel of an accepting run and that T ( x ) is well-defined.Some frequencies are obtained as stationary distributions of Markov chains[7, Thm 4.1]. For that purpose, we associate a Markov chain M to each stronglyconnected automaton A . For simplicity, we assume that the state set Q of A is the set { , . . . , Q } . The state set of the Markov chain is the same set { , . . . , Q } . The transition matrix of the Markov chain is the matrix P =( p i,j ) ≤ i,j ≤ Q where each entry p i,j is equal to { a : i a −→ j } / A . Note that { a : i a −→ j } is the number of transitions from i to j . Since the automatonis assumed to be deterministic and complete, the matrix P is stochastic. Ifthe automaton A is strongly connected, the Markov chain is irreducible and ithas therefore a unique stationary distribution π such that πP = π . Note thatall entries of P and π are rational numbers which can be effectively computedfrom A . The vector π is called the distribution of A . This definition as well asLemmas 5 and 6 below also apply to input-deterministic transducers by ignoringthe output labels.Each run of either an automaton or a transducer can be seen as a sequenceof transitions. Therefore, the notion of the frequency freq( ρ, γ ) of a finite run γ in an infinite run ρ is defined as in Section 2. Note that freq( ρ, γ ) is a limit andmight not exist. This notion applies also to states seen as runs of length 0.The following lemma which is Lemma 4.5 in [14] states that if the automa-ton A is strongly connected, then the run on a normal input visits each statewith a frequency. Moreover, these frequencies are independent of the input aslong as it is normal. Lemma 5.
Let A be a deterministic and complete automaton which is stronglyconnected and let ρ be a run in A labeled by a normal word. Then the frequency freq( ρ, q ) is equal to π q for each state q where π is the stationary distribution ofthe Markov chain associated to A . Let γ be a finite run whose first state is p and let ρ be an infinite run. Wecall conditional frequency of γ in ρ the ratio freq( ρ, γ ) / freq( ρ, p ). It is definedas soon as both frequencies freq( ρ, γ ) and freq( ρ, p ) do exist. Lemma 6.
Let A be a deterministic and complete automaton which is stronglyconnected and let ρ be a run in A labeled by a normal word. The conditionalfrequency of each run of length k is / ( A ) k .Proof. Let p ∗ w denote the unique run starting in state p with label w . Wenow define an automaton whose states are the runs of length n in A . We let A n denote the automaton whose state set is { p ∗ w : p ∈ Q, w ∈ A n } and whoseset of transitions is defined by (cid:8) ( p ∗ bw ) a −→ ( q ∗ wa ) : p b −→ q in A , a, b ∈ A and w ∈ A n − (cid:9) The Markov chain associated with the automaton A n is called the snake Markovchain. See Problems 2.2.4, 2.4.6 and 2.5.2 (page 90) in [7] for more details. Itis pure routine to check that the stationary distribution ξ of A n is given by ξ p ∗ w = π p / ( A ) n for each state p and each word w of length n and where π
6s the stationary distribution of A . To prove the statement, apply Lemma 5 tothe automaton A n .The output labels of the transitions in T may have arbitrary lengths. We firstdescribe the construction of an equivalent transducer T ′ such that all outputlabels in T ′ have length at most 1. We call this transformation normalization and it consists in replacing each transition p a | v −−→ q in T such that | v | ≥ n transitions: p a | b −−→ q λ | b −−→ q · · · q n − λ | b n −−−→ q where q , q , . . . , q n − are new states and v = b · · · b n . We refer to p as theparent of q , · · · , q n − .To illustrate the construction, the normalized transducer obtained from thetransducer of Figure 1 is pictured in Figure 4.1 234 5 a | a b | λ a | λb | bλ | b a | λb | bλ | a Figure 4: The normalized transducer of the transducer pictured in Fig. 1From the normalized transducer T ′ we construct a weighted automaton A with the same state set as T ′ . We define transitions between every pair of states p, q for each symbol b in B , that is the transition p b −→ q is defined for all states p, q and for every symbol b ∈ B . To assign weight to transitions in A , we firstassign weights to transitions in T ′ as follows. Each transition starting from astate in T (and having a symbol as input label) has weight 1 / A and eachtransition starting from a newly added state (and having the empty word asinput label) has weight 1. Note that for each state p in T ′ , the sum of weightsof transitions starting from p is 1. We now consider separately transitions thatgenerate empty output from those that do not. Consider the Q × Q matrix E whose ( p, q )-entry is given for each pair ( p, q ) of states by E p,q = X a ∈ A weight T ′ ( p a | λ −−→ q ) . Let E ∗ be the matrix defined by E ∗ = P k ≥ E k . Hence the entry E ∗ p,q is the sumof weights of all finite runs with empty output going from p to q . The matrix E ∗ can be computed because it is the solution of the system E ∗ = EE ∗ + I where I is the identity matrix. This proves in particular that all its entries are rationalnumbers. 7or each symbol b ∈ B consider the Q × Q matrix N b whose ( p, q )-entry isgiven for each pair ( p, q ) of states by( N b ) p,q = X a ∈ A ∪{ λ } weight T ′ ( p a | b −−→ q ) . We define the weight of a transition p b −→ q in A asweight A ( p b −→ q ) = ( E ∗ N b ) p,q . To assign initial weight to states we consider the Markov chains M whosetransition matrix is the stochastic matrix P = P b ∈ B E ∗ N b . The fact that thismatrix is indeed stochastic follows from the observation that, for each state p ,the set of input labels of the runs in S q ∈ Q,b ∈ B Γ p,b,q (see below for the definitionof Γ p,b,q ) is a maximal prefix code and Proposition 3.8 in [5, Chap. II.3]. Theinitial vector of A is the stationary distribution π of M , that is, the line vector π such that πP = π . We assign to each state q the initial weight π q . Finally weassign final weight 1 to all states.We give below the matrices E , E ∗ , N a , N b and P and the initial vector π ofthe weighted automaton obtained from the transducer pictured in Figure 4. E = / / / and E ∗ = / / N a = / and N b = /
20 0 0 1 / P = / / /
40 0 0 1 / /
20 0 0 0 11 0 0 0 01 0 0 0 0 and π = (cid:2) / / / (cid:3) Proposition 7.
The automaton A computes frequencies, that is, for everynormal word x and any finite word w in B ∗ , weight A ( w ) = freq( T ( x ) , w ) . The proof of the proposition requires some preliminary results. The followinglemma is straightforward and follows directly from the normalization of T into T ′ . Lemma 8.
Both transducers T and T ′ realize the same function, that is, T ( x ) = T ′ ( x ) for any infinite word x . Let us recall that a set of words L (resp. runs) is called prefix-free if noword in L is a proper prefix of another word in L . Let Γ be a set of finite runsand ρ be an infinite run. The limit freq( ρ, Γ) is defined as the limit as n goes8o infinity of the ratio between the number of occurrences of runs in Γ in theprefix of length n of ρ and n . If Γ is prefix-free (not to count twice the samestart), the following equality holdsfreq( ρ, Γ) = X γ ∈ Γ freq( ρ, γ )assuming that each term of the right-hand sum does exist. If Γ is a set of finiteruns starting from the same state p , the conditional frequency of Γ in a run ρ is defined as the ratio between the frequency of Γ in ρ and the frequency of p in ρ , that is, freq( ρ, Γ) / freq( ρ, p ). Furthermore if Γ is prefix-free, the conditionalfrequency of Γ is the sum of the conditional frequencies of its elements.Let x be a fixed normal word and let ρ and ρ ′ be respectively the runs in T and T ′ with label x . By Lemma 5, the frequency freq( ρ, q ) of each state q is π q where is the stationary distribution of T . The following lemma gives thefrequency of states in ρ ′ . Lemma 9.
There exists a constant C such that if q is a state of T , then freq( ρ ′ , q ) = freq( ρ, q ) /C and if q is newly created, then freq( ρ ′ , q ) = freq( ρ ′ , p ) / A where p is the parent of q .Proof. Observe that there is a one-to-one relation between runs labeled withnormal words in T and in T ′ . More precisely, each transition τ in ρ is replacedby max(1 , | v τ | ) transitions in ρ ′ (where v τ is the output label of τ ).By combining Lemmas 5 and 6, each transition of T has a frequency in ρ .The first result follows by taking C = P τ freq( ρ, τ ) · max(1 , | v τ | ) where thesummation is taken over all transitions τ of T and v τ is implicitly the outputlabel of τ . The second result follows from Lemma 6 stating that transitionshave a conditional frequency of 1 / A in ρ .For each pair ( p, q ) of states and each symbol b ∈ B , consider the set Γ p,b,q of runs from p to q in T ′ that have empty output labels for all their transitionsbut the last one, which has b as output label.Γ p,b,q = { p a | λ −−−→ · · · a n | λ −−−→ q n a n +1 | b −−−−→ q : n ≥ , q i ∈ Q, a i ∈ A ∪ { λ }} and let Γ be the union S p,q ∈ Q,b ∈ B Γ p,b,q . Note that the set Γ is prefix-free.Therefore, the run ρ ′ has a unique factorization ρ = γ γ γ · · · where each γ i is a finite run in Γ and the ending state of γ i is the starting state of γ i +1 . Let( p i ) i ≥ and ( b i ) i ≥ be respectively the sequence of states and the sequence ofsymbols such that γ i belongs to Γ p i ,b i ,p i +1 for each i ≥
0. Let us call ρ ′′ thesequence p p p · · · of states of T ′ . Lemma 10.
For each state q of T ′ , the frequency freq( ρ ′′ , q ) does exist.Proof. The sequence ρ ′′ is a subsequence of the sequence of states in the run ρ ′ .An occurrence of a state q in ρ ′ is removed whenever the output of the previoustransition is empty.Consider the transducer ˆ T obtained by splitting each state q of T into twostates q λ and q o in such a way that transitions with an empty output labelend in a state q λ and other transitions end in a state q o . Then each transitiontransition p a | v −−→ q is replaced by either the two transitions p λ a | v −−→ q λ and9 o a | v −−→ q λ if v is empty or by the two transitions p λ a | v −−→ q o and p o a | v −−→ q o otherwise. The state q λ becomes the new initial state and non reachable statesare removed. Let ˆ ρ be the run in ˆ T labeled by x . By Lemma 5, the frequenciesfreq(ˆ ρ, q λ ) and freq(ˆ ρ, q o ) do exist. Now consider the normalization ˆ T ′ of ˆ T andthe run ˆ ρ ′ in ˆ T ′ labeled by x . By a lemma similar to Lemma 9, the frequenciesfreq(ˆ ρ ′ , q λ ) and freq(ˆ ρ ′ , q o ) do exist. The sequence ρ ′′ is obtained from ˆ ρ ′ byremoving each occurrence of states q λ and keeping occurrences of states q o . Itfollows that the frequency of each state does exist in ρ ′′ . Proof of Proposition 7.
By Lemma 6, the conditional frequency of each finiterun γ of length n in ρ is 1 / ( A ) n . It follows that the conditional frequency ofeach finite run γ ′ in ρ ′ is equal to its weight in T ′ as defined auxiliary whendefining the weights of transitions in the weighted automaton A . This provesthat the weight of the transition p b −→ q in A is exactly the conditional frequencyof the set Γ p,b,q for each triple ( p, b, q ) in Q × B × Q . More generally, theproduct of the weights of the transitions p b −→ p · · · p n − b n −→ p n is equal tothe conditional frequency of the set Γ p ,b ,p · · · Γ p n ,b n ,p n +1 in ρ ′ .It remains to prove that the frequency of each state q in ρ ′′ is indeed itsinitial weight in the automaton A . Let us recall that the initial vector of A isthe stationary distribution of the stochastic matrix P whose ( p, q )-entry is thesum P b ∈ B weight A ( p b −→ q ), which is the conditional frequency of pq (as a wordof length 2) in ρ ′′ . It follows that the frequencies of states in ρ ′′ must be thestationary of the matrix P .Since the frequency of a word v = b · · · b n in T ′ ( x ) is the same as the sumover all sequences p , p . . . , p n +1 of the frequencies of Γ p ,b ,p · · · Γ p n ,b n ,p n +1 in ρ ′ , it is the weight of the word v in the automaton A . Proofs of Theorems 1 and 2.
To complete the proof of Theorems 1 and 2, weexhibit an algorithm deciding in cubic time whether an input-deterministictransducer preserves normality. Let T be the transducer h Q, A, B, δ, { q }i . Bydefinition, its size is the sum P τ ∈ δ | τ | , where the size of a single transition τ = p a | v −−→ q is | τ | = | av | . We consider the alphabets to be fixed so they are nottaken into account when calculating complexity.111 b : 1 /n ... b n : 1 /n Figure 5: Weighted automaton B such that weight B ( w ) = 1 / ( A ) | w | By Proposition 3, the algorithm decomposes the transducer into stronglyconnected components and checks that each recurrent one does preserve nor-mality. This is achieved by computing the weighted automaton A and checkingthat the weight of each word w is 1 / ( A ) | w | . This latter step is performedby comparing A with the weighted automaton B such that weight B ( w ) =1 / ( A ) | w | . The automaton B is pictured in Figure 5.10 nput: T = h Q, A, B, δ, q i an input-deterministic complete transducer. Output:
True if T preserves normality and False otherwise.
Procedure:
I. Compute the strongly connected components of T II. For each recurrent strongly connected component S i of T :1. Compute the normalized transducer T ′ , equivalent to S i .2. Use T ′ to build the weighted automaton A :a. Compute the weights of the transitions of A .Compute the matrix E Compute the matrix E ∗ solving ( I − E ) X = I For each b ∈ B , for each p, q ∈ Q :compute the matrix N b define the transition p b −→ q with weight ( E ∗ N b ) p,q .b. Compute the stationary distribution π of the Markov chaininduced by A .c. Assign initial weight π [ i ] to each state i , and let final weightbe 1 for all states.3. Compare A against the automaton B using Sch¨utzenberger’salgorithm [9] to check whether they realize the same function.4. If they do not compute the same function, return False .III. Return
True
Now we analyze the complexity of the algorithm. Computing recurrentstrongly connected components can be done in O ( | Q | ) ≤ O ( n ) using Kosaraju’salgorithm is if the transducer is implemented with an adjacency matrix [11,Section 22.5].We refer to the size of the component | S i | as n i . The cost of normalizing thecomponent is O ( n i ), mainly from filling the new adjacency matrix. The mostexpensive step when computing the transitions and their weight is to compute E ∗ . The cost is O ( n i ) to solve the system of linear equations. To compute theweights of the states we have O ( n i ) to solve the system of equations to find thestationary distribution. Comparing the automaton to the one computing theexpected frequencies can be done in time O ( n i ) [9] since the coefficients of bothautomata are in Q . Acknowledgements
The authors would like to thank Ver´onica Becher for many fruitful discussionsand suggestions. Both authors are members of the Laboratoire InternationalAssoci´e INFINIS, CONICET/Universidad de Buenos Aires–CNRS/Universit´eParis Diderot and they are supported by the ECOS project PA17C04. Cartonis also partially funded by the DeLTA project (ANR-16-CE40-0007).11 eferences [1] V. N. Agafonov. Normal sequences and finite automata.
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