Table Of ContentIndex appearance record for transforming
Rabin automata into parity automata
Jan Křetínský, Tobias Meggendorfer, Clara Waldmann, and
Maximilian Weininger
7
1 Technical Universityof Munich
0
2
Abstract. Transformingdeterministicω-automataintodeterministicpar-
n
ity automata is traditionally done using variants of appearance records.
a
We present a more efficient variant of this approach, tailored to Rabin
J
automata,andseveraloptimizationsapplicabletoallappearancerecords.
0
We compare the methods experimentally and find out that our method
2
produces smaller automata than previous approaches. Moreover, the ex-
periments demonstrate the potential of our method for LTL synthesis,
]
O using LTL-to-Rabin translators. It leads to significantly smaller parity
automata when compared to state-of-the-art approaches on complex for-
L
mulae.
.
s
c
[
1 Introduction
1
v Constructing correct-by-design systems from specifications given in linear tem-
8
poral logic (LTL) [Pnu77] is a classical problem [PR89], called LTL synthesis.
3
The automata-theoretic solution to this problem is to translate the LTL for-
7
5 mula to a deterministic automaton and solve the corresponding game on the
0 automaton. Although different kinds of automata can be used, a reasonable
. choice would be parity automata (DPA) due to the practical efficiency of par-
1
0 ity game solvers [FL09,ML16] and the fact they allow for optimal memoryless
7 strategies. The bottleneck is thus to create a reasonably small DPA. The classi-
1 calwaytotransformLTLformulaeintoDPAistofirstcreateanon-deterministic
:
v Büchi automaton (NBA) and then determinize it, as implemented in ltl2dstar
i [KB06].Sincedeterminizationprocedures[Pit06,Sch09]basedonSafra’sconstruc-
X
tion [Saf88] are practically inefficient, many alternative approaches to LTL syn-
r
thesis arose, trying to avoid determinization and/or focusing on fragments of
a
LTL,e.g.[KV05,PPS06,AL04].However,newresultsontranslatingLTLdirectly
and efficiently into deterministic automata [KE12,EK14] open new possibilities
for the automata-theoretic approach. Indeed, tools such as Rabinizer [KK14]
or LTL3DRA [BBKS13] can produce practically small deterministic Rabin au-
tomata(DRA).Consequently,thetaskistoefficientlytransformDRAintoDPA,
which is the aim of this paper.
Transformations of deterministic automata into DPA are mostly based on
appearance records [GH82]. For instance, for deterministic Muller automata, we
want to track which states appear infinitely often and which do not. In order to
do that, the state appearance record keeps a permutation of the states, ordered
accordingtotheirmostrecentvisits,seee.g.[Sch01].Incontrast,fordeterministic
Streett automata (DSA) we only want to track which sets of states are visited
infinitelyoftenandwhichnot.Consequently,indexappearancerecord (IAR)keeps
apermutationofthesesetsofinterestinstead,whicharetypicallyveryfew.Such
a transformation has been given first in [Saf92] from DSA to DRA only (not
DPA, which is a subclass of DRA). Fortunately, this constructioncan be further
modified into a transformation of DSA to DPA, as shown in [Löd99b].
Since 1) DRA and DSA are syntactically the same, recognizing the comple-
ment languages of each other, and 2) DPA can be complemented without any
cost, one can apply the IAR of [Löd99b] to DRA, too. However, we design an-
otherIAR,whichismorenaturalfromtheDRApointofview,asopposedtothe
DSA perspective taken in [Löd99b]. This is in spirit more similar to a sketch of
a construction suggested in [FEK11]. Surprisingly, we have found that the DRA
perspective yields an algorithm producing considerably smaller automata than
the DSA perspective.
Our contribution in this paper is as follows:
– We provide an IAR construction transforming DRA to DPA.
– We present optimizations applicable to all appearance records.
– We evaluate all the unoptimized and optimized versions of our IAR and the
IARof[Löd99b]experimentally,incomparisontotheprocedureimplemented
in GOAL [TTH13].
Rabinizer optimizedIAR
– We compare our approachLTL−−−−−−→DRA−−−−−−−−−→DPA to the state-
of-the-art translation of LTL to DPA by Spot 2.1 [DLLF+16], which mixes
the construction of [Red12] with some optimizations of ltl2dstar [KB06]
andof their own.The experiments show thatfor more complex formulaeour
method produces smaller automata.
2 Preliminaries on ω-automata
We recall basic definitions of ω-automata and establish some notation.
2.1 Alphabets and words
An alphabet is any finite set Σ. The elements of Σ are called letters. A word is
a (possibly infinite) sequence of letters. The set of all infinite words is denoted
by Σω. A set of words L ⊆ Σω is called (infinite) language. The i-th letter of a
word w∈Σω is denoted by w , i.e. w =w w ....
i 0 1
2.2 Transition systems
Adeterministic transition system (DTS)T isgivenbyatuple(Q,Σ,δ,q )where
0
Q is a set of states, Σ is an alphabet, δ is a transition function δ : Q×Σ → Q
which may be partial (due to technical reasons) and q ∈ Q is the initial state.
0
The transition function induces the set of transitions ∆={hq,a,q′i|q ∈Q,a∈
Σ,q′ = δ(p,a)}. For a transition t = hq,a,q′i ∈ ∆ we say that t starts at q,
moves under a and ends in q′. A sequence of transitions ρ is a run of a DTS T
2
on a word w ∈ Σω if ρ starts at q , ρ moves under w for each i ≥ 0 and ρ
0 0 i i i+1
starts at the same state as ρ ends for each i ≥0. We write T(w) to denote the
i
uniquerunofT onw,ifitexists.Atransitiontoccurs inρifthereissomeiwith
ρ = t. By Inf(ρ) we denote the set of all transitions occurring infinitely often
i
in ρ. Additionally, we extend Inf to words by defining Inf (w)=Inf(T(w)) if T
T
has a run on w. If T is clear from the context, we write Inf(w) for Inf (w).
T
2.3 Acceptance conditions and ω-automata
An acceptance condition for T is a positive Boolean formula over the formal
variables V ={Inf(T),Fin(T)|T ⊆∆}. Acceptance conditions are interpreted
∆
over runs as follows. Given a run ρ of T and such an acceptance condition α,
we consider the truth assignment that sets the variable Inf(T) to true iff ρ visits
(some transition of) T infinitely often, i.e. Inf(ρ)∩T 6= ∅. Dually, Fin(T) is set
to true iff ρ visits every transition in T finitely often, i.e. Inf(ρ)∩T = ∅. A run
ρ satisfies α if this truth-assignment evaluates α to true.
A deterministic ω-automaton over Σ is a tuple A = (Q,Σ,δ,q ,α), where
0
(Q,Σ,δ,q ) is a DTS and α is an acceptance condition for it. An automaton A
0
accepts awordw ∈Σω iftherunoftheautomatononwsatisfiesα.Thelanguage
ofA,denotedbyL(A),isthesetofwordsacceptedbyA.Anacceptancecondition
α is a
k k
– Rabin condition {(F ,I )} if α = (Fin(F )∧Inf(I )). Each (F ,I ) is
i i i=1 i=1 i i i i
called a Rabin pair, where the F and I are called the prohibited set and the
i i
W
required set respectively.
– generalized Rabin condition {(F ,{Ij}ki )}k if the acceptance condition is
i i j=1 i=1
oftheformα= n (Fin(F )∧ ki Inf(Ik)).ThisgeneralizestheRabincon-
i=1 i j=1 j
dition, where each k =1. Furthermore, every generalized Rabin automaton
i
W V
can be de-generalized into an equivalent Rabin automaton, which however
may incur an exponential blow-up [KE12].
k k
– Streett condition {(F ,I )} if α = (Inf(F )∨Fin(I )). Note that the
i i i=1 i=1 i i
Streett condition is exactly the negation of the Rabin condition and thus an
V
automatonwithaRabinconditioncanbeinterpretedasaStreettautomaton
recognizing exactly the complement language.
k
– Rabin chain condition {(F ,I )} if it is a Rabin condition and F ⊆ I ⊆
i i i=1 1 1
··· ⊆ F ⊆ I . A Rabin chain condition is equivalent to a parity condition,
k k
specified by a priority assignment λ : ∆ → N. Such a parity condition is
satisfied by a run ρ iff the maximum priority of all infinitely often visited
transitions max{λ(q)|q ∈Inf(ρ)} is even.
A deterministic Rabin, generalizedRabin, Street or parity automaton is a deter-
ministic ω-automaton with an acceptance condition of the corresponding kind.
In the rest of the paper we use the corresponding abbreviations DRA, DGRA,
DSA and DPA.
k
Furthermore, given a DRA with an acceptance set {(F ,I )} and a word
i i i=1
w ∈Σω, we write F ={F |F ∩Inf(w)6=∅} and I ={I |I ∩Inf(w)6=∅}
inf i i inf i i
to denote the set of all infinitely often visited prohibited and required sets, re-
spectively.
3
3 Index appearance record
In order to translate (state-based acceptance) Muller automata to parity au-
tomata, a construction called latest appearance record has been devised1. In
essence, the constructed state space consists of permutations of all states in the
original automaton. In each transition, the state which has just been visited is
moved to the front of the permutation. From this, one can deduce the set of all
infinitely often visited states by investigating which states change their position
in the permutation infinitely often along the run of the word. Such a constraint
can be encoded as parity condition.
However, this approach comes with a very fast growing state space, as the
amount of permutations grows exponentially. Moreover, applying this idea to
transition based acceptance leads to even faster growth, as there usually are a
lotmoretransitionsthanstates.Incontrastto Mullerautomata,the exactsetof
infinitelyoftenvisitedtransitionsisnotneededtodecideacceptanceofawordby
a Rabinautomaton.It is sufficient to know whichofthe prohibitedand required
sets are visited infinitely often. Hence, index appearance record uses the indices
of the Rabin pairs instead of particular states in the permutation construction.
This provides enough information to decide acceptance.
Weintroducesomeformalitiesregardingpermutations:Foragivenn∈N,we
use Πn to denote the set ofall permutations of N ={1,...,n}, i.e. the set of all
bijective functions π : N → N. We identify π with its canonical representation
as a vector (π(1),...,π(n)). In the following, we will often say “the position of
F inπ”orsimilarto refertothe positionofiina particularπ,i.e.π−1(i).With
i
this, we define our variant of the index appearance record construction. Note
that in contrast to previous constructions, ours is transition based, which also
has a positive effect on the size of the produced automata, as discussed in our
experimental results.
Definition 1 (Transition-based index appearance record for Rabin au-
tomata). Let R = (Q,Σ,δ,q ,{(F ,I )}k ) be a Rabin automaton. Then the
0 i i i=1
index appearance record automaton IAR(R) = (Q˜,Σ,δ˜,q˜ ,λ) is defined as the
0
parity automaton with
– Q˜ =Q×Πk.
– q˜ =(q ,(1,...,k)).
0 0
– δ˜((q,π),a) = (δ(q,a),π′) where π′ is the permutation obtained from π by
movingallindicesof prohibitedsetsvisitedbythetransitiont=hq,a,δ(q,a)i
to the front. Formally, let Move = {i | t ∈ F } be the set of positions of
π(i)
currently visited prohibited sets. If Move = ∅, define π′ = π, otherwise let
n=|Move| and Move={i ,...,i }. With this
1 n
i if j ≤n
π′(j)= j
(π(j−n+|{i∈Move|i≤j}|) otherwise.
1 Originally, it appeared in an unpublished report of McNaughton under the name
“order vector with hit”
4
a5
p p
a12 p a4
12 21
5
a
b1 a2 b3 a3
5
b
q q
q
12 21
1
a
b2 a
b 3
5 b
r
b1 r b4
21
Fig.1: An example DRA andits resulting IAR DPA. For the Rabin automaton,
a number in a white box next to a transition indicates that this transition is a
required one of that Rabin pair. A black shape dually indicates membership in
the corresponding prohibited set. For example, with t = hp,a,pi we have t ∈ F
1
and t ∈ I . In the IAR construction, we shorten the notation for permutations
2
to save space, so p,12 corresponds to (p,(1,2)). The priority of a transition is
written next to the transitions letter.
– To define the priority assignment, we first introduce some auxiliary nota-
tion. For a transition t˜= h(q,π),a,(q′,π′)i and its corresponding transition
hq,a,q′i in the original automaton, let
maxInd t˜ =max π−1(i)|t∈F ∪I ∪{0}
i i
(cid:0) (cid:1) (cid:0)(cid:8) (cid:9) (cid:1)
be the maximal position of acceptance pair in π visited by t (or 0 if none is
visited). Using this, define the priority assignment as follows:
1 if maxInd t˜ =0,
λ t˜ :=2·maxInd(t˜) if t∈Iπ(m(cid:0)ax(cid:1)Ind(t˜))\Fπ(maxInd(t˜))
(cid:0) (cid:1) 2·maxInd(t˜)+1 otherwise, i.e. if t∈Fπ(maxInd(t˜)).
Whenatransitionvisits multiple prohibitedsets,they canbe movedtothe front
oftheappearancerecordinarbitraryorder.Asanoptimizationwechooseexisting
states as successors whenever possible.
Before formally proving correctness, i.e. that IAR(R) recognizes the same
language as R, we provide a small example in Figure 1 and explain the general
intuition behind the construction. For a given run, all prohibited sets which are
visited infinitely often will eventually be “in front” of all those only seen finitely
often: After some finite number of steps, none of the finitely often visited ones
will be seen any more. Taking another sufficiently large amount of steps, every
infinitely often visited set has been seen again and all their indices have been
moved to the front.
5
Lemma 1. Let w ∈ Σω be a word on which IAR(R) has a run ρ˜. Then, the
positions of all finitely often visited prohibited sets stabilize after a finite num-
ber of steps, i.e. their positions are identical in all infinitely often visited states.
Moreover, for any i,j with F ∈ F , F ∈/ F we have that the position of F
i inf j inf i
is smaller than the position of F in every infinitely often visited state.
j
Proof. The position of any F only changes in two different ways:
i
– Either F itself has been visited and thus is moved to the front,
i
– or some Fi′ with a position greater than the one of Fi has been visited and
is moved to the front, increasing the position of F .
i
Let ρ be the run of R on w. (We prove the existence of such a run in Lemma 3.)
AssumethatF isvisitedfinitelyofteninsomerunρ,i.e.thereisastepintherun
i
from which on F is never visited again. As the amount of positions is bounded,
i
the second case may only occur finitely often after this step and the position of
F eventually remains constant. As F was chosen arbitrarily, we conclude that
i i
allfinitely oftenvisitedF areeventuallymovedto the rightandremainontheir
i
position. Trivially, all infinitely often visited F move to the left, proving the
i
claim. ⊓⊔
As an immediate consequence we see that if some transition (q,a,q′) ∈ F is
i
visited infinitely often, then everyF with a smaller position than F in q is also
j i
visited infinitely often:
Corollary 1. Let t˜∈ Inf (w) be an infinitely often visited transition with
IAR(R)
its corresponding transition t∈F for some i. Then ∀j ≤i.F ∈F .
π(i) π(j) inf
Looking back at the definition of the priority function, the central idea of cor-
rectness can be outlined as follows. For every I which is visited infinitely often
i
we can distinguish two cases:
– F isvisitedfinitelyoften.Thenthepositionofthepairisgreaterthantheone
i
ofeveryF ∈F .Hencethepriorityofeverytransitiont˜withcorresponding
j inf
transitiont∈I isbothevenandbiggerthaneveryoddpriorityseeninfinitely
i
often along the run.
– F is visited infinitely often, i.e. after eachvisit of I , F is eventually visited.
i i i
As argued in the proof of Lemma 1, the position of F can only increase
i
until it is visited again.Hence every visit of I which yields an evenparity is
i
followed by a visit of F yielding an odd parity which is strictly greater.
i
Using this intuition, we formally show correctness of the construction in Ap-
pendix A.1.
Theorem 1. For any DRA R we have that L(IAR(R))=L(R).
Proposition 1 (Complexity). For every DRA R with n states and k Rabin
pairs, the constructed automaton IAR(R) has at most n·k! states and 2k+1
priorities.
6
Moreover,usingthe[Löd99a],onecanshowthatthisisessentiallyoptimal.There
existsafamily{L } oflanguagessuchthatforeverynthelanguageL canbe
n n≥2 n
recognizedbyaDRA withO(n) statesandO(n) pairs,butcannotberecognized
by a DPA with less than n! states. For details, see Appendix A.2.
Remark 1 (Comparison to previous IAR). Our construction is similar to the in-
dex appearance record of [Löd99b] in that it keeps the information about the
current state and a permutation of pairs, implementing the appearance record.
However, from the point of view of Streett automata, it is very natural to keep
two pointers into the permutation, indicating the currently extreme positions of
both types of sets in the accpetance condition. Indeed, this way we can keep
track of all conjuncts of the form Inf(I ) =⇒ Inf(F ). This is also the approach
j j
that [Löd99b] takes. In contrast, we have no pointers at all. From the Rabin
pointofview,itismorenaturaltokeeptrackofthe prohibitedsets onlyandthe
respective pointer is hidden in the information about the current state together
with the current permutation. Additionally, the pointer for the required set is
hidden into the acceptance status of transitions. In the transition-based setting,
it is not necessary to remember the visit of a required set in the state-space; it
is sufficient to emit the respective priorityupon seeing this during the transition
when we know both the source and target states. The absence of these pointers
results in better performance.
Remark 2 (UsingIARforDGRA).ThestraightforwardwaytotranslateaDGRA
to DPA is to first de-generalize the DGRA and then apply the presented IAR
construction.However,onecanalsoapplytheIARideatodirectlytranslatefrom
DGRA to DPA: Instead of only tracking the pair indices, one could incorporate
allF andIj intotheappearancepermutation.Withthesamereasoningasabove,
i i
a parity condition can be used to decide acceptance.
This approach yields a correct algorithm, but compared to de-generalization
combined with IAR, the state space grows much larger. Indeed, given a DGRA
with n states and k accepting pairswith l requiredsets each,the de-generalized
i
DRAhasatmostn· k l statesandkpairs,hencetheresultingparityautoma-
i=1 i
k
tonhasatmostk!·n· l statesand2k+1priorities.Applyingthementioned
Q i=1 i
k k
specific constructiongives n·( (l +1))! states and2·( (l +1))+1 pri-
Q i=1 i i=1 i
orities.Asimpleinductiononk sufficestoshowthatthe worstcaseupperbound
P P
for the specific construction is always larger. We conjecture that this behaviour
also shows in real-worldapplications.
4 Optimizations
In general, many states generated by the IAR procedure are often superfluous
and could be omitted. In the following, we present several optimizations of our
construction, which aim to do so. Moreover, these optimizations can be applied
alsototheIARconstructionof[Löd99b]andinaslighlyadjustedwayalsotothe
standard SAR [Sch01]. Further, although the optimizations are transition-based,
they can be of course easily adapted to the state-based setting. Due to space
constraints, the correctness proofs can be found in Appendix A.3.
7
a,b3
p q r
a2 b1
b1 a2
a,b7
p q r q p
312 231 123 213 123
a7 b7 b5 a5
b4 a2 a2 b2
Fig.2: Example of a suboptimal initial permutation, using the same notation
as in Figure 1. Only the shaded states are constructed when choosing a better
initial permutation.
4.1 Choosing an initial permutation
The first observation is that the arbitrary choice of (1,...,k) as initial permu-
tation can lead to suboptimal results. It may happen that several states of the
resulting automaton are visited at most once by every run before some “recur-
rent” permutation is reached. These states enlarge the state-space unnecessar-
ily, as demonstrated in Figure 2. Indeed, when choosing (p,(3,1,2)) instead of
(p,(1,2,3)) as the initial state in the example, only the shaded states are built
during the construction, while the language of the resulting automaton is still
equal to that of the input DRA.
We overload the IAR algorithm to be parametrized by the starting permu-
tation, i.e. we write IAR(R,π ) to denote the IAR construction applied to the
0
DRA R starting with permutation π .
0
Theorem 2. For an arbitrary Rabin automaton R with k pairs we have that
L(IAR(R))=L(IAR(R,π )) for all π ∈Πk.
0 0
How to choose a “good” initial permutation is deferred to Section 4.3, as it is
intertwined with the algorithm presented in the following section.
4.2 SCC decomposition
Acceptanceofawordbyanω-automatononlydependsonthesetofstatesvisited
infinitelyoftenbyitsrun.Thissetofstatesisstronglyconnectedontheunderlying
graph structure, i.e. starting from any state in the set, any other state can be
reached with finitely many steps. In general, any strongly connected set belongs
8
to exactly one strongly connected component (SCC). Therefore, for a fixed SCC,
only the Rabin pairs with required sets intersecting this SCC are relevant.
Using this we can restrict ourselves to the Rabin pairs that can possibly
accept in that SCC while processing it. This reduces the number of indices we
needtotrackintheappearancerecordforeachSCC,whichcanleadtosignificant
savings.
For readability, we introduce some abbreviations. Given a DRA R =
(Q,Σ,δ,q ,{(F ,I )}k ) and a set of states S ⊆ Q we write δ ↾ S : S ×Σ → S
0 i i j=1
to denote the restriction of δ to S, i.e. δ ↾S(q,a) = δ(q,a) if δ(q,a) ∈ S and
undefined otherwise.Analogously,we define∆↾S =∆∩S×Σ×S asthe set of
transitions in the restricted automaton. Consequently, we define the restriction
of the whole automaton R to the set of states S using q ∈S as initial state by
R↾ S =(S,Σ,δ ↾S,q,{(F ∩(∆↾S),I ∩(∆↾S))|I ∩(∆↾S)6=∅}).
q i i i
Furthermore,wecallaSCCofanautomatontransient,ifitisasingletonsetwith-
outaself-loop.Thismeansthatitisvisitedatmostoncebyanyrunanditisnot
of interest for acceptance. Finally, we use ε to denote the “empty” permutation
(of length 0).
Using this notation, we describe the optimized IAR construction, denoted
IAR∗ inAlgorithm1.The algorithmdecomposestheDRAintoitsSCCs,applies
the formerly introduced IAR procedure to each sub-automaton separately and
finally connects the resulting DPAs back together.
Aswe applythe IARconstructionto eachSCC separately,we haveto choose
the initial permutation for each state of those SCCs. Theorem 2 shows that
for a particular initial state, correctness of IAR does not depend on the chosen
permutation.WethereforedelegatethechoicetoafunctionpickPermandprove
correctnessof the optimized algorithmindependent of this function, allowing for
furtheroptimizations.WepresentanoptimaldefinitionofpickPerminthenext
subsection.
Figure 3 shows an example application and the obtained savings of the con-
struction. Pair 1 is only relevant for acceptance in the SCC {p}, but in the
unoptimized construction it still changes the permutations in the part of the au-
tomatonconstructedfrom{q,r}, ase.g.the transitionhr,b,qiis containedin F .
1
Similarly,pair2istrackedin{p}whileactuallynotbeingrelevant.Theoptimized
version yields improvements in both state-space size and amount of priorities.
Theorem 3. For any DRA R we have that L(IAR∗(R)) = L(R), independent
of pickPerm.
4.3 Optimal choice of the initial permutation
In Figure 2 we provided a scalable example where the choice of the initial per-
mutation can significantly reduce the size of the generated automaton. In this
subsection, we explain a procedure yielding a permutation which minimizes the
state space of the automaton generated by IAR∗.
9
Input :A DRA R=(Q,Σ,δ,q0,{(Fi,Ii)}kj=1)
Output:A DPA recognizing thesame language as R
1 Q∗←{}, δ∗ ←{}, λ∗ ≡1
2 foreachSCC S in R do
3 if S transient or {i|Ii∩∆↾S 6=∅}=∅ then // SCC not relevant
4 Add S×{ε} to Q∗
5 foreach q∈S, a∈Σ such that (δ↾S)(q,a) is defined do
6 Let q′=δ(q,a)
7 Set δ∗((q,ε),a)=(q′,ε) and λ∗(h(q,ε),a,(q′,ε)i)=1
8 end
9 else // SCC relevant, apply IAR to the sub-automaton
10 Pick a starting state q∈S
11 (QS,Σ,δS,(q,π),λS)←IAR(R↾q S,pickPerm(q,S))
12 UpdateQ∗, δ∗ and λ∗ with QS, δS and λS, respectively
13 end
14 end
// Connect all SCCs
15 foreach(q,π)∈Q∗ and a∈Σ s.t. q′ =δ(q,a) in different SCC of R than q do
16 Pick a π′ with (q′,π′)∈Q∗
17 Set δ∗((q,π),a)=(q′,π′)
18 end
Algorithm 1: The optimized IAR construction IAR∗
a5
p p p
a12 p a4 a2
12 21 1
b b1 b1 b1
q q q
b2 q b4 b2 b2
12 21 2
a
5
b12 a2 b4 a3 b3 a3
b
5
r r r
a1 r a3 a3
12 21 2
a5
Example DRA, Result of unoptimized Outputof theop-
SCCs shaded IAR applied to the DRA timized IAR∗.
Fig.3: Example application of Algorithm 1
10
