Games, logic and Automata Seminar Chapter 1 in “Automata, Logic and infinite games”, edited by Gradel, Thomas and Wilke Games, logic and Automata Seminar Assaf Ben Shimon

Intro The main topic covered in this chapter is the question how to define acceptance of infinite words by finite automata. In contrast to the case of finite words, there are many possibilities, and it is a nontrivial problem to compare them with respect to expressive power. Connections were established with other specification formalisms, e.g. regular expressions, grammars, and logical systems. In this chapter, we confine ourselves to the automata theoretic view.

Intro Automata on infinite words have gained a great deal of importance since their first definition some forty years ago. Apart from the interests from a theoretical point of view, they have practical importance for the specification and verification of reactive systems that are not supposed to terminate at some point of time. Operating systems are an example of such systems, as they should be ready to process any user input as it is entered, without terminating after or during some task.

Topics of this lecture Formal definition of ω-Automata Nondeterministic models Deterministic models Lower bounds for transformations Weak acceptance conditions With the different acceptance conditions defined in the following sections the question arises how they are related in expressive power, i.e. whether there exist transformations from one acceptance condition to another. If such transformation can be established another question naturally arises: what is the complexity for the respective translations?

Some Notations ω := {0, 1, 2, 3, . . . } Σ* – the set of finite words over Σ Σ ω – the set of infinite words over Σ 𝑢,𝑣,𝑤,… – finite words 𝛼,𝛽,𝛾… – infinite words 𝛼 𝑎 – number of a’s in 𝛼 𝛼(𝑖) – the I’th letter of 𝛼 REG – the class of regular languages

Some Notations Given an ω-word 𝛼∈ Σ 𝜔 , we will also define: 𝑂𝑐𝑐 𝛼 ={𝑎∈Σ|∃𝑖. 𝛼 𝑖 =𝑎} 𝐼𝑛𝑓 𝛼 ={𝑎∈Σ|∀𝑖 ∃𝑗>𝑖 . 𝛼 𝑗 =𝑎}

ω-Automata In the present context, we are interested only in the acceptance of words by automata (and not in generation of ω-words by grammars). We only consider finite automata. The usual definitions of deterministic and nondeterministic automata are adapted to the case of ω-input-words by the introduction of new acceptance conditions. For this purpose one introduces an “acceptance component” in the specification of automata, which will arise in different formats.

ω-Automata Definition: An ω-automaton is a quintuple (𝑄, Σ, 𝛿, 𝑞 𝐼 , 𝐴𝑐𝑐), where: Q is a finite set of states Σ is a finite alphabet 𝛿 : 𝑄×Σ → 2 𝑄 is the state transition function 𝑞 𝐼 ∈ 𝑄 is the initial state Acc is the acceptance component In a deterministic ω-automaton, a transition function δ : Q × Σ → Q is used. The acceptance component can be given as a set of states, as a set of statesets, or as a function from the set of states to a finite set of natural numbers. Instances of all these case will be presented below.

ω-Automata Let A = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐴𝑐𝑐) be an ω-automaton. A run of A on an ω-word 𝛼= 𝑎 1 𝑎 2 … ∈ Σ 𝜔 is an infinite state sequence 𝜚=𝜚(0)𝜚(1) 𝜚 2 …∈ 𝑄 𝜔 , such that the following conditions hold: 𝜚 0 = 𝑞 𝐼 𝜚 𝑖 ∈𝛿 𝜚 𝑖−1 , 𝑎 𝑖 for 𝑖≥1 if A is nondeterministic, 𝜚 𝑖 =𝛿 𝜚 𝑖−1 , 𝑎 𝑖 for 𝑖≥1 if A is deterministic

ω-Automata The size of an automaton 𝐴, denoted by |𝐴| , is measured by the number of its states, i.e. for 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐴𝑐𝑐) the size is |𝐴| = 𝑄 . In addition to the number of states of an automaton the size of the acceptance condition is also of some importance for the efficiency of the transformation. This is usually measured by the number of designated sets or pairs of such. Details are given in the respective sections.

Nondeterministic Models

Büchi Acceptance Acceptance component – a set of states Definition: An ω-automaton 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐹) with acceptance component 𝐹 ⊆ 𝑄 is called Büchi automaton if it is used with the following acceptance condition (Büchi acceptance): A word 𝛼 ∈ Σ 𝜔 is accepted by A ⟷ There exists a run 𝜚 of 𝐴 on 𝛼 satisfying the condition: 𝐼𝑛𝑓 𝜚 ∩ 𝐹≠ ∅ i.e. at least one of the states in F has to be visited infinitely often during the run.

Büchi Acceptance For example, which language does the following automaton accepts? (The states from F are drawn with a double circle, i.e 𝐹={ 𝑞 1 , 𝑞 3 } ) This Büchi automaton accepts the words from 𝑎+𝑏 ∗ 𝑎 𝜔 + 𝑎+𝑏 ∗ 𝑎𝑏 𝜔

Büchi Acceptance Consider a Büchi automaton 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐹). Using this automaton with initial state 𝑝 and final state 𝑞 we obtain a regular language 𝑊(𝑝, 𝑞) of finite words. An ω-word 𝛼 is accepted by 𝐴 ⟷ some run of 𝐴 on 𝛼 visits some final state 𝑞∈𝐹 infinitely often. This is equivalent to 𝛼∈𝑊 𝑞 0 , 𝑞 ⋅𝑊 𝑞, 𝑞 𝜔

Büchi Acceptance Taking the union over these sets for 𝑞∈𝐹, we obtain the following representation result for Büchi recognizable ω-languages: Theorem: The Büchi recognizable ω-languages are the ω-languages of the form L = 𝑖=1 𝑘 𝑈 𝑖 𝑉 𝑖 𝜔 𝑤𝑖𝑡ℎ 𝑘∈𝜔 𝑎𝑛𝑑 𝑈 𝑖 , 𝑉 𝑖 ∈𝑅𝐸𝐺 𝑓𝑜𝑟 𝑖=1,…,𝑘 This family of ω-languages is also called the ω-Kleene closure of the class of regular languages. - From this remark one concludes immediately that each nonempty Büchi recognizable ω-language contains an ultimately periodic word. - Let us also note that the emptiness problem is decidable for Büchi automata, i.e. there exists an algorithm that decides whether the language recognized by an arbitrary (nondeterministic) Büchi automaton is empty. Given a Büchi automaton A, one computes the set of reachable states, and for each reachable state q from F checks whether q is reachable from q by a nonempty path. Such a loop exists if and only if there exists an infinite word α and a run of A on α such that q is a recurring state in this run.

Muller Acceptance Acceptance component – a set of state sets (𝐹⊆ 2 𝑄 ) Definition: An ω-automaton 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐹) with acceptance component 𝐹⊆ 2 𝑄 is called Muller automaton if it is used with the following acceptance condition (Muller acceptance): A word 𝛼∈ Σ 𝜔 is accepted by A ⟷ There exists a run 𝜚 of 𝐴 on 𝛼 satisfying the condition: 𝐼𝑛𝑓 𝜚 ∈𝐹 i.e. the set of infinitely recurring states of 𝜚 is exactly one of the sets in F.

Muller Acceptance For example, which language does the following automaton accepts, When 𝐹= 𝑞 𝑎 , 𝑞 𝑏 ? And when 𝐹= 𝑞 𝑎 , q b ? When 𝐹= 𝑞 𝑎 , 𝑞 𝑏 - words with infinitely many a’s and b’s. When 𝐹= 𝑞 𝑎 , q b - Words with finitely many a’s and infinitely many b’s, or finitely many b’s and infinitely many a’s

Büchi and Muller Equivalence We will show Büchi and Muller automaton are equivalent in terms of expressive power. Buchi  Muller: Let 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐹) be a Büchi automaton. Define the Muller automaton 𝐴′= (𝑄,Σ, 𝛿, 𝑞 𝐼 ,𝐹) with 𝐹 := {𝐺∈ 2 𝑄 |𝐺∩𝐹= ∅}. Then 𝐿(𝐴) = 𝐿(𝐴′). define the family F of sets of states by collecting all subsets of Q which contain a state from F.

Büchi and Muller Equivalence Muller  Buchi : The idea: We will guess the set 𝐺∈𝐹 which should turn out to be 𝐼𝑛𝑓 𝜚 nondeterministically For each set, we will create a unique copy of the states We will guess the time during the run from which we will no longer see any state that is seen only finite amount of times We will simulate a “memory” that remembers all the states we have been to, and reset it every time we seen all the states in G ( = 𝐼𝑛𝑓 𝜚 ) If all of our guesses were correct, we will visit the reset state infinitely many times.

Büchi and Muller Equivalence Let 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝐹) be a Muller automaton. For each set 𝐺∈𝐹, we introduce a separate copy of 𝑄∩𝐺. We’ll indicate it as 𝑞 𝐺 . We will define 𝑄 ′ =𝑄∪ 𝐺∈𝐹 (𝐺× 2 𝐺 ) , where 𝑞 𝐺 ,𝑅 codes that q is the current state of A and R is the set of accumulated states since the last reset The automaton will make the 2 nondeterministic guesses at once – and switch from a state 𝑝∈𝑄 to a state 𝑞 𝐺 corresponding to some group G, with memory =∅ (reset state)

Büchi and Muller Equivalence We will define 𝐹 ′ to contain all the reset states- states of the form ( 𝑞 𝐺 ,∅) The Büchi automaton will be defines as 𝐴 ′ =(𝑄′,Σ, 𝛿′, 𝑞 𝐼 , 𝐹′), With 𝑄 ′ , 𝛿 ′ ,𝐹′ defined as above. Then 𝐿(𝐴) = 𝐿(𝐴′). We do not give a formal definition of the transitions, which should be clear from the description above

Büchi and Muller Equivalence If 𝑄 has n states, and 𝐹 contains m sets, then 𝑄 ′ ≤𝑛+𝑚𝑛 2 𝑛 = 2 𝑂 𝑛 Summarizing- Theorem: nondeterministic Büchi automaton with 𝑛 states can be converted into an equivalent Muller automaton of equal size, and a nondeterministic Muller automaton with 𝑛 states and 𝑚 accepting sets can be transformed into an equivalent Büchi automaton with ≤ 𝑛 + 𝑚𝑛 2 𝑛 states. The transformation sketched above transforms nondeterministic Büchi automata into nondeterministic Muller automata and conversely. For a given deterministic Büchi automaton, the translation yields a deterministic Muller automaton. On the other hand, a deterministic Muller automaton is converted into a nondeterminsitic Büchi automaton. As we shall see later, this nondeterminism cannot in general be avoided.

So far: Nondeterministic models Büchi Automaton = Muller Automaton

Rabin Acceptance Acceptance component – a finite family Ω of pairs ( 𝐸 𝑖 , 𝐹 𝑖 ) of state sets Definition: An ω-automaton 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) with acceptance component Ω={ 𝐸 1 , 𝐹 1 ,…,( 𝐸 𝑘 , 𝐹 𝑘 )} with 𝐸 𝑖 , 𝐹 𝑖 ⊆𝑄 is called Rabin automaton if it is used with the following acceptance condition (Rabin acceptance): A word 𝛼 ∈ Σ 𝜔 is accepted by A ⟷ there exists a run 𝜚 of 𝐴 on 𝛼 such that ∃ 𝐸,𝐹 ∈Ω . 𝐼𝑛𝑓 𝜚 ∩𝐸=∅ ∧(𝐼𝑛𝑓 𝜚 ∩𝐹≠∅)

Rabin Acceptance For example, which language does the following Rabin automaton accepts, When Ω={( 𝑞 𝑏 , 𝑞 𝑎 )}? This Rabin automaton accepts all words that consist of infinitely many a’s but only finitely many b’s.

Rabin Acceptance If we want to specify the language consisting of all words that contain infinitely many b’s only if they also contain infinitely many a’s, using this state graph, which Ω should we choose? Ω={ ∅, 𝑞 𝑎 , ( 𝑞 𝑎 , 𝑞 𝑏 , 𝑞 𝑐 )} each word in the accepted language has either infinitely many a’s or it has neither infinitely many a’s nor infinitely many b’s.

Streett Acceptance Dual to the Rabin condition. Acceptance component – same as in Rabin. Definition: An ω-automaton 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) with acceptance component Ω={ 𝐸 1 , 𝐹 1 ,…,( 𝐸 𝑘 , 𝐹 𝑘 )} with 𝐸 𝑖 , 𝐹 𝑖 ⊆𝑄 is called Streett automaton if it is used with the following acceptance condition (Streett acceptance): A word 𝛼 ∈ Σ 𝜔 is accepted by A ⟷ there exists a run 𝜚 of 𝐴 on 𝛼 such that ∀ 𝐸,𝐹 ∈Ω . 𝐼𝑛𝑓 𝜚 ∩𝐸≠∅ ∨(𝐼𝑛𝑓 𝜚 ∩𝐹=∅) (equivalently: if 𝐼𝑛𝑓 𝜚 ∩𝐹≠∅ then 𝐼𝑛𝑓 𝜚 ∩𝐸≠∅ )

Streett Acceptance For example, which language does the following Streett automaton accepts, When Ω={( 𝑞 𝑏 , 𝑞 𝑎 )}? This Streett automaton accepts all words that contain infinitely many b’s if they contain infinitely many a’s.

Rabin+Streett and Muller Equivalance Rabin/Streett  Muller: Let 𝐴= (𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) be a Rabin automaton (respectively, Streett automaton) Define a Muller automaton 𝐴′ = (𝑄,Σ, 𝛿, 𝑞 𝐼 ,𝐹) with 𝐹 := 𝐺 ∈ 2 𝑄 ∃ 𝐸,𝐹 ∈Ω .(𝐺∩𝐸=∅) ∧(𝐺∩𝐹≠∅)} , (respectively, with 𝐹 := 𝐺∈ 2 𝑄 ∀ 𝐸,𝐹 ∈Ω .(𝐺∩ 𝐸≠∅) ∨(𝐺∩𝐹=∅)} ) Then 𝐿(𝐴) = 𝐿(𝐴′).

Rabin+Streett and Muller Equivalance Muller  Rabin/Streett: Perform the transition from Muller automaton to Büchi Automaton. Observe that Büchi Acceptance can be viewed as a special case of Rabin Acceptance (with Ω={(∅,𝐹)}) As well as Streett Acceptance (with Ω={(𝐹,𝑄)}

So far: = = = Nondeterministic models Büchi Automaton Muller Automaton Rabin Automaton Streett Automaton

So far: = Nondeterministic models Büchi Automaton Muller Automaton Rabin Automaton Streett Automaton

Parity Condition A different formalization for the Rabin Acceptance, if the special case where the accepting pair 𝐸 1 , 𝐹 1 ,…,( 𝐸 𝑚 , 𝐹 𝑚 ) form a chain with respect to set inclusion- meaning 𝐸 1 ⊂ 𝐹 1 ⊂ 𝐸 2 ⊂…⊂ 𝐸 𝑚 ⊂ 𝐹 𝑚 . We will associate indices (colors) with states as follows: states of 𝐸 1 receive color 1, states of 𝐹 1 \ 𝐸 1 receive color 2, and so on with the rule that states of 𝐸 𝑖 \ 𝐹 𝑖−1 have color 2𝑖−1 and states of 𝐹 𝑖 \ 𝐸 𝑖 have color 2𝑖. An ω-word 𝛼 is then accepted by the parity automaton ⟷ the least color occurring infinitely often in a run on 𝛼 is even (hence the term “parity condition”).

Parity Condition Definition: An ω-automaton 𝐴 = (𝑄,Σ, 𝛿, 𝑞 𝐼 ,c) with acceptance component 𝑐:𝑄→ 1,…,𝑘 (𝑤ℎ𝑒𝑟𝑒 𝑘∈𝜔) is called parity automaton if it is used with the following acceptance condition (parity condition): A word 𝛼 ∈ Σ 𝜔 is accepted by A ⟷ there exists a run 𝜚 of 𝐴 where min 𝑐 𝑞 |𝑞∈𝐼𝑛𝑓(𝜚) is even Sometimes it is more convenient to work with the condition that the maximal color occurring infinitely often in the run under consideration is even. This applies to some constructions in later chapters of this book.

Parity Condition For example, in the following parity automaton, what language is accepted when: 𝑐 𝑞 𝑖 =𝑖 ? 𝑎𝑏 𝑎 ∗ 𝑐 𝑏 ∗ 𝑐 𝜔 ∨ 𝑎𝑏 𝑎 ∗ 𝑐 𝑏 ∗ 𝑐 ∗ 𝑎 𝜔 𝑐 𝑞 𝑖 =𝑖+1 ? 𝑎𝑏 𝑎 ∗ 𝑐 𝑏 ∗ 𝑐 ∗ 𝑐 𝑏 𝜔

Parity and Rabin Equivalence Parity Rabin: Let 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 ,c) be a parity automaton with 𝑐:𝑄→{0,…,k}. An equivalent Rabin automaton 𝐴′ =(𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) has the acceptance component Ω := {( 𝐸 0 , 𝐹 0 ), . . . , ( 𝐸 𝑟 , 𝐹 𝑟 )} with 𝑟 := 𝑘 2 , 𝐸 𝑖 :={𝑞∈𝑄 | 𝑐(𝑞)<2𝑖} and 𝐹 𝑖 := {𝑞∈𝑄 | 𝑐(𝑞) ≤ 2𝑖}. Rabin  Parity : Can just Rabin  Büchi, and then define 𝑐 𝑞 =2 if 𝑞∈𝐹, 𝑐 𝑞 =3 otherwise. Rabin  Parity - not in book

So far: = Nondeterministic models Büchi Automaton Muller Automaton Rabin Automaton Streett Automaton Parity Automaton

Result Theorem: Nondeterministic Büchi automata, Muller automata, Rabin automata, Streett automata, and parity automata are all equivalent in expressive power, i.e. they recognize the same ω-languages. The ω-languages recognized by these ω-automata form the class ω-KC(REG), i.e. the ω-Kleene closure of the class of regular languages. The ω-languages in this class are commonly referred to as the regular ω- languages, denoted by ω-REG. At this point two fundamental questions arise: • Are there types of deterministic ω-automata which recognize precisely the ω-languages in ω-REG? • Is the class ω-REG closed under complementation?

Deterministic Models Note that the equivalence proof given above for the nondeterministic case cannot be copied: We proceeded via nondeterministic Büchi automata and thus, even from deterministic Muller automata, would obtain nondeterministic Rabin, Streett, and parity automata.

Büchi automaton We will see that a deterministic Büchi automata are too weak to recognize even very simple ω-languages from ω-REG. Let’s look at the following nondeterministic automaton: According to Büchi Acceptance with 𝐹= 𝑞 1 , it recognizes the language 𝑎+𝑏 ∗ 𝑎 𝜔

Büchi automaton It is easy to provide an equivalent deterministic Muller automaton, using the following automaton with 𝐹 = {{ 𝑞 𝑎 }} as acceptance component. If one would work with the Büchi acceptance condition, using a set 𝐹 of accepting states, then one has a specification of states which should be visited infinitely often, but it is not directly possible to specify which states should be seen only finitely often.

Büchi automaton We will show that deterministic Büchi automata are too weak for recognizing the language 𝐿= 𝑎+𝑏 ∗ 𝑏 𝜔 , by contradiction: Assuming that the deterministic Büchi automaton 𝐴 with final state set 𝐹 recognizes 𝐿, it will on input 𝑏 𝜔 visit an 𝐹-state after a finite prefix, say after the 𝑛 0 -th letter. It will also accept b n 0 𝑎 𝑏 𝜔 , visiting 𝐹-states infinitely often and hence after the 𝑎, say when finishing the prefix 𝑏 𝑛 0 𝑎 𝑏 𝑛 1 . Continuing this construction the ω-word 𝑏 𝑛 0 𝑎 𝑏 𝑛 1 𝑎 𝑏 𝑛 2 𝑎 . . . is generated which causes A to pass through an F-state before each letter a, but which should of course be rejected.

So far: = Nondeterministic models Deterministic models ≠ Büchi Automaton Büchi Automaton ≠ Muller Automaton Muller Automaton Rabin Automaton Rabin Automaton = Let us now show that deterministic Muller automata, Rabin automata, Streett automata, and parity automata all have the same expressive power. Streett Automaton Streett Automaton Parity Automaton Parity Automaton

Muller  Rabin We use a technique called latest appearance record (LAR). In a list of (distinct) states, we use the last entry for the current state in the run on the given Muller automaton. The hit position (the position of the marker ♮) indicates where the last change occurred in the record. For every transition from one state p to q in the original automaton, the state q is moved to the last position of the record while the symbols which were to the right of q are shifted one position to the left (so the previous place of q is filled again). The marker is inserted in front of the position where q was taken from. The idea is to use permutations of the states of the given Muller automaton as new states, extended by a hit position. So the memory of the new automaton stores lists of states from the original automaton; this is in contrast to the construction of Theorem 1.10 which produced a nondeterministic Büchi automaton from a Muller automaton; in that case we stored sets of states of the original automaton in the memory of the constructed one.

𝑞 3 𝑞 7 𝑞 2 𝑞 5 ♮ 𝑞 4 𝑞 7 𝑞 1 𝑞 6 𝑞 3

Muller  Rabin Formally: Let 𝐴=(Σ,𝑄, 𝛿, 𝑞 𝐼 ,𝐹) be a deterministic Muller automaton. Assume w.l.o.g. that 𝑄= {1, . . . , 𝑘} and 𝑞 𝐼 = 1. Let ♮ be a new symbol, i.e. ♮∉𝑄. An equivalent Rabin automaton A is given by the following definition: 𝑄 is the set of all order vector words with hit position over 𝑄, i.e. 𝑄:={𝑤∈ 𝑄 ∪ ♮ ∗ | ∀𝑞∈𝑄 ∪ ♮ . 𝑤 𝑞 = 1} The initial state is 𝑞 𝐼 ′ ≔♮k…1

Muller  Rabin The transition function 𝛿 is constructed as follows: Assume 𝑖, 𝑖′∈𝑄, 𝑎 ∈ Σ, and 𝛿 𝑖, 𝑎 =𝑖′. Then 𝛿′ is defined for any word 𝑚 1 . . . 𝑚 𝑟 ♮ 𝑚 𝑟+1 . . . 𝑚 𝑘 ∈𝑄 with 𝑚 𝑘 =𝑖. Supposing that 𝑖′= 𝑚 𝑠 , define 𝛿’( 𝑚 1 . . . 𝑚 𝑟 ♮ 𝑚 𝑟+1 . . . 𝑚 𝑘 , 𝑎) := 𝑚 1 . . . 𝑚 𝑠−1 ♮ 𝑚 𝑠+1 . . . 𝑚 𝑘 𝑚 𝑠 The acceptance component is given by Ω={( 𝐸 1 , 𝐹 1 ), . . . , ( 𝐸 𝑘 , 𝐹 𝑘 )}, defined as follows: - 𝐸 𝑗 :={𝑢♮𝑣 ||𝑢|<𝑗} - 𝐹𝑗 :={𝑢♮𝑣 ||𝑢|<𝑗} ∪ {𝑢♮𝑣 | |𝑢|=𝑗∧ {𝑚∈𝑄 | 𝑚⊑𝑣} ∈ 𝐹} Here the infix relation m⊑v should be read as “m occurs in v”, since m is a single letter.

Muller  Parity In this transformation, we have 𝐸 1 ⊆ 𝐹 1 ⊆ 𝐸 2 …⊆ 𝐸 𝑘 ⊆ 𝐹 𝑘 We can remove pairs where 𝐸 𝑖 = 𝐹 𝑖 , And present the automaton as a Parity automaton Theorem: By this transformation, a deterministic Muller automaton with 𝑛 states is transformed into a deterministic Rabin automaton with 𝑛⋅𝑛! states and 𝑛 accepting pairs, and also into a deterministic parity automaton with 𝑛⋅𝑛! states and 2𝑛 colors. The transformation is given here for deterministic automata, but it works analogously for nondeterministic automata.

Rabin  Streett Note that the negation of the Rabin acceptance condition: ∃ 𝐸,𝐹 ∈Ω . 𝐼𝑛𝑓 𝜚 ∩𝐸=∅ ∧(𝐼𝑛𝑓 𝜚 ∩𝐹≠∅) Is equivalent to the Streett condition: ∀ 𝐸,𝐹 ∈Ω . 𝐼𝑛𝑓 𝜚 ∩𝐸≠∅ ∨(𝐼𝑛𝑓 𝜚 ∩𝐹=∅) Hence, when we transform a deterministic Rabin automaton recognizing L into a Streett automaton by keeping all its components, including the acceptance component, but using it in the form (∗∗) instead of (∗), then the resulting Streett automaton recognizes the complement of L.

Rabin  Streett Let 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 ,𝐹) be a deterministic Muller automaton. Then the Muller automaton 𝐴 ′ :=(𝑄,Σ, 𝛿, 𝑞 𝐼 , 2 𝑄 \𝐹) recognizes the complement of 𝐿(𝐴). From Rabin to Streett: (Rabin automaton 𝐴 with language 𝐿) From A construct an equivalent Muller automaton Complement the Muller automaton Transform it back to a Rabin automaton 𝐴 ′ , recognizing the complement of L Used as Streett automaton, 𝐴′ recognizes L The converse transformation from Streett to Rabin automata works analogously.

So far: = Nondeterministic models Deterministic models ≠ Büchi Automaton Büchi Automaton ≠ Muller Automaton Muller Automaton Rabin Automaton Rabin Automaton = * We showed how to get from Muller to Rabin and to Parity * We also showed how to get from Rabin to Streett and vice versa through Muller * Transformation from Parity to Rabin is same to the nondeterministic case Streett Automaton Streett Automaton Parity Automaton Parity Automaton

So far: = = Nondeterministic models Deterministic models ≠ Büchi Automaton Büchi Automaton ≠ Muller Automaton Muller Automaton Rabin Automaton Rabin Automaton = = Streett Automaton Streett Automaton Parity Automaton Parity Automaton

Complement We showed Muller automaton is closed under complement, and therefor Rabin and Streett are too. It is not hard to show this for Parity as well (directly): Let 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 , 𝑐) be a deterministic ω-automaton with parity condition. Then the complement of 𝐿(𝐴) is recognized by the parity automaton 𝐴′:= (𝑄,Σ, 𝛿, 𝑞𝐼, 𝑐′) where 𝑐’ 𝑞 =𝑐 𝑞 +1. For showing that the complement of a language accepted by an ω-automaton with parity condition is also acceptable by a parity automaton, the color function has to be modified such that henceforth every word previously not accepted has even parity in its minimal color value and uneven parity for all previously accepted words.

Deterministic Models Theorem: Deterministic Muller automata, Rabin automata, Streett automata and parity automata recognize the same ω-languages, and the class of ω-languages recognized by any of these types of ω-automata is closed under complementation.

Lower Bounds In this section we establish two lower bounds of rate 2 𝑂 𝑛 log 𝑛 for the transformation of ω-automata: (1) from nondeterministic Büchi automata to deterministic Rabin automata, (2) from deterministic Streett to deterministic Rabin automata. The first lower bound will useful in Chapter 3, where a transformation from Büchi automata to deterministic Rabin automata is presented, using the construction of Safra [158]. The lower bound will show that Safra’s construction is optimal. The second lower bound is of interest in connection with the conversion of Streett automata into Rabin automata (or conversely) presented above. The lower bound result will be taken up again in Chapter 5, where Streett automata are studied in more depth.

Lower Bounds Lemma: Let 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) be an ω-automaton with Rabin condition, and assume 𝜚 1 , 𝜚 2 are two nonaccepting runs. Then any run with 𝐼𝑛𝑓(𝜚)=𝐼𝑛𝑓( 𝜚 1 )∪𝐼𝑛𝑓( 𝜚 2 ) is also non-accepting. Proof: Assume that 𝜚 1 , 𝜚 2 are non accepting, but 𝜚 with 𝐼𝑛𝑓 𝜚 =𝐼𝑛𝑓 𝜚 1 ∪𝐼𝑛𝑓 𝜚 2 𝑖𝑠. There is a pair (𝐸,𝐹) s.t. 𝐼𝑛𝑓 𝜚 ∩𝐸=∅ and 𝐼𝑛𝑓 𝜚 ∩𝐹≠∅ 𝐼𝑛𝑓 𝜚 =𝐼𝑛𝑓 𝜚 1 ∪𝐼𝑛𝑓 𝜚 2  𝐼𝑛𝑓 𝜚 1 ∩𝐸=∅ and 𝐼𝑛𝑓 𝜚 2 ∩𝐸=∅ , And also 𝐼𝑛𝑓 𝜚 1 ∩𝐹≠∅ or 𝐼𝑛𝑓 𝜚 2 ∩𝐹≠∅ One of 𝜚 1 , 𝜚 2 would be accepting, in contradiction.

Lower Bounds Duality of Rabin and Streett also gives us: Lemma: Let 𝐴=(𝑄,Σ, 𝛿, 𝑞 𝐼 ,Ω) be an ω-automaton with Streett condition, and assume 𝜚 1 , 𝜚 2 are two accepting runs. Then any run with 𝐼𝑛𝑓(𝜚)=𝐼𝑛𝑓( 𝜚 1 )∪𝐼𝑛𝑓( 𝜚 2 ) is also accepting.

Büchi  Rabin We will show a lower bound on the transformation from nondeterministic Büchi automata to deterministic Rabin automata We will use the languages of the following automata family: (𝑛≥2) A word α is accepted by 𝐴 𝑛 iff there exists k and 𝑖 1 , . . . , 𝑖 𝑘 ∈{1, . . . , 𝑛} such that each pair 𝑖 𝑗 𝑖 𝑗+1 for j <𝑘 and i 𝑘 𝑖 1 appears infinitely often in α.

Büchi  Rabin We can encode the symbols 1,…,𝑛 by words over 0,1 ∗ such that 𝑖 is encoded by 0 𝑖 1 𝑖𝑓 𝑖<𝑛 0 𝑖 0 ∗ 1 𝑖𝑓 𝑖=𝑛 Now we can specify the same family of languages w.r.t. the encoding by the family of automata 𝐴 ′ 𝑛 𝑛≥2 over the fixed alphabet {0, 1,#} The size of 𝐴 𝑛 (in either of the two versions) is 𝑂(𝑛).

Büchi  Rabin Lemma: There exists a family of languages 𝐿 𝑛 𝑛≥2 over the alphabet {0, 1,#} recognizable by nondeterministic Büchi automata of size 𝑂(𝑛) such that any nondeterministic Streett automaton accepting the complement language of 𝐿 𝑛 has at least 𝑛! states. Proof: Let ( 𝑖 1 , . . . , 𝑖 𝑛 ), ( 𝑗 1 , . . . , 𝑗 𝑛 ) be different permutations of {1, . . . , 𝑛} Words 𝛼:= 𝑖 1 . . . 𝑖 𝑛 # 𝜔 and 𝛽:= 𝑗 1 . . . 𝑗 𝑛 # 𝜔 are not accepted by 𝐴 𝑛 . For any Streett automaton 𝐴′ accepting 𝐿 𝐴 ′ =𝐿′, there have to exist accepting runs 𝜚 𝛼 and 𝜚 𝛽 with 𝑅≔𝐼𝑛𝑓( 𝜚 𝛼 ) and 𝑆≔𝐼𝑛𝑓( 𝜚 𝛽 ) . L’ = the complement language to L

Büchi  Rabin We want to show that 𝑅∩𝑆=∅. Assume in the contrary that there is a state 𝑞∈𝑅∩𝑆 Then there has to exist an accepting run 𝜚 𝛾 of 𝐴′ on a word 𝛾=𝑢 𝑣𝑤 𝜔

Büchi  Rabin The run is indeed accepting, thanks to the Lemma we proved earlier. But 𝛾∈𝐿( 𝐴 𝑛 ), in contradiction. (On Board) Thus, 𝑅∩𝑆=∅ for every two words. There are 𝑛! different permutations of {1, . . . , 𝑛}, thus 𝑛! words, with no intersection of their 𝐼𝑛𝑓 sets. 𝐴′ has at least 𝑛! states.

Büchi  Rabin From the duality of Rabin and Streett condition, If there is a deterministic Rabin automaton of size <𝑛! that accepts 𝐿 𝑛 then there also exists a deterministic Streett automaton that accepts the complement language with same number of states. Therefor- Theorem: There exists a family of languages 𝐿 𝑛 𝑛≥2 over the alphabet {0, 1,#} recognizable by nondeterministic Büchi automata of size 𝑂(𝑛) such that any equivalent deterministic Rabin automaton must be of size 𝑛! or larger. By the duality of Rabin and Streett conditions it is obvious that if there exists an ω-automaton of size less than n! with Rabin condition that accepts 𝐿 𝑛 then there also exists a deterministic Streett automaton that accepts the complement language Σ 𝑛 𝜔 \ 𝐿 𝑛 with less than n! states. Thus from Lemma 1.29 we conclude the theorem.

Streett  Rabin We can show a lower bound on the transformation from deterministic Streett automata to deterministic Rabin automata The proof uses the languages of the following automata family (𝑛≥2), when the acceptance component is Ω 𝑛 ={ 𝐸 1 , 𝐹 1 ,…, 𝐸 𝑛 , 𝐹 𝑛 } and 𝐸 𝑖 = 𝑖 , 𝐹 𝑖 = 𝑖 ′ : Each wordα in 𝐿( 𝐴 𝑛 ) satisfies the condition that each symbol occurring infinitely often in an odd position must also occur infinitely often in an even position of α.

Streett  Rabin Like before, we can encode it with 0,1 ∗ Theorem: There exists a family of languages 𝐿 𝑛 𝑛≥2 over the alphabet {0, 1} recognizable by deterministic Streett automata with 𝑂(𝑛) states and 𝑂(𝑛) pairs of designated state sets such that any deterministic Rabin automaton accepting 𝐿 𝑛 requires at least 𝑛! states. We will only show the general guidelines.

Streett  Rabin Proof: By induction 𝑛=2: trivial Step: Any given deterministic Rabin automaton 𝐴 accepting L≔𝐿( 𝐴 𝑛 ) can be modified to a deterministic automaton 𝐴 𝑖 𝑞 over {1, . . . , 𝑛}\{𝑖} by removing all arcs labelled by 𝑖. It has at least 𝑛−1 ! states. Moreover, has a reachable SCC of size 𝑛−1 ! For each 𝑖 we can construct 𝛼 𝑖 ∈ Σ 𝑛 𝜔 with run 𝜚 𝑖 s.t. 𝛼 𝑖 ∉𝐿 and 𝐼𝑛𝑓 𝜚 𝑖 ≥ 𝑛−1 ! We can show that for each 𝑖≠𝑗, 𝐼𝑛𝑓 𝜚 𝑖 ∩𝐼𝑛𝑓 𝜚 𝑗 =∅ Therefor we have at least 𝑛 𝑛−1 !=𝑛! states The base case for the induction is obvious: Any (Rabin) automaton accepting a proper subset of the infinite words over a 2-letter alphabet with some word having occurrences of both letters needs at least two states. The proof uses the fact that for any finite word 𝑢∈ 1, . . ., 𝑛 ∗ of even length, the word 𝑢𝛼 is accepted by 𝐴 𝑛 iff α is accepted by 𝐴 𝑛 .

Weak Acceptance Conditions In the previous sections we have defined acceptance by a reference to those states in a run which occur infinitely often. For some purposes a “weak acceptance condition” is appropriate. This is a condition on the set of states that occur at least once (but maybe only finitely often) in a run.

Weak Acceptance Instead of 𝐼𝑛𝑓(𝜚), we will use the set 𝑂𝑐𝑐(𝜚) for acceptance. Staiger and Wagner acceptance: 𝑂𝑐𝑐 𝜚 ∈𝐹 1-acceptance: 𝑂𝑐𝑐 𝜚 ∩𝐹≠∅ 1’-acceptance: 𝑂𝑐𝑐 𝜚 ⊆𝐹 Last two are special cases of Staiger and Wagner acceptance Staiger and Wagner acceptance - Analogue to Muller Condition. In the first case one collects in F all sets X with X ∩F =∅, in the second case the sets X with X ⊆ F. In later chapters of the book also the parity condition will be used in the weak sense. The requirement for acceptance is that the minimal (or maximal) color occurring in a run is even.

Weak Acceptance For example, in this automaton, what language is accepted When 𝐹= 𝑞 𝑎 , with 1-acceptance? When 𝐹= 𝑞 𝑏 , with 1′-acceptance? When 𝐹= 𝑞 𝑎 , with 1-acceptance – words that have at list one ‘a’. When 𝐹= 𝑞 𝑏 , with 1′-acceptance – only 𝑏 𝜔

Staiger-Wagner  Büchi Let 𝐴=(𝑄,Σ,𝛿, 𝑞 𝐼 ,𝐹). The language 𝐿(𝐴) recognized by 𝐴 with the Staiger- Wagner acceptance condition is recognized by a Büchi automaton 𝐴′=(𝑄× 2 𝑄 ,Σ, 𝛿′, ( 𝑞 𝐼 , { 𝑞 𝐼 }), 𝐹′) where 𝛿′((𝑝, 𝑃), 𝑎) contains all states (𝑝′, 𝑃′) with 𝑝′∈𝛿(𝑝) and 𝑃 ′ =𝑃 ∪ 𝑝 ′ 𝐹 contains all states (𝑝, 𝑃) with 𝑃∈𝐹. The exponential blow-up can be avoided if only 1-acceptance or 1′-acceptance are involved. In order to capture 1-acceptance via a set F by Büchi acceptance, one introduces a transition from each F-state to a new state 𝑞 𝑓 , with a transition back to 𝑞 𝑓 , which serves as only final state in the Büchi automaton. For 1’-acceptance, it suffices to take the given automaton and use it as a Büchi automaton (with the same set of designated states). – I don’t think that’s right, but we can just delete every state which is not in F.

Büchi  Staiger-Wagner Not Possible! For example, the set L of words with infinitely many b’s cannot be recognized by an ω-automaton with Staiger-Wagner acceptance. Proof: assume otherwise, and be 𝑛 the number of states of this automaton. It should have an accepting run on the word 𝑎 𝑛+1 𝑏 𝜔 There is a 𝑘 s.t. after 𝑎 𝑛+1 𝑏 𝑘 it went through all the states to be visited. In the next 𝑎 𝑛+1 there must be a loop; which can also be taken in 𝑎 𝑛+1 𝑏 𝑘 𝑎 𝜔 Total, in the run on 𝑎 𝑛+1 𝑏 𝑘 𝑎 𝜔 same states will be visited like in 𝑎 𝑛+1 𝑏 𝜔 Hence 𝑎 𝑛+1 𝑏 𝑘 𝑎 𝜔 would be accepted- a contradiction. it should be obvious that an infinity condition in the definition of an ω-language cannot in general be simulated by an occurrence condition.

Conclusion We saw today: Expressive equivalence of nondeterministic Büchi, Muller, Rabin, Streett, and parity automata Expressive equivalence of deterministic Muller, Rabin, Streett, and parity automata Some lower bounds on the transformations between them Weak acceptance condition and their inferiority in expressive power.

Spoiler to chapter 3 = = = Nondeterministic models Büchi Automaton Büchi Automaton = Muller Automaton Muller Automaton Rabin Automaton Rabin Automaton = in Chapter 3 : Nondeterministic Büchi automata accept the same ω-languages as deterministic Muller automata. = Streett Automaton Streett Automaton Parity Automaton Parity Automaton

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