ΜΕΣΑΠΣΤΧΙΑΚΗ ΔΙΠΛΩΜΑΣΙΚΗ ΕΡΓΑΙΑ

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1 Αυτόματα σε άπειρες λέξεις και άπειρα δένδρα ΜΕΣΑΠΣΤΧΙΑΚΗ ΔΙΠΛΩΜΑΣΙΚΗ ΕΡΓΑΙΑ Μαρία Γ. Τζιτζιλή Επιβλέπων: Γεϊργιοσ Ραχϊνθσ Επίκουροσ Κακθγθτισ Α.Π.Θ. Θεςςαλονίκθ, Δεκζμβριοσ 2009

2 Automata on infinite words and infinite trees MASTER THESIS Maria G. Tzitzili Supervisor: George Rahonis Assistant Professor Aristotle University of Thessaloniki Thessaloniki, December 2009

3 Αυτόματα σε άπειρες λέξεις και άπειρα δένδρα ΜΕΣΑΠΣΤΧΙΑΚΗ ΔΙΠΛΩΜΑΣΙΚΗ ΕΡΓΑΙΑ Μαρία Γ. Τζιτζιλή Επιβλέπων: Γεϊργιοσ Ραχϊνθσ Επίκουροσ Κακθγθτισ Α.Π.Θ. Εγκρίκθκε από τθν τριμελι εξεταςτικι επιτροπι τθν 21 θ Δεκεμβρίου 2009: Δ. Πουλάκθσ Α. Πάπιςτασ Γ. Ραχϊνθσ Κακθγθτισ Α.Π.Θ. Επ. Κακθγθτισ Α.Π.Θ. Επ. Κακθγθτισ Α.Π.Θ. Θεςςαλονίκθ, Δεκζμβριοσ 2009

4 Μαρία Γ. Σηιτηιλι Πτυχιοφχοσ Μακθματικόσ Α.Π.Θ. Copyright Μαρία Γ. Σηιτηιλι, Με επιφφλαξθ παντόσ δικαιϊματοσ. All rights reserved. Απαγορεφεται θ αντιγραφι, αποκικευςθ και διανομι τθσ παροφςασ εργαςίασ, εξ ολοκλιρου τμιματοσ αυτισ, για εμπορικό ςκοπό. Επιτρζπεται θ ανατφπωςθ, αποκικευςθ και διανομι για ςκοπό μθ κερδοςκοπικό, εκπαιδευτικισ ι ερευνθτικισ φφςθσ, υπό τθν προχπόκεςθ να αναφζρεται θ πθγι προζλευςθσ και να διατθρείται το παρόν μινυμα. Ερωτιματα που αφοροφν τθ χριςθ τθσ εργαςίασ για κερδοςκοπικό ςκοπό πρζπει να απευκφνονται προσ τον ςυγγραφζα. Οι απόψεισ και τα ςυμπεράςματα που περιζχονται ςε αυτό το ζγγραφο εκφράηουν τον ςυγγραφζα και δεν πρζπει να ερμθνευτεί ότι εκφράηουν τισ επίςθμεσ κζςεισ του Α.Π.Θ.

5 ΠΕΡΙΛΗΨΗ Γύν βαζηθά ραξαθηεξηζηηθά ησλ πξνγξακκάησλ πνπ απαηηνύλ ηα ζύγρξνλα ππνινγηζηηθά ζπζηήκαηα, είλαη ε αδηάθνπε αιιειεπίδξαζε κε ην πεξηβάιινλ ηνπο θαη ε κε-ηεξκαηίδνπζα ζπκπεξηθνξά ηνπο. Γηα ηελ θαηαζθεπή θαη ηελ αλάιπζε ησλ παξαπάλσ πξνγξακκάησλ έρεη αλαπηπρζεί κηα ηζρπξή ζεσξεηηθή βάζε κε ηε βνήζεηα ηεο ζεσξίαο ησλ απηνκάησλ ζε άπεηξα αληηθείκελα. Τα θύξηα ζπζηαηηθά απηήο ηεο ζεσξίαο είλαη: απηόκαηα σο θπζηθό «κνληέιν» ησλ ζπζηεκάησλ πνπ βαζίδνληαη ζε θαηαζηάζεηο, ινγηθά ζπζηήκαηα γηα ηνλ πξνζδηνξηζκό ηεο κε-ηεξκαηίδνπζαο ζπκπεξηθνξάο, άπεηξα παίγληα δύν παηρηώλ σο πιαίζην γηα ην ζρεδηαζκό ηεο δηαξθνύο αιιειεπίδξαζεο κεηαμύ ελόο πξνγξάκκαηνο θαη ηνπ πεξηβάιινληόο ηνπ. Απηή ε ζεσξία ησλ απηνκάησλ, ηεο ινγηθήο θαη ησλ άπεηξσλ παηρληδηώλ έρεη ελ ησ κεηαμύ παξάγεη έλα κεγάιν αξηζκό από βαζηά θαη από καζεκαηηθήο άπνςεο ειθπζηηθά απνηειέζκαηα. Αθόκα πην ζεκαληηθό είλαη ην γεγνλόο όηη ε ζεσξία απηή είλαη ζηελά ζπλδεδεκέλε κε ηελ αλάπηπμε αιγνξίζκσλ γηα η νλ έιεγρν κνληέισλ (model checking) θαη ηε ζύλδεζε ησλ ππνινγηζηώλ κε ην ινγηζκηθό. Πιήζνο εξγαιείσλ ινγηζκηθνύ πνπ έρνπλ αλαπηπρζεί ζηε βάζε απηή, βξίζθνπλ ζήκεξα πξαθηηθή εθαξκνγή ζηε βηνκεραλία. Από ηελ άιιε, όιν θαη πεξηζζόηεξν απμάλεηαη ε αλάγθε γηα αθόκα πην ηζρπξά ζεσξεηηθά απνηειέζκαηα γηα ηε ζπλερή βειηίσζε απηώλ ησλ εξγαιείσλ θαη ηελ επέθηαζε ηνπ πεδίνπ εθαξκνγήο ηνπο. I

6 Σε απηή ηελ έξεπλα, ηεξάζηηα πξόνδνο έρεη ζεκεησζεί ηα ηειεπηαία δέθα ρξόληα, ηόζν ράξε ζηηο λέεο γλώζεηο ζρεηηθά κε ηα πην θιαζηθά απνηειέζκαηα, όζν θαη ράξε ζηε δεκηνπξγία λέσλ κεζόδσλ θαη θαηαζθεπώλ. Η ζεσξία ησλ απηνκάησλ πάλσ ζηηο άπεηξεο ιέμεηο, ηα ιεγόκελα σ-απηόκαηα, μεθίλεζε κε νξόζεκν ηελ εξγαζία [2] ηνπ Büchi, όπνπ πξνηάζεθε ε «ζπλζήθε αλαγλσξηζηκόηεηαο ή απνδνρήο» ηνπ Büchi γηα ηα κεπξνζδηνξηζηά απηόκαηα. Πεξαηηέξσ ζπλζήθεο αλαγλσξηζηκόηεηαο εηζήρζεζαλ αξγόηεξα από ηνλ Muller ζηε [15], ηνλ Rabin ζηε [19] θαη ηνλ Mostowski ζηε [14]. Ο ηειεπηαίνο ήηαλ απηόο πνπ εηζήγαγε ηε ιεγόκελε «parity ζπλζήθε». Θα πξέπεη επίζεο λα αλαθεξζνύλ νη έξεπλεο ησλ Steiger θαη W agner ζηηο [24,25,26] ζρεηηθά κε ηελ «εθθξαζηηθή δύλακε» ησλ δηαθόξσλ ζπλζεθώλ απνδνρήο ησλ σ-απηνκάησλ. Ο Thomas ζηηο [28,29] εμήγεζε ηε ζύλδεζε ησλ σ-απηνκάησλ κε ηε κνλαδηαία δεπηεξνβάζκηα ινγηθή θαη ηα άπεηξα παίγληα. Η ηδέα ηνπ άπεηξνπ παηγλίνπ δηείζδπζε ζησπεξά ζηε κειέηε ηεο ζύλζεζεο ησλ ςεθηαθώλ θπθισκάησλ. Ο Church ζηελ [8], έδσζε κηα ζαθή δηαηύπσζε ησλ θεληξηθώλ πξνβιεκάησλ. To ζεκειηώδεο απνηέιεζκα πνπ παξείρε κηα ιύζε ζην πξόβιεκα ζύλζεζεο πνπ είρε ζέζεη ν Church, (πνπ ηζνδπλακεί κε ηελ επίιπζε ησλ Muller παηγλίσλ πάλσ ζε πεπεξαζκέλα γξαθήκαηα), απνδείρζεθε ηειηθά από ηνπο Büchi θαη Landweber ζηελ [6]. Ο Rabin ζηε [17] εηζήγαγε ηα κεπξνζδηνξηζηά πεπεξαζκέλα απηόκαηα πνπ απνδέρνληαη labeled άπεηξα πιήξε δπαδηθά δέλδξα. Ο Rabin μεθίλεζε κε ηελ απνδνρή Muller ζηε [15] θαη ζηε [17] όζνλ αθνξά ηα πεπεξαζκέλα απηόκαηα ζε άπεηξα δέλδξα, κειέηεζε θαη έδσζε απαληήζεηο ζην «πξόβιεκα ηνπ ζπκπιεξώκαηνο» θαη ζην «πξόβιεκα ηνπ θελνύ». Γελ πξέπεη λα μεράζνπκε λα αλαθέξνπκε ηνλ Safra, ηνπ νπνίνπ ν θαηαζθεπαζηηθόο αιγόξηζκνο ζηελ [22] ζπλέβαιε ζηνλ έιεγρν θελόηεηαο ησλ parity δελδξν-απηνκάησλ. II

7 Γύν ζεκαληηθά πιαίζηα ηεο ζεσξίαο καο παξνπζηάδνληαη ζηηο Δλόηεηεο 1 θαη 2 αληίζηνηρα απηήο ηεο δηπισκαηηθήο εξγαζίαο: απηόκαηα ζε άπεηξεο ιέμεηο (σ-απηόκαηα) θαη άπεηξα παίγληα δύν παηρηώλ. Απηό έρεη δύν πηπρέο: Πξώηνλ, έλα ζύζηεκα κεηάβαζεο πνπ ρξεζηκνπνηείηαη γηα λα θαζνξίζεη ηα ζηνηρεηώδε βήκαηα πνπ πξαγκαηνπνηνύληαη ζε κηα κεηεξκαηίδνπζα δηαδηθαζία. Γεύηεξνλ, ηα άπεηξα runs, αληίζηνηρα plays, πνπ παξάγνληαη κε απηόλ ηνλ ηξόπν ζα ειέγρνληαη κε κηα "ζπλζήθε απνδνρήο" ή " ζπλζήθε λίθεο". Σπλήζσο νη ζπλζήθεο απηέο αλαθέξνληαη ζε απηέο ηηο θαηαζηάζεηο ή ηηο θνξπθέο πνπ επηζθέπηνληαη απείξσο ζπρλά ζε έλα run. Οη δηαθνξεηηθέο ρξήζεηο ησλ ελ ιόγσ θαηαζηάζεσλ νδεγνύλ ζε έλα επξύ θάζκα ζπλζεθώλ απνδνρήο θαη λίθεο. Ο θεληξηθόο ζθνπόο ησλ πξώησλ δύν ελνηήησλ είλαη λα απνζαθελίζνπλ ηελ εθθξαζηηθή δύλακε ησλ παξαπάλσ ζπλζεθώλ θαη όζνλ αθνξά ηα άπεηξα παίγληα, λα δείμεη πώο ην είδνο ηεο ζπλζήθεο λίθεο επεξεάδεη ην είδνο ησλ πηζαλώλ ζηξαηεγηθώλ λίθεο. Η Δλόηεηα 3 αζρνιείηαη κε έλα ζεκειηώδεο αιγνξηζκηθό εξώηεκα, πνπ είλαη ε κεηαηξνπή ησλ απηνκάησλ ζύκθσλα κε ινγηθέο πξάμεηο, ηδίσο κε ηελ πξάμε ηνπ ζπκπιεξώκαηνο. Έλαο βαζηθόο ηξόπνο πξνζέγγηζεο, πξνθεηκέλνπ λα επηηεπρζεί ε κεηάθξαζε ησλ ηύπσλ ζε απηόκαηα, είλαη λα πξνρσξήζνπκε επαγσγηθά ζηελ θαηαζθεπή ησλ ηύπσλ. Τέηνηεο θαηαζθεπέο παξνπζηάδνληαη γηα ηελ πεξίπησζε πνπ ηα άπεηξα δέληξα είλαη ηα ππό εμέηαζε «κνληέια». Σηελ Δλόηεηα 3 ινηπόλ, αλαπηύζζεηαη ε ζεσξία πάλσ ζηα άπεηξα δέλδξα, πξνεξρόκελε από ην έξγν ηνπ Rabin ζηε [17]. Οηηδήπνηε παξνπζηάδεηαη ζε απηέο ηηο ηξεηο ελόηεηεο βαζίδεηαη ζηηο κειέηεο δηαθόξσλ επηζηεκόλσλ, κεξηθέο από ηηο νπνίεο αλαθέξνληαη αλσηέξσ. Η παξνπζίαζε πνπ αθνινπζεί απεπζύλεηαη ζε αλαγλώζηεο πνπ έρνπλ γλώζεηο Λνγηθήο θαη Θεσξίαο Απηνκάησλ. Παξαθάησ ζα δώζνπκε πην αλαιπηηθά ην III

8 πεξηερόκελν ηεο θάζε ελόηεηαο ηεο παξνύζαο δηπισκαηηθήο εξγαζίαο. Σηελ Δλόηεηα 1 παξνπζηάδνπκε ηα απηόκαηα πάλσ ζε άπεηξεο ιέμεηο πνπ ζα ηα απνθαινύκε ζην εμήο σ -απηόκαηα. Δπίζεο νξίδνπκε ηηο αθόινπζεο ζπλζήθεο απνδνρήο: ηελ Büchi Απνδνρή, ηελ Muller Απνδνρή, ηελ Rabin Απνδνρή, ηελ Parity Σπλζήθε θαη ηα αληίζηνηρα κε-πξνζδηνξηζηά απηόκαηα. Με ηε βνήζεηα ελόο ζεσξήκαηνο, ζύκθσλα κε ην νπνίν, ε νηθνγέλεηα ησλ σ-γισζζώλ πνπ αλαγλσξίδνπλ ηα Büchi απηόκαηα είλαη ην σ- Kleene closure ηεο θιάζεο ησλ θαλνληθώλ γισζζώλ, θαηαιήγνπκε ζην ζπκπέξαζκα όηη θάζε κε θελή Büchi αλαγλσξίζηκε γιώζζα πεξηέρεη κηα ηειηθά πεξηνδηθή ιέμε. Δπίζεο ζπκπεξαίλνπκε όηη ππάξρεη αιγόξηζκνο πνπ λα απνθαζίδεη γηα η ν αλ κηα Büchi αλαγλσξίζηκε γιώζζα είλαη θελή ή όρη. Σηε ζπλέρεηα δείρλνπκε πώο κπνξεί έλα Büchi απηόκαην κε πιήζνο θαηαζηάζεσλ n λα κεηαζρεκαηηζηεί ζε έλα ηζνδύλακν Muller απηόκαην κε ην ίδην πιήζνο θαηαζηάζεσλ. Γηα ην αληίζηξνθν, δείρλνπκε πώο έλα Muller απηόκαην κε n θαηαζηάζεηο θαη m ζύλνια απνδνρήο κπνξεί λα κεηαζρεκαηηζηεί ζε έλα ηζνδύλακν Büchi κε n mn2 θαηαζηάζεηο. Όκνηα, δείρλνπκε ην κεηαζρεκαηηζκό ελόο Rabin απηνκάηνπ ζε έλα ηζνδύλακν Muller θαη ελόο parity απηνκάηνπ ζε έλα ηζνδύλακν Rabin. Τα ζπκπεξάζκαηά καο σο εδώ ζπλνςίδνληαη ζηα εμήο: (1) Τα κε-πξνζδηνξηζηά Büchi, Muller, Rabin θαη parity απηόκαηα είλαη όια ηζνδύλακα ζε εθθξαζηηθή δύλακε, δειαδή αλαγλσξίδνπλ ηηο ίδηεο σ -γιώζζεο θαη (2) Οη σ- γιώζζεο πνπ αλαγλσξίδνληαη από ηα παξαπάλσ σ -απηόκαηα απνηεινύλ ηελ θιάζε σ-kc(reg), δειαδή ην σ-kleene closure ηεο θιάζεο ησλ θαλνληθώλ γισζζώλ. Σηε ζπλέρεηα αζρνινύκαζηε κε ηα αληίζηνηρα πξνζδηνξηζηά απηόκαηα, όπνπ εμεγνύκε γηαηί ηα Büchi απηόκαηα είλαη πνιύ αδύλακα ζην λα αλαγλσξίδνπλ αθόκα θαη πνιύ απιέο σ-γιώζζεο. Έπεηηα παξαζέηνπκε ηε δηαδηθαζία κεηαζρεκαηηζκνύ ελόο πξνζδηνξηζηνύ Muller απηνκάηνπ κε n IV n

9 θαηαζηάζεηο ζε έλα ηζνδύλακν πξνζδηνξηζηό Rabin κε n n! θαηαζηάζεηο θαη ζε έλα ηζνδύλακν πξνζδηνξηζηό parit y κε n n! θαηαζηάζεηο θαη 2n ρξώκαηα. Καηαιήγνπκε ινηπόλ ζην όηη ηα πξνζδηνξηζηά Muller, Rabin θαη parity απηόκαηα αλαγλσξίδνπλ ηηο ίδηεο σ-γιώζζεο θαη επηπιένλ ε θιάζε ηνπο είλαη θιεηζηή σο πξνο ην ζπκπιήξσκα. Δλώ κέρξη απηό ην ζεκείν νη «απνδεθηέο» θαηαζηάζεηο ήηαλ κόλν απηέο πνπ εκθαλίδνληαλ άπεηξεο θνξέο ζε έλα run, ηώξα νξίδνπκε ηηο «αζζελείο ζπλζήθεο απνδνρήο» έηζη ώζηε λα απνδέρνληαη θαηαζηάζεηο πνπ εκθα λίδνληαη ηνπιάρηζηνλ κία θνξά (νπόηε ίζσο όρη άπεηξ εο θνξέο). Τέηνηεο ζπλζήθεο είλαη απηή ησλ Steiger-W agner θαη νη εηδηθέο πεξηπηώζεηο ηεο: ε 1- απνδνρή θαη ε 1 ' -απνδνρή. Η ρξεζηκόηεηα ησλ ηειεπηαίσλ έγθεηηαη κεηαμύ άιισλ ζηελ απνθπγή εθζεηηθήο έθξεμεο θαηά ην κεηαζρεκαηηζκό άιινπ πξνζδηνξηζηνύ απηνκάηνπ ζε Βüchi απηόκαην. Σηελ Δλόηεηα 2 νξίδνπκε ηα άπεηξα παίγληα δύν παηρηώλ πάλσ ζε θαηεπζπλόκελα γξαθήκαηα κε ηε βνήζεηα ησλ ελλνηώλ «αξέλα», play θαη «ζύλνιν λίθεο». Αλαινγηθά κε ηηο ζπλζήθεο απνδνρήο ζηηο νπνίεο αλαθεξζήθακε ζηελ Δλόηεηα 1, νξίδνπκε ηώξα ηηο «ζπλζήθεο λίθεο». Σπλεπώο, έρνπκε ηηο εμήο ζπλζήθεο λίθεο: Τελ Muller, ηελ Rabin, ηελ Rabin chain, ηελ parity, ηελ Büchi θαη ηελ 1-winning ζπλζήθε λίθεο θαη κε παξόκνην ηξόπν ραξαθηεξίδνπκε ηα αληίζηνηρα παίγληα. Έηζη γηα παξάδεηγκα κηιάκε γηα Muller, Βüchi παίγληα. Παξαθάησ εηζάγνπκε ηηο έλλνηεο ηεο «ζηξαηεγηθήο», ηεο «ζηξαηεγηθήο λίθεο» θαη ηεο «πεξηνρήο λίθεο». Σρεηηθά κε ηε κεηαηξνπή ζπλζεθώλ λίθεο από ηε κία κνξθή ζηελ άιιε απνδεηθλύνπκε ην εμήο: Γηα θάζε Muller παίγλην (Α,χ,F) ππάξρεη έλα parity παίγλην (Α,χ,Acc ) θαη κηα ζπλάξηεζε r : V V' ηέηνηα ώζηε γηα θάζε v V ν Παίρηεο ζ λα θεξδίδεη ην ((Α,χ,F),v) αλ θαη κόλν αλ θεξδίδεη ην ((Α,χ,Acc ),r(v)). Οξίδνπκε έπεηηα πόηε έλα παίγλην είλαη «θαζνξηζκέλν» θαζώο θαη ηηο έλλνηεο forgetful θαη memoryless ζηξαηεγηθή παξαζέηνληαο θαη V

10 ζρεηηθά παξαδείγκαηα. Σεκεηώλνπκε εδώ όηη θάζε parity παίγλην είλαη θαζνξηζκέλν θαη ζπλεπώο θάζε θαλνληθό παίγλην είλαη θαζνξηζκέλν. Απνδεηθλύνπκε αθόκα ηνλ memoryless θαζνξηζκό ησλ parity παηγλίσλ θαη όηη ζε θάζε Rabin παίγλην ν Παίρηεο 0 έρεη κία memoryless ζηξαηεγηθή λίθεο ζηελ πεξηνρή λίθεο ηνπ. Σην ηέινο ηεο ελόηεηαο απηήο απνδεηθλύνπκε θάπνηεο εηδηθέο πεξηπηώζεηο ησλ παξαπάλσ ζεσξεκάησλ θαη πνξηζκάησλ νξίδνληαο πξώηα ηα reachability παίγληα θαη απνδεηθλύνληαο όηη ραίξνπλ memoryless θαζνξηζκνύ. Έπεηηα απνδεηθλύνπκε ηνλ memoryless θαζνξηζκό ηόζν γηα ηα 1-παίγληα όζν θαη γηα ηα Büchi παίγληα. Η Δλόηεηα 3 κεηαθέξεη ηνλ αλαγλώζηε από ηα απηόκαηα ιέμεσλ, (πνπ είλαη απηά πνπ θαηαλαιώλνπλ άπεηξεο ιέμεηο), ζηα «δελδξναπηόκαηα», δειαδή ηα πεπεξαζκέλσλ θαηαζηάζεσλ απηόκαηα, ηα νπνία θαηαλαιώλνπλ άπεηξα δέλδξα θαη αλαθεξόκαζηε θπξίσο ζε άπεηξα δπαδηθά δέλδξα. Γίλνληαο θάπνηεο εηζαγσγηθέο έλλνηεο, όπσο απηή ηνπ «κνλνπαηηνύ» πεξλάκε ζηνλ νξηζκό ησλ ζπλζεθώλ απνδνρήο, αιιά γηα δελδξναπηόκαηα απηή ηε θνξά. Έηζη νξίδνπκε αλαιπηηθά ηα Muller θαη ηα parity δελδξναπηόκαηα θαη εμεγνύκε όηη κε αλάινγν ηξόπν κπνξνύλ λα νξηζηνύλ θαη ηα Büchi θαη Rabin δελδξναπηόκαηα. Αθνινπζνύλ δύν ζεκαληηθά ζεσξήκαη α. Τν πξώην απνδεηθλύεη όηη ηα Büchi δελδξναπηόκαηα είλαη απζηεξώο αζζελέζηεξα από ηα Muller δελδξναπηόκαηα ππό ηελ έλλνηα όηη ππάξρεη γιώζζα αλαγλσξίζηκε από Muller δελδξναπηόκαην πνπ όκσο δελ είλαη αλαγλσξίζηκε από Büchi δελδξναπηόκαην. Τν δεύηεξν ζεώξεκα, κε κηα απόδεημε βαζηζκέλε ζε αληίζηνηρνπο κεηαζρεκαηηζκνύο ηεο πξώηεο ελόηεηαο, δείρλεη όηη ηα Muller, ηα parity θαη ηα Rabin δελδξναπηόκαηα αλαγλσξίδνπλ όια ηηο ίδηεο δελδξνγιώζζεο. Σηηο ηειεπηαίεο δύν παξαγξάθνπο ηεο Δλόηεηαο 3 αζρνινύκαζηε κε ηα πνιπζπδεηεκέλα πξνβιήκαηα, απηά ηνπ ζπκπιεξώκαηνο θαη ηεο θελόηεηαο. Γηα ην πξώην εξγαδόκαζηε σο εμήο: Οξίδνπκε έλα VI

11 parity δελδξναπηόκαην Α θη έλα input δέλδξν κε έλα άπεηξν παίγλην δύν παηρηώλ G A, t έρνληαο ηνλ Παίρηε 0 θαη ηνλ Παίρηε 1 λα παίδνπλ ην παίγλην ζην t. Μεηά από καθξνζθειή δηαδηθαζία θαηαιήγνπκε ζην όηη έλα δελδξναπηόκαην Α δέρεηαη έλα input δέλδξν t αλ θαη κόλν αλ ππάξρεη κηα ζηξαηεγηθή λίθεο γηα ηνλ Παίρηε 0 από ηε ζέζε (ε,q I ) ζην παίγλην G A, t. Σηε ζπλέρεηα γηα έλα parity δελδξναπηόκαην Α όπσο παξαπάλσ θαη έλα σ-απηόκαην Μ κε parity ζπλζήθε απνδνρήο, ην νπνίν ειέγρεη γηα θάζε κνλνπάηη π ηνπ t θαη γηα θάζε πηζαλή θίλεζε ηνπ Παίρηε 0 μερσξηζηά αλ ζπλαληάηαη ε ζπλζήθε απνδνρήο ηνπ Α, απνδεηθλύνπκε έλα ιήκκα ηδηαίηεξα βνεζεηηθό ζην λα απνθαλζνύκε γηα ην πξόβιεκα ηνπ ζπκπιεξώκαηνο. Σύκθσλα ινηπόλ κε ην ιήκκα απηό, ην δέληξν s είλαη έλα δέληξν λίθεο γηα ην t αλ θαη κόλν αλ L(s,t) L(Μ)=. Η νξηζηηθή απάληεζε δίλεηαη κε ην ζεώξεκα ζύκθσλα κε ην νπνίν ε θιάζε ησλ γισζζώλ πνπ αλαγλσξίδνληαη από πεπεξαζκέλσλ θαηαζηάζεσλ δελδξναπηόκαηα είλαη θιεηζηή σο πξνο ην ζπκπιήξσκα. Η Δλόηεηα 3 θιείλεη ινηπόλ κε ηε κειέηε ηνπ πξνβιήκαηνο θελόηεηαο γηα ηα απηόκαηα ζε άπεηξα δέλδξα. Γηα ηε βνήζεηα ηεο απόδεημεο ηεο απνθαζηζηκόηεηαο ηεο θελόηεηαο εηζάγνπκε ηηο έλλνηεο input-free δελδξναπηόκαην θαη «θαλνληθό» δέλδξν. Καηαιήγνπκε κε ηε ρξήζε απνηειεζκάησλ από ηα άπεηξα παίγληα ζην όηη γηα ηα parity δελδξναπηόκαηα κπνξεί λα απνθαλζεί θαλείο αλ ε γιώζζα πνπ αλαγλσξίδνπλ είλαη θελή ή όρη. Μάιηζηα θαηαιήγνπκε όηη ην ηεζη θελόηεηαο γηα ηα parity δελδξναπηόκαηα κπνξεί λα γίλεη ζε ρξόλν rn. [d / 2] 2 [d/2] O(d r m( ) ) VII

12 Abstract We introduce automata on in nite words (!-Automata) and de ne the following acceptance conditions: Büchi Acceptance, Muller Acceptance, Rabin Acceptance, the parity condition and the respective nondeterministic automata. Then we show how to transform these automata from one form to another and we prove that all nondeterministic automata are equivalent in expressive power. Next, we study deterministic models of such automata. Deterministic Büchi automata are too weak for recognizing even very simple!-languages. We transform deterministic Muller into deterministic Rabin and parity automata and show that deterministic Muller, Rabin and parity automata recognize the same class of!-languages which is closed under complementation. We introduce also 1 and acceptance, which are weak acceptance conditions. Then, we de ne di erent types of in nite games and explain what a strategy and a winning condition is. We de ne winning conditions analogously to the above acceptance conditions and show how to transform from one to another. The notions of forgetful, memoryless strategies and determinacy are also introduced and help us solve games with simple winning conditions. In the end we study nondeterninistic tree automata and especially Muller, parity and Rabin tree automata and we prove that they all recognize the same tree languages. The only exception are Büchi tree automata, which are much weaker in recognizing. We study the complementation and the emptiness problem for automata on in nite trees and show: (1) closure under complementation for tree languages acceptable by nite-state tree automata and (2) decicability of the emptiness problem for parity tree automata by utilizing results about in nite games on nite graphs. 1

13 CONTENTS "{ I-VII Abstract 1 Contents 2 Preface 3 1.!-Automata Introduction and Notation Notation 5 1.2!-Automata Nondeterministic Models Büchi Acceptance Muller Acceptance Rabin Acceptance The Parity Condition Discussion Deterministic Models The Büchi Condition for Deterministic!-Automata Transforming Muller Automata to Rabin Automata Weak Acceptance conditions In nite Games Introduction Games Arenas Plays Games and Winning Sets Winning Conditions Strategies and Determinacy Strategies Transforming Winning Conditions Determinacy Forgetful and Memoryless Strategies Solving Games with Simple Winning Conditions Reachability Games and Attractors Acceptance Büchi Acceptance Nondeterministic Tree Automata Introduction Preliminaries Finite-State Tree Automata The Complementation Problem for Automata on In nite Trees The Emptiness Problem for Automata on In nite Trees Conclusions 44 References 44 2

14 Preface A central aim of computer science is to put the development of hardware and software systems on a mathematical basis which is both rm and practical. Such a scienti c foundation is necessary especially in the construction of reactive programs, like communication protocols or control systems. Characteristic features of such programs are the perpetual interaction with their environment as well as their nonterminating behaviour. For the construction and analysis of reactive programs an elegant and powerful theoretical basis has been developed with the theory of automata on in nite objects. The main ingredients of this theory are: automata as a natural model of state-based systems, logical systems for the speci cation of nonterminating behaviour, in nite two-person games as a framework to model the ongoing interaction between a program and its environment. This theory of automata, logics and in nite games has meanwhile produced a large number of deep and mathematically appealing results. More important, this theory is intimately connected with the development of algorithms for the automatic veri cation ("model-checking") and synthesis of hardware and software systems. Numerous software tools have been developed on this basis, which are now used in industrial practice. On the other hand, more powerful theoretical results are needed for the continuous improvement of these tools and the extension of their scope. In this research, enormous progress was achieved over the past ten years, both by new insights regarding the more classical results and by the creation of new methods and constructions. The theory of automata on in nite words, so-called!- automata started with Büchi s landmark paper [2], where the Büchi acceptance condition for nondeterministic automata was proposed. Further acceptance conditions were introduced later by Muller in [15], Rabin in [19] and Mostowski in [14] who introduced the parity condition. We should also mention the surveys of Staiger and Wagner [24,25,26] on the expressive power of the diverse acceptance conditions of!-automata. Thomas in [28,29] discussed the connections of the!-automata to monadic second-order logic and in nite games. The idea of in nite games arose implicitly in the study of synthesis of digital circuits. Church in [8] gave a clear formulation of the central problems. The fundamental result providing a solution of Church s synthesis problem (which amounts to solving Muller games over nite graphs) was obtained by Büchi and Landweber in [6]. Rabin in [17] introduced nondeterministic nite automata accepting labeled in nite complete binary trees. Rabin started out with Muller s acceptance in [15] and in [17] with respect to nite automata on in nite trees he showed that complementation is e ective and emptiness is decidable. We should not forget to mention Safra, whose construction algorithm in [22] helped in checking emptiness of parity tree automata. Two frameworks of the theory are introduced in Section 1 and Section 2 respectively: automata over in nite words (!-automata) and in nite two-person games. This involves two aspects: First, a transition system is used to specify elementary steps which are carried out in a nonterminating process. Secondly, the in nite runs, respectively plays, generated 3

15 in this way are checked with an "acceptance condition" or "winning condition". Usually these conditions refer to those states or vertices which are visited in nitely often in a run. Di erent uses of these states lead to a broad spectrum of acceptance and winning conditions. A central purpose of the two sections is to clarify the expressive power of these conditions and for in nite games, to show how the form of a winning condition a ects the form of possible winning strategies. Section 3 deals with a fundamental algorithmic question, that is the transformation of automata according to logical operations in particular complementation. In order to obtain a translation of formulas into automata, the standard approach is to proceed by induction on the construction of formulas. Such constructions are presented for the case that in nite trees are the models under consideration. In this section is developed the theory over in nite trees, originating in the work of Rabin in [17]. Everything presented in these three sections is based on the studies of several scientists, some of them mentioned above. The presentation that follows is directed at readers who have a knowledge of automata theory and logic. 4

16 1!-Automata 1.1 Introduction and Notation Automata on in nite words gain more and more importance during the latest years. This interest about this kind of automata is not only theoritical but also practical, as for example to specify and verify reactive systems that are not supposed to terminate at some point of time. Such systems are operating systems, which need to be able to process any user input as it is entered, without terminating after or during some task. Basically, there is one question to be answered; how to de ne acceptance of in nite words by nite automata. Though in the case of nite words this is a hard question, in this case the possibilities are many and it is a nontrivial problem to compare them with respect to expressive power. Büchi and Muller made the rst publications referring to! - languages. Büchi obtained a decision procedure for a restricted second-order theory of classical logic by using nite automata with in nite inputs [1]. Muller [15] de ned a similar concept in a totally di erent domain, namely in asynchronous switching network theory. Starting from these studies, a theory of automaton de nable! - languages emerged Notation By symbol! we denote the non-negative integers, i.e.,! := f0; 1; 2; 3; : : :g. By = fa; b; c : : :g we denote a nite alphabet, is the set of nite words over, while! denotes the set of in nite words (or!-words) over, where each word 2! has length jj = 1. Finite words are denoted by letters u; v; w; : : :, while in nite words are denoted by small greek letters ; ; ; : : :. We write = (0) (1) : : : with (i) 2. The symbols %; ; : : : are often used to indicate in nite runs of automata. A set of!-words over a given alphabet is called an! - language. The number of occurences of a letter a in words and w is denoted by jj a and jwj a respectively. Given an!-word 2! let Occ() = fa 2 j 9i (i) = ag be the ( nite) set of letters occurring in, and Inf() = fa 2 j 8i 9j > i (j) = ag be the ( nite) set of letters occurring in nitely often in. 2 M denotes the powerset of a set M and jmj denotes the cardinality of M. The i-th projection of an ordered tuple or vector a = (a 1; : : : ; a k ) is de ned for i k and is written i (a) = i. The class of regular languages is denoted by REG. 1.2!-Automata Acceptance of a word by an automaton is a very popular and well-known issue in classical formal theory. Finite computation or nite run of an automaton on a given input word is 5

17 de ned. Control states or control states together with memory contents are used to specify the con gurations, which are considered to be (so-called) " nal". An input is declared an "accepted" one if there exists a run in the input, which terminates in a nal con guration. The entire interest now lies in the acceptance of words by automata and not in generation of!-words by grammars.in addition, we now consider only nite automata. Even for in nite words, we use nite automata. But from now on, we will use!-automaton synonymously for nite!-automaton. Also, we adapt the de nitions of acceptors and generators for contextfree languages and more general language classes to suit the case of in nite words too (see for example [9] and the survey [25]). We use the usual de nitions of deterministic and non-deterministic automata to the case of!-input words by the introduction of new acceptance conditions. For this purpose, we need to introduce an "acceptance component" in the speci cation of automata, which will arise in di erent formats. De nition 1 An!-automaton is a quintiple (Q; ; ; q I ; Acc) ; where Q is a nite set of states, is a nite alphabet, : Q! 2 Q is the state transition function, q I 2 Q is the initial state, and Acc is the acceptance component. (In deterministic!-automata we use the transition function : Q! Q). The acceptance component can be given in many di erent ways; sometimes as a set of states, other times as a set of state-sets or even as a function from the set of states to a nite set of natural numbers. Each case will be presented in the following pages. De nition 2 Let A = (Q; ; ; q I ; Acc) be an!-automaton. A run of A on an!-word = 1 2 : : : 2! is an in nite state sequence % = %(0)%(1)%(2) : : : 2 Q!, such that the following conditions hold: %(0) = q I %(i) 2 (% (i %(i) = (% (i 1) ; a i ) for i 1 ifa is nondeterministic, 1) ; a i ) for i 1 if A is deterministic. The question that needs to be answered now is how each one of the di erent acceptance conditions are related to expressive power. Are there any transformations from one acceptance condition to another and if we establish such transformations, what is their complexity? By jaj we denote the size of an automaton A, which is measured by the number of its states, i.e., for A = (Q; ; ; q I ; Acc) the size is jaj = jqj. Apart from the number of its states, the size of the acceptance condition is also important for the e ciency of the transformation. This size is usually measured by the number of designated sets or pairs of such sets. 1.3 Nondeterministic Models Büchi acceptance The Büchi acceptance condition has originally been introduced for nondeterministic!-automata, where the acceptance component is a set of states. 6

18 De nition 3 An!-automaton A = (Q; ; ; 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 2! is accepted by A i there exists a run % of A on satisfying the condition: Inf(%) \ F 6=? i.e., at least one of the states in F has to be visited in nitely often during the run. L(A) := f 2! j A accepts g is the! - language recognized by A. Example 4 Consider the! - language L over the alphabet fa; b; g de ned by L := f 2 fa; bg! j ends with a! or ends with (ab)! g L is recognized by the nondeterministic Büchi automaton given by the state transition diagram from Figure 1 below, where the states from F are drawn with a double circle. Figure 1: A Büchi automaton accepting words from (a [ b) a! [ (a [ b) (ab)! Now consider a Büchi automaton A = (Q; ; ; q I; F ). If p is the initial state and q a nal state of this automaton, we obtain a regular language W (p; q) of nite words. As we already know, an!-word is accepted by A i some run of A on A visits some nal state q 2 F in nitely often. Then obviously A 2 W (q 0 ; q) W (q; q)!. The union over these sets for q 2 F leads us to the following representation result for Büchi recognizable! - languages. Theorem 5 The Büchi recognizable! - languages are the! - languages of the form L = k[ U i V i! with k 2! and U i; V i 2 REG for i = 1; : : : ; k i=1 This family of! - languages is also called the!-kleene closure of the class of regular languages. 7

19 The conclusion is that each nonempty Büchi recognizable! - language contains an ultimately periodic word. As for the emptiness problem,it is decidable for Büchi automata. This means that there exists an algorithm able to decide whether the language recognized by a nondeterministic Büchi automaton is empty. We can compute the set of reachable states by a given Büchi automaton A. For each reachable state q from F, we can check whether q is reachable from q by a nonempty path. Such a loop exists if and only if there exists an in nite word and a run of A on such that q is a recurring state in this run Muller acceptance The Muller acceptance condition does not refer to an acceptance component which is a set of states like in Büchi case, but to one that is set of state sets F 2 Q. De nition 6 An!-automaton A = (Q; ; ; 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 2! is accepted by A i there exists a run % of A on satisfying the condition: Inf(%) 2 F i.e., the set of in nitely recurring states of % is exactly one of the sets in F. Example 7 Consider again the!-language over fa; bg consisting of the!-words which end with a! or with (ab)!. The deterministic Muller automaton of Figure 2 below recognizes L, where the acceptance component consists of the two sets fq a g; fq a ; q b g. Figure 2: A state transition diagram where the state q a is reached after reading a and q b after reading b. Now we need to verify that nondeterministic Büchi automata are equivalent in expressive power to nondeterministic Muller automata. We can convert a Büchi automaton A = (Q; ; ; q I; F ) to an equivalent Muller automaton by de ning the family F of sets of states by collecting all subsets of Q which contain a state from F. Transformation 1. Let A = (Q; ; ; q I; F ) be a Büchi automaton. De ne the Muller automaton A 0 = (Q; ; ; q I; F) with F := fg 2 2 Q j G \ F 6=?g 8

20 Then L(A) = L(A 0 ). The question now is how we can convert a Muller automaton to a Büchi automaton. Consider a Muller automaton A = (Q; ; ; q I; F). The desired Büchi automaton A 0 simulates A and it guesses the set G 2 F which should turn out to be Inf(%) for the run % to be pursued. A 0 makes another guess during the run to check that the guess is correct. Namely it guesses from which position onwards exactly the states from G will be seen again and again. The visited states are accumulated in memory until the set G is complete, then the memory is being reset to? and the accumulation starts again and so on. If after many resets no state outside G is visited, the automaton A 0 should accept. We obtain the required Büchi automaton by declaring the "reset states" as accepting ones. To make this work, we introduce for each set G 2 F a separate copy of Q \ G: Such states are indicated with index G and we write q G : The automaton A 0 does the two guesses at the same moment, at which time it switches from a state p of Q to a state from q G 2 G and initializes the accumulation component to?. So the new states from the accepting set G will be from 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 (where the R-value is?). So the set Q 0 of states of A 0 is Q 0 = Q [ [ (G 2 G ) G2F and the set F 0 of nal states of A 0 consists of the states (q G,?) for G 2 F. Transformation 2. Let A = (Q; ; ; q I; F) be a Muller automaton. De ne the Büchi automaton A 0 = (Q 0 ; ; 0 ; q I; F 0 ) with Q 0 ; 0 ; F 0 de ned as described above. Then. L(A) = L(A 0 ) If Q has n states and F contains m sets then jq 0 j has at most n + mn2 n = 2 O(n) states. Theorem 8 A 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 + mn2 n states. The transformation schetched in Figure 3 transforms nondeterministic Büchi automata into nondeterministic Muller automata and conversely. For a given deterministic Büchi automaton the transformation yields a deterministic Muller automaton. On the other hand, a deterministic Muller automaton is converted into a nondeterministic Büchi automaton. This nondeterminism cannot in general be avoided. 9

21 Figure 3: A state diagram where q x is reached after reading x Rabin Acceptance Apart from the cases of Büchi and Muller automata, there are also formalisms specifying acceptance and rejection criteria separately. The Rabin condition (also called pairs condition) is such a condition. The acceptance component is given by a nite family of pairs (E i; F i ) of designated state sets with the understanding that the sets E i should be excluded from an accepting run after a nite initial segment, while at least one state in F i has to be visited in nitely often. De nition 9 An!-automaton A = (Q; ; ; q I; ) with acceptance component = f(e 1; F 1 ) ; : : : ; (E k; F k )g with E i; F i Q is called Rabin automaton if it is used with the following acceptance condition (Rabin acceptance) : A word 2! is accepted by A i there exists a run % of A on satisfying the condition: 9 (E; F ) 2 : (Inf(%) \ E =?) ^ (Inf(%) \ F 6=?) Example 10 The Rabin automaton with state transition diagram from Figure 2 and Rabin condition = f(fq b g; fq a g)g accepts all words that contain in nitely many a s but only nitely many b s. We have to use = f(?; fq a g); (fq a ; q b g;?)g to specify the language consisting of all words that contain in nitely many a s only if they also contain in nitely many b s with a Rabin automaton based on the state graph from Figure 2. This means that each word in the accepted language has either in nitely many b s or it has neither in nitely many b s nor in nitely many a s. Obviously, in the latter case no!-word can be accepted and the condition could be simpli ed to = f(?; fq a g)g: 10

22 Transformation 3. Let A = (Q; ; ; q I; ) be a Rabin automaton. De ne a Muller automaton A 0 = (Q; ; ; q I; F) with F := fg 2 2 Q j 9 (E;F ) 2 : G \ E =? ^ G \ F 6=?g Then L(A) = L(A 0 ). For the converse we have to convert the Muller automata to Büchi automata and to observe that the Büchi acceptance can be viewed as a special case of Rabin acceptance (for the set F of nal states we take = f(?; F )g: The Parity Condition The parity condition can be viewed as a special case of the Rabin condition, where the accepting pairs (E 1; F 1 ) ; : : : ; (E m; F m ) form a chain with respect to set inclusion. We consider the case of an increasing chain E 1 F 1 E 2 : : : E m F m : Now we shall associate indices (called colours) with states as follows: states of E 1 receive colour 1, states of F 1 ne 1 receive colour 2, and so on with the rule that states of E i nf i 1 have colour 2i-1 and states of F i ne i have colour 2i. An!-word A is then accepted by the Rabin automaton i the least colour occurring in nitely often in a run on A is even. De nition 11 An!-automaton A = (Q; ; ; q I; c) with acceptance component c : Q! f0; : : : ; kg where k 2 is called a parity automaton if it is used with the following acceptance condition (parity condition) : A word 2! is accepted by A i there exists a run % of A on satisfying the condition: minfc(q) j q 2 Inf(%)g is even Example 12 Consider the parity automaton from Figure 4 with colouring function c de ned by c(q i ) = i. It accepts the!-words which start with ab, continue with (a cb c) and end with a! : So L(A) = ab(a cb c) a! : Transformation 4. Let A = (Q; ; ; q I; c) be a parity automaton with c : Q! f0; : : : ; kg: An equivalent Rabin automaton A 0 = (Q; ; ; q I; ) has the acceptance component = f(e 0; F 0 ) ; : : : ; (E r; F r )g with r := k 2, Ei : = fq 2 Q j c(q) < 2ig and F i : = fq 2 Q j c(q) 2ig 11

23 Figure 4: A parity automaton accepting words from ab(a cb c) a! Discussion Theorem 13 (1) Nondeterministic Büchi automata, Muller automata, Rabin automata and parity automata are all equivalent in expressive power, i.e., they recognize the same!-languages. (2) The!-languages recognized by these!-automata form the class! KC(REG), i.e., the!- Kleene closure of the class of regular languages. Now two questions arise: Are there types of deterministic!-automata which recognize precisely the!-languages in! REG? Is the class! REG closed under complementation? The answer is a rmative in both questions. 1.4 Deterministic Models It will be discussed below the relation between deterministic Muller automata and other deterministic!-automata and will be given some remarks on the complementation problem. We will see that deterministic Muller, Rabin and parity automata are all equivalent in expressive power and that it is too hard to sharpen the construnction of a Büchi automaton to obtain a deterministic one The Büchi Condition for Deterministic!-Automata Büchi automata are too weak to recognize even very simple!-languages from! REG. The Büchi automaton A in Figure 5 with F = fq 1 g accepts those!-words over the alphabet fa; bg that have only nitely many b s. We can easily provide an equivalent deterministic Muller automaton, using two states, q a ; q b which are visited after reading a; b respectively. As acceptance component we declare F = ffq a gg. Then a run is accepting i the input word ends with a! : 12

24 Figure 5: An automaton recognizing (a [ b)a! If we had tried to work with the Büchi acceptance condition using a set F of accepting states, we would have a speci cation of states which should be visited in nitely often, but we would not be able to specify directly which states should be seen only nitely often. It is now necessary to explain why deterministic Büchi automata are too weak to recognize L = (a [ b) b! and we will show it by contradiction: Assume that the deterministic Büchi automaton A with nal state set F recognizes L, then on input b! it will visit an F -state after a nite pre x, say after the n 0 -th letter. It will also accept b n 0 ba!, visiting F -states in nitely often and hence after the a, say when nishing the pre x b n 0 ab n 1. Continuing this construnction, the!-word b n 0 ab n 1 ab n 2 a... is generated. But this causes A to pass through an F -state before each letter a, which should of course be rejected Transforming Muller Automata to Rabin Automata For this transformation we use a technique called latest appearance record (LAR). For this purpose we use permutations of the states of the given Muller automaton as new states, extended by a hit position, so that the memory of the new automaton stores lists of states from the original automaton. Speci cally, 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 (which is the position of the marker \) indicates where the last change occurred in the record. Every time in the original automaton we move from a state p to a state q, 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 that the previous place of q is lled again). We insert the marker in front of the position where q was taken from, so the positions before the marker are untouched by the transition under consideration. Transformation 5. Let A = (Q; ; ; q I; F) be a deterministic Muller automaton. Assume (without loss of generality) that Q = f1; : : : ; kg and q I = 1. Let \ be a new symbol i.e., \ =2 Q. An equivalent Rabin automaton A 0 is given by the following de nition. eq is the set of all order vector words with hit position over Q, i.e., fq := fw 2 (Q [ f\g) j 8q 2 Q [ f\g jwj q = 1 The initial state is q 0 I := \k : : : 1 The transition function 0 is constructed as follows: Assume i; i 0 2 Q; a 2 and (i; a) = i 0. Then 0 is de ned for any word m 1 : : : m r 1 \m r+1 : : : m k 2 e Q; with m k = i: 13

25 Supposing that i 0 = m s, de ne 0 (m 1 : : : m r 1 \m r+1 : : : m k ; a) := (m 1 : : : m s 1 \m s+1 : : : m k i 0 ) The acceptance component is given by = f(e 1 ; F 1 ); : : : ; (E k ; F k )g, de ned as follows: - E j := fu\v j juj < jg - F j := fu\v j juj < jg [ fu\v j juj = j ^ fm 2 Q j m v vg 2 F g (The in x relation m v v should be read "m occurs in v", since m is a single letter) Consider the run of a Muller automaton A, where J = fm 1; : : : ; m j g is the set of in nitely often visited states. This means that in the corresponding run of the Rabin automaton A 0, the states of QnJ will eventually reach the rst positions and then stay inde nitely in front of the marker. So, nally the A 0 -states will be of the form u\v where the (QnJ)-elements occur at the beginning of u (or constitute the whole word u). Hence, we will have juj jqnjj, in other words jvj jjj = j. Clearly, in nitely often we have jvj = jjj = j, since otherwise, from some point onwards we would have jvj < j and thus less than j states would be visited in nitely often. So, in nitely often a state u\v with v > j. Moreover, the states which constitute the word v in the rst case jvj = j form precisely the set J. We can summarize this as follows: Lemma 14 Let % be an in nite run of the deterministic Muller automaton A with state set Q = f1; : : : ; kg and let u 0 \v 0 ; u 1 \v 1 ; u 2 \v 2 ; : : : be the corresponding sequence of order vectors with hit, according to the transformation above. Then Inf(%) = J with jjj = j i the following conditions hold: (1) for only nitely many i we have jv i j > j (and hence ju i j k j) (2)for in nitely many i we have jv i j = j (and hence ju i j = k j) and J = fm 2 Q j m v v i g The Muller automaton A accepts by the run % if the set J considered in the Lemma belongs to F. So, the run will visit a state in the set F k j in nitely often, but will visit states u\v with juj < k j i.e., states from E k j only nitely often. Hence, the Rabin condition of A 0 is satis ed and A 0 accepts in this case. Symmetrically is shown the converse (if A 0 accepts an input word, then A does). From the de nition of the sets E j ; F j we see that they are arranged in a chain: E 1 F 1 E 2 : : : E k F k : We can shorten the chain by admitting only pairs where E j 6= F j, without altering the set of accepting runs. Then we have a strictly increasing chain of sets and thus have de ned an!-automaton which is presentable as a parity automaton. We obtain the following result: 14

26 Theorem 15 By Transformation 5, 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 colours. Transformation 6. Let A = (Q; ; ; q I; F) be a deterministic Muller automaton. Then, the Muller automaton A 0 := Q; ; ; q I ; 2 Q nf recognizes the complement of L(A). Theorem 16 Deterministic Muller automata, Rabin automata and parity automata recognize the same!-languages and the class of the!-languages recognized by any of these types of!- automata is closed under complementation. In order to show that the complement of a language accepted by an!-automaton with parity condition is also acceptable by a parity automaton, we have to modify the colour function, such that henceforth every word previously not accepted has even parity in its minimal colour value and uneven parity for all previously accepted words. Transformation 7. Let A = (Q; ; ; q I; c) be a deterministic!-automaton with parity condition. Then the complement of L(A) is recognized by the parity automaton A 0 := (Q; ; ; q I; c 0 ) where c 0 (q) = c(q) + 1. As it is obvious, the process of the complementation is easy and does not a ect the number of the states of the automata, if we deal with deterministic Muller or parity automata. Lemma 17 Let A = (Q; ; ; q I; ) be an!-automaton with Rabin condition and assume % 1 ; % 2 are two non-accepting runs. Then any run % with Inf(%) = Inf(% 1 ) [ Inf(% 2 ) is also non-accepting. For the proof assume that % 1 ; % 2 are non-accepting but % with Inf(%) = Inf(% 1 )[Inf(% 2 ) is accepting. Then for some accepting pair (E; F ) we have Inf(%) \ E =? and Inf(%) \ F 6=?. By Inf(%) = Inf(% 1 ) [ Inf(% 2 ) we must have Inf(% 1 ) \ E = Inf(% 2 ) \ E =? and also Inf(% 1 ) \ F 6=? or Inf(% 2 ) \ F 6=?: So one of the two runs would be accepting, contradicting the assumption. 1.5 Weak Acceptance Conditions Until now we have only de ned acceptance by a reference to those states in a run which occur in nitely often. But for some purposes a "weak acceptance condition" is appropriate. This is a condition on the set of states that occur at least once in a run, but maybe only nitely often. Let Occ(%) := fq 2 Q j j% 1 (q)j 1 15

27 be the set of states that occur at least once in % and let A = (Q; ; ; q I; Acc) be an!- automaton. The set Occ(%) can be used for acceptance in many di erent ways. Similar to the Muller condition, we can use a family F of state sets and declare the run accepting if Occ(%) 2 F This mode was introduced by Staiger and Wagner [25]. Other acceptance modes refer to a set F of designated states and require (also called 1-acceptance), and Occ(%) \ F 6=? Occ(%) F,(also called 1 0 -acceptance). These acceptance modes are special cases of Staiger and Wagner acceptance. In the rst case one collects in F all sets X with X \ F 6=?, while in the second case the sets X with X F: Example 18 We take an automaton A = (fq a ; q b g; fa; bg; ; q b ; F ) to accept the!-words over the alphabet fa; bg that have at least one symbol a. We have F = fq a g and is de ned according to the state transition graph of Figure 2 and 1-acceptance is used. The requirement that only the word b! should be accepted can be speci ed with the same transition graph, now with 1 0 -acceptance using the set F = fq b g i.e., the only state that may be visited in any successful run is q b. The parity condition can be also used in the weak sense. The requirement for acceptance is that the minimal (or maximal) colour occuring in a run is even. Acceptance by an occurrence set can be simulated by Büchi acceptance. The idea is to simulate A and to accumulate the visited states in a separate component of the state, signalling acceptance whenever this componenet is a set from F. Transformation 8. Let A = (Q; ; ; q I; F). The language L(A) recognized by A with the Staiger-Wagner acceptance condition is recognized by a Büchi automaton A 0 = (Q 2 Q ; ; 0 ; (q I ; fq I g); F 0 ) where 0 ((p; P ); a) contains all states (p 0 ; P 0 ) with p 0 2 (p) and P 0 = P [fp 0 g and F 0 contains all states (p; P ) with P 2 F. The exponential blow-up can be avoided only by the use of 1-acceptance or 1 0 -acceptance. To help us capture 1-acceptance via a set F by Büchi acceptance, we introduce a transition from each F -state to a new state q f, with a transition back to q f, which serves as the only 16

28 nal state in the Büchi automaton. For 1 0 -acceptance, it su ces to take the given automaton and use it as a Büchi automaton (with the same set of designated states). The reverse transformations are not possible. An in nity condition in the de nition of an!-language cannot in general be simulated by an occurence condition. As a conclusion, we can note that we have shown the expressive equivalence of nondeterministic Büchi, Muller, Rabin and parity automata deterministic Muller, Rabin and parity automata. 2 In nite Games 2.1 Introduction We will introduce now in nite two-person games on directed graphs. We will de ne what they are, how they are played, what exactly a strategy is, what we mean when we say a game is won by a certain player, etc. We will also introduce fundamental notions such as determinacy, forgetful strategies, memoryless strategies and we will state fundamental results. 2.2 Games A game is composed by an arena and a winning condition. We will begin from arenas and then we will add the winning conditions Arenas An arena is a triple A = (V 0 ; V 1; E) where V 0 is a set of 0-vertices, V 1 is a set of 1-vertices, disjoint from V 0 and E (V 0 [ V 1 ) (V 0 [ V 1 ) is the edge relation, sometimes also called the set of moves. The union of V 0 and V 1 is denoted V. The edge relation now turns to be: E V V. The set of successors of v 2 V is de ned by ve = fv 0 2 V j (v; v 0 ) 2 Eg We are interested only in games played by two players called Player 0 and Player 1. We will often say Player, where 2 f0; 1g. If we then want to refer to his opponent, we will call him Player. Formally we set = 1 for 2 f0; 1g. Note here that there is no restriction on the number of the successors of a vertex in an arena and that (V ; E) is not required to be a bi-partite graph with corresponding partition fv 0 ; V 1 g. 17

29 2.2.2 Plays Let us now describe a play of a game with an arena as above. A token is palced on some initial vertex v 2 V. If v is a 0-vertex, then Player 0 moves the token from v to a successor v 0 2 ve of v. Symmetrically, if v is a 1-vertex, then Player 1 moves the token from v to a successor v 0 2 ve of v: More clearly, if v is a -vertex, then Player moves the token from v to a successor v 0 2 ve of v: Now, when v 0 is a -vertex, then Player moves the token from v 0 to a successor v 00 2 v 0 E of v 0 : This is repeated either in nitely often or until a vertex v without successors, a dead end, is reached. Formally, a vertex v is called a dead end if ve =?. We de ne a play in the arena A as above as being either an in nite path = v 0 v 1 v 2 : : : 2 V! with v i+1 2 v i E for all i 2! (in nite play) or a nite path = v 0 v 1 : : : v l 2 V + with v i+1 2 v i E for all i < l but v l E =? ( nite play) A pre x of a play is de ned in the obvious way. Now that we have explained what arenas and plays are, it is necessary to explain what kind of winning conditions we are going to use and how arenas together with winning conditions make games Games and Winning Sets Let A be an arena as above and W in V!. The pair (A; W in) is then called a game, where A is the arena of the game and W in is its winning set. The plays of the game are the plays in the arena A. Player 0 is declared the winner of the play in the game G i is a nite play = v 0 v 1 : : : v l 2 V + and v l is a 1-vertex where Player 1 can t move anymore (when v l is a dead end) or is an in nite play and 2 W in Player 1 wins if Player 0 does not win Winning Conditions We will be interested only in winning sets that can be described using the acceptance conditions that we have already studied. Remember here that these acceptance conditions were used only with automata with a nite state space (as a run of an in nite-state automaton might have no recurring state). Therefore we will colour the vertices of an arena and apply the acceptance conditions from the previous chapter on colour sequences. Let A be as above and assume := V! C is some function mapping the vertices of the arena to a nite set C of so-called colours. Such a function is called a colouring function. The colouring function is extended to plays in a straightforward way. When = v 0 v 1 : : : is 18

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

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