Tree Languages Definable with One Quantifier Alternation Mikołaj Bojańczyk (Warszawa) Luc Segoufin (Paris)
e following problem is decidable: Input : A regular tree language L , given by a tree automaton. Question : Is L definable by a formula with quantifier prefix ∃ * ∀ * and also by a formula with quantifier prefix ∀ * ∃ *
is talk is about understanding the expressive power of logics on words and trees. e logics involved can only define (some) regular languages.
is talk is about understanding the expressive power of logics on words and trees. e logics involved can only define (some) regular languages. Understand logic X = give na algorithm to decide if a language L is definable in X all regular languages languages definable in logic X
Why this notion of understanding? ere is a rich theory connecting logic, regular languages, and algebra.
Why this notion of understanding? ere is a rich theory connecting logic, regular languages, and algebra. eorem. (Schützenberger, McNaughton/Papert) e following are equivalent for a word language: – L is definable in first-order logic – L is star-free – the syntactic monoid of L is group-free
Why this notion of understanding? ere is a rich theory connecting logic, regular languages, and algebra. eorem. (Schützenberger, McNaughton/Papert) e following are equivalent for a word language: – L is definable in first-order logic – L is star-free – the syntactic monoid of L is group-free ... more results, including modulo quantifiers, the quantifier alternation hierarchy, etc.
Why this notion of understanding? is paper is part of a program investigating the algebra- logic connection for trees. Eventually, we want to answer ere is a rich theory connecting logic, regular languages, and algebra. questions such as: – what is the expressive power of first-order logic on trees? – what is a tree group? – is there a Krohn-Rhodes decomposition theory? eorem. (Schützenberger, McNaughton/Papert) e following are equivalent for a word language: – L is definable in first-order logic – L is star-free – the syntactic monoid of L is group-free ... more results, including modulo quantifiers, the quantifier alternation hierarchy, etc.
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } *
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic � � ∃ x. a ( x ) ∧ ∀ y < x. b ( y )
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 a ∧ ¬ ( F − 1 c ) � � F
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ *
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * � � ∀ x ∃ y. c ( x ) ⇒ y < x ∧ a ( y )
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata a ⇾ b ⇾
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata “go right to first a; go left to first c ” fails “go right to first a ” works
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata 6. Monoids in DA
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata 6. Monoids in DA 7. A type of unambiguous expression
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * What about trees? 4. Two – way ordered deterministic automata 5. Turtle automata 6. Monoids in DA 7. A type of unambiguous expression
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata 6. Monoids in DA 7. A type of unambiguous expression
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 4. Two – way ordered deterministic automata 5. Turtle automata 6. Monoids in DA
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 5. Turtle automata 6. Monoids in DA
eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · { a,b,c } * 1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 6. Monoids in DA
1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 6. Monoids in DA
1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 6. Monoids in DA
1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 6. Monoids in DA
1. Two-variable first-order logic 2. Temporal logic with operators F and F -1 3. Languages definable with prefix ∃ * ∀ * and also with prefix ∀ * ∃ * 6. Monoids in DA
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