Tree Languages Definable with One Quantifier Alternation Mikoaj - - PowerPoint PPT Presentation

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Tree Languages Definable with One Quantifier Alternation Mikoaj - - PowerPoint PPT Presentation

Nov 14, 2022 464 likes •1.17k views

Tree Languages Definable with One Quantifier Alternation Mikoaj Bojaczyk (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

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Tree Languages Definable with One Quantifier Alternation

Mikołaj Bojańczyk (Warszawa) Luc Segoufin (Paris)

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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 ∀*∃*

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is talk is about understanding the expressive power of logics on words and trees. e logics involved can only define (some) regular languages.

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is talk is about understanding the expressive power of logics on words and trees. e logics involved can only define (some) regular languages.

all regular languages languages definable in logic X

Understand logic X = give na algorithm to decide if a language L is definable in X

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Why this notion of understanding?

ere is a rich theory connecting logic, regular languages, and algebra.

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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

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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.

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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. is paper is part of a program investigating the algebra- logic connection for trees. Eventually, we want to answer 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?

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages: b* · a · {a,b,c}*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic

b* · a · {a,b,c}*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic

∃x. a(x) ∧

  • ∀y < x. b(y)
  • b* · a · {a,b,c}*
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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1

b* · a · {a,b,c}*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1

b* · a · {a,b,c}* F

  • a ∧ ¬(F−1c)
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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1

b* · a · {a,b,c}* ∀x∃y. c(x) ⇒

  • y < x ∧ a(y)
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 4. Two–way ordered deterministic automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 4. Two–way ordered deterministic automata

b* · a · {a,b,c}* b⇾ a⇾

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}* “go right to first a; go left to first c” fails “go right to first a” works

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 7. A type of unambiguous expression
  • 6. Monoids in DA
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 7. A type of unambiguous expression
  • 6. Monoids in DA
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃* What about trees?

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 7. A type of unambiguous expression
  • 6. Monoids in DA
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 4. Two–way ordered deterministic automata
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 5. Turtle automata

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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eorem. (Etessami, Schützenberger, Schwentick, érien, Vollimer, Wilke) e following formalisms define the same word languages:

  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA

b* · a · {a,b,c}*

  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“two a’s”

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“two a’s”

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“all children of root have label a” “two a’s”

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“all children of root have label a” “two a’s” “three a’s”

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“all children of root have label a” “two a’s” “three a’s”

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  • 1. Two-variable first-order logic
  • 2. Temporal logic with operators F and F-1
  • 6. Monoids in DA
  • 3. Languages definable with prefix ∃* ∀*

and also with prefix ∀* ∃*

“all children of root have label a” “two a’s” “three a’s”

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We consider forest languages instead of tree languages (a forest is a sequence of trees)

a b a a a b a a b a b

We use first-order formulas to describe properties of forests. Variables quantify over nodes. Predicates allowed are: “x ancestor of y” “x lexicographically before y” “label of x is a”

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We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2.

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We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2. E.g. Trees ∈ ∆2

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We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2. E.g. Trees ∈ ∆2 every two nodes have a common ancestor ∀x∀y∃z. z ≤ x ∧ z ≤ y

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∀* ∃* We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2. E.g. Trees ∈ ∆2 every two nodes have a common ancestor ∀x∀y∃z. z ≤ x ∧ z ≤ y

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∀* ∃* We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2. E.g. Trees ∈ ∆2 ∃x∀y. x ≤ y some node is ancestor to every node every two nodes have a common ancestor ∀x∀y∃z. z ≤ x ∧ z ≤ y

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∃* ∀* ∀* ∃* We are interested in forest languages that can be defined in both ∀* ∃* and ∃* ∀*. We call this class ∆2. E.g. Trees ∈ ∆2 ∃x∀y. x ≤ y some node is ancestor to every node every two nodes have a common ancestor ∀x∀y∃z. z ≤ x ∧ z ≤ y

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Question: What forest languages can be defined in ∆2 ? Preferably, give an algorithm that decides if L ∈ ∆2.

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a word w can have two encodings:

a b b a a b a

hor(w)

a b b a a b a

ver(w)

  • Fact. if L is a word language definable in ∆2, then both

hor(L) and ver(L) are forest languages definable in ∆2.

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Our characterization is stated as an identity. Intuitively, a forest language is definable in ∆2 iff it admits a certain pumping lemma.

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A context is a forest with a hole in a leaf

a b a b a

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A context is a forest with a hole in a leaf

a b a b a a a b b a b a a a b b a b

=

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A context is a forest with a hole in a leaf

a b a b a a a b b a b a a a b b a b

=

Notion of piece for contexts. is a piece of

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Myhill-Nerode congruence for a forest language L.

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Myhill-Nerode congruence for a forest language L. Two contexts and are called L-equivalent if

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Myhill-Nerode congruence for a forest language L. Two contexts and are called L-equivalent if for every context and every forest

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Myhill-Nerode congruence for a forest language L. Two contexts and are called L-equivalent if for every context and every forest ∈ L iff ∈ L

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Main eorem. A forest language is definable in ∆2 iff the following holds for all sufficiently large n

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Main eorem. A forest language is definable in ∆2 iff the following holds for all sufficiently large n is a piece of if , then

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Main eorem. A forest language is definable in ∆2 iff the following holds for all sufficiently large n is equivalent to is a piece of if , then

n times{ n times{ n times

}

n times

}

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Main eorem. A forest language is definable in ∆2 iff the following holds for all sufficiently large n is equivalent to is a piece of if , then

n times{ n times{ n times

}

n times

}

is criterion is decidable. We also have variants of the theorem for unordered trees / forests.

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Application. e set of binary trees (every node has zero or two children) is not definable in ∆2 is confused with

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Big project: understand the expressive power of first-order logic on trees.

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FO(≤) Big project: understand the expressive power of first-order logic on trees. regular languages

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FO(≤) Σ1(≤) Π1(≤) Big project: understand the expressive power of first-order logic on trees. regular languages

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FO(≤) Σ1(≤) Π1(≤) Big project: understand the expressive power of first-order logic on trees. regular languages Easy excercise

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FO(≤) Σ1(≤) Π1(≤) Bool(Σ1(≤)) Big project: understand the expressive power of first-order logic on trees. regular languages Easy excercise

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FO(≤) Σ1(≤) Π1(≤) Bool(Σ1(≤)) Big project: understand the expressive power of first-order logic on trees. regular languages BSS LICS  Easy excercise

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FO(≤) Σ1(≤) Π1(≤) Bool(Σ1(≤)) Σ2(≤) Π2(≤) ∆2(≤) Big project: understand the expressive power of first-order logic on trees. regular languages BSS LICS  Easy excercise

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FO(≤) Σ1(≤) Π1(≤) Bool(Σ1(≤)) Σ2(≤) Π2(≤) ∆2(≤) Big project: understand the expressive power of first-order logic on trees. regular languages BSS LICS  this paper Easy excercise

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FO(≤) Σ1(≤) Π1(≤) Bool(Σ1(≤)) Σ2(≤) Π2(≤) ∆2(≤) Big project: understand the expressive power of first-order logic on trees. regular languages =? =? =? BSS LICS  this paper Easy excercise