An automaton model for forest algebras. Internship conditions - PowerPoint PPT Presentation

An automaton model for forest algebras. The word case Antoine ◮ Language over A = subset of the free monoid Delignat-Lavaud over A ◮ Universal property of the free monoid: any map A → S is uniquely extended to a morphism A ∗ → S Internship conditions ◮ In a complete DFA accepting L , each letter Goals and context a ∈ A defines a transformation f a ∈ Q Q Forest algebras Forest automata ◮ ϕ : a �→ f a is uniquely extended to a morphism Algebraic model from A ∗ to M = � f a � . L = ϕ − 1 ( X ⊆ M ) . Bottom-up deterministic forest automata Transition forest algebra ◮ Syntactic congruence: w ≡ L w ′ if Minimization Syntactic forest algebra xwy ∈ L ⇔ xw ′ y ∈ L Monadic second order logic Implementation of ◮ Any morphism accepting L factors through BUDFA Efficient minimization A ∗ → A ∗ / ≡ L : there is a unique syntactic Examples Conclusion and monoid M L . perspectives ∨ 4 / 33 /25

An automaton model Back to trees for forest algebras. Antoine ◮ What is a prefix, suffix, factor of a tree? Delignat-Lavaud b b c a c y c x c Internship conditions Goals and context a a a z t Forest algebras Forest automata ◮ Contexts: trees in wich one leaf is labelled by Algebraic model Bottom-up deterministic forest a variable ∗ . automata Transition forest algebra Minimization c Syntactic forest algebra Monadic second order logic Implementation of BUDFA c c Efficient minimization Examples Conclusion and a ∗ perspectives ∨ 5 / 33 /25

An automaton model Back to trees for forest algebras. Antoine ◮ What is a prefix, suffix, factor of a tree? Delignat-Lavaud b b c a c y c x c Internship conditions Goals and context a a a z t Forest algebras Forest automata ◮ Contexts: trees in wich one leaf is labelled by Algebraic model Bottom-up deterministic forest a variable ∗ . automata Transition forest algebra Minimization c Syntactic forest algebra Monadic second order logic Implementation of BUDFA c c Efficient minimization Examples Conclusion and a ∗ perspectives ∨ 5 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimization and syntactic object ◮ Myhill-Nerode congruence: t ≡ L t ′ if for all contexts p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L . ◮ Equivalently, coarsest congruence that refines L . Internship conditions Goals and context ◮ Minimal automaton: Q = T ( A ) / ≡ L , F = [ L ] . Forest algebras a i ([ t 1 ] , ..., [ t k i ]) → [ a i ( t 1 · · · t k i )] . Forest automata ◮ Minimal �⇒ unique up to isomorphism! Algebraic model Bottom-up deterministic forest automata ◮ Syntactic structure: finite set Q with maps Transition forest algebra Minimization f a i : Q k i → Q Syntactic forest algebra Monadic second order logic ◮ Well understood if k i ≤ 2, impractical Implementation of BUDFA otherwise. Efficient minimization Examples Conclusion and perspectives ∨ 6 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model for forest algebras. Unranked trees Antoine Delignat-Lavaud ◮ Alphabet A is a finite set of symbols ◮ Forest: t ::= ε | a ∈ A | t + t | at ◮ Unranked tree: at , t a forest ◮ Important case for many applications (HTML, Internship conditions XML, databases) Goals and context ◮ New horizontal dimension: t + t | at Forest algebras Forest automata Algebraic model Bottom-up deterministic forest automata c a c a b b Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic a c a c c c Implementation of BUDFA Efficient minimization Figure: Concatenation of forests Examples Conclusion and perspectives ∨ 7 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Usefulness of a simple syntactic object for forest algebras. Antoine A tool to study properties of languages Delignat-Lavaud ◮ Schützenberger, McNaughton: L star-free ⇔ M L aperiodic ⇔ L FO [ < ] -definable ◮ Simon: L piecewise-testable (union of A ∗ a 1 A ∗ · · · A ∗ a n A ∗ ) ⇔ M L J -trivial, i.e. Internship conditions M L mM L = M L m ′ M L ⇒ m = m ′ Goals and context Forest algebras Varieties Forest automata Algebraic model ◮ Variety of monoids: family closed under Bottom-up deterministic forest automata Transition forest algebra submonoid, quotients and direct products Minimization Syntactic forest algebra ◮ Pseudovariety: variety V of finite monoids Monadic second order logic Implementation of ◮ Variety of languages: family V (Σ) closed BUDFA Efficient minimization under booleans operations, inverses of Examples morphisms and residuals Conclusion and perspectives ◮ Eilemberg’s theorem: V → V is bijective ∨ 8 / 33 /25

An automaton model Forest algebras for forest algebras. Antoine Delignat-Lavaud Definition A forest algebra is a tuple ( H , V , · , ✐♥ L , ✐♥ R ) such that: Internship conditions ◮ H is the horizontal monoid denoted ( H , 0 , +) . Goals and context ◮ V is the vertical monoid denoted ( V , 1 , ◦ ) . Forest algebras ◮ · is a left monoidal action of V on H . Forest automata Algebraic model ◮ ✐♥ L , ✐♥ R : H → V are such that ✐♥ L ( g ) · h = g + h Bottom-up deterministic forest automata Transition forest algebra and ✐♥ R ( g ) · h = h + g for all g , h ∈ H . Minimization Syntactic forest algebra ◮ · is faithful, ∀ v , w ∈ V , ∃ h ∈ H , v · h � = w · h . Monadic second order logic Implementation of BUDFA Forest algebras were proposed by Walukiewicz Efficient minimization Examples and Bojańczyk in a 2007 paper. Conclusion and perspectives ∨ 9 / 33 /25

An automaton model Forest algebra morphism for forest algebras. A morphism of forest algebras from ( H , V ) to Antoine Delignat-Lavaud ( G , W ) is a pair of monoid morphisms ( α : H → G , β : V → W ) such that ∀ h ∈ H , v ∈ V , α ( v · h ) = β ( v ) · α ( h ) Internship conditions Example Goals and context ( F ( A ) , C ( A ) , · , ✐♥ L , ✐♥ R ) where Forest algebras Forest automata ◮ F ( A ) is the monoid of forests over A with Algebraic model Bottom-up deterministic forest concatenation automata Transition forest algebra ◮ C ( A ) is the monoid of contexts over A with Minimization Syntactic forest algebra composition Monadic second order logic Implementation of ◮ · is forest substitution in contexts BUDFA Efficient minimization is the free forest algebra denoted A ∆ . Universal Examples Conclusion and property: any map A → V extends uniquely to a perspectives morphism A ∆ → ( H , V ) such that β ( a ( ∗ )) = f ( a ) ∨ 10 / 33 /25

An automaton model Forest algebra morphism for forest algebras. A morphism of forest algebras from ( H , V ) to Antoine Delignat-Lavaud ( G , W ) is a pair of monoid morphisms ( α : H → G , β : V → W ) such that ∀ h ∈ H , v ∈ V , α ( v · h ) = β ( v ) · α ( h ) Internship conditions Example Goals and context ( F ( A ) , C ( A ) , · , ✐♥ L , ✐♥ R ) where Forest algebras Forest automata ◮ F ( A ) is the monoid of forests over A with Algebraic model Bottom-up deterministic forest concatenation automata Transition forest algebra ◮ C ( A ) is the monoid of contexts over A with Minimization Syntactic forest algebra composition Monadic second order logic Implementation of ◮ · is forest substitution in contexts BUDFA Efficient minimization is the free forest algebra denoted A ∆ . Universal Examples Conclusion and property: any map A → V extends uniquely to a perspectives morphism A ∆ → ( H , V ) such that β ( a ( ∗ )) = f ( a ) ∨ 10 / 33 /25

An automaton model Recognizability for forest algebras. Antoine Delignat-Lavaud Definition L ⊆ F ( A ) is a recognized by a morphism ( α, β ) : A ∆ → ( H , V ) if L = α − 1 ( F ⊆ H ) . It is recognizable if there exists such a morphism such that ( H , V ) is finite. Internship conditions Goals and context Forest algebras Syntactic congruence Forest automata We define the relation ≡ L on forests and contexts: Algebraic model Bottom-up deterministic forest automata t ≡ L t ′ ⇐ ⇒ ∀ p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L Transition forest algebra Minimization p ≡ L p ′ ⇐ ⇒ ∀ t ∈ F ( A ) , p · t ∈ L ⇔ p ′ · t ∈ L Syntactic forest algebra Monadic second order logic Implementation of The forest algebra ( H L = F ( A ) / ≡ L , V L = C ( A ) / ≡ L ) BUDFA Efficient minimization Examples recognizes L and any morphism recognizing L factors through ( α L , β L ) : A ∆ → ( H L , V L ) . Conclusion and perspectives ∨ 11 / 33 /25

An automaton model Recognizability for forest algebras. Antoine Delignat-Lavaud Definition L ⊆ F ( A ) is a recognized by a morphism ( α, β ) : A ∆ → ( H , V ) if L = α − 1 ( F ⊆ H ) . It is recognizable if there exists such a morphism such that ( H , V ) is finite. Internship conditions Goals and context Forest algebras Syntactic congruence Forest automata We define the relation ≡ L on forests and contexts: Algebraic model Bottom-up deterministic forest automata t ≡ L t ′ ⇐ ⇒ ∀ p ∈ C ( A ) , p · t ∈ L ⇔ p · t ′ ∈ L Transition forest algebra Minimization p ≡ L p ′ ⇐ ⇒ ∀ t ∈ F ( A ) , p · t ∈ L ⇔ p ′ · t ∈ L Syntactic forest algebra Monadic second order logic Implementation of The forest algebra ( H L = F ( A ) / ≡ L , V L = C ( A ) / ≡ L ) BUDFA Efficient minimization Examples recognizes L and any morphism recognizing L factors through ( α L , β L ) : A ∆ → ( H L , V L ) . Conclusion and perspectives ∨ 11 / 33 /25

An automaton model Algebraic automaton model for forest algebras. Antoine Definition Delignat-Lavaud A forest algebra automaton is a tuple: A = ( � Q , 0 , + � , A , δ : A × Q → Q , F ⊆ Q ) ( Q , 0 , +) is the finite state monoid, δ is the transition Internship conditions Goals and context function and F the set of accepting states. Forest algebras Forest automata Run of the automaton Algebraic model Bottom-up deterministic forest A induces a map F ( A ) → Q written · A defined by automata Transition forest algebra Minimization induction on the structure of forests: Syntactic forest algebra Monadic second order logic ◮ 0 A = 0 Implementation of BUDFA ◮ ( t 1 + t 2 ) A = t A 1 + t A Efficient minimization 2 Examples ◮ ( a · t ) A = δ ( a , t A ) Conclusion and perspectives L ( A ) = { t ∈ F ( A ) | t A ∈ F } . ∨ 12 / 33 /25

An automaton model Algebraic automaton model for forest algebras. Antoine Definition Delignat-Lavaud A forest algebra automaton is a tuple: A = ( � Q , 0 , + � , A , δ : A × Q → Q , F ⊆ Q ) ( Q , 0 , +) is the finite state monoid, δ is the transition Internship conditions Goals and context function and F the set of accepting states. Forest algebras Forest automata Run of the automaton Algebraic model Bottom-up deterministic forest A induces a map F ( A ) → Q written · A defined by automata Transition forest algebra Minimization induction on the structure of forests: Syntactic forest algebra Monadic second order logic ◮ 0 A = 0 Implementation of BUDFA ◮ ( t 1 + t 2 ) A = t A 1 + t A Efficient minimization 2 Examples ◮ ( a · t ) A = δ ( a , t A ) Conclusion and perspectives L ( A ) = { t ∈ F ( A ) | t A ∈ F } . ∨ 12 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Transition forest algebra Transition forest algebra ( H , V ) of A = � ( Q , 0 , +) , A , δ : A × Q → Q , F ⊆ Q � : ◮ Horizontal monoid H = ( Q , 0 , +) Internship conditions ◮ Vertical monoid V = ( H H , ✐❞ H , ◦ ) Goals and context Forest algebras ◮ Action is function application: v · h = v ( h ) . Forest automata ◮ Insertion functions are uniquely defined by Algebraic model Bottom-up deterministic forest automata the action. Transition forest algebra Minimization Syntactic forest algebra The morphism defined by β ( a ( ∗ )) = δ ( a ) ∈ H H Monadic second order logic recognizes L . Implementation of BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 13 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Transition forest algebra Transition forest algebra ( H , V ) of A = � ( Q , 0 , +) , A , δ : A × Q → Q , F ⊆ Q � : ◮ Horizontal monoid H = ( Q , 0 , +) Internship conditions ◮ Vertical monoid V = ( H H , ✐❞ H , ◦ ) Goals and context Forest algebras ◮ Action is function application: v · h = v ( h ) . Forest automata ◮ Insertion functions are uniquely defined by Algebraic model Bottom-up deterministic forest automata the action. Transition forest algebra Minimization Syntactic forest algebra The morphism defined by β ( a ( ∗ )) = δ ( a ) ∈ H H Monadic second order logic recognizes L . Implementation of BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 13 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Transition forest algebra Transition forest algebra ( H , V ) of A = � ( Q , 0 , +) , A , δ : A × Q → Q , F ⊆ Q � : ◮ Horizontal monoid H = ( Q , 0 , +) Internship conditions ◮ Vertical monoid V = ( H H , ✐❞ H , ◦ ) Goals and context Forest algebras ◮ Action is function application: v · h = v ( h ) . Forest automata ◮ Insertion functions are uniquely defined by Algebraic model Bottom-up deterministic forest automata the action. Transition forest algebra Minimization Syntactic forest algebra The morphism defined by β ( a ( ∗ )) = δ ( a ) ∈ H H Monadic second order logic recognizes L . Implementation of BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 13 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Transition forest algebra Transition forest algebra ( H , V ) of A = � ( Q , 0 , +) , A , δ : A × Q → Q , F ⊆ Q � : ◮ Horizontal monoid H = ( Q , 0 , +) Internship conditions ◮ Vertical monoid V = ( H H , ✐❞ H , ◦ ) Goals and context Forest algebras ◮ Action is function application: v · h = v ( h ) . Forest automata ◮ Insertion functions are uniquely defined by Algebraic model Bottom-up deterministic forest automata the action. Transition forest algebra Minimization Syntactic forest algebra The morphism defined by β ( a ( ∗ )) = δ ( a ) ∈ H H Monadic second order logic recognizes L . Implementation of BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 13 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Transition forest algebra Transition forest algebra ( H , V ) of A = � ( Q , 0 , +) , A , δ : A × Q → Q , F ⊆ Q � : ◮ Horizontal monoid H = ( Q , 0 , +) Internship conditions ◮ Vertical monoid V = ( H H , ✐❞ H , ◦ ) Goals and context Forest algebras ◮ Action is function application: v · h = v ( h ) . Forest automata ◮ Insertion functions are uniquely defined by Algebraic model Bottom-up deterministic forest automata the action. Transition forest algebra Minimization Syntactic forest algebra The morphism defined by β ( a ( ∗ )) = δ ( a ) ∈ H H Monadic second order logic recognizes L . Implementation of BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 13 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Definition Antoine Delignat-Lavaud A bottom-up deterministic forest automaton (BUDFA) is a tuple: A = � S , Q , s 0 , γ : S × Q → S , λ : S × A → Q , F � ◮ S is the finite set of horizontal states ◮ s 0 ∈ S is the initial state Internship conditions Goals and context ◮ F ⊆ S is the set of accepting states Forest algebras ◮ Q is the finite set of vertical states Forest automata Algebraic model ◮ γ is a semiautomaton on S over the alphabet Bottom-up deterministic forest automata Q Transition forest algebra Minimization Syntactic forest algebra ◮ λ is the output function Monadic second order logic γ defines a left monoidal action of Q ∗ on S , we Implementation of BUDFA Efficient minimization will write s · q 1 . . . q n instead of Examples γ ( γ ( · · · γ ( s , q 1 ) · · · ) , q n − 1 ) , q n ) . Conclusion and perspectives ∨ 14 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata a a Transition forest algebra b b Minimization a : q f s f Syntactic forest algebra b : q f Monadic second order logic Implementation of a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b a Forest automata q a a : q a a : q f Algebraic model s 0 s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata a a Transition forest algebra b b b Minimization a : q f s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata a a Transition forest algebra b b b Minimization a : q f s 1 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata a a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b b a Forest automata q a a : q a a : q f Algebraic model s 0 s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a q a a q a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b b a Forest automata q a a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras b b a Forest automata q a s 1 a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a b a Forest automata q a s 1 a : q a a : q f Algebraic model s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a b a a Forest automata q a s 1 a : q a a : q f Algebraic model s 0 s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a q b q b Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a b a a Forest automata q a s 1 s 0 a : q a a : q f Algebraic model s 0 s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a q b Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a q a b q a a Forest automata q a s 1 s 0 a : q a a : q f Algebraic model s 0 s 0 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a q b Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a b q a q a a Forest automata q a s 1 s 0 a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a q b Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Run on a BUDFA A defines a map · A : F ( A ) → S defined by: ◮ 0 A = s 0 . ◮ ( t 1 + a · t 2 ) A = t A 1 · λ ( t A 2 , a ) . Internship conditions Goals and context q b q a , q b Forest algebras q a b q a a Forest automata q a s 1 s 0 a : q a a : q f Algebraic model s 0 s 1 s 1 b : q b b : q a Bottom-up deterministic forest q f q f automata q a a q a q a a q b Transition forest algebra b b Minimization a : q f s 0 s 1 s 0 s 0 s f Syntactic forest algebra b : q f Monadic second order logic Implementation of q a a q a , q b , q f BUDFA s 0 Efficient minimization Examples Conclusion and perspectives ∨ 15 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model for forest algebras. Antoine Transition forest algebra Delignat-Lavaud Let A = � S , Q , s 0 , γ, λ, F � . ◮ ( S , Q , s 0 , γ, F ) is the horizontal DFA: let H be its transition monoid ◮ For all q ∈ Q , let γ q = s �→ γ ( s , q ) . H is the Internship conditions submonoid of ( S S , ✐❞ S , ∗ ) generated by ( γ q ) q ∈ Q Goals and context Forest algebras ◮ X = { f w ∈ H | f w ( s 0 ) ∈ F } Forest automata ◮ For all a ∈ A , let v a : h �→ γ λ ( h ( s 0 ) , a ) Algebraic model Bottom-up deterministic forest automata ◮ For all t ∈ F ( A ) , let v ✐♥ L ( t ) : h �→ γ t ∗ h , Transition forest algebra Minimization v ✐♥ R ( t ) : h �→ h ∗ γ t Syntactic forest algebra Monadic second order logic ◮ V is the submonoid of ( H H , ✐❞ H , ◦ ) generated Implementation of BUDFA by � v a , v ✐♥ R ( t ) , v ✐♥ L ( t ) � . Efficient minimization Examples Conclusion and perspectives ∨ 16 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model Minimization of BUDFA for forest algebras. Antoine Delignat-Lavaud Myhill-Nerode congruence ◮ Left contexts C ℓ ( A ) : the hole has no sibling on its left ◮ Relation on F ( A ) : t ∼ L t ′ if ∀ p ∈ C ℓ ( A ) , Internship conditions p · t ∈ L ⇔ p · t ′ ∈ L . Goals and context Forest algebras ◮ ≡ L ⊆∼ L : ∼ L is of finite index Forest automata Algebraic model ◮ Restriction of ≡ L on F ( A ) to T ( A ) is an Bottom-up deterministic forest automata equivalence still denoted ≡ L Transition forest algebra Minimization ◮ A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � Syntactic forest algebra Monadic second order logic ◮ λ ([ t ] ∼ L , a ) = [ at ] ≡ L Implementation of BUDFA ◮ γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Efficient minimization Examples Conclusion and perspectives ∨ 17 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimal automaton A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Internship conditions Myhill-Nerode congruence Goals and context Forest algebras ◮ t 1 ∼ L t 2 ⇒ at 1 ≡ L at 2 : Forest automata p ∈ C ( A ) ⇒ p ◦ ( a ( ∗ )) ∈ C ℓ ( A ) . Algebraic model Bottom-up deterministic forest automata ◮ t 1 ∼ L t ′ 1 and t 2 ≡ L t ′ 2 ⇒ t 1 + t 2 ∼ L t ′ 1 + t ′ 2 : Transition forest algebra Minimization ∀ p ∈ C ℓ ( A ) , since p ◦ ✐♥ R ( t 2 ) ∈ C ℓ ( A ) : Syntactic forest algebra Monadic second order logic ( p ◦ ✐♥ R ( t 2 )) · t 1 ∈ L ⇔ ( p ◦ ✐♥ R ( t 2 )) · t ′ 1 ∈ L Implementation of p · ( t 1 + t 2 ) ∈ L ⇔ p · ( t ′ 1 + t 2 ) ∈ L BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 18 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Minimal automaton A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Internship conditions Myhill-Nerode congruence Goals and context Forest algebras ◮ t 1 ∼ L t 2 ⇒ at 1 ≡ L at 2 : Forest automata p ∈ C ( A ) ⇒ p ◦ ( a ( ∗ )) ∈ C ℓ ( A ) . Algebraic model Bottom-up deterministic forest automata ◮ t 1 ∼ L t ′ 1 and t 2 ≡ L t ′ 2 ⇒ t 1 + t 2 ∼ L t ′ 1 + t ′ 2 : Transition forest algebra Minimization ∀ p ∈ C ℓ ( A ) , since p ◦ ✐♥ R ( t 2 ) ∈ C ℓ ( A ) : Syntactic forest algebra Monadic second order logic ( p ◦ ✐♥ R ( t 2 )) · t 1 ∈ L ⇔ ( p ◦ ✐♥ R ( t 2 )) · t ′ 1 ∈ L Implementation of p · ( t 1 + t 2 ) ∈ L ⇔ p · ( t ′ 1 + t 2 ) ∈ L BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 18 / 33 /25

An automaton model for forest algebras. Minimal automaton Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Minimality proof Internship conditions T s = { t ∈ F ( A ) | t A = s } ⊆ F ( A ) Goals and context Forest algebras T q = { at ∈ F ( A ) | λ ( t A , a ) = q } ⊆ T ( A ) Forest automata ◮ If t s , t ′ s ∈ T s then t s ∼ L t ′ Algebraic model s . Bottom-up deterministic forest automata ◮ If t q , t ′ q ∈ T q then t q ≡ L t ′ q . Transition forest algebra Minimization Syntactic forest algebra ◮ s �→ [ T s ] ∼ L , q �→ [ T q ] ≡ L defines a surjective Monadic second order logic morphism from A = � S , s 0 , Q , γ, λ, F � to A L Implementation of BUDFA ◮ s 0 is mapped to [ 0 ] ∼ L Efficient minimization Examples ◮ F is mapped to [ L ] ≡ L Conclusion and perspectives ∨ 19 / 33 /25

An automaton model for forest algebras. Minimal automaton Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Minimality proof Internship conditions T s = { t ∈ F ( A ) | t A = s } ⊆ F ( A ) Goals and context Forest algebras T q = { at ∈ F ( A ) | λ ( t A , a ) = q } ⊆ T ( A ) Forest automata ◮ If t s , t ′ s ∈ T s then t s ∼ L t ′ Algebraic model s . Bottom-up deterministic forest automata ◮ If t q , t ′ q ∈ T q then t q ≡ L t ′ q . Transition forest algebra Minimization Syntactic forest algebra ◮ s �→ [ T s ] ∼ L , q �→ [ T q ] ≡ L defines a surjective Monadic second order logic morphism from A = � S , s 0 , Q , γ, λ, F � to A L Implementation of BUDFA ◮ s 0 is mapped to [ 0 ] ∼ L Efficient minimization Examples ◮ F is mapped to [ L ] ≡ L Conclusion and perspectives ∨ 19 / 33 /25

An automaton model for forest algebras. Minimal automaton Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Minimality proof Internship conditions T s = { t ∈ F ( A ) | t A = s } ⊆ F ( A ) Goals and context Forest algebras T q = { at ∈ F ( A ) | λ ( t A , a ) = q } ⊆ T ( A ) Forest automata ◮ If t s , t ′ s ∈ T s then t s ∼ L t ′ Algebraic model s . Bottom-up deterministic forest automata ◮ If t q , t ′ q ∈ T q then t q ≡ L t ′ q . Transition forest algebra Minimization Syntactic forest algebra ◮ s �→ [ T s ] ∼ L , q �→ [ T q ] ≡ L defines a surjective Monadic second order logic morphism from A = � S , s 0 , Q , γ, λ, F � to A L Implementation of BUDFA ◮ s 0 is mapped to [ 0 ] ∼ L Efficient minimization Examples ◮ F is mapped to [ L ] ≡ L Conclusion and perspectives ∨ 19 / 33 /25

An automaton model for forest algebras. Minimal automaton Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Minimality proof Internship conditions T s = { t ∈ F ( A ) | t A = s } ⊆ F ( A ) Goals and context Forest algebras T q = { at ∈ F ( A ) | λ ( t A , a ) = q } ⊆ T ( A ) Forest automata ◮ If t s , t ′ s ∈ T s then t s ∼ L t ′ Algebraic model s . Bottom-up deterministic forest automata ◮ If t q , t ′ q ∈ T q then t q ≡ L t ′ q . Transition forest algebra Minimization Syntactic forest algebra ◮ s �→ [ T s ] ∼ L , q �→ [ T q ] ≡ L defines a surjective Monadic second order logic morphism from A = � S , s 0 , Q , γ, λ, F � to A L Implementation of BUDFA ◮ s 0 is mapped to [ 0 ] ∼ L Efficient minimization Examples ◮ F is mapped to [ L ] ≡ L Conclusion and perspectives ∨ 19 / 33 /25

An automaton model for forest algebras. Minimal automaton Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Minimality proof Internship conditions T s = { t ∈ F ( A ) | t A = s } ⊆ F ( A ) Goals and context Forest algebras T q = { at ∈ F ( A ) | λ ( t A , a ) = q } ⊆ T ( A ) Forest automata ◮ If t s , t ′ s ∈ T s then t s ∼ L t ′ Algebraic model s . Bottom-up deterministic forest automata ◮ If t q , t ′ q ∈ T q then t q ≡ L t ′ q . Transition forest algebra Minimization Syntactic forest algebra ◮ s �→ [ T s ] ∼ L , q �→ [ T q ] ≡ L defines a surjective Monadic second order logic morphism from A = � S , s 0 , Q , γ, λ, F � to A L Implementation of BUDFA ◮ s 0 is mapped to [ 0 ] ∼ L Efficient minimization Examples ◮ F is mapped to [ L ] ≡ L Conclusion and perspectives ∨ 19 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Minimal automaton for forest algebras. Antoine Delignat-Lavaud A L = �F ( A ) / ∼ L , T ( A ) / ≡ L , [ 0 ] ∼ L , γ, λ, [ L ] ∼ L � λ ([ t ] ∼ L , a ) = [ at ] ≡ L , γ ([ t 1 ] ∼ L , [ t 2 ] ≡ L ) = [ t 1 + t 2 ] ∼ L Transition forest algebra Internship conditions ◮ γ t instead of γ [ t ] ≡ L if t ∈ T ( A ) Goals and context ◮ γ at instead of γ λ ( γ t ([ 0 ] ∼ L ) , a ) Forest algebras Forest automata ◮ γ t 1 ∗ γ t 2 = γ t 1 + t 2 Algebraic model Bottom-up deterministic forest ◮ [ t 1 ] ≡ L = [ t 2 ] ≡ L ⇐ ⇒ γ t 1 = γ t 2 automata Transition forest algebra Minimization ◮ v a : γ s �→ γ as , v ✐♥ L ( t ) : γ s �→ γ t + s , v ✐♥ R ( t ) : γ s �→ γ s + t Syntactic forest algebra Monadic second order logic ◮ If p ∈ C ( A ) , v p is defined on the Implementation of BUDFA decomposition of p using a ( ∗ ) , ✐♥ R ( t ) , ✐♥ L ( t ) Efficient minimization Examples ◮ ∀ t ∈ F ( A ) , p ∈ C ( A ) , γ p · t = v p ( γ t ) Conclusion and perspectives ◮ [ p 1 ] ≡ L = [ p 2 ] ≡ L ⇐ ⇒ v p 1 = v p 2 ∨ 20 / 33 /25

An automaton model Theorem for forest algebras. Antoine Every recognizable forest language is accepted Delignat-Lavaud by an unique (up to isomorphism) BUDFA whose transition forest algebra is equal to the syntactic forest algebra of the language. Internship conditions Corollary Goals and context Every forest algebra has a canonical Forest algebras representation in which the vertical monoid is a Forest automata Algebraic model transformation monoid over the horizontal monoid Bottom-up deterministic forest automata and the action is function application. Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic Remark Implementation of BUDFA If A is a BUDFA such that | S | = n , | V | ≤ n 2 n ( 1 + n n 2 ) Efficient minimization Examples In other words, | V L | = O ( | H L | ❞✐♠ ( H L ) ) , a much lower Conclusion and perspectives bound than the expected | H L | | H L | ∨ 21 / 33 /25

An automaton model Theorem for forest algebras. Antoine Every recognizable forest language is accepted Delignat-Lavaud by an unique (up to isomorphism) BUDFA whose transition forest algebra is equal to the syntactic forest algebra of the language. Internship conditions Corollary Goals and context Every forest algebra has a canonical Forest algebras representation in which the vertical monoid is a Forest automata Algebraic model transformation monoid over the horizontal monoid Bottom-up deterministic forest automata and the action is function application. Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic Remark Implementation of BUDFA If A is a BUDFA such that | S | = n , | V | ≤ n 2 n ( 1 + n n 2 ) Efficient minimization Examples In other words, | V L | = O ( | H L | ❞✐♠ ( H L ) ) , a much lower Conclusion and perspectives bound than the expected | H L | | H L | ∨ 21 / 33 /25

An automaton model Theorem for forest algebras. Antoine Every recognizable forest language is accepted Delignat-Lavaud by an unique (up to isomorphism) BUDFA whose transition forest algebra is equal to the syntactic forest algebra of the language. Internship conditions Corollary Goals and context Every forest algebra has a canonical Forest algebras representation in which the vertical monoid is a Forest automata Algebraic model transformation monoid over the horizontal monoid Bottom-up deterministic forest automata and the action is function application. Transition forest algebra Minimization Syntactic forest algebra Monadic second order logic Remark Implementation of BUDFA If A is a BUDFA such that | S | = n , | V | ≤ n 2 n ( 1 + n n 2 ) Efficient minimization Examples In other words, | V L | = O ( | H L | ❞✐♠ ( H L ) ) , a much lower Conclusion and perspectives bound than the expected | H L | | H L | ∨ 21 / 33 /25

An automaton model Logical characterization for forest algebras. Antoine Delignat-Lavaud FO [ N , C ] ϕ ::= x = y | N ( x , y ) | C ( x , y ) | L a ( x ) , a ∈ A | ϕ ∧ ϕ | ¬ ϕ | ∃ x ϕ ◮ Forest over A : P ♦s ( t ) ⊆ N ∗ → A Internship conditions ◮ Domain of interpretation I is N ∗ Goals and context Forest algebras ◮ ∀ p , p ′ ∈ N ∗ , N I ( p , p ′ ) ⇔ ∃ p ′′ ∈ N ∗ , ∃ i ∈ N , p = Forest automata p ′′ · i ∧ p ′ = p ′′ · ( i + 1 ) Algebraic model Bottom-up deterministic forest ◮ ∀ p , p ′ ∈ N ∗ , C I ( p , p ′ ) ⇔ ∃ i ∈ N , p = p ′ · i automata Transition forest algebra Minimization ◮ V -forests: forests over A × 2 V given V finite set Syntactic forest algebra Monadic second order logic of first-order variables Implementation of BUDFA ◮ { ( a i , U i ) , i ≤ n } set of labels in t : Efficient minimization Examples ∀ i � = j , U i ∩ U j = ∅ and � i ≤ n U i = V Conclusion and perspectives ∨ 22 / 33 /25

An automaton model for forest algebras. Antoine Delignat-Lavaud Model using V -forests ◮ t | = I L a ( x ) if and only if ∃ p ∈ P ♦s ( t ) , t ( p ) = ( a , U ) with x ∈ U . ◮ If P is a binary predicate, t | = I P ( x , y ) if and Internship conditions only if P I ( p x , p y ) , where p x is the position such Goals and context that t ( p x ) = ( a i , U i ) with x ∈ U i . Forest algebras ◮ t | = I ϕ 1 ∧ ϕ 2 if and only if t | = I ϕ 1 and t | = I ϕ 2 . Forest automata Algebraic model ◮ t | = I ¬ ϕ if and only if t �| = I ϕ . Bottom-up deterministic forest automata Transition forest algebra ◮ t | = I ∃ x ϕ if and only if ∃ i ≤ n , t i | = I ϕ where t i is Minimization Syntactic forest algebra the V ∪ { x } -forest obtained by replacing the Monadic second order logic Implementation of label ( a i , U i ) in t by ( a i , U i ∪ { x } ) . BUDFA Efficient minimization Examples Conclusion and perspectives ∨ 23 / 33 /25

An automaton model for forest algebras. MSO [ N , C ] Antoine Delignat-Lavaud Φ ::= ϕ ∈ FO [ N , C ] | X ( x ) | ∃ X Φ ◮ ( V , W ) -forests over A × 2 V × 2 W ◮ Restriction to labels in A × 2 V is a V -forest ◮ W is a finite set of second-order variables Internship conditions Goals and context ◮ t | = I ϕ ∈ FO [ N , C ] if and only if the restriction of t to labels in A × 2 V is a model of ϕ . Forest algebras Forest automata ◮ t | = I X ( x ) ⇔ ∃ p ∈ P ♦s ( t ) such that Algebraic model Bottom-up deterministic forest automata t ( p ) = ( a i , V i , W i ) with x ∈ V i and X ∈ W i Transition forest algebra Minimization ◮ t | = I ∃ X Φ ⇔ ∃ P ⊆ P ♦s ( t ) such that the Syntactic forest algebra Monadic second order logic ( V , W ∪ { X } ) -forest obtained by replacing all Implementation of BUDFA labels t ( p ) = ( a p , V p , W p ) for p ∈ P by Efficient minimization ( a p , V p , W p ∪ { X } ) satisfies Φ Examples Conclusion and perspectives ∨ 24 / 33 /25