Outline Lattices and fixed points 1 The varieties D n 2 Results - PowerPoint PPT Presentation

Lattices and fixed points The varieties D n Results Fixed-point theory in the varieties D n Sabine Frittella and Luigi Santocanale Laboratoire d’Informatique Fondamentale de Marseille, France May 1, 2014 RAMiCS 2014, Marienstatt im Westerwald, Germany 1 1/17

Lattices and fixed points The varieties D n Results Outline Lattices and fixed points 1 The varieties D n 2 Results 3 2 2/17

Lattices and fixed points The varieties D n Results Lattices ( L , ⊥ , ⊤ , ∧ , ∨ ) Let L be an ordered set s.t. : ∀ x , y ∈ L ∃ u , v ∈ L s.t. u = x ∨ y = the least upper bound = supremum v = x ∧ y = the greatest lower bound = infimum . ⊥ : the smallest element of L . ⊤ : the largest element of L . 3 3/17

Lattices and fixed points The varieties D n Results Fixed points and lattices Let • ( L , � ) be a lattice, • f : L → L increasing . • Fix ( f ) = { x ∈ L | f ( x ) = x } Let’s note : µ x . f ( x ) = � Fix ( f ) , ν x . f ( x ) = � Fix ( f ) . Theorem (Tarski ’55) If L is a complete lattice and f is increasing, then µ x . f ( x ) = min Fix ( f ) = the least fixed point of f ν x . f ( x ) = max Fix ( f ) = the greatest fixed point of f 4 4/17

Lattices and fixed points The varieties D n Results Algorithm to calculate fixed points L complete lattice, ⊥ the least element, ⊤ the largest element, f : L → L increasing. • The least fixed point : ⊥ � f ( ⊥ ) � f 2 ( ⊥ ) � ... � f k ( ⊥ ) � ... � f n ( ⊥ ) = f n + 1 ( ⊥ ) Here we have : f n ( ⊥ ) = µ x . f ( x ) . • The greatest fixed point : ⊤ � f ( ⊤ ) � f 2 ( ⊤ ) � ... � f k ( ⊤ ) � ... � f n ( ⊤ ) = f n + 1 ( ⊤ ) Here we have : f n ( ⊤ ) = ν x . f ( x ) . 5 5/17

Lattices and fixed points The varieties D n Results µ -calculus Lattices µ -calculus : = x | ⊥ | ⊤ | φ ∧ φ | φ ∨ φ | µ x φ ( x ) | ν x φ ( x ) φ • µ x ν y φ ( x , y ) : difficult to calculate ψ = µ x d .ν y d .µ x d − 1 .ν y d − 1 . ... .µ x 1 .ν y 1 .ϕ ( x 1 , y 1 , x 2 , y 2 , ..., x d , y d ) with ϕ containing neither µ nor ν , complexity ( ψ ) = d . Complexity of a formula = number of blocks µν . 6 6/17

Lattices and fixed points The varieties D n Results expressiveness : the alternation hierarchy of µ -calculus The hierarchy . . . is strict : for all d there exists ψ with complexity ( ψ ) = d such that if complexity ( φ ) < d then φ �≡ ψ . . . . is degenerate : there exists d such that if ψ verifies complexity ( ψ ) > d then there exists φ with complexity ( φ ) ≤ d and φ ≡ ψ . ψ = µ x d .ν y d .µ x d − 1 .ν y d − 1 . ... .µ x 1 .ν y 1 .ϕ ( x 1 , y 1 , x 2 , y 2 , ..., x d , y d ) Can we simplify ψ ? 7 7/17

Lattices and fixed points The varieties D n Results Motivations Lattices µ -calculus : φ = x | ⊥ | ⊤ | φ ∧ φ | φ ∨ φ | µ x φ ( x ) | ν x φ ( x ) ψ = µ x d .ν y d .µ x d − 1 .ν y d − 1 . ... .µ x 1 .ν y 1 .ϕ ( x 1 , y 1 , x 2 , y 2 , ..., x d , y d ) The alternation hierarchy of µ -calculus : • strict: lattices [San02] • degenerate: distributive lattices µ x φ ( x ) = φ ( ⊥ ) and ν x φ ( x ) = φ ( ⊤ ) • Varieties of lattices D n , with n ∈ N [Nat90], [Sem05]. Examples : • D 0 = distributive lattices • lattices of permutations: S n ∈ D n − 2 8 8/17

Lattices and fixed points The varieties D n Results Characterization of the varieties D n D n : defined via equations of a weaker version of distributivity Lattices in D n are locally finite D n ∩ finite lattices: combinatorial characterization OD-graph of a finite lattice L : G ( L ) := � J ( L ) , ≤ , M� 1 J ( L ) : join-irreducible elements ( j = a ∨ b iff j = a or j = b ) 2 ≤ : order restricted to J ( L ) 3 M : J ( L ) − → PP J ( L ) : minimal covers 9 9/17

Lattices and fixed points The varieties D n Results The OD-graph of a lattice � J ( L ) , ≤ , M� with M : J ( L ) → P ( P ( J ( L ))) 1 C is a cover of j : j ≤ � C 2 order ≪ ⊆ P L × P L : A ≪ B iff ↓ A ⊆ ↓ B . 3 C is a minimal cover of j if • C is a cover of j , • C is a ≤ -antichain, • for any ≤ -antichain D ⊆ L , ( j ≤ � D and D ≪ C ) imply D = C . 4 M ( j ) = minimal covers of j 1 b ≤ � { b } trivial cover 2 b ≤ � { d } = � { c , b } minimal covers are subsets of J ( L ) 3 b ≤ � { c , e } minimal 10 10/17

Lattices and fixed points The varieties D n Results Finite lattices and their OD-graphs L a finite lattice and G ( L ) := � J ( L ) , ≤ , M� its OD-graph. ( L , � ) � G ( L ) = � J ( L ) , � J ( L ) , M� � ( L ( G ( L )) , ⊆ ) G ( L ) is similar to a neighborhood frame. • Language on L : φ := ⊥ | ⊤ | φ ∧ φ | φ ∨ φ • Logic on the frame G ( L ) : φ := ⊥ | ⊤ | φ ∧ φ | ( ∃∀ )( φ ∨ φ ) � monotone modal logic, let a ∈ L and v an assignment, let the valuation v ′ be as follows: v ′ ( j ) = ↓ j , we have: a ≤ v ( φ ) iff ∀ j ≤ a , G ( L ) , j ⊢ v ′ τ ( φ ) G ( L ) , j ⊢ v ′ ( ∃∀ )( φ ∨ ψ ) iff ∃ C ∈ M ( j ) , ∀ c ∈ C , G ( L ) , c ⊢ v ′ φ or G ( L ) , c ⊢ v ′ ψ 11 11/17

Lattices and fixed points The varieties D n Results Finite lattices in D n L a finite lattice in D n and G ( L ) := � J ( L ) , ≤ , M� its OD-graph. The relation D Let j , k ∈ J ( L ) , jDk if j � = k and ∃ C ∈ M ( j ) s.t. k ∈ C the class of finite lattices in D n A finite lattice L ( � J ( L ) , ≤ , M� ) belongs to the class D n iff any path j 0 Dj 1 D ... Dj k has length at most n . 12 12/17

Lattices and fixed points The varieties D n Results Results for each variety D n with n ∈ N : 1 Upper bound on the approximations chain : The µ -calculus hierarchy on D n is degenerate D n � µ x .φ ( x ) = φ n + 1 ( ⊥ ) and D n � ν x .φ ( x ) = φ n + 1 ( ⊤ ) 2 Lower bound : On the lattices in D n the value n + 1 is optimal. 3 Lower bound : On the atomistic lattices in D n the value n + 1 is optimal. 4 Lower bound : On the lattices in D n ∩ D op the value n + 1 is optimal. n 13 13/17

Lattices and fixed points The varieties D n Results Upper bound for the operator ν on the varieties D n Upper bound = n + 1 For the variety D n with n ∈ N , the hierarchy of the µ -calculus is degenerated (upper bound) : D n � µ x .φ ( x ) = φ n + 1 ( ⊥ ) and D n � ν x .φ ( x ) = φ n + 1 ( ⊤ ) Sketch of proof: D n � ν x .φ ( x ) = φ n + 1 ( ⊤ ) ⇔ D n ∩ finite � ν x .φ ( x ) = φ n + 1 ( ⊤ ) (Nation ’90 : locally finite) ⇔ D n ∩ finite � φ n + 1 ( ⊤ ) = φ n + 2 ( ⊤ ) ⇔ D n ∩ finite � φ n + 2 ( ⊤ ) � φ n + 1 ( ⊤ ) and φ n + 1 ( ⊤ ) � φ n + 2 ( ⊤ ) ⇔ D n ∩ finite � φ n + 1 ( ⊤ ) � φ n + 2 ( ⊤ ) Tool: game semantic on the OD-graph 14 14/17

Lattices and fixed points The varieties D n Results Game semantic D n ∩ finite � φ n + 1 ( ⊤ ) � φ n + 2 ( ⊤ ) L � φ n + 1 ( ⊤ ) � φ n + 2 ( ⊤ ) ⇔ for any finite lattice L in D n , ⇔ for any finite lattice L in D n , for any closed valuation v , for any j ∈ J ( L ) , G ( L ) , j � v τ ( φ n + 1 ( ⊤ )) implies G ( L ) , j � v τ ( φ n + 2 ( ⊤ )) we define a finite 2 player game such that: player A has a winning strategy from the position ( j , ψ ) iff G ( L ) , j � ψ . 15 15/17

Lattices and fixed points The varieties D n Results Results For variety D n with n ∈ N : 1 The hierarchy of the µ -calculus is degenerated (upper bound) : D n � µ x .φ ( x ) = φ n + 1 ( ⊥ ) and D n � ν x .φ ( x ) = φ n + 1 ( ⊤ ) 2 Optimality : D n � µ x .φ ( x ) = φ n ( ⊥ ) and D n � ν x .φ ( x ) = φ n ( ⊤ ) open problems and outlook • ∃ ? a term t φ “simpler” than φ n + 1 ( ⊥ ) s.t. D n � µ x φ ( x ) = t φ • links between lattice theory and modal logic ? • similar results on fixed points for modal logic ? 16 16/17

Lattices and fixed points The varieties D n Results Références I J. B. Nation. An approach to lattice varieties of finite height. Algebra Universalis , 27(4):521–543, 1990. Luigi Santocanale. The alternation hierarchy for the theory of µ -lattices. Theory Appl. Categ. , 9:166–197, 2001/02. CT2000 Conference (Como). M. V. Semënova. On lattices that are embeddable into lattices of suborders. Algebra Logika , 44(4):483–511, 514, 2005. 17 17/17