Plan A primer on mu-calculi The intuitionistic -calculus The - PowerPoint PPT Presentation

Fixed-point elimination in Heyting algebras 1 Silvio Ghilardi, Universit` a di Milano Maria Jo˜ ao Gouveia, Universidade de Lisboa Luigi Santocanale, Aix-Marseille Universit´ e TACL@Praha, June 2017 1 See [Ghilardi et al., 2016] 1/32

Plan A primer on mu-calculi The intuitionistic µ -calculus The elimination procedure Bounding closure ordinals 2/32

µ -calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors. 3/32

µ -calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors. E.g., the propositional modal µ -calculus: φ := x | ¬ x | ⊤ | φ ∧ φ | ⊥ | φ ∨ φ | � φ | ⋄ φ | µ x .φ | ν x .φ , when x is positive in φ . 3/32

µ -calculi Add to a given algebraic framework syntactic least and greatest fixed-point constructors. E.g., the propositional modal µ -calculus: φ := x | ¬ x | ⊤ | φ ∧ φ | ⊥ | φ ∨ φ | � φ | ⋄ φ | µ x .φ | ν x .φ , when x is positive in φ . Interpret the syntactic least (resp. greatest) fixed-point as expected. � µ x .φ � v := least fixed-point of the monotone mapping X �→ � φ � v , X / x 3/32

Alternation hierarchies in µ -calculi Let ♯ count the number of alternating blocks of fixed-points. 4/32

Alternation hierarchies in µ -calculi Let ♯ count the number of alternating blocks of fixed-points. Problem. For a given µ -calculus, does there exist n such that, for each φ with ♯φ > n , there exists ψ with γ ≡ ψ and ♯ψ ≤ n ? 4/32

Alternation hierarchies, facts ◮ The alternation hierarchy for the modal µ -calculus is infinite (there exists no such n ) [Lenzi, 1996, Bradfield, 1998]. ◮ Idem for the lattice µ -calculus [Santocanale, 2002]. ◮ The alternation hierarchy for the linear µ -calculus ( ⋄ x = � x ) is reduced to the B¨ uchi fragment (here n = 2) . ◮ The alternation hierarchy for the modal µ -calculus on transitive frames collapses to the alternation free fragment (here n = 1 . 5) [Alberucci and Facchini, 2009]. ◮ The alternation hierarchy for the distributive µ -calculus is trivial (here n = 0) [Kozen, 1983]. 5/32

Alternation hierarchies, facts ◮ The alternation hierarchy for the modal µ -calculus is infinite (there exists no such n ) [Lenzi, 1996, Bradfield, 1998]. ◮ Idem for the lattice µ -calculus [Santocanale, 2002]. ◮ The alternation hierarchy for the linear µ -calculus ( ⋄ x = � x ) is reduced to the B¨ uchi fragment (here n = 2) . ◮ The alternation hierarchy for the modal µ -calculus on transitive frames collapses to the alternation free fragment (here n = 1 . 5) [Alberucci and Facchini, 2009]. ◮ The alternation hierarchy for the distributive µ -calculus is trivial (here n = 0) [Kozen, 1983]. A research plan: Develop a theory explaining why alternation hierarchies collapses. 5/32

µ -calculi on generalized distributive lattices Theorem. [Frittella and Santocanale, 2014] There are lattice varieties (Nation’s varieties) D 0 ⊆ D 1 ⊆ . . . ⊆ D n ⊆ . . . with D 0 the variety of distributive lattices, such that, on D n and for any lattice term φ , φ n +2 ( ⊥ ) = φ n +1 ( ⊥ ) (= µ x .φ ) , φ n ( ⊥ ) � = φ n +1 ( ⊥ ) . 6/32

µ -calculi on generalized distributive lattices Theorem. [Frittella and Santocanale, 2014] There are lattice varieties (Nation’s varieties) D 0 ⊆ D 1 ⊆ . . . ⊆ D n ⊆ . . . with D 0 the variety of distributive lattices, such that, on D n and for any lattice term φ , φ n +2 ( ⊥ ) = φ n +1 ( ⊥ ) (= µ x .φ ) , φ n ( ⊥ ) � = φ n +1 ( ⊥ ) . Corollary. The alternation hierarchy of the lattice µ -calculus is trivial on D n , for each n ≥ 0. 6/32

Plan A primer on mu-calculi The intuitionistic µ -calculus The elimination procedure Bounding closure ordinals 7/32

The intuitionistic µ -calculus After the distributive µ -calculus, the next on the list—by Pitt’s quantifiers, we knew that least fixed-points and greatest fixed-points are definable. We extend the signature of Heyting algebras (i.e. Intuitionistic Logic) with least and greatest fixed-point constructors. Intuitionistic µ -terms are generated by the grammar: φ := x | ⊤ | φ ∧ φ | ⊥ | φ ∨ φ | φ → φ | µ x .φ | ν x .φ , when x is positive in φ . 8/32

Heyting algebra semantics We take any provability semantics of IL with fixed points: ◮ (Complete) Heyting algebras. ◮ Kripke frames. ◮ Any sequent calculus for Intuitionisitc Logic (e.g. LJ) plus Park/Kozen’s rules for least and greatest fixed-points: φ [ ψ/ x ] ⊢ ψ Γ ⊢ φ ( µ x .φ ) φ ( ν x .φ ) ⊢ δ ψ ⊢ φ [ ψ/ x ] µ x .φ ⊣ ψ Γ ⊢ µ x .φ ν x .φ ⊢ δ ψ ⊢ ν x .φ Definition. A Heyting algebra is a bounded lattice H = � H , ⊤ , ∧ , ⊥ , ∨� with an additional binary operation → satisfying x ∧ y ≤ z iff x ≤ y → z . 9/32

Ruitenburg’s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ , there exists n ≥ 0 such that φ n ( x ) ≡ φ n +2 ( x ). 10/32

Ruitenburg’s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ , there exists n ≥ 0 such that φ n ( x ) ≡ φ n +2 ( x ). Then φ n ( ⊥ ) ≤ φ n +1 ( ⊥ ) ≤ φ n +2 ( ⊥ ) = φ n ( ⊥ ) , so φ n ( ⊥ ) is the least fixed-point of φ . 10/32

Ruitenburg’s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ , there exists n ≥ 0 such that φ n ( x ) ≡ φ n +2 ( x ). Then φ n ( ⊥ ) ≤ φ n +1 ( ⊥ ) ≤ φ n +2 ( ⊥ ) = φ n ( ⊥ ) , so φ n ( ⊥ ) is the least fixed-point of φ . Corollary. The alternation hierarchy for the intuitionistic µ -calculus is trivial. 10/32

Ruitenburg’s theorem [Ruitenburg, 1984] Theorem. For each intuitionistic formula φ , there exists n ≥ 0 such that φ n ( x ) ≡ φ n +2 ( x ). Then φ n ( ⊥ ) ≤ φ n +1 ( ⊥ ) ≤ φ n +2 ( ⊥ ) = φ n ( ⊥ ) , so φ n ( ⊥ ) is the least fixed-point of φ . Corollary. The alternation hierarchy for the intuitionistic µ -calculus is trivial. NB : Ruitenburg’s n might not be the closure ordinal of µ x .φ . 10/32

Peirce, compatibility, strenghs and strongness Proposition. Peirce’s theorem for Heyting algebras. Every term φ is compatible. In particular, for ψ, χ arbitrary terms, the equation φ [ ψ/ x ] ∧ χ = φ [ ψ ∧ χ/ x ] ∧ χ . holds on Heyting algebras. Corollary. Every term φ monotone in x is strong in x . That is, any the following equivalent conditions φ [ ψ/ x ] ∧ χ ≤ φ [ ψ ∧ χ/ x ] , ψ → χ ≤ φ [ ψ/ x ] → φ [ χ/ x ] , hold, for any terms ψ and χ . 11/32

Plan A primer on mu-calculi The intuitionistic µ -calculus The elimination procedure Bounding closure ordinals 12/32

Greatest fixed-points Proposition. On Heyting algebras, we have ν x .φ = φ ( ⊤ ) . 13/32

Greatest fixed-points Proposition. On Heyting algebras, we have ν x .φ = φ ( ⊤ ) . Using the deduction theorem and Pitts’ quantifiers: ν x .φ ( x ) = ∃ x . ( x ∧ x → φ ( x )) = ∃ x . ( x ∧ φ ( x )) = φ ( ⊤ ) . 13/32

Greatest fixed-points Proposition. On Heyting algebras, we have ν x .φ = φ ( ⊤ ) . Using the deduction theorem and Pitts’ quantifiers: ν x .φ ( x ) = ∃ x . ( x ∧ x → φ ( x )) = ∃ x . ( x ∧ φ ( x )) = φ ( ⊤ ) . Using strongness: φ ( ⊤ ) = φ ( ⊤ ) ∧ φ ( ⊤ ) ≤ φ ( ⊤ ∧ φ ( ⊤ )) = φ 2 ( ⊤ ) . 13/32

Greatest solutions of systems of equations Proposition. On Heyting algebras, a system of equations   x 1 = φ 1 ( x 1 , . . . , x n )   .   . .   x n = φ n ( x 1 , . . . , x n )   has a greatest solution obtained by iterating φ := � φ 1 , . . . , φ n � n times from ⊤ . Proof . Using the Bekic property. 14/32

Least fixed-points: splitting the roles of variables Due to µ x .φ ( x , x ) = µ x .µ y .φ ( x , y ) 15/32

Least fixed-points: splitting the roles of variables Due to µ x .φ ( x , x ) = µ x .µ y .φ ( x , y ) we can separate computing the least fixed-points w.r.t: weakly negative variables: variables that appear within the left-hand-side of an implication, fully positive variables: those appearing only within the right-hand-side of an implication. 15/32

Weakly negative least fixed-points: an example Use µ x . ( f ◦ g )( x ) = f ( µ y . ( g ◦ f )( y ) ) 16/32

Weakly negative least fixed-points: an example Use µ x . ( f ◦ g )( x ) = f ( µ y . ( g ◦ f )( y ) ) to argue that: µ x . [ ( x → a ) → b ] = [ ν y . ( y → b ) → a ] → b = [ ( ⊤ → b ) → a ] → b = [ b → a ] → b . 16/32

Weakly negative least fixed-points: reducing to greatest fixed-points If each occurrence of x in φ is weakly negative, then φ ( x ) = φ 0 [ φ 1 ( x ) / y 1 , . . . , φ n ( x ) / y n ] with φ 0 ( y 1 , . . . , y n ) negative in each y j . Due to µ x . ( f ◦ g )( x ) = f ( µ y . ( g ◦ f )( y ) ) 17/32

Weakly negative least fixed-points: reducing to greatest fixed-points If each occurrence of x in φ is weakly negative, then φ ( x ) = φ 0 [ φ 1 ( x ) / y 1 , . . . , φ n ( x ) / y n ] with φ 0 ( y 1 , . . . , y n ) negative in each y j . Due to µ x . ( f ◦ g )( x ) = f ( µ y . ( g ◦ f )( y ) ) we have µ x .φ ( x ) = µ x . ( φ 0 ◦ � φ 1 , . . . , φ n � )( x ) = φ 0 ( ν y 1 ... y n . ( � φ 1 , . . . φ n � ◦ φ 0 )( y 1 , . . . y n )) . 17/32