Obstacle 1: computational danger zone Example ◮ Language: � R , � U ◮ Intended Semantics: N × N ◮ ( m , n ) R ( m ′ , n ′ ) iff m ′ = m + 1 and n ′ = n ◮ ( m , n ) U ( m ′ , n ′ ) iff m ′ = m and n ′ = n + 1 ◮ Logic K G := K + ◮ functionality: � R p ↔ � R p and � U p ↔ � U p ◮ confluence: � R � U p → � U � R p ◮ K G is sound and complete with respect to its Kripke frames ◮ Add master modality, �∗� p := µ x . p ∨ � R x ∨ � U x ◮ µ K G is sound but incomplete with respect to its Kripke frames ◮ Proof:
Obstacle 1: computational danger zone Example ◮ Language: � R , � U ◮ Intended Semantics: N × N ◮ ( m , n ) R ( m ′ , n ′ ) iff m ′ = m + 1 and n ′ = n ◮ ( m , n ) U ( m ′ , n ′ ) iff m ′ = m and n ′ = n + 1 ◮ Logic K G := K + ◮ functionality: � R p ↔ � R p and � U p ↔ � U p ◮ confluence: � R � U p → � U � R p ◮ K G is sound and complete with respect to its Kripke frames ◮ Add master modality, �∗� p := µ x . p ∨ � R x ∨ � U x ◮ µ K G is sound but incomplete with respect to its Kripke frames ◮ Proof: Use recurrent tiling problem to show that
Obstacle 1: computational danger zone Example ◮ Language: � R , � U ◮ Intended Semantics: N × N ◮ ( m , n ) R ( m ′ , n ′ ) iff m ′ = m + 1 and n ′ = n ◮ ( m , n ) U ( m ′ , n ′ ) iff m ′ = m and n ′ = n + 1 ◮ Logic K G := K + ◮ functionality: � R p ↔ � R p and � U p ↔ � U p ◮ confluence: � R � U p → � U � R p ◮ K G is sound and complete with respect to its Kripke frames ◮ Add master modality, �∗� p := µ x . p ∨ � R x ∨ � U x ◮ µ K G is sound but incomplete with respect to its Kripke frames ◮ Proof: Use recurrent tiling problem to show that ◮ the � R , � U , �∗� -logic of Fr ( K G ) is not recursively enumerable
Obstacle 2: compactness failure n ∈ ω � n p ◮ Example: �∗� p := � ◮ {�∗� p } ∪ { � n ¬ p | n ∈ ω } is finitely satisfiable but not satisfiable
Obstacle 2: compactness failure n ∈ ω � n p ◮ Example: �∗� p := � ◮ {�∗� p } ∪ { � n ¬ p | n ∈ ω } is finitely satisfiable but not satisfiable ◮ Fixpoint logics have no nice Stone-based duality
Obstacle 3: fixpoint alternation ◮ tableaux: fixpoint unfolding ◮ ν -fixpoints may be unfolded infinitely often ◮ µ -fixpoints may only be unfolded finitely often
Obstacle 3: fixpoint alternation ◮ tableaux: fixpoint unfolding ◮ ν -fixpoints may be unfolded infinitely often ◮ µ -fixpoints may only be unfolded finitely often ◮ with every branch of tableau associate a trace graph
Obstacle 3: fixpoint alternation ◮ tableaux: fixpoint unfolding ◮ ν -fixpoints may be unfolded infinitely often ◮ µ -fixpoints may only be unfolded finitely often ◮ with every branch of tableau associate a trace graph ◮ obstacle 3a: conjunctions cause trace proliferation
Obstacle 3: fixpoint alternation ◮ tableaux: fixpoint unfolding ◮ ν -fixpoints may be unfolded infinitely often ◮ µ -fixpoints may only be unfolded finitely often ◮ with every branch of tableau associate a trace graph ◮ obstacle 3a: conjunctions cause trace proliferation ◮ obstacle 3b: fixpoint alternations cause intricate combinatorics
What to do?
What to do? ◮ consider simple frame conditions only (if at all)
What to do? ◮ consider simple frame conditions only (if at all) ◮ restrict language to fixpoints of simple formulas (avoid alternation)
What to do? ◮ consider simple frame conditions only (if at all) ◮ restrict language to fixpoints of simple formulas (avoid alternation) ◮ allow alternation, but develop suitable combinatorical framework
Overview ◮ Introduction ◮ Obstacles ◮ A general result ◮ A general framework ◮ Frame conditions ◮ Conclusions
Flat Modal Fixpoint Logics: Syntax ◮ Fix a basic modal formula γ ( x ,� p ), positive in x
Flat Modal Fixpoint Logics: Syntax ◮ Fix a basic modal formula γ ( x ,� p ), positive in x ◮ Add a fixpoint connective ♯ γ to the language of ML (arity of ♯ γ depends on γ but notation hides this)
Flat Modal Fixpoint Logics: Syntax ◮ Fix a basic modal formula γ ( x ,� p ), positive in x ◮ Add a fixpoint connective ♯ γ to the language of ML (arity of ♯ γ depends on γ but notation hides this) ◮ Example: Upq := µ x . p ∨ ( q ∧ � x ), now: Upq := ♯ γ ( p , q ) with γ = p ∨ ( q ∧ � x ) ◮ Intended reading: ♯ γ ( � ϕ ) ≡ µ x .γ ( x , � ϕ ) for any � ϕ = ( ϕ 1 , . . . , ϕ n ).
Flat Modal Fixpoint Logics: Syntax ◮ Fix a basic modal formula γ ( x ,� p ), positive in x ◮ Add a fixpoint connective ♯ γ to the language of ML (arity of ♯ γ depends on γ but notation hides this) ◮ Example: Upq := µ x . p ∨ ( q ∧ � x ), now: Upq := ♯ γ ( p , q ) with γ = p ∨ ( q ∧ � x ) ◮ Intended reading: ♯ γ ( � ϕ ) ≡ µ x .γ ( x , � ϕ ) for any � ϕ = ( ϕ 1 , . . . , ϕ n ). ◮ Obtain language ML γ : ϕ ::= p | ¬ p | ⊥ | ⊤ | ϕ 1 ∨ ϕ 2 | ϕ 1 ∧ ϕ 2 | � i ϕ | � i ϕ | ♯ γ ( � ϕ )
Flat Modal Fixpoint Logics: Syntax ◮ Fix a basic modal formula γ ( x ,� p ), positive in x ◮ Add a fixpoint connective ♯ γ to the language of ML (arity of ♯ γ depends on γ but notation hides this) ◮ Example: Upq := µ x . p ∨ ( q ∧ � x ), now: Upq := ♯ γ ( p , q ) with γ = p ∨ ( q ∧ � x ) ◮ Intended reading: ♯ γ ( � ϕ ) ≡ µ x .γ ( x , � ϕ ) for any � ϕ = ( ϕ 1 , . . . , ϕ n ). ◮ Obtain language ML γ : ϕ ::= p | ¬ p | ⊥ | ⊤ | ϕ 1 ∨ ϕ 2 | ϕ 1 ∧ ϕ 2 | � i ϕ | � i ϕ | ♯ γ ( � ϕ ) ◮ Examples: CTL, LTL, (PDL), . . .
Flat Modal Fixpoint Logics: Kripke Semantics ◮ Kripke frame S = � S , R � with R ⊆ S × S . ◮ Complex algebra: S + := � ℘ ( S ) , ∅ , S , ∼ S , ∪ , ∩ , � R �� , � R � : ℘ ( S ) → ℘ ( S ) given by � R � ( X ) := { s ∈ S | Rst for some t ∈ X }
Flat Modal Fixpoint Logics: Kripke Semantics ◮ Kripke frame S = � S , R � with R ⊆ S × S . ◮ Complex algebra: S + := � ℘ ( S ) , ∅ , S , ∼ S , ∪ , ∩ , � R �� , � R � : ℘ ( S ) → ℘ ( S ) given by � R � ( X ) := { s ∈ S | Rst for some t ∈ X } ◮ Every modal formula ϕ ( p 1 , . . . , p n ) corresponds to a term function ϕ S : ℘ ( S ) n → ℘ ( S ) . ◮ γ positive in x , hence γ S order preserving in x .
Flat Modal Fixpoint Logics: Kripke Semantics ◮ Kripke frame S = � S , R � with R ⊆ S × S . ◮ Complex algebra: S + := � ℘ ( S ) , ∅ , S , ∼ S , ∪ , ∩ , � R �� , � R � : ℘ ( S ) → ℘ ( S ) given by � R � ( X ) := { s ∈ S | Rst for some t ∈ X } ◮ Every modal formula ϕ ( p 1 , . . . , p n ) corresponds to a term function ϕ S : ℘ ( S ) n → ℘ ( S ) . ◮ γ positive in x , hence γ S order preserving in x . ◮ By Knaster-Tarski we may define ♯ S : ℘ ( S ) n → ℘ ( S ) by ♯ S ( � B ) := LFP .γ S ( − , � B ) .
Flat Modal Fixpoint Logics: Kripke Semantics ◮ Kripke frame S = � S , R � with R ⊆ S × S . ◮ Complex algebra: S + := � ℘ ( S ) , ∅ , S , ∼ S , ∪ , ∩ , � R �� , � R � : ℘ ( S ) → ℘ ( S ) given by � R � ( X ) := { s ∈ S | Rst for some t ∈ X } ◮ Every modal formula ϕ ( p 1 , . . . , p n ) corresponds to a term function ϕ S : ℘ ( S ) n → ℘ ( S ) . ◮ γ positive in x , hence γ S order preserving in x . ◮ By Knaster-Tarski we may define ♯ S : ℘ ( S ) n → ℘ ( S ) by ♯ S ( � B ) := LFP .γ S ( − , � B ) . ◮ Kripke ♯ -algebra S ♯ := � ℘ ( S ) , ∅ , S , ∼ S , ∪ , ∩ , � R � , ♯ S � .
Candidate Axiomatization K γ := K extended with ◮ prefixpoint axiom: γ ( ♯ ( � ϕ ) , � ϕ ) ⊢ ♯ ( � ϕ ) ◮ Park’s induction rule: from γ ( ψ, � ϕ ) ⊢ ψ infer ♯ γ ( � ϕ ) ⊢ ψ.
Flat Modal Fixpoint Logics: Algebraic completeness proof
Flat Modal Fixpoint Logics: Algebraic completeness proof ◮ Modal ♯ -algebra: A = � A , ⊥ , ⊤ , ¬ , ∧ , ∨ , � , ♯ � with ♯ : A n → A satisfying ♯ ( � b ) = LFP .γ A b , � b ( a ) := γ A ( a ,� where γ A b : A → A is given by γ A b ). � �
Flat Modal Fixpoint Logics: Algebraic completeness proof ◮ Modal ♯ -algebra: A = � A , ⊥ , ⊤ , ¬ , ∧ , ∨ , � , ♯ � with ♯ : A n → A satisfying ♯ ( � b ) = LFP .γ A b , � b ( a ) := γ A ( a ,� where γ A b : A → A is given by γ A b ). � � ◮ Axiomatically: modal ♯ -algebras satisfy ◮ γ ( ♯ ( � y ) ,� y ) ≤ ♯ ( � y ) ◮ if γ ( x ,� y ) ≤ x then ♯ ( � y ) ≤ x . ? ◮ Completeness for flat fixpoint logics: Equ(MA ♯ ) = Equ(KA ♯ )
Flat Modal Fixpoint Logics: Algebraic completeness proof ◮ Modal ♯ -algebra: A = � A , ⊥ , ⊤ , ¬ , ∧ , ∨ , � , ♯ � with ♯ : A n → A satisfying ♯ ( � b ) = LFP .γ A b , � b ( a ) := γ A ( a ,� where γ A b : A → A is given by γ A b ). � � ◮ Axiomatically: modal ♯ -algebras satisfy ◮ γ ( ♯ ( � y ) ,� y ) ≤ ♯ ( � y ) ◮ if γ ( x ,� y ) ≤ x then ♯ ( � y ) ≤ x . ? ◮ Completeness for flat fixpoint logics: Equ(MA ♯ ) = Equ(KA ♯ ) ◮ Two key concepts:
Flat Modal Fixpoint Logics: Algebraic completeness proof ◮ Modal ♯ -algebra: A = � A , ⊥ , ⊤ , ¬ , ∧ , ∨ , � , ♯ � with ♯ : A n → A satisfying ♯ ( � b ) = LFP .γ A b , � b ( a ) := γ A ( a ,� where γ A b : A → A is given by γ A b ). � � ◮ Axiomatically: modal ♯ -algebras satisfy ◮ γ ( ♯ ( � y ) ,� y ) ≤ ♯ ( � y ) ◮ if γ ( x ,� y ) ≤ x then ♯ ( � y ) ≤ x . ? ◮ Completeness for flat fixpoint logics: Equ(MA ♯ ) = Equ(KA ♯ ) ◮ Two key concepts: ◮ constructiveness ◮ O -adjointness
Constructiveness ◮ An MA ♯ -algebra A is constructive if ♯ ( � � γ n b ) = b ( ⊥ ) . � n ∈ ω
Constructiveness ◮ An MA ♯ -algebra A is constructive if ♯ ( � � γ n b ) = b ( ⊥ ) . � n ∈ ω Note: we do not require A to be complete!
Constructiveness ◮ An MA ♯ -algebra A is constructive if ♯ ( � � γ n b ) = b ( ⊥ ) . � n ∈ ω Note: we do not require A to be complete! Theorem (Santocanale & Venema) Let A be a countable, residuated, modal ♯ -algebra. If A is constructive, then A can be embedded in a Kripke ♯ -algebra.
Constructiveness ◮ An MA ♯ -algebra A is constructive if ♯ ( � � γ n b ) = b ( ⊥ ) . � n ∈ ω Note: we do not require A to be complete! Theorem (Santocanale & Venema) Let A be a countable, residuated, modal ♯ -algebra. If A is constructive, then A can be embedded in a Kripke ♯ -algebra. Proof Via a step-by-step construction/generalized Lindenbaum Lemma. Alternatively, use Rasiowa-Sikorski Lemma.
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map.
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map. ◮ f is a (left) adjoint or residuated if it has a residual g : Q → P with fp ≤ q ⇐ ⇒ p ≤ gq .
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map. ◮ f is a (left) adjoint or residuated if it has a residual g : Q → P with fp ≤ q ⇐ ⇒ p ≤ gq . ◮ f is a (left) O -adjoint if it has an O -residual G f : Q → ℘ ω ( P ) with fp ≤ q ⇐ ⇒ p ≤ y for some y ∈ G f q .
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map. ◮ f is a (left) adjoint or residuated if it has a residual g : Q → P with fp ≤ q ⇐ ⇒ p ≤ gq . ◮ f is a (left) O -adjoint if it has an O -residual G f : Q → ℘ ω ( P ) with fp ≤ q ⇐ ⇒ p ≤ y for some y ∈ G f q . Proposition (Santocanale 2005) ◮ f is a left adjoint iff f is a join-preserving O -adjoint
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map. ◮ f is a (left) adjoint or residuated if it has a residual g : Q → P with fp ≤ q ⇐ ⇒ p ≤ gq . ◮ f is a (left) O -adjoint if it has an O -residual G f : Q → ℘ ω ( P ) with fp ≤ q ⇐ ⇒ p ≤ y for some y ∈ G f q . Proposition (Santocanale 2005) ◮ f is a left adjoint iff f is a join-preserving O -adjoint ◮ O -adjoints are Scott continuous
O -adjoints Let f : ( P , ≤ ) → ( Q , ≤ ) be an order-preserving map. ◮ f is a (left) adjoint or residuated if it has a residual g : Q → P with fp ≤ q ⇐ ⇒ p ≤ gq . ◮ f is a (left) O -adjoint if it has an O -residual G f : Q → ℘ ω ( P ) with fp ≤ q ⇐ ⇒ p ≤ y for some y ∈ G f q . Proposition (Santocanale 2005) ◮ f is a left adjoint iff f is a join-preserving O -adjoint ◮ O -adjoints are Scott continuous ◮ ∧ is continuous but not an O -adjoint.
Finitary O -adjoints Let f : A n → A be an O -adjoint with O -residual G .
Finitary O -adjoints Let f : A n → A be an O -adjoint with O -residual G . ◮ Inductively define G n : A → ℘ ( A ) G 0 ( a ) := { a } G n +1 ( a ) G [ G n ( a )] :=
Finitary O -adjoints Let f : A n → A be an O -adjoint with O -residual G . ◮ Inductively define G n : A → ℘ ( A ) G 0 ( a ) := { a } G n +1 ( a ) G [ G n ( a )] := ◮ Call f finitary if G ω ( a ) := � n ∈ ω G n ( a ) is finite.
Finitary O -adjoints Let f : A n → A be an O -adjoint with O -residual G . ◮ Inductively define G n : A → ℘ ( A ) G 0 ( a ) := { a } G n +1 ( a ) G [ G n ( a )] := ◮ Call f finitary if G ω ( a ) := � n ∈ ω G n ( a ) is finite. Theorem (Santocanale 2005) If f : A → A is a finitary O -adjoint, then LFP . f , if existing, is constructive.
Adjoints on free algebras
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints!
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras,
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras, ◮ Whitman’s rule for free lattices, . . .
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras, ◮ Whitman’s rule for free lattices, . . . ◮ Call a modal formula γ untied in x if it belongs to γ ::= x | ⊤ | γ ∨ γ | ψ ∧ γ | ∇{ γ 1 , . . . , γ n } where ψ does not contain x
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras, ◮ Whitman’s rule for free lattices, . . . ◮ Call a modal formula γ untied in x if it belongs to γ ::= x | ⊤ | γ ∨ γ | ψ ∧ γ | ∇{ γ 1 , . . . , γ n } where ψ does not contain x ◮ Examples: � x , � x , � x ∧ �� x ∧ � p , � x ∧ �� x ∧ � ( � x ∨ �� x ), . . .
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras, ◮ Whitman’s rule for free lattices, . . . ◮ Call a modal formula γ untied in x if it belongs to γ ::= x | ⊤ | γ ∨ γ | ψ ∧ γ | ∇{ γ 1 , . . . , γ n } where ψ does not contain x ◮ Examples: � x , � x , � x ∧ �� x ∧ � p , � x ∧ �� x ∧ � ( � x ∨ �� x ), . . . ◮ Counterexamples: � ( x ∧ � x ), � x ∧ �� x
Adjoints on free algebras ◮ Free modal ( ♯ -)algebras have many O -adjoints! ◮ cf. free distributive lattice are Heyting algebras, ◮ Whitman’s rule for free lattices, . . . ◮ Call a modal formula γ untied in x if it belongs to γ ::= x | ⊤ | γ ∨ γ | ψ ∧ γ | ∇{ γ 1 , . . . , γ n } where ψ does not contain x ◮ Examples: � x , � x , � x ∧ �� x ∧ � p , � x ∧ �� x ∧ � ( � x ∨ �� x ), . . . ◮ Counterexamples: � ( x ∧ � x ), � x ∧ �� x Theorem (Santocanale & YV 2010) Untied formulas are finitary O -adjoints.
A general result
A general result Theorem (Santocanale & YV 2010) Let γ be untied wrt x . Then K γ is sound and complete wrt its Kripke semantics.
A general result Theorem (Santocanale & YV 2010) Let γ be untied wrt x . Then K γ is sound and complete wrt its Kripke semantics. Notes
A general result Theorem (Santocanale & YV 2010) Let γ be untied wrt x . Then K γ is sound and complete wrt its Kripke semantics. Notes ◮ Santocanale & YV have fully general result for extended axiom system.
A general result Theorem (Santocanale & YV 2010) Let γ be untied wrt x . Then K γ is sound and complete wrt its Kripke semantics. Notes ◮ Santocanale & YV have fully general result for extended axiom system. ◮ Schr¨ oder & YV have similar results for wider coalgebraic setting.
Overview ◮ Introduction ◮ Obstacles ◮ A general result ◮ A general framework ◮ Frame conditions ◮ Conclusions
The modal µ -calculus ◮ [+] natural extension of basic modal logic with fixpoint operators ◮ [+] expressive: LTL, CTL, PDL, CTL*, . . . ⊆ µ ML ◮ [+] good computational properties ◮ [+] nice meta-logical theory
The modal µ -calculus ◮ [+] natural extension of basic modal logic with fixpoint operators ◮ [+] expressive: LTL, CTL, PDL, CTL*, . . . ⊆ µ ML ◮ [+] good computational properties ◮ [+] nice meta-logical theory ◮ [ – ] hard to understand (nested) fixpoint operators
The modal µ -calculus ◮ [+] natural extension of basic modal logic with fixpoint operators ◮ [+] expressive: LTL, CTL, PDL, CTL*, . . . ⊆ µ ML ◮ [+] good computational properties ◮ [+] nice meta-logical theory ◮ [ – ] hard to understand (nested) fixpoint operators ◮ [ – ] theory of µ ML isolated from theory of ML
The modal µ -calculus ◮ [+] natural extension of basic modal logic with fixpoint operators ◮ [+] expressive: LTL, CTL, PDL, CTL*, . . . ⊆ µ ML ◮ [+] good computational properties ◮ [+] nice meta-logical theory ◮ [ – ] hard to understand (nested) fixpoint operators ◮ [ – ] theory of µ ML isolated from theory of ML ◮ this applies in particular to the completeness result
The modal µ -calculus ◮ [+] natural extension of basic modal logic with fixpoint operators ◮ [+] expressive: LTL, CTL, PDL, CTL*, . . . ⊆ µ ML ◮ [+] good computational properties ◮ [+] nice meta-logical theory ◮ [ – ] hard to understand (nested) fixpoint operators ◮ [ – ] theory of µ ML isolated from theory of ML ◮ this applies in particular to the completeness result Most results on µ ML use automata . . .
Logic & Automata
Logic & Automata Automata in Logic ◮ long & rich history (B¨ uchi, Rabin, . . . ) ◮ mathematically interesting theory ◮ many practical applications ◮ automata for µ ML: ◮ Janin & Walukiewicz (1995): µ -automata (nondeterministic) ◮ Wilke (2002): modal automata (alternating)
Modal automata Fix a set X of proposition letters; PX is a set of colours ◮ A modal automaton is a triple A = ( A , Θ , Acc ), where ◮ A is a finite set of states ◮ Θ : A × PX → 1ML( A ) is the transition map ◮ Acc ⊆ A ω is the acceptance condition
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