Completeness for coalgebraic µ-calculus: part 2. Fatemeh Seifan (Joint work with Sebastian Enqvist and Yde Venema)

Transcription

1 Completeness for coalgebraic µ-calculus: part 2 Fatemeh Seifan (Joint work with Sebastian Enqvist and Yde Venema)

2 Overview

3 Overview Completeness of Kozen s axiomatisation of the propositional µ-calculus

4 Overview Completeness of Kozen s axiomatisation of the propositional µ-calculus Completeness for the modal µ-calculus: Separating the combinatorics from the dynamics (TCS 2018)

5 Overview Completeness of Kozen s axiomatisation of the propositional µ-calculus Completeness for the modal µ-calculus: Separating the combinatorics from the dynamics (TCS 2018) Completeness for coalgebraic fixpoint logic (CSL 2016)

6 Overview Completeness of Kozen s axiomatisation of the propositional µ-calculus Completeness for the modal µ-calculus: Separating the combinatorics from the dynamics (TCS 2018) Completeness for coalgebraic fixpoint logic (CSL 2016) Completeness for µ-calculi: a coalgebraic approach

7 Coalgebraic µ-calculus

8 Coalgebraic µ-calculus Given a set functor T and n ω, an n-place predicate lifting λ for T is an assignment, to each set S, of a map λ S : (PS) n PTS, such that for any map f : S S and any n-tuple Z = (Z 1,..., Z n ) (PS) n we have, for all σ TS: σ λ S (f 1 [Z]) iff Tf (σ) λ S (Z) where f 1 [Z] abbreviates (f 1 [Z 1 ],..., f 1 [Z n ]).

9 Coalgebraic µ-calculus A predicate lifting λ : P n PT is monotone if for every set S, the map λ S : (PS) n PTS is order-preserving in each coordinate (with respect to the subset order).

10 Coalgebraic µ-calculus A predicate lifting λ : P n PT is monotone if for every set S, the map λ S : (PS) n PTS is order-preserving in each coordinate (with respect to the subset order). The induced predicate lifting λ : P n PT, given by λ S(X 1,..., X n ) := TS \ λ S (S \ X 1,..., S \ X 1 ), is called the (Boolean) dual of λ.

11 Coalgebraic µ-calculus

12 Coalgebraic µ-calculus To each predicate lifting λ we associate a modality λ with the same arity as λ. The semantics of λ in a T-model S = (S, σ, V ) is: S, s λ (ϕ) if σ(s) λ S ([[ϕ 1 ]] S,..., [[ϕ n ]] S ).

13 Coalgebraic µ-calculus To each predicate lifting λ we associate a modality λ with the same arity as λ. The semantics of λ in a T-model S = (S, σ, V ) is: S, s λ (ϕ) if σ(s) λ S ([[ϕ 1 ]] S,..., [[ϕ n ]] S ). Example: λ, λ : PS PPS: λ : U {T PS T U } λ : U {T PS T U}.

14 Coalgebraic µ-calculus To each predicate lifting λ we associate a modality λ with the same arity as λ. The semantics of λ in a T-model S = (S, σ, V ) is: S, s λ (ϕ) if σ(s) λ S ([[ϕ 1 ]] S,..., [[ϕ n ]] S ). Example: λ, λ : PS PPS: λ : U {T PS T U } λ : U {T PS T U}. We can then formulate the semantics of via the map λ : S, s ϕ iff σ R (s) λ ([[ϕ]] S )

15 Coalgebraic µ-calculus Example

16 Coalgebraic µ-calculus Example For the monotone neighbourhood functor M, we define two unary predicate liftings, ɛ and ɛ : ɛ S : U { α MS U α } ɛ S : U { α MS S \ U α }.

17 Coalgebraic µ-calculus Example For the monotone neighbourhood functor M, we define two unary predicate liftings, ɛ and ɛ : ɛ S : U { α MS U α } ɛ S : U { α MS S \ U α }. The modal operators ɛ and ɛ modalities and : coincide with the standard monotone S, s ϕ iff U [[ϕ]] S, for some U σ(s) S, s ϕ iff U [[ϕ]] S, for all U σ(s).

18 Coalgebraic µ-calculus Example

19 Coalgebraic µ-calculus Example Consider the finitary multiset or bag functor B:

20 Coalgebraic µ-calculus Example Consider the finitary multiset or bag functor B: S BS := {σ : S ω σ has a finite support}

21 Coalgebraic µ-calculus Example Consider the finitary multiset or bag functor B: S BS := {σ : S ω σ has a finite support} On arrows: given a map f : S S and σ BS, we define the weight function (Bf )σ : S ω by: ((Bf )σ)(s ) := {σ(s) f (s) = s }.

22 Coalgebraic µ-calculus Example Consider the finitary multiset or bag functor B: S BS := {σ : S ω σ has a finite support} On arrows: given a map f : S S and σ BS, we define the weight function (Bf )σ : S ω by: ((Bf )σ)(s ) := {σ(s) f (s) = s }. For a natural number k, we define the predicate liftings k and k by putting k S : U { σ BS u U σ(u) k} k S : U {σ BS u U σ(u) < k},

23 Coalgebraic µ-calculus Given a set Λ of predicate liftings, the modal fixpoint language µml Λ is defined as follows: ϕ ::= p ϕ ϕ 0 ϕ 1 λ (ϕ 1,..., ϕ n ) µx.ϕ

24 Coalgebraic µ-calculus Given a set Λ of predicate liftings, the modal fixpoint language µml Λ is defined as follows: ϕ ::= p ϕ ϕ 0 ϕ 1 λ (ϕ 1,..., ϕ n ) µx.ϕ [[µx.ϕ]] S := { U PS [[ϕ]] (S,R,V [x U]) U }

25 Coalgebraic µ-calculus Given a set Λ of predicate liftings, the modal fixpoint language µml Λ is defined as follows: ϕ ::= p ϕ ϕ 0 ϕ 1 λ (ϕ 1,..., ϕ n ) µx.ϕ [[µx.ϕ]] S := { U PS [[ϕ]] (S,R,V [x U]) U } Given a set Λ of predicate liftings and a set X of proposition letters, we let µml Λ (X) denote the set of µml Λ -formulas ϕ of which all free variables belong to X.

26 One-step logic Given a set of predicate liftings Λ, and two disjoint sets A, X of variables, we define the set Bool(A) of boolean formulas over A and the set ML 1 Λ (A, X) of one-step Λ-formulas over A and parameters X, by the following grammars: Bool(A) π ::= a π π π π π ML 1 Λ(A, X) α ::= p λ π α α α α α

27 One-step logic Given a set of predicate liftings Λ, and two disjoint sets A, X of variables, we define the set Bool(A) of boolean formulas over A and the set ML 1 Λ (A, X) of one-step Λ-formulas over A and parameters X, by the following grammars: Bool(A) π ::= a π π π π π ML 1 Λ(A, X) α ::= p λ π α α α α α The A-positive fragment of ML 1 Λ (A, X) is denoted by ML1 Λ (A, X)+.

28 One-step logic Given a set of predicate liftings Λ, and two disjoint sets A, X of variables, we define the set Bool(A) of boolean formulas over A and the set ML 1 Λ (A, X) of one-step Λ-formulas over A and parameters X, by the following grammars: Bool(A) π ::= a π π π π π ML 1 Λ(A, X) α ::= p λ π α α α α α The A-positive fragment of ML 1 Λ (A, X) is denoted by ML1 Λ (A, X)+. The negation-free fragment of Bool(A) is denoted by Latt(A) and refer to its elements as lattice formulas over A.

29 One-step logic A one-step T-frame over A is a triple (X, Y, ξ) where ξ TX and Y X, or equivalently, a pair (X, ξ) with ξ T X X. A one-step model (X, ξ, m) is a one-step frame (X, ξ) together with a marking m : X PA.

30 One-step logic A one-step T-frame over A is a triple (X, Y, ξ) where ξ TX and Y X, or equivalently, a pair (X, ξ) with ξ T X X. A one-step model (X, ξ, m) is a one-step frame (X, ξ) together with a marking m : X PA. Given a marking m : X PA, by [[π]] 0 X we denote the 0-step interpretation of π Bool(A).

31 One-step logic A one-step T-frame over A is a triple (X, Y, ξ) where ξ TX and Y X, or equivalently, a pair (X, ξ) with ξ T X X. A one-step model (X, ξ, m) is a one-step frame (X, ξ) together with a marking m : X PA. Given a marking m : X PA, by [[π]] 0 X we denote the 0-step interpretation of π Bool(A). The 1-step interpretation [[α]] 1 m of α ML 1 Λ (A, X) is defined as a subset of T X X, with [[p]] 1 m := {(Y, ξ) p Y}, [[ λ (π 1,..., π n )]] 1 m := {(Y, ξ) ξ λ X ([[π 1 ]] 0 m,..., [[π n ]] 0 m)}, and standard clauses for,, and.

32 One-step logic A one-step T-frame over A is a triple (X, Y, ξ) where ξ TX and Y X, or equivalently, a pair (X, ξ) with ξ T X X. A one-step model (X, ξ, m) is a one-step frame (X, ξ) together with a marking m : X PA. Given a marking m : X PA, by [[π]] 0 X we denote the 0-step interpretation of π Bool(A). The 1-step interpretation [[α]] 1 m of α ML 1 Λ (A, X) is defined as a subset of T X X, with [[p]] 1 m := {(Y, ξ) p Y}, [[ λ (π 1,..., π n )]] 1 m := {(Y, ξ) ξ λ X ([[π 1 ]] 0 m,..., [[π n ]] 0 m)}, and standard clauses for,, and. We write X, ξ, m 1 α for ξ [[α]] 1 m.

33 One-step logic A monotone modal signature for T is expressively complete if, for every monotone n-place predicate lifting λ Λ and variables a 1,..., a n there is a formula α ML 1 Λ ({a 1,..., a n }) which is equivalent to λ a.

34 Coalgebraic automa A Λ-automaton over X is a quadruple (A, Θ, Ω, a I ) where A is a finite set of states, Ω : A ω is the priority map, while the transition map Θ : A ML 1 Λ(A, X) + maps states to (positive) one-step formulas over X and A. Let A = (A, Θ, Ω, a I ) be a Λ-automaton, and let S = (S, σ, V ) be a T-model, both over the set X of proposition letters. The acceptance game A(A, S) for A with respect to S is defined as in the following table: Position Pl r Admissible moves (a, s) {m : S PA (S, σ(s), m) 1 Θ(a)} m {(b, t) b m(t)}

35 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that:

36 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that: (1) X, ξ, m 0 1 α,

37 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that: (1) X, ξ, m 0 1 α, (2) Tf (ξ ) = ξ,

38 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that: (1) X, ξ, m 0 1 α, (2) Tf (ξ ) = ξ, and (3) m 0 (v) 1 and m 0 (v) m(f (v)) for each v X.

39 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that: (1) X, ξ, m 0 1 α, (2) Tf (ξ ) = ξ, and (3) m 0 (v) 1 and m 0 (v) m(f (v)) for each v X. Example: The cover modality of standard modal logic: {a 1,..., a n } a 1... a n (a 1... a n ).

40 Disjunctive bases A formula α ML 1 Λ (A)+ is said to be disjunctive if, for every one-step frame (X, ξ) and every marking m : X PA, if X, ξ, m 1 α then there is a one-step frame (X, ξ ), a map f : X X and a marking m 0 : X P(A) such that: (1) X, ξ, m 0 1 α, (2) Tf (ξ ) = ξ, and (3) m 0 (v) 1 and m 0 (v) m(f (v)) for each v X. Example: The cover modality of standard modal logic: {a 1,..., a n } a 1... a n (a 1... a n ). Example: Moss modality T provides disjunctive formulas for every weak-pullback preserving functor T.

41 Disjunctive bases For each finite set A let χ A : P(A) Latt(A) be the substitution defined by B B. Similarly, let θ A,B : A B Latt(A B) be defined by (a, b) a b.

42 Disjunctive bases For each finite set A let χ A : P(A) Latt(A) be the substitution defined by B B. Similarly, let θ A,B : A B Latt(A B) be defined by (a, b) a b. Let D be an assignment of a set of positive one-step formulas D(A) ML 1 Λ (A)+ for all finite sets A. Then D is called a disjunctive basis for Λ if each formula in D(A) is disjunctive, and the following conditions hold:

43 Disjunctive bases For each finite set A let χ A : P(A) Latt(A) be the substitution defined by B B. Similarly, let θ A,B : A B Latt(A B) be defined by (a, b) a b. Let D be an assignment of a set of positive one-step formulas D(A) ML 1 Λ (A)+ for all finite sets A. Then D is called a disjunctive basis for Λ if each formula in D(A) is disjunctive, and the following conditions hold: (1) D(A) is closed under finite disjunctions.

44 Disjunctive bases For each finite set A let χ A : P(A) Latt(A) be the substitution defined by B B. Similarly, let θ A,B : A B Latt(A B) be defined by (a, b) a b. Let D be an assignment of a set of positive one-step formulas D(A) ML 1 Λ (A)+ for all finite sets A. Then D is called a disjunctive basis for Λ if each formula in D(A) is disjunctive, and the following conditions hold: (1) D(A) is closed under finite disjunctions. (2) D is distributive over Λ: for every one-step formula λ π there is a formula δ D(P(A)) such that λ π 1 δ[χ A ].

45 Disjunctive bases For each finite set A let χ A : P(A) Latt(A) be the substitution defined by B B. Similarly, let θ A,B : A B Latt(A B) be defined by (a, b) a b. Let D be an assignment of a set of positive one-step formulas D(A) ML 1 Λ (A)+ for all finite sets A. Then D is called a disjunctive basis for Λ if each formula in D(A) is disjunctive, and the following conditions hold: (1) D(A) is closed under finite disjunctions. (2) D is distributive over Λ: for every one-step formula λ π there is a formula δ D(P(A)) such that λ π 1 δ[χ A ]. (3) D admits a distributive law: for any two formulas α D(A) and β D(B), there is a formula γ D(A B) such that α β 1 γ[θ A,B ].

46 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis.

47 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary?

48 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary? NO!

49 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary? NO! Consider L 3 2

50 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary? NO! Consider L 3 2 Question: What about preservation of finite sets and expressively completeness?

51 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary? NO! Consider L 3 2 Question: What about preservation of finite sets and expressively completeness? Not necessary!

52 Disjunctive bases Proposition. Let Λ be an expressively complete signature for a weak pullback-preserving functor T and suppose that T preserves finite sets. Then Λ admits a disjunctive basis. Question: Preservation of weak pullbacks is necessary? NO! Consider L 3 2 Question: What about preservation of finite sets and expressively completeness? Not necessary! Proposition. B admits a disjunctive basis.

53 One-step soundness and completeness A one-step formula α is one-step valid, notation 1 α, if [[α]] 1 m = T X X for all sets X and markings m : X PA,

54 One-step soundness and completeness A one-step formula α is one-step valid, notation 1 α, if [[α]] 1 m = T X X for all sets X and markings m : X PA, and we say that β is a one-step consequence of α (written α 1 β) if [[α]] 1 m [[β]] 1 m for all X, m.

55 A general completeness theorem The one-step derivation system H 1 : (H) All formulas in H are axioms of H 1. (MP) From α β and α, derive β, where α, β ML 1 Λ (X). (CT) The formula α ML 1 Λ (X) is an axiom if it is a substitution instance of some propositional tautology. (Cg) For all π 1, π 2 Bool(X), if π 1 π 2 is a substitution instance of a propositional tautology then λ π 1 λ π 2 is an axiom. (US) Given any map τ : X Bool(X) and α ML 1 Λ (X), derive α[τ] from α. (Du) the formula λ a λ a is an axiom, for all λ Λ and a X; (Mon) For all λ Λ and a, b X, the formula λ a λ (a b) is an axiom.

56 A general completeness theorem We add the pre-fixpoint schema and the Kozen-Park induction rule to obtain a define a Hilbert system µh for µ-calculus. ϕ[µp.ϕ/p] µp.ϕ ϕ[ψ/p] ψ µp.ϕ ψ

57 A general completeness theorem We add the pre-fixpoint schema and the Kozen-Park induction rule to obtain a define a Hilbert system µh for µ-calculus. ϕ[µp.ϕ/p] µp.ϕ ϕ[ψ/p] ψ µp.ϕ ψ We write H ϕ to say that ϕ is provable in the system µh, ϕ H ψ for H ϕ ψ and ϕ H ψ for H ϕ ψ.

58 A general completeness theorem A one-step axiomatization H is said to be one-step sound if 1 α whenever 1 H α, for α ML1 Λ (A).

59 A general completeness theorem A one-step axiomatization H is said to be one-step sound if 1 α whenever 1 H α, for α ML1 Λ (A). The system H is said to be one-step complete if 1 H α whenever 1 α, for α ML 1 Λ (A).

60 A general completeness theorem A one-step axiomatization H is said to be one-step sound if 1 α whenever 1 H α, for α ML1 Λ (A). The system H is said to be one-step complete if 1 H α whenever 1 α, for α ML 1 Λ (A). The system H is said to be one-step complete if 1 H α whenever 1 α, for α ML 1 Λ (A).

61 A general completeness theorem Theorem Let T be a set functor, let Λ be a monotone modal signature for T, and let H be a one-step axiomatization for Λ and T. If H is one-step sound and complete and Λ admits a disjunctive basis, then µh is a sound and complete axiom system for the µml Λ -formulas that are valid in the class of all T-coalgebras.

62 Applications Theorem The proof system µh is sound and complete for validity over T-models, where: (1) T = Id and H = I, (2) T = Id k and H = I k, (3) T = P and H = K, (4) T = B and H = B.

63 Applications Theorem The proof system µh is sound and complete for validity over T-models, where: (1) T = Id and H = I, (2) T = Id k and H = I k, (3) T = P and H = K, (4) T = B and H = B. Uniform Interpolation

64 Applications Theorem The system µm is sound and complete for validity over M-models.

65 Applications Theorem The system µm is sound and complete for validity over M-models. We define the supported companion functor M s of M as the subfunctor of P M, given, on objects, by M s S := {(S 0, γ) PS MS S 0 supports γ}.

66 Applications Theorem The system µm is sound and complete for validity over M-models. We define the supported companion functor M s of M as the subfunctor of P M, given, on objects, by M s S := {(S 0, γ) PS MS S 0 supports γ}. A subset S 0 S supports an object γ MS whenever U γ iff U S 0 γ, for all U PS.