Many-valued modal logics A bit on what, why and how Amanda Vidal - - PowerPoint PPT Presentation

Many-valued modal logics

A bit on what, why and how

Amanda Vidal PhDs in Logic 2019 Bern, 24-26 April

Institute of Computer Science, Czech Academy of Sciences

Introduction

Modal logics

• Modal logics expand CPL with non “truth-functional” operators
• K models naturally notions like ”possibly/necessarily”,

”sometimes/always”, and many other modal operators/logics are considered in the literature (deontic logics, doxastic logics)

• One of the first, best known, more studied, and more applied

non-classical logics.

(partially) why? offer a much higher expressive power than CPL and (generally) much lower complexity than FOL (most well-known and used modal logics are decidable).

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Many-valued logics

• Many-valued logics valuate the formulas out of {0, 1}(⊤, ⊥) and

enrich the set of operation, to richer algebraic structures than 2.

• Huge family of logics (different classes of algebras for evaluation).

Allow modeling vague/uncertain/incomplete knowledge and probabilistic notions

• Very developed theory (via algebraic logic and development in AAL)
• Applications in industry/AI etc. + (classical) mathematical interest

for its relation with Universal Algebra and particular algebraic areas.

• Many well-known infinitely-valued cases still decidable (

L, G¨

• del,

Product, H-BL...).

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Many-valued modal logics

• Natural idea: expansion of MV logics with modal-like
• perators/interaction (or of modal-logics with wider algebraic

evaluations/operations)

• Intuitionistic modal logics are particularly ”nice”: they naturally

enjoy a relational semantics with an intuitive meaning.

• what about the rest? a seemingly reasonable approach: valuation of

Kripke models/frames over classes of algebras

• Some modal MV logics have been axiomatised, but most have not.

[Many usual intuitions fail, and usual constructions need to be adapted to get completeness.]

• Relation to purely relational semantics is unknown.
• Tools from classical modal logic like Sahlqvist theory have not been

developed (wider set of operations + more specific semantics...)

• ...

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Some definitions (aka whats and whys)

The non-modal part

Definition A Residuated Lattice A is A, ⊙, →, ∧, ∨, 0, 1 such that

• A, ∧, ∨ is a lattice,
• A, ⊙, 1 is a commutative monoid
• x ⊙ y ≤ z ⇐

⇒ x ≤ y → z (residuation law)

• 0 ≤ x ≤ 1 ∀x ∈ A.

Γ | =C ϕ iff for any A ∈ C and any h ∈ Hom(Fm, A), if h(Γ) ⊆ {1} then h(ϕ) = 1. Well known examples

• Heyting algebras,
• [0, 1]G,
• [0, 1]

L ( x ⊙ y = max{0, x + y − 1})

• [0, 1]Π (⊙ = · )

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From Classical modal logic...

• (minimal)Modal logic K = CPC +
• K : ✷(ϕ → ψ) → (✷ϕ → ✷ψ),
• N✷ : from ϕ infer ✷ϕ obs: (over theorems/over deductions ⇒

local(≡ theorems via D.T)/global logic).

• ✸ := ¬✷¬

Definition A Kripke model M is a K. Frame F = W , R (W set, R ⊆ W 2) together with an evaluation e : V → P(W ). M, v p iff v ∈ e(p), M, v ¬ϕ iff v ∈ e(ϕ) M, v ϕ{∧, ∨}ψ iff M, v ϕ {and, or} M, v ψ M, v ✷ϕ iff for all w ∈ W s.t. R(v, w), M, w ϕ M, v ✸ϕ iff there is w ∈ W s.t. R(v, w) and M, w ϕ

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From Classical modal logic...

• (minimal)Modal logic K = CPC +
• K : ✷(ϕ → ψ) → (✷ϕ → ✷ψ),
• N✷ : from ϕ infer ✷ϕ) obs: over theorems/over deductions ⇒

local(≡ theorems via D.T)/global logic.

• ✸ := ¬✷¬

Definition A Kripke model M is a K. Frame F = W , R (W set, R : W 2 → {0, 1}) together with an evaluation e : W × V → {0, 1}. e(v, ¬p) = ¬e(v, p), e(v, ϕ{∧, ∨}ψ) = e(v, ϕ){∧, ∨}e(v, ψ) e(v, ✷ϕ) =

• 1

if for all w ∈ W s.t. R(v, w), e(u, ϕ) = 1

• therwise

e(v, ✸ϕ) =

• 1

if there is w ∈ W s.t. R(v, w) and e(w, ϕ) = 1

• therwise

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From Classical modal logic...

• (minimal)Modal logic K = CPC +
• K : ✷(ϕ → ψ) → (✷ϕ → ✷ψ),
• N✷ : from ϕ infer ✷ϕ) obs: over theorems/over deductions ⇒

local(≡ theorems via D.T)/global logic.

• ✸ := ¬✷¬

Definition A Kripke model M is a K. Frame F = W , R (W set, R : W 2 → {0, 1}) together with an evaluation e : W × V → {0, 1}. e(v, ¬p) = ¬e(v, p), e(v, ϕ{∧, ∨}ψ) = e(v, ϕ){∧, ∨}e(v, ψ) e(v, ✷ϕ) =

• w∈W

{Rvw → e(w, ϕ)} e(v, ✸ϕ) =

• w∈W

{Rvw ∧ e(w, ϕ)}

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From (Classical) modal logic...

• (Local): Γ K ϕ iff for all M K-model and for all w ∈ W ,

M, w Γ ⇒ M, w ϕ e(w, [Γ]) ⊆ {1} ⇒ e(w, ϕ) = 1

• (Global): Γ g

K ϕ iff for all M K-model,

M, w Γ for all w ∈ W ⇒ M, w ϕ for all w ∈ W e(w, [Γ]) ⊆ {1} for all w ∈ W ⇒ e(u, ϕ) = 1 for all w ∈ W Completeness: Γ ⊢K ϕ ⇔ Γ K ϕ

• proven via a canonical model:
• W = maximally consistent theories,
• RTQ ⇔ ✷−1T ⊆ Q,
• e(p) = {T : p ∈ T}. e(T, p) =

   1 if p ∈ T

• therwise

Truth Lemma: e(ϕ) = {T : ϕ ∈ T}. e(T, ϕ) =    1 if ϕ ∈ T

• therwise

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...to MV-modal logics

A residuated lattice. Definition A A-Kripke model M is an A- K.Frame F = W , R (W set, R : W 2 → A) together with an evaluation e : W × V → A. e(v, ϕ{∧, ∨}ψ) = e(v, ϕ){∧, ∨}e(v, ψ) e(v, ϕ ⊙ ψ) = e(v, ϕ) ⊙ e(v, ψ) e(v, ϕ → ψ) = e(v, ϕ) → e(v, ψ) e(v, ✷ϕ) =

• w∈W

{R(v, w) → e(w, ϕ)} e(v, ✸ϕ) =

• w∈W

{R(v, w) ⊙ e(w, ϕ)} safe whenever e(u, ✷ϕ), e(u, ✸ϕ) are defined in every world.

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Modal logics over residuated lattices

Let A be a class of RLs, and K be a class of A-Kripke models for A ∈ A .

• (Local -over K): Γ K ϕ iff for all M ∈ K and for all w ∈ W ,

e(w, [Γ]) ⊆ {1} ⇒ e(w, ϕ) = 1

• (Global -over K): Γ g

K ϕ iff for all M ∈ K,

e(w, [Γ]) ⊆ {1} for all w ∈ W ⇒ e(u, ϕ) = 1 for all w ∈ W

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Comparaisons

Some initial observations

• K is a theorem (Axiom!) from (Classical) modal logic. No more:

K is not necessarily valid Over [0, 1]

L consider the model W = {a, b}, R(a, b) = 0.8,

e(b, x) = 0.7, e(b, y) = 0.5. Then

• ✷(x → y) = 0.8 → (0.7 → 0.5) = 0.8 → 0.8 = 1, but
• ✷x → ✷y = (0.8 → 0.7) → (0.8 → 0.5) = 0.9 → 0.7 < 1.
• If ⊙ is idempotent over the values taken by R, K is valid in the

model (eg., over Heyting and G¨

• del algebras, or with R crisp).
• In (c.) modal logic, the D.T. holds (Γ, γ ⊢K ϕ ⇔ Γ ⊢ γ → ϕ).
• In (non-modal) MV-logics in general, this D.T already fails. At mot

weaker versions will be attainable, but still unclear (by semantic methods-only is not easy to see). Over order-preserving logics (eg. [0, 1]G) D.T. naturally still holds.

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Some initial observations

• In (c.) modal logic ✸ can be given as an abbreviation of ✷ (or

vice-versa).

• In the general case this approach has some flaws (eg. cancelative

negations give boolean ✸). The semantic definition based on and seems reasonable, but

• Only very particular cases allow for the above inter-definability of

✷ − ✸ (eg. chains with an involutive negation like [0, 1]

L)

• (enough) Constants in the language allow certain level of

expressability, but as for now, quite ad hoc.

• In general, 3 minimal modal logics: ✷-fragment, ✸-fragment,

bi-modal logic (both ✷ and ✸)

• Axioms relating ✷ and ✸ are crucial to get both of them over the

same accessibility relation (eg. also intutionistic Modal logics have faced this in different ways)

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Decidability/FMP

• (c.) modal logic (both local and global) are decidable. Follow (eg.)

from the Finite Model Property. No longer the case: FMP (as a K.model) is not necessarily valid Over [0, 1]G consider the formula ¬✷x → ✸¬x. Then

• In any [0, 1]G model with finite W, finite model the formula is true

(infima/suprema turn to minimum and maximum),

• The model {a, bi : i ∈ ω+}, R(a, bi) = 1 for all i, e(bi, x) = 1/i falsifies

the formula.

• Even in cases where the underlying MV-logic is decidable, the

decidability of the MV-modal logics is unclear.

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On the methodology for proving completeness

• Recall the canonical model from (c) modal logic.
• We could move from having Theories (as worlds) to have values on

the algebra because we are working in 2.

• Richer algebras (and operations) need finer definition of the

canonical model in order to prove completeness.

• Up to now, the C.M in MV-modal logics is based on letting W to be

the set of homomorphisms into the algebra (preserving the modal theorems). Observe in the cases when all -or enough- constants are added to the language, this is equivalent to ”the Theories” approach).

• This highly complicates the Truth-lemma proof.

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What is known (aka some more whats and

hows)

Finite algebras and the constants issue

• Early works on Many-valued modal logics focus on some

finitely-valued MV logics

• Segerberg (1967) and Thomason (1978) work over some particular

finitely-valued cases + classical frames.

• Fitting (1992a,b) studies Modal MV logics over finite lattices and

Heyting algebras, considering also non-classical frames.

• Axiomatizations are based on adding all constants from the algebra,

and consider the ✷-fragment.

• ✷ fragment, over arbitrary frames and finite RLs, is axiomatized with

constants, and studied, in Bou et al. (2011). To get completeness

• wrt. models with {0, 1} R, needed to move to chains only.

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A (particularly) well-behaved case: G¨

• del logics
• Considering models over [0, 1]G, the three minimal modal logics are

different.

• All basic ones, and all basic ones over models with crisp accessibility

(R in {0, 1}) have been finitely axiomatized. Caicedo and Rodr´ ıguez (2010); Caicedo and Rodriguez (2015),Metcalfe and Olivetti (2011)[Rodriguez and V., 2019?]

• The ✷ fragment over arbitrary models collapses to the one over

crisp-accessibility models, and does not have the FMP. The ✸ fragments are different, but they have the FMP.

• The bi-modal logics over arbitrary Rs or over crisp R are different,

and they do not have the FMP.

• But! Caicedo et al. (2013) provided an alternative semantics for

those logics WITH bounded size FMP = ⇒ decidability!

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• Lukasiewicz and others: the infinitarity issue
• Considering models over [0, 1]

L, ✷ and ✸ are inter-definable.

• The logic with crisp accessibility relation has been axiomatized

Hansoul and Teheux (2013).

• They need an infinitary inference rule.
• what about the finitary logic? Local one is decidable, but what

about a finitely generated axiomatization?

• Considering models over [0, 1]Π (and many other RLs), the

fragments differ again.

• Some of these have been axiomatized and studied Vidal et al. (2017)

but they need a dense set of constants and an infinitary inference rule.

• any better?

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Undecidability and unaxiomatizability

• Turns out some of the previous global logics are undecidable [V.

2018]. In particular, the crisp accessibility ones over [0, 1]

L and

[0, 1]Π.

• Even if we restrain the logic to that of the finite models,

undecidability still arises.

• The undecidability can be used to prove those logics are not R.E.

(and so, not axiomatizable in the usual sense).

• ps: also transitive modal logics are problematic, which gives idea on

the complexity of formalizing transitive-relation environments.

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References

Bou, F., Esteva, F., Godo, L., and Rodr´ ıguez, R. (2011). On the minimum many-valued modal logic over a finite residuated lattice. Journal of Logic and Computation, 21(5):739–790. Caicedo, X., Metcalfe, G., Rodr´ ıguez, R., and Rogger, J. (2013). A finite model property for G¨

• del modal logics. In Libkin, L., Kohlenbach, U.,

and de Queiroz, R., editors, Logic, Language, Information, and Computation, volume 8071 of Lecture Notes in Computer Science. Springer Berlin Heidelberg. Caicedo, X. and Rodr´ ıguez, R. O. (2010). Standard G¨

• del modal logics.

Studia Logica, 94(2):189–214. Caicedo, X. and Rodriguez, R. O. (2015). Bi-modal G¨

• del logic over

[0, 1]-valued Kripke frames. Journal of Logic and Computation, 25(1):37–55.

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Fitting, M. (1992a). Many-valued modal logics. Fundamenta Informaticae, 15:235–254. Fitting, M. (1992b). Many-valued modal logics, II. Fundamenta Informaticae, 17:55–73. Hansoul, G. and Teheux, B. (2013). Extending lukasiewicz logics with a modality: Algebraic approach to relational semantics. Studia Logica, 101(3):505–545. Metcalfe, G. and Olivetti, N. (2011). Towards a proof theory of G¨

• del

modal logics. Logical Methods in Computer Science, 7(2):27. Vidal, A., Esteva, F., and Godo, L. (2017). On modal extensions of product fuzzy logic. Journal of Logic and Computation, 27(1):299–336.

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