Order-Based and Continuous Modal Logics

George Metcalfe

Mathematical InstituteUniversity of Bern

Joint research with Xavier Caicedo, Denisa Diaconescu,Ricardo Rodrıguez, Jonas Rogger, and Laura Schnuriger

SYSMICS 2016, Barcelona, 5-9 September 2016

George Metcalfe (University of Bern) Order-Based and Continuous Modal Logics SYSMICS 2016 1 / 21

Modalities meet Degrees

“Eat before shopping. If you go to the store hungry,you are likely to make unnecessary purchases.”

American Heart Association Cookbook

Many-Valued Modal Logics

Many-valued modal logics with values in R fall loosely into two families:

Order-based modal logics (e.g., Godel modal logics)

Continuous modal logics (e.g., Lukasiewicz modal logics)

Key problems include finding axiomatizations and algebraic semantics,and establishing decidability and complexity results.

Order-Based Algebras

Let us say that an algebra A = 〈A,∧,∨, 0, 1, . . .〉 is order-based if

(a) 〈A,∧,∨, 0, 1〉 is a complete sublattice of 〈[0, 1],min,max, 0, 1〉.

(b) Each operation of A is definable by a quantifier-free first-orderformula in a language with operations ∧, ∨, and constants of A.

A Definable Operation

The Godel implication

a→ b =

{1 if a ≤ b

b otherwise

can always be defined by the quantifier-free first-order formula

F→(x , y , z) = ((x ≤ y)⇒ (z ≈ 1)) & ((y < x)⇒ (z ≈ y)).

That is, for all a, b, c ∈ A,

A |= F→(a, b, c) ⇔ a→ b = c .

Note also that we can also define ¬a := a→ 0.

a→ b =

{1 if a ≤ b

b otherwise

F→(x , y , z) = ((x ≤ y)⇒ (z ≈ 1)) & ((y < x)⇒ (z ≈ y)).

A |= F→(a, b, c) ⇔ a→ b = c .

a→ b =

{1 if a ≤ b

b otherwise

F→(x , y , z) = ((x ≤ y)⇒ (z ≈ 1)) & ((y < x)⇒ (z ≈ y)).

A |= F→(a, b, c) ⇔ a→ b = c .

a→ b =

{1 if a ≤ b

b otherwise

F→(x , y , z) = ((x ≤ y)⇒ (z ≈ 1)) & ((y < x)⇒ (z ≈ y)).

A |= F→(a, b, c) ⇔ a→ b = c .

Frames and Formulas

An A-frame F = 〈W ,R〉 consists of

a non-empty set of states W

an A-valued accessibility relation R : W ×W → A.

F is called crisp if also Rxy ∈ {0, 1} for all x , y ∈W .

We extend the language of A with unary (modal) connectives �,♦and define the set of formulas Fm inductively as usual.

Frames and Formulas

Models

An A-model M = 〈W ,R,V 〉 adds a map V : Fm×W → A satisfying

V (?(ϕ1, . . . , ϕn), x) = ?A(V (ϕ1, x), . . . ,V (ϕn, x))

for each operation symbol ? of A, and

V (�ϕ, x) =∧{Rxy → V (ϕ, y) : y ∈W }

V (♦ϕ, x) =∨{Rxy ∧ V (ϕ, y) : y ∈W }.

M is called crisp if 〈W ,R〉 is crisp, in which case,

V (�ϕ, x) =∧{V (ϕ, y) : Rxy}

V (♦ϕ, x) =∨{V (ϕ, y) : Rxy}.

Models

V (?(ϕ1, . . . , ϕn), x) = ?A(V (ϕ1, x), . . . ,V (ϕn, x))

V (�ϕ, x) =∧{Rxy → V (ϕ, y) : y ∈W }

V (♦ϕ, x) =∨{Rxy ∧ V (ϕ, y) : y ∈W }.

V (�ϕ, x) =∧{V (ϕ, y) : Rxy}

V (♦ϕ, x) =∨{V (ϕ, y) : Rxy}.

Models

V (?(ϕ1, . . . , ϕn), x) = ?A(V (ϕ1, x), . . . ,V (ϕn, x))

V (�ϕ, x) =∧{Rxy → V (ϕ, y) : y ∈W }

V (♦ϕ, x) =∨{Rxy ∧ V (ϕ, y) : y ∈W }.

V (�ϕ, x) =∧{V (ϕ, y) : Rxy}

V (♦ϕ, x) =∨{V (ϕ, y) : Rxy}.

Validity

A formula ϕ is called

valid in an A-model 〈W ,R,V 〉 if V (ϕ, x) = 1 for all x ∈W

K(A)-valid if it is valid in all A-models

K(A)C-valid if it is valid in all crisp A-models.

Validity

Standard Godel Modal Logics

Consider the standard algebra for Godel logic

G = 〈[0, 1],∧,∨,→, 0, 1〉.

An axiomatization for K(G) is obtained by adding the prelinearity axiomschema (ϕ→ ψ) ∨ (ψ → ϕ) to the intuitionistic modal logic IK.

X. Caicedo and R. Rodrıguez.Bi-modal Godel logic over [0,1]-valued Kripke frames.Journal of Logic and Computation, 25(1) (2015), 37–55.

However, no axiomatization has yet been found for K(G)C.

G = 〈[0, 1],∧,∨,→, 0, 1〉.

Other Godel Modal Logics

More generally, we may consider (expansions of) Godel modal logicsK(A) and K(A)C where A is any complete subalgebra of G; e.g.,

A = {0} ∪ { 1n+1 | n ∈ N} or A = {1− 1

n+1 | n ∈ N} ∪ {1}.

Indeed, there are countably infinitely many different infinite-valued Godelmodal logics (considered as sets of valid formulas).

A = {0} ∪ { 1n+1 | n ∈ N} or A = {1− 1

n+1 | n ∈ N} ∪ {1}.

A = {0} ∪ { 1n+1 | n ∈ N} or A = {1− 1

n+1 | n ∈ N} ∪ {1}.

Failure of the Finite Model Property

The following formula is valid in all finite K(G)-models

�¬¬p → ¬¬�p

but not in the infinite K(G)-model 〈N,N2,V 〉 where V (p, x) = 1x+1 .

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

�¬¬p → ¬¬�p

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

�¬¬p → ¬¬�p

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

�¬¬p → ¬¬�p

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

�¬¬p → ¬¬�p

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

�¬¬p → ¬¬�p

(V (�¬¬p → ¬¬�p, 0) = (

∧x∈N

V (¬¬p, x))→ (¬¬∧x∈N

V (p, x))

= (∧x∈N

1)→ (¬¬∧x∈N

1x+1 )

= 1→ 0 = 0.)

Towards Decidability

We prove decidability (indeed PSPACE-completeness) for order-basedmodal logics satisfying a certain topological property by providing newsemantics that admit the finite model property.

X. Caicedo, G. Metcalfe, R. Rodrıguez, and J. Rogger.Decidability of Order-Based Modal Logics.Journal of Computer System Sciences, to appear.

The idea is to restrict the values at each state that can be taken by boxand diamond formulas; �ϕ and ♦ϕ can then be “witnessed” at stateswhere the value of ϕ is “sufficiently close” to the value of �ϕ or ♦ϕ.

Towards Decidability

We prove decidability (indeed PSPACE-completeness) for order-basedmodal logics satisfying a certain topological property by providing newsemantics that admit the finite model property.

X. Caicedo, G. Metcalfe, R. Rodrıguez, and J. Rogger.Decidability of Order-Based Modal Logics.Journal of Computer System Sciences, to appear.

The idea is to restrict the values at each state that can be taken by boxand diamond formulas; �ϕ and ♦ϕ can then be “witnessed” at stateswhere the value of ϕ is “sufficiently close” to the value of �ϕ or ♦ϕ.

A New Semantics

We augment G-frames with a map T from states to finite subsets of [0, 1]containing 0 and 1, and G-models are defined as before except that

V (�ϕ, x) = max{r ∈ T (x) : r ≤∧

y∈W(Rxy → V (ϕ, y))}

V (♦ϕ, x) = min{r ∈ T (x) : r ≥∨

y∈Wmin(Rxy ,V (ϕ, y))}.

A New Semantics

We augment G-frames with a map T from states to finite subsets of [0, 1]containing 0 and 1, and G-models are defined as before except that

V (�ϕ, x) = max{r ∈ T (x) : r ≤∧

y∈W(Rxy → V (ϕ, y))}

V (♦ϕ, x) = min{r ∈ T (x) : r ≥∨

y∈Wmin(Rxy ,V (ϕ, y))}.

A Finite Counter Model

We find a finite counter-model for �¬¬p → ¬¬�p:

〈{a}, {(a, a)},T ,V 〉 where V (p, a) = 12 and T (a) = {0, 1}.(

V (�¬¬p, a) = max{r ∈ T (a) : r ≤ V (¬¬p, a)} = 1

V (¬¬�p, a) = ¬¬max{r ∈ T (a) : r ≤ V (p, a)} = 0

V (�¬¬p → ¬¬�p, a) = 1→ 0 = 0.)

V (�¬¬p, a) = max{r ∈ T (a) : r ≤ V (¬¬p, a)} = 1

V (¬¬�p, a) = ¬¬max{r ∈ T (a) : r ≤ V (p, a)} = 0

V (�¬¬p → ¬¬�p, a) = 1→ 0 = 0.)

V (�¬¬p, a) = max{r ∈ T (a) : r ≤ V (¬¬p, a)} = 1

V (¬¬�p, a) = ¬¬max{r ∈ T (a) : r ≤ V (p, a)} = 0

V (�¬¬p → ¬¬�p, a) = 1→ 0 = 0.)

V (�¬¬p, a) = max{r ∈ T (a) : r ≤ V (¬¬p, a)} = 1

V (¬¬�p, a) = ¬¬max{r ∈ T (a) : r ≤ V (p, a)} = 0

V (�¬¬p → ¬¬�p, a) = 1→ 0 = 0.)

V (�¬¬p, a) = max{r ∈ T (a) : r ≤ V (¬¬p, a)} = 1

V (¬¬�p, a) = ¬¬max{r ∈ T (a) : r ≤ V (p, a)} = 0

V (�¬¬p → ¬¬�p, a) = 1→ 0 = 0.)

More Generally. . .

We consider an order-based algebra A that is “locally homogeneous”;roughly, for any right (or left) accumulation point a of A, there is aninterval [a, c) (or (c , a]) that can be squeezed without changing the order.

We augment an A-frame 〈W ,R〉 with maps

T� : W → P(A) and T♦ : W → P(A)

such that for each x ∈W ,

the constants of A are contained in both T�(x) and T♦(x)

T�(x) = A \⋃

i∈I (ai , ci ) for some finite I , where each ci ∈ Awitnesses homogeneity at a right accumulation point ai of A

T♦(x) = A \⋃

j∈J(dj , bj) for some finite J, where each dj ∈ Awitnesses homogeneity at a left accumulation point bj of A.

More Generally. . .

T� : W → P(A) and T♦ : W → P(A)

T�(x) = A \⋃

T♦(x) = A \⋃

More Generally. . .

T� : W → P(A) and T♦ : W → P(A)

T�(x) = A \⋃

T♦(x) = A \⋃

More Generally. . .

T� : W → P(A) and T♦ : W → P(A)

T�(x) = A \⋃

T♦(x) = A \⋃

More Generally. . .

T� : W → P(A) and T♦ : W → P(A)

T�(x) = A \⋃

T♦(x) = A \⋃

More Generally. . .

T� : W → P(A) and T♦ : W → P(A)

T�(x) = A \⋃

T♦(x) = A \⋃

Decidability and Complexity

For any locally homogeneous order-based algebra A:

K(A) and K(A)C are sound and complete with respect to the newsemantics.

The new semantics has the finite model property.

If there is an oracle for checking consistency with finite models, thenvalidity in K(A) and K(A)C are both decidable.

In particular, validity in K(G) and K(G)C are both PSPACE-complete.

Beyond the Basic Modal Logics

We have also obtained decidability (indeed, co-NP completeness) fororder-based modal logics S5(A)C based on crisp K(A)-models whereR is an equivalence relation.

This provides co-NP completeness also for one-variable fragments offirst-order order-based logics (in particular, first-order Godel logic).

Extending these results to a general theory seems to be difficult. . .

Continuous Modal Logics

Lukasiewicz modal logics are defined with connectives on [0, 1]

x → y = min(1, 1− x + y) ¬x = 1− x

x ⊕ y = min(1, x + y) x � y = max(0, x + y − 1).

Lukasiewicz (multi-)modal logics can also be viewed as fragments ofcontinuous logic and studied as fuzzy description logics.

Using the fact that Lukasiewicz modal logics enjoy the finite modelproperty, it can be shown that validity in these logics is in 2EXPTIME .

x → y = min(1, 1− x + y) ¬x = 1− x

x ⊕ y = min(1, x + y) x � y = max(0, x + y − 1).

x → y = min(1, 1− x + y) ¬x = 1− x

x ⊕ y = min(1, x + y) x � y = max(0, x + y − 1).

An Axiomatization Problem

Hansoul and Teheux (2013) axiomatize Lukasiewicz modal logic over crispKripke frames by adding to an axiomatization of Lukasiewicz logic

�(ϕ→ ψ)→ (�ϕ→ �ψ)

�(ϕ⊕ ϕ)→ (�ϕ⊕�ϕ)

�(ϕ� ϕ)→ (�ϕ��ϕ)

and a rule with infinitely many premises

ϕ⊕ ϕ ϕ⊕ ϕ2 ϕ⊕ ϕ3 . . .ϕ

But is this infinitary rule really necessary?

�(ϕ→ ψ)→ (�ϕ→ �ψ)

�(ϕ⊕ ϕ)→ (�ϕ⊕�ϕ)

�(ϕ� ϕ)→ (�ϕ��ϕ)

ϕ⊕ ϕ ϕ⊕ ϕ2 ϕ⊕ ϕ3 . . .ϕ

�(ϕ→ ψ)→ (�ϕ→ �ψ)

�(ϕ⊕ ϕ)→ (�ϕ⊕�ϕ)

�(ϕ� ϕ)→ (�ϕ��ϕ)

ϕ⊕ ϕ ϕ⊕ ϕ2 ϕ⊕ ϕ3 . . .ϕ

Towards a Solution. . .

We have axiomatized a modal logic over R with abelian group operations(extending the multiplicative fragment of Abelian logic), whose validityproblem is in EXPTIME.

D. Diaconescu, G. Metcalfe, and L. Schnuriger.Axiomatizing a Real-Valued Modal Logic.Proceedings of AiML 2016, King’s College Publications (2016), 236–251.

Extending this system with the additive (lattice) connectives would providethe basis for a finitary axiomatization for Lukasiewicz modal logic.

Towards a Solution. . .

We have axiomatized a modal logic over R with abelian group operations(extending the multiplicative fragment of Abelian logic), whose validityproblem is in EXPTIME.

D. Diaconescu, G. Metcalfe, and L. Schnuriger.Axiomatizing a Real-Valued Modal Logic.Proceedings of AiML 2016, King’s College Publications (2016), 236–251.

Extending this system with the additive (lattice) connectives would providethe basis for a finitary axiomatization for Lukasiewicz modal logic.

Challenges

Find an axiomatization of crisp Godel modal logic.

Develop a robust algebraic theory for order-based modal logics.

Prove decidability for guarded fragments of order-based modal logics.

Prove (un)decidability of two-variable fragments of first-orderorder-based modal logics.

Find an axiomatization of crisp Lukasiewicz modal logic andinvestigate its algebraic semantics.

Establish the complexity of validity in Lukasiewicz modal logics.

Challenges

Order-Based and Continuous Modal Logics -...

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