Relation-Changing Logics as Fragments of Hybrid Logics Carlos Areces - - PowerPoint PPT Presentation

Relation-Changing Logics as Fragments of Hybrid Logics

Carlos Areces1, Raul Fervari1, Guillaume Hoffmann1, Mauricio Martel2

1 FaMAF, Universidad Nacional de Córdoba & CONICET, Argentina 2 Fachbereich Mathematik und Informatik, Universität Bremen, Germany

GandALF 2016 - Catania, Italy

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.
• Choose the right paintbrush:
• ♦ϕ, ♦−ϕ
• Eϕ
• ♦≥nϕ
• ♦∗ϕ
• . . .

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.
• Choose the right paintbrush:
• ♦ϕ, ♦−ϕ
• Eϕ
• ♦≥nϕ
• ♦∗ϕ
• . . .
• Now, what about operators that can modify models?

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.
• Choose the right paintbrush:
• ♦ϕ, ♦−ϕ
• Eϕ
• ♦≥nϕ
• ♦∗ϕ
• . . .
• Now, what about operators that can modify models?
• Change the domain of the model.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.
• Choose the right paintbrush:
• ♦ϕ, ♦−ϕ
• Eϕ
• ♦≥nϕ
• ♦∗ϕ
• . . .
• Now, what about operators that can modify models?
• Change the domain of the model.
• Change the properties of the elements of the domain while we

are evaluating a formula.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Modal logics from a semantic perspective

• Modal logics are known to describe models.
• Choose the right paintbrush:
• ♦ϕ, ♦−ϕ
• Eϕ
• ♦≥nϕ
• ♦∗ϕ
• . . .
• Now, what about operators that can modify models?
• Change the domain of the model.
• Change the properties of the elements of the domain while we

are evaluating a formula.

• Change the accessibility relation of a model while we are

evaluating a formula.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Logics that can change the model

What about a swapping modal operator? w sw♦⊤ v w v ♦⊤ What happens when you add that to the basic modal logic?

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Logics that can change the model

What about a swapping modal operator? w sw♦⊤ v w v ♦⊤ What about

• an edge-deleting modality?
• an edge-adding modality?

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Sabotage Modal Logic [van Benthem 05]

M, w | = gsbϕ iff ∃ pair (u, v) of M such that M−

{(u,v)}, w |

= ϕ, where M−

{(u,v)} is M without the edge (u, v).

Note: (u, v) can be anywhere in the model.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Sabotage Modal Logic [van Benthem 05]

M, w | = gsbϕ iff ∃ pair (u, v) of M such that M−

{(u,v)}, w |

= ϕ, where M−

{(u,v)} is M without the edge (u, v).

Note: (u, v) can be anywhere in the model. We are interested in operators that can modify the accessibility relation of a model.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Relation-Changing Logics

Remember the Basic Modal Logic (ML)

• Syntax: propositional language + a modal operator ♦.
• Semantics of ♦ϕ: traverse some edge, then evaluate ϕ.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Relation-Changing Logics

Remember the Basic Modal Logic (ML)

• Syntax: propositional language + a modal operator ♦.
• Semantics of ♦ϕ: traverse some edge, then evaluate ϕ.

Now add new dynamic operators (Sabotage, Bridge, and Swap):

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Relation-Changing Logics

Remember the Basic Modal Logic (ML)

• Syntax: propositional language + a modal operator ♦.
• Semantics of ♦ϕ: traverse some edge, then evaluate ϕ.

Now add new dynamic operators (Sabotage, Bridge, and Swap):

• sbϕ: traverse some edge, delete it, then evaluate ϕ.
• brϕ: add a new edge, traverse it, then evaluate ϕ.
• swϕ: traverse some edge, turn it around, then evaluate ϕ.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Relation-Changing Logics

Remember the Basic Modal Logic (ML)

• Syntax: propositional language + a modal operator ♦.
• Semantics of ♦ϕ: traverse some edge, then evaluate ϕ.

Now add new dynamic operators (Sabotage, Bridge, and Swap):

• sbϕ: traverse some edge, delete it, then evaluate ϕ.
• brϕ: add a new edge, traverse it, then evaluate ϕ.
• swϕ: traverse some edge, turn it around, then evaluate ϕ.
• gsbϕ: delete some edge anywhere, then evaluate ϕ.
• gbrϕ: add a new edge anywhere, then evaluate ϕ.
• gswϕ: swap an edge anywhere, then evaluate ϕ.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Relation-Changing Logics

Remember the Basic Modal Logic (ML)

• Syntax: propositional language + a modal operator ♦.
• Semantics of ♦ϕ: traverse some edge, then evaluate ϕ.

Now add new dynamic operators (Sabotage, Bridge, and Swap):

• sbϕ: traverse some edge, delete it, then evaluate ϕ.
• brϕ: add a new edge, traverse it, then evaluate ϕ.
• swϕ: traverse some edge, turn it around, then evaluate ϕ.
• gsbϕ: delete some edge anywhere, then evaluate ϕ.
• gbrϕ: add a new edge anywhere, then evaluate ϕ.
• gswϕ: swap an edge anywhere, then evaluate ϕ.

We call this family of logics Relation-Changing Logics.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

• Incomparable among them in expressive power (even between

local and global cases of same modification).

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

• Incomparable among them in expressive power (even between

local and global cases of same modification).

• Model checking is PSpace-complete (via QBF reduction).

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

• Incomparable among them in expressive power (even between

local and global cases of same modification).

• Model checking is PSpace-complete (via QBF reduction).
• The satisfiability problem is undecidable (via spy points and

memory logic reduction).

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

• Incomparable among them in expressive power (even between

local and global cases of same modification).

• Model checking is PSpace-complete (via QBF reduction).
• The satisfiability problem is undecidable (via spy points and

memory logic reduction).

• Sound and complete (but non-terminating) tableaux methods;

Standard translations into FOL.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Some results about RCL

• Lack of tree-model property and finite model property (more

expressivity than ML).

• Incomparable among them in expressive power (even between

local and global cases of same modification).

• Model checking is PSpace-complete (via QBF reduction).
• The satisfiability problem is undecidable (via spy points and

memory logic reduction).

• Sound and complete (but non-terminating) tableaux methods;

Standard translations into FOL. We now provide translations into hybrid logics.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Hybrid Logics

• The basic hybrid logic HL is obtained by adding a set NOM of

nominals to ML. For n ∈ NOM, its valuation is a singleton set V (n) = {w}, for some state w.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Hybrid Logics

• The basic hybrid logic HL is obtained by adding a set NOM of

nominals to ML. For n ∈ NOM, its valuation is a singleton set V (n) = {w}, for some state w.

• We have a satisfaction operator n : ϕ with the usual semantics:

M, w | = n : ϕ iff M, v | = ϕ, where V (n) = {v}.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Hybrid Logics

• The basic hybrid logic HL is obtained by adding a set NOM of

nominals to ML. For n ∈ NOM, its valuation is a singleton set V (n) = {w}, for some state w.

• We have a satisfaction operator n : ϕ with the usual semantics:

M, w | = n : ϕ iff M, v | = ϕ, where V (n) = {v}.

• And we also consider the down-arrow binder operator ↓:

W , R, V , w | = ↓n.ϕ iff W , R, V w

n , w |

= ϕ, where V w

n (n) = {w} and V w n (m) = V (m), when n = m.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Translations to Hybrid Logics

• The translations are parametrized over a set of pair of nominals

S ⊆ NOM × NOM that simulates the modification of edges.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Translations to Hybrid Logics

• The translations are parametrized over a set of pair of nominals

S ⊆ NOM × NOM that simulates the modification of edges. Sabotage to Hybrid Logic We define the translation ( )′

S from formulas of ML(sb) to

formulas of HL(:, ↓) as: (♦ϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ (ϕ)′

S)

(sbϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ ↓m.(ϕ)′

S∪nm)

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Translations to Hybrid Logics

• The translations are parametrized over a set of pair of nominals

S ⊆ NOM × NOM that simulates the modification of edges. Sabotage to Hybrid Logic We define the translation ( )′

S from formulas of ML(sb) to

formulas of HL(:, ↓) as: (♦ϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ (ϕ)′

S)

(sbϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ ↓m.(ϕ)′

S∪nm)

And for ML(gsb) we translate into HL(E, ↓): (gsbϕ)′

S =

↓k.E↓n.♦(¬(

xy∈S

¬(y ∧ n:x) ∧ ↓m.k:(ϕ)′

S∪nm)

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Translations to Hybrid Logics

• The translations are parametrized over a set of pair of nominals

S ⊆ NOM × NOM that simulates the modification of edges. Sabotage to Hybrid Logic We define the translation ( )′

S from formulas of ML(sb) to

formulas of HL(:, ↓) as: (♦ϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ (ϕ)′

S)

(sbϕ)′

S =

↓n.♦(

xy∈S

¬(y ∧ n:x) ∧ ↓m.(ϕ)′

S∪nm)

And for ML(gsb) we translate into HL(E, ↓): (gsbϕ)′

S =

↓k.E↓n.♦(¬(

xy∈S

¬(y ∧ n:x) ∧ ↓m.k:(ϕ)′

S∪nm)

The translations for Bridge and Swap follow similar ideas.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Comparing Expressive Power

Theorem

ML(1) < HL(:, ↓), for 1 ∈ {sb, sw}. ML(2) < HL(E, ↓), for 2 ∈ {gsb, gsw, br, gbr}.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Comparing Expressive Power

Theorem

ML(1) < HL(:, ↓), for 1 ∈ {sb, sw}. ML(2) < HL(E, ↓), for 2 ∈ {gsb, gsw, br, gbr}.

Proof.

• To prove that ML(1) < HL(:, ↓) it suffices to find two

1-bisimilar models distinguishable by HL(:, ↓)

• To prove that ML(2) < HL(E, ↓) it suffices to find two

2-bisimilar models distinguishable by HL(E, ↓)

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Comparing Expressive Power

M, w M′, w′ Bisimilar for w w′ ML(sw) ML(br) ML(gsw) ML(gbr) w w′ ML(sb) ML(gsb) The formula ↓n.n can distinguish the models in the first row. The formula ↓n.♦↓m.n:♦♦m can distinguish the models in the second row.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.

Since the translations preserve equivalence, we get:

1 The satisfiability problem for ML(sb) and ML(sw) over

models of bounded width is decidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.
• HL(E, ↓) over linear frames (i.e., irreflexive, transitive, and

trichotomous frames). Since the translations preserve equivalence, we get:

1 The satisfiability problem for ML(sb) and ML(sw) over

models of bounded width is decidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.
• HL(E, ↓) over linear frames (i.e., irreflexive, transitive, and

trichotomous frames).

• HL(E, ↓) over models with a single, transitive tree relation.

Since the translations preserve equivalence, we get:

1 The satisfiability problem for ML(sb) and ML(sw) over

models of bounded width is decidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.
• HL(E, ↓) over linear frames (i.e., irreflexive, transitive, and

trichotomous frames).

• HL(E, ↓) over models with a single, transitive tree relation.
• HL(E, ↓) over models with a single, S5 relation.

Since the translations preserve equivalence, we get:

1 The satisfiability problem for ML(sb) and ML(sw) over

models of bounded width is decidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Semantic Restrictions)

The hybrid logic fragments considered in the translations are known to be decidable over the indicated classes:

• HL(:, ↓) over models with a single relation of bounded width.
• HL(E, ↓) over linear frames (i.e., irreflexive, transitive, and

trichotomous frames).

• HL(E, ↓) over models with a single, transitive tree relation.
• HL(E, ↓) over models with a single, S5 relation.

Since the translations preserve equivalence, we get:

1 The satisfiability problem for ML(sb) and ML(sw) over

models of bounded width is decidable.

2 The satisfiability problem for all relation-changing logics over

linear, transitive trees, and S5 frames is decidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Syntactic Restrictions)

HL(:, ↓) \ ↓ is the fragment obtained by removing formulas that contain a nesting of , ↓, and again . This fragment is known to be decidable [B. ten Cate & M. Franceschet 05].

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Syntactic Restrictions)

HL(:, ↓) \ ↓ is the fragment obtained by removing formulas that contain a nesting of , ↓, and again . This fragment is known to be decidable [B. ten Cate & M. Franceschet 05]. Let ∈ {sb, sw} and ∈ {[sb], [sw]}. The following patterns are produced by the translations: RC Pattern Hybrid Pattern

• ↓↓
• ↓

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Decidable Fragments (Syntactic Restrictions)

HL(:, ↓) \ ↓ is the fragment obtained by removing formulas that contain a nesting of , ↓, and again . This fragment is known to be decidable [B. ten Cate & M. Franceschet 05]. Let ∈ {sb, sw} and ∈ {[sb], [sw]}. The following patterns are produced by the translations: RC Pattern Hybrid Pattern

• ↓↓
• ↓

The following fragments are decidable on the class of all models:

1 ML(sb) \ {, , , } 2 ML(sw) \ {, , , }

where is either or .

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Implementation in HTab

• We implemented the translations as a new feature of the

tableaux-based theorem prover HTab [G. Hoffmann & C. Areces 09].

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Implementation in HTab

• We implemented the translations as a new feature of the

tableaux-based theorem prover HTab [G. Hoffmann & C. Areces 09].

• HTab originally handles the hybrid logic HL(E, ↓).

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Implementation in HTab

• We implemented the translations as a new feature of the

tableaux-based theorem prover HTab [G. Hoffmann & C. Areces 09].

• HTab originally handles the hybrid logic HL(E, ↓).
• We added a flag --translate that interprets the input

formula as a relation-changing one. It first translates it to an HL(E, ↓)-formula and then runs its internal hybrid tableaux calculus on the translation.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Implementation in HTab

• We implemented the translations as a new feature of the

tableaux-based theorem prover HTab [G. Hoffmann & C. Areces 09].

• HTab originally handles the hybrid logic HL(E, ↓).
• We added a flag --translate that interprets the input

formula as a relation-changing one. It first translates it to an HL(E, ↓)-formula and then runs its internal hybrid tableaux calculus on the translation.

• This implementation is useful to check the correctness of the

translations and for checking that RC formulas build models in the expected way.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.
• We provided and implementation in HTab.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.
• We provided and implementation in HTab.
• Further work using hybrid logic techniques:

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.
• We provided and implementation in HTab.
• Further work using hybrid logic techniques:
• Find axiomatizations.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.
• We provided and implementation in HTab.
• Further work using hybrid logic techniques:
• Find axiomatizations.
• Compute interpolants.

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics

Conclusions

• Relation-changing logics are very expressive:
• Model checking is PSpace-complete.
• Satisfiability is undecidable.
• We defined translations into hybrid logics:
• They are useful to analyze expressive power.
• They allow us to identify some decidable fragments.
• We provided and implementation in HTab.
• Further work using hybrid logic techniques:
• Find axiomatizations.
• Compute interpolants.

Thanks!

Mauricio Martel Relation-Changing Logics as Fragments of Hybrid Logics