[PPT] - Conservative Extensions in Guarded and Two-Variable Fragments PowerPoint Presentation

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Conservative Extensions in Guarded and Two-Variable Fragments

Mauricio Martel

Universität Bremen, Germany

Joint work with Jean Christoph Jung, Carsten Lutz, Thomas Schneider, and Frank Wolter

Logic Tea ILLC, Amsterdam

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments

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Conservative Extensions

Fundamental notion in mathematical logic to relate different theories

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

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Conservative Extensions

Fundamental notion in mathematical logic to relate different theories Also used in computer science:

for formalizing modularity in software specification
for composing subgoals in higher-order theorem proving
for formalizing various notions in ontology engeneering

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

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Conservative Extensions

Fundamental notion in mathematical logic to relate different theories Also used in computer science:

for formalizing modularity in software specification
for composing subgoals in higher-order theorem proving
for formalizing various notions in ontology engeneering

Remarkably positive results for modal logics and description logics:

turn out to be decidable in many relevant cases
often possible to provide natural and insightful model-theoretic

characterizations

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

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Conservative Extensions

Fundamental notion in mathematical logic to relate different theories Also used in computer science:

for formalizing modularity in software specification
for composing subgoals in higher-order theorem proving
for formalizing various notions in ontology engeneering

Remarkably positive results for modal logics and description logics:

turn out to be decidable in many relevant cases
often possible to provide natural and insightful model-theoretic

characterizations Natural question: how far do these extend? To GF? To FO2?

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

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Overview

It is known that conservative extensions are

undecidable in FO
decidable in many modal and description logics

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Overview

It is known that conservative extensions are

undecidable in FO
decidable in many modal and description logics

We show that conservative extensions are

undecidable in GF
undecidable in FO2
decidable and 2ExpTime-complete in GF2 = GF ∩ FO2

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Decidable Fragments of FO

FO2 is the two-variable fragment of FO Example: ∀xyRxy

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Decidable Fragments of FO

FO2 is the two-variable fragment of FO Example: ∀xyRxy In GF quantification is restricted to the pattern ∀ y(α( x, y) → ϕ( x, y)) ∃ y(α( x, y) ∧ ϕ( x, y)) Example: ∃xy(Rxyz ∧ Pxy ∧ Qyz)

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Decidable Fragments of FO

FO2 is the two-variable fragment of FO Example: ∀xyRxy In GF quantification is restricted to the pattern ∀ y(α( x, y) → ϕ( x, y)) ∃ y(α( x, y) ∧ ϕ( x, y)) Example: ∃xy(Rxyz ∧ Pxy ∧ Qyz) GF2 = GF ∩ FO2 Example: ∀x(Rxy → Pxy)

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Conservative Extensions

A theory T2 is said to be a (deductive) conservative extension of a theory T1 if the language of T2 extends the language of T1:

Any consequence of T1 is also a consequence of T2; and
Any consequence of T2, which uses only symbols from T1, is

also a consequence of T1.

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Conservative Extensions

A theory T2 is said to be a (deductive) conservative extension of a theory T1 if the language of T2 extends the language of T1:

Any consequence of T1 is also a consequence of T2; and
Any consequence of T2, which uses only symbols from T1, is

also a consequence of T1.

Observation. Conservative extensions can be used to define the

notion of a module for ontologies: a subtheory is a module if the whole ontology is a conservative extension of the subtheory

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Conservative Extensions

A theory T2 is said to be a (deductive) conservative extension of a theory T1 if the language of T2 extends the language of T1:

Any consequence of T1 is also a consequence of T2; and
Any consequence of T2, which uses only symbols from T1, is

also a consequence of T1.

Observation. Conservative extensions can be used to define the

notion of a module for ontologies: a subtheory is a module if the whole ontology is a conservative extension of the subtheory Similar for other applications such as ontology versioning and forgetting a set of symbols Σ

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Conservative Extensions in Fragments of FO

Sentence ϕ1 extended to ϕ1 ∧ ϕ2, and ϕ2 may use fresh symbols

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

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Conservative Extensions in Fragments of FO

Sentence ϕ1 extended to ϕ1 ∧ ϕ2, and ϕ2 may use fresh symbols

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

Equivalently, ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if,

ϕ1 ∧ ϕ2 ∧ ¬ψ is UNSAT ⇒ ϕ1 ∧ ¬ψ is UNSAT

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Conservative Extensions in Fragments of FO

Sentence ϕ1 extended to ϕ1 ∧ ϕ2, and ϕ2 may use fresh symbols

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

Equivalently, ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if,

ϕ1 ∧ ϕ2 ∧ ¬ψ is UNSAT ⇒ ϕ1 ∧ ¬ψ is UNSAT ϕ1 ∧ ψ is SAT ⇒ ϕ1 ∧ ϕ2 ∧ ψ is SAT

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Conservative Extensions in Fragments of FO

Sentence ϕ1 extended to ϕ1 ∧ ϕ2, and ϕ2 may use fresh symbols

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

Equivalently, ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if,

ϕ1 ∧ ϕ2 ∧ ¬ψ is UNSAT ⇒ ϕ1 ∧ ¬ψ is UNSAT ϕ1 ∧ ψ is SAT ⇒ ϕ1 ∧ ϕ2 ∧ ψ is SAT

Deciding conservative extensions mean, given two sentences

ϕ1, ϕ2, to decide whether ϕ1 ∧ ϕ2 is a conservative extension

f ϕ1

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Σ-Entailment

In applications, it is useful to consider a vocabulary Σ of interest

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Σ-Entailment

In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form:

ϕ1 |

=Σ ϕ2 (“Σ-entails”), if ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from Σ

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Σ-Entailment

In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form:

ϕ1 |

=Σ ϕ2 (“Σ-entails”), if ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from Σ

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

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Σ-Entailment

In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form:

ϕ1 |

=Σ ϕ2 (“Σ-entails”), if ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from Σ

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1

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Σ-Entailment

In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form:

ϕ1 |

=Σ ϕ2 (“Σ-entails”), if ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from Σ

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 if

ϕ1 ∧ ϕ2 | = ψ ⇒ ϕ1 | = ψ for all sentences ψ that use symbols from ϕ1 Then

ϕ1 ∧ ϕ2 is a conservative extension of ϕ1 iff

ϕ1 | =Σ=sig(ϕ1) ϕ1 ∧ ϕ2

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Examples

Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R}

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Examples

Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∀x∃y(Rxy → x = y)

R

R . . .

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Examples

Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∀x∃y(Rxy → x = y)

R

R . . . ϕ2 = ∀x((∃yRxy ∧ Ay) ∧ (∃yRxy ∧ ¬Ay))

R

R . . . . . .

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Examples

Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∀x∃y(Rxy → x = y)

R

R . . . ϕ2 = ∀x((∃yRxy ∧ Ay) ∧ (∃yRxy ∧ ¬Ay))

R

R . . . . . . ϕ1 | =Σ ϕ2 since GF2 cannot count the number of successors

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Examples

Consider sentences ϕ′

1, ϕ2 in GF2 and Σ = {R}

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Examples

Consider sentences ϕ′

1, ϕ2 in GF2 and Σ = {R}

ϕ′

1 = ∀x∃yRxy

R

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Examples

Consider sentences ϕ′

1, ϕ2 in GF2 and Σ = {R}

ϕ′

1 = ∀x∃yRxy

R

|

R

R . . .

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Examples

Consider sentences ϕ′

1, ϕ2 in GF2 and Σ = {R}

ϕ′

1 = ∀x∃yRxy

R

|

R

R . . . ϕ2 = ∀x((∃yRxy ∧ Ay) ∧ (∃yRxy ∧ ¬Ay))

R

R . . . . . .

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Examples

Consider sentences ϕ′

1, ϕ2 in GF2 and Σ = {R}

ϕ′

1 = ∀x∃yRxy

R

|

R

R . . . ϕ2 = ∀x((∃yRxy ∧ Ay) ∧ (∃yRxy ∧ ¬Ay))

R

R . . . . . . ϕ′

1 |

=Σ ϕ2 because there is a sentence ψ = ∀xy(Rxy → x = y) s.t. ϕ′

1 ∧ ψ is SAT but ϕ2 ∧ ψ is UNSAT

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Modal and Description Logics

Both modal and description logics are fragments of FO

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Modal and Description Logics

Both modal and description logics are fragments of FO The description logic ALC (or the modal logic K) is defined as ∀xϕ(x) with ϕ(x) built from ¬, ∧, ∃y(Rxy ∧ ϕ(y))

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Modal and Description Logics

Both modal and description logics are fragments of FO The description logic ALC (or the modal logic K) is defined as ∀xϕ(x) with ϕ(x) built from ¬, ∧, ∃y(Rxy ∧ ϕ(y)) Conservative extensions can be characterized model-theoretically using Σ-bisimulations: (A, a) ∼Σ (B, b)

a• a′•

b
b′

∼ ∼

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Modal and Description Logics

Consider the description logic ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation

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Modal and Description Logics

Consider the description logic ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Similar characterizations for many other modal/description logics

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Modal and Description Logics

Consider the description logic ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Similar characterizations for many other modal/description logics This model-theoretic characterization enables decision procedures based on tree automata:

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Modal and Description Logics

Consider the description logic ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Similar characterizations for many other modal/description logics This model-theoretic characterization enables decision procedures based on tree automata: Given two sentences ϕ1 and ϕ2 and a signature Σ we can construct a tree automaton A such that ϕ1 | =Σ ϕ2 iff L(A) = ∅

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Modal and Description Logics

Consider the description logic ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Similar characterizations for many other modal/description logics This model-theoretic characterization enables decision procedures based on tree automata: Given two sentences ϕ1 and ϕ2 and a signature Σ we can construct a tree automaton A such that ϕ1 | =Σ ϕ2 iff L(A) = ∅ The problem turns out to be 2ExpTime-complete

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Conservative Extensions vs. Satisfiability

Regarding decidability, conservative extensions seem to behave similarly to the satisfiability problem:

given a sentence ϕ, is ϕ satisfiable in some model A?

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Conservative Extensions vs. Satisfiability

Regarding decidability, conservative extensions seem to behave similarly to the satisfiability problem:

given a sentence ϕ, is ϕ satisfiable in some model A?

Satisfiability is undecidable in FO and decidable in

modal and description logics (PSpace-complete)
guarded fragment (2ExpTime-complete)
two-variable fragment (NExpTime-complete)

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Conservative Extensions vs. Satisfiability

Regarding decidability, conservative extensions seem to behave similarly to the satisfiability problem:

given a sentence ϕ, is ϕ satisfiable in some model A?

Satisfiability is undecidable in FO and decidable in

modal and description logics (PSpace-complete)
guarded fragment (2ExpTime-complete)
two-variable fragment (NExpTime-complete)

Conservative extensions are undecidable in FO and decidable in

modal and description logics (2ExpTime-complete)
guarded fragment and two-variable fragment?

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Conservative Extensions vs. Satisfiability

Regarding decidability, conservative extensions seem to behave similarly to the satisfiability problem:

given a sentence ϕ, is ϕ satisfiable in some model A?

Satisfiability is undecidable in FO and decidable in

modal and description logics (PSpace-complete)
guarded fragment (2ExpTime-complete)
two-variable fragment (NExpTime-complete)

Conservative extensions are undecidable in FO and decidable in

modal and description logics (2ExpTime-complete)
guarded fragment and two-variable fragment?

Why are conservative extensions in modal and description logics decidable and how far does the decidability extend?

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Guarded and Two-Variable Fragments

Theorem[JungLutzMartelSchneiderWolterICALP17]

Conservative extensions are undecidable in ever logic that

contains GF3 or FO2 (such as the guarded negation fragment)

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Guarded and Two-Variable Fragments

Theorem[JungLutzMartelSchneiderWolterICALP17]

Conservative extensions are undecidable in ever logic that

contains GF3 or FO2 (such as the guarded negation fragment)

Conservative extensions and Σ-entailment are decidable and

2ExpTime-complete in GF2 = GF ∩ FO2

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Guarded and Two-Variable Fragments

Theorem[JungLutzMartelSchneiderWolterICALP17]

Conservative extensions are undecidable in ever logic that

contains GF3 or FO2 (such as the guarded negation fragment)

Conservative extensions and Σ-entailment are decidable and

2ExpTime-complete in GF2 = GF ∩ FO2 The GF2 result is based on a model-theoretic characterization, but it is much more complex than for modal and description logics

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Undecidability in GF

Reduction from the halting problem of two-register machines M

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Undecidability in GF

Reduction from the halting problem of two-register machines M Construct ϕ1, ϕ2 such that

1 If M halts, then ϕ1 ∧ ϕ2 is not a GF2-conservative extensions

f ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then M halts

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Undecidability in GF

Reduction from the halting problem of two-register machines M Construct ϕ1, ϕ2 such that

1 If M halts, then ϕ1 ∧ ϕ2 is not a GF2-conservative extensions

f ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then M halts Important: ϕ2 uses ternary guard that is not in Σ

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Undecidability in GF

Reduction from the halting problem of two-register machines M Construct ϕ1, ϕ2 such that

1 If M halts, then ϕ1 ∧ ϕ2 is not a GF2-conservative extensions

f ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then M halts Important: ϕ2 uses ternary guard that is not in Σ Theorem[JungLutzMartelSchneiderWolterICALP17] In any fragment of FO that extends GF3, conservative extensions and Σ-entailment are undecidable

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Undecidability in FO2

Reduction from an NxN tiling problem D

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Undecidability in FO2

Reduction from an NxN tiling problem D Construct ϕ1, ϕ2 such that

1 D has a solution iff ϕ1 ∧ ϕ2 is not an FO2-conservative

extension of ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then D has a solution

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Undecidability in FO2

Reduction from an NxN tiling problem D Construct ϕ1, ϕ2 such that

1 D has a solution iff ϕ1 ∧ ϕ2 is not an FO2-conservative

extension of ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then D has a solution Important: ability to use FO2 expressive power in consequences ψ

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Undecidability in FO2

Reduction from an NxN tiling problem D Construct ϕ1, ϕ2 such that

1 D has a solution iff ϕ1 ∧ ϕ2 is not an FO2-conservative

extension of ϕ1

2 If there exists a Σ-structure that satisfies ϕ1 which cannot be

extended to a model of ϕ2, then D has a solution Important: ability to use FO2 expressive power in consequences ψ Theorem[JungLutzMartelSchneiderWolterICALP17] In any fragment of FO that extends FO2, conservative extensions and Σ-entailment are undecidable

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Guarded Fragment with Two-Variables GF2

We show that conservative extensions are decidable in GF2

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Guarded Fragment with Two-Variables GF2

We show that conservative extensions are decidable in GF2 Recall that in ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation

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Guarded Fragment with Two-Variables GF2

We show that conservative extensions are decidable in GF2 Recall that in ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation In GF2 we have Σ-GF2 bisimulations: (A, a) ∼Σ (B, b)

a• a′•

b
b′

τ Σ

A(a)

τ Σ

A(a, a′)

τ Σ

B(b)

τ Σ

B(b, b′)

∼

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Guarded Fragment with Two-Variables GF2

We show that conservative extensions are decidable in GF2 Recall that in ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation In GF2 we have Σ-GF2 bisimulations: (A, a) ∼Σ (B, b)

a• a′•

b
b′

τ Σ

A(a)

τ Σ

A(a, a′)

τ Σ

B(b)

τ Σ

B(b, b′)

∼

τ Σ

A(a, a′) = τ Σ B(b, b′)

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Guarded Fragment with Two-Variables GF2

We show that conservative extensions are decidable in GF2 Recall that in ALC: Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation In GF2 we have Σ-GF2 bisimulations: (A, a) ∼Σ (B, b)

a• a′•

b
b′

τ Σ

A(a)

τ Σ

A(a, a′)

τ Σ

B(b)

τ Σ

B(b, b′)

∼ ∼

τ Σ

A(a, a′) = τ Σ B(b, b′)

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Can’t we replace Σ-bisimulations with Σ-GF2-bisimulations?

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Can’t we replace Σ-bisimulations with Σ-GF2-bisimulations? No: Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R}

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Can’t we replace Σ-bisimulations with Σ-GF2-bisimulations? No: Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∃xAx ∧ ∀x(Ax → ∃y(Rxy ∧ Ay))

R

R . . .

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Can’t we replace Σ-bisimulations with Σ-GF2-bisimulations? No: Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∃xAx ∧ ∀x(Ax → ∃y(Rxy ∧ Ay))

R

R . . . ϕ2 = ϕ1 ∧ ∃x(Ax ∧ Bx) ∧ ∀x(Bx → ∃y(Ryx ∧ By))

R

R . . . . . .

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Model-Theoretic Characterization

Theorem[LutzWolterIJCAI11] ϕ1 | =Σ ϕ2 iff every model of ϕ1 can be extended to a model of ϕ2, up to Σ-bisimulation Can’t we replace Σ-bisimulations with Σ-GF2-bisimulations? No: Consider sentences ϕ1, ϕ2 in GF2 and Σ = {R} ϕ1 = ∃xAx ∧ ∀x(Ax → ∃y(Rxy ∧ Ay))

R

R . . . ϕ2 = ϕ1 ∧ ∃x(Ax ∧ Bx) ∧ ∀x(Bx → ∃y(Ryx ∧ By))

R

R . . . . . . Then ϕ1 | =Σ ϕ2, but Σ-GF2-bisimulations fail

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 16

SLIDE 66

Bounded Bisimulations

k-bounded bisimulations: (A, a) ∼k

Σ (B, b) iff (A, a) and (B, b) are Σ-GF2-bisimilar up to

depth k

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 17

SLIDE 67

Bounded Bisimulations

k-bounded bisimulations: (A, a) ∼k

Σ (B, b) iff (A, a) and (B, b) are Σ-GF2-bisimilar up to

depth k Theorem[JungLutzMartelSchneiderWolterICALP17] ϕ1 | =Σ ϕ2 iff for every model A of ϕ1 and every k ≥ 0, there is a model B of ϕ2 such that

1 for every a ∈ A, there is a b ∈ B such that (A, a) ∼Σ (B, b) 2 for every b ∈ B, there is an a ∈ A such that (A, a) ∼k Σ (B, b)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 17

SLIDE 68

Bounded Bisimulations

k-bounded bisimulations: (A, a) ∼k

Σ (B, b) iff (A, a) and (B, b) are Σ-GF2-bisimilar up to

depth k Theorem[JungLutzMartelSchneiderWolterICALP17] ϕ1 | =Σ ϕ2 iff for every model A of ϕ1 and every k ≥ 0, there is a model B of ϕ2 such that

1 for every a ∈ A, there is a b ∈ B such that (A, a) ∼Σ (B, b) 2 for every b ∈ B, there is an a ∈ A such that (A, a) ∼k Σ (B, b)

A

R

R . . .

R

R . . . . . . B

R

R . . .

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 17

SLIDE 69

Bounded Bisimulations

k-bounded bisimulations: (A, a) ∼k

Σ (B, b) iff (A, a) and (B, b) are Σ-GF2-bisimilar up to

depth k Theorem[JungLutzMartelSchneiderWolterICALP17] ϕ1 | =Σ ϕ2 iff for every model A of ϕ1 and every k ≥ 0, there is a model B of ϕ2 such that

1 for every a ∈ A, there is a b ∈ B such that (A, a) ∼Σ (B, b) 2 for every b ∈ B, there is an a ∈ A such that (A, a) ∼k Σ (B, b)

A

R

R . . .

R

R . . . . . . B

R

R . . . But it is not so easy to deal with bounded bisimulations when using tree automata or related techniques

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 17

SLIDE 70

"Marker-Delimited" Bisimulations

Substitute for k-bounded bisimulation between A and B:

decorate A with unary predicate X such that
on every infinite path there are infinitely many X
the distance between two X is at least k
break off bisimulations at second X seen (both back and forth)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 18

SLIDE 71

"Marker-Delimited" Bisimulations

Substitute for k-bounded bisimulation between A and B:

decorate A with unary predicate X such that
on every infinite path there are infinitely many X
the distance between two X is at least k
break off bisimulations at second X seen (both back and forth)

Problem: decoration does not exist when k ≥ 3, not even in forest models

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 18

SLIDE 72

"Marker-Delimited" Bisimulations

Substitute for k-bounded bisimulation between A and B:

decorate A with unary predicate X such that
on every infinite path there are infinitely many X
the distance between two X is at least k
break off bisimulations at second X seen (both back and forth)

Problem: decoration does not exist when k ≥ 3, not even in forest models Solution: we need bounded bisimulations only when traveling upwards, but not when traveling downwards

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 18

SLIDE 73

"Marker-Delimited" Bisimulations

Substitute for k-bounded bisimulation between A and B:

decorate A with unary predicate X such that
on every infinite path there are infinitely many X
the distance between two X is at least k
break off bisimulations at second X seen (both back and forth)

Problem: decoration does not exist when k ≥ 3, not even in forest models Solution: we need bounded bisimulations only when traveling upwards, but not when traveling downwards Then the distance between markers only matters on upwards paths

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 18

SLIDE 74

Automata-Friendly Characterization

Theorem[JungLutzMartelSchneiderWolterICALP17] ϕ1 | =Σ ϕ2 iff for every model A of ϕ1 and every marking X ⊆ A, there is a model B of ϕ2 such that

1 for every a ∈ A, there is a b ∈ B such that (A, a) ∼Σ (B, b) 2 for every b ∈ B, there is an a ∈ A such that (A, a) ∼k Σ (B, b)

where a ∼k

Σ b satifies bisimulation conditions when “going down” in

A, but breaks off after seeing second X when “going up” in A

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 19

SLIDE 75

Automata Constructions

We use two-way alternating parity automata over trees

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 76

Automata Constructions

We use two-way alternating parity automata over trees
Input: sentences ϕ1, ϕ2 such that

ϕ1 | =Σ ϕ2 iff L(A1) ∩ L(A2) ∩ L(A3) = ∅

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 77

Automata Constructions

We use two-way alternating parity automata over trees
Input: sentences ϕ1, ϕ2 such that

ϕ1 | =Σ ϕ2 iff L(A1) ∩ L(A2) ∩ L(A3) = ∅

A1: accepts all models of ϕ1 of the right shape
A2: accepts all proper markings X ⊆ A
A3: accepts (A, X) ∈ L(A1) ∩ L(A2) iff there is a B satisfying

conditions 1. and 2. of the characterization

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 78

Automata Constructions

We use two-way alternating parity automata over trees
Input: sentences ϕ1, ϕ2 such that

ϕ1 | =Σ ϕ2 iff L(A1) ∩ L(A2) ∩ L(A3) = ∅

A1: accepts all models of ϕ1 of the right shape
A2: accepts all proper markings X ⊆ A
A3: accepts (A, X) ∈ L(A1) ∩ L(A2) iff there is a B satisfying

conditions 1. and 2. of the characterization

Theorem. Deciding Σ-entailment in GF2 is in 2ExpTime

(Exponential in |ϕ1| and double exponential in |ϕ2|)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 79

Automata Constructions

We use two-way alternating parity automata over trees
Input: sentences ϕ1, ϕ2 such that

ϕ1 | =Σ ϕ2 iff L(A1) ∩ L(A2) ∩ L(A3) = ∅

A1: accepts all models of ϕ1 of the right shape
A2: accepts all proper markings X ⊆ A
A3: accepts (A, X) ∈ L(A1) ∩ L(A2) iff there is a B satisfying

conditions 1. and 2. of the characterization

Theorem. Deciding Σ-entailment in GF2 is in 2ExpTime

(Exponential in |ϕ1| and double exponential in |ϕ2|)

Theorem. Deciding conservative extensions is 2ExpTime-hard

(Reduction from exponentially space-bounded, alternating TMs)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 80

Automata Constructions

We use two-way alternating parity automata over trees
Input: sentences ϕ1, ϕ2 such that

ϕ1 | =Σ ϕ2 iff L(A1) ∩ L(A2) ∩ L(A3) = ∅

A1: accepts all models of ϕ1 of the right shape
A2: accepts all proper markings X ⊆ A
A3: accepts (A, X) ∈ L(A1) ∩ L(A2) iff there is a B satisfying

conditions 1. and 2. of the characterization

Theorem. Deciding Σ-entailment in GF2 is in 2ExpTime

(Exponential in |ϕ1| and double exponential in |ϕ2|)

Theorem. Deciding conservative extensions is 2ExpTime-hard

(Reduction from exponentially space-bounded, alternating TMs)

Corollary. Deciding conservative extensions is 2ExpTime-complete

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 20

SLIDE 81

Uniform Interpolation

A logic has Craig interpolation if for all sentences ψ1, ψ2 with

ψ1 | = ψ2 there exists an sentence ψ such that ψ1 | = ψ | = ψ2 and sig(ψ) ⊆ sig(ψ1) ∩ sig(ψ2)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 21

SLIDE 82

Uniform Interpolation

A logic has Craig interpolation if for all sentences ψ1, ψ2 with

ψ1 | = ψ2 there exists an sentence ψ such that ψ1 | = ψ | = ψ2 and sig(ψ) ⊆ sig(ψ1) ∩ sig(ψ2)

A logic has uniform interpolation if the interpolant does not

depend on ψ2, but only on ψ1 and sig(ψ1) ∩ sig(ψ2)

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 21

SLIDE 83

Uniform Interpolation

A logic has Craig interpolation if for all sentences ψ1, ψ2 with

ψ1 | = ψ2 there exists an sentence ψ such that ψ1 | = ψ | = ψ2 and sig(ψ) ⊆ sig(ψ1) ∩ sig(ψ2)

A logic has uniform interpolation if the interpolant does not

depend on ψ2, but only on ψ1 and sig(ψ1) ∩ sig(ψ2)

Observation. Let L be a logic that has Craig interpolation. Then ψ

with ϕ1 | = ψ is a uniform Σ-interpolant of ϕ1 iff ψ | =Σ ϕ1

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 21

SLIDE 84

Uniform Interpolation

A logic has Craig interpolation if for all sentences ψ1, ψ2 with

ψ1 | = ψ2 there exists an sentence ψ such that ψ1 | = ψ | = ψ2 and sig(ψ) ⊆ sig(ψ1) ∩ sig(ψ2)

A logic has uniform interpolation if the interpolant does not

depend on ψ2, but only on ψ1 and sig(ψ1) ∩ sig(ψ2)

Observation. Let L be a logic that has Craig interpolation. Then ψ

with ϕ1 | = ψ is a uniform Σ-interpolant of ϕ1 iff ψ | =Σ ϕ1 GF2 has Craig interpolation, thus

Corollary. The uniform interpolation recognition problem is

2ExpTime-complete in GF2

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 21

SLIDE 85

Uniform Interpolation

A logic has Craig interpolation if for all sentences ψ1, ψ2 with

ψ1 | = ψ2 there exists an sentence ψ such that ψ1 | = ψ | = ψ2 and sig(ψ) ⊆ sig(ψ1) ∩ sig(ψ2)

A logic has uniform interpolation if the interpolant does not

depend on ψ2, but only on ψ1 and sig(ψ1) ∩ sig(ψ2)

Observation. Let L be a logic that has Craig interpolation. Then ψ

with ϕ1 | = ψ is a uniform Σ-interpolant of ϕ1 iff ψ | =Σ ϕ1 GF2 has Craig interpolation, thus

Corollary. The uniform interpolation recognition problem is

2ExpTime-complete in GF2 Another consequence regarding uniform interpolation:

Corollary. There is no decidable extension of FO2 and of GF3 that

has effective uniform interpolation

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 21

SLIDE 86

Conclusions

Our results: Conservative extensions and Σ-entailment are

undecidable in GF3 and extensions
undecidable in FO2 and extensions
decidable and 2ExpTime-complete in GF2 = GF ∩ FO2

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 22

SLIDE 87

Conclusions

Our results: Conservative extensions and Σ-entailment are

undecidable in GF3 and extensions
undecidable in FO2 and extensions
decidable and 2ExpTime-complete in GF2 = GF ∩ FO2

plus: implications for uniform interpolation

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 22

SLIDE 88

Conclusions

Our results: Conservative extensions and Σ-entailment are

undecidable in GF3 and extensions
undecidable in FO2 and extensions
decidable and 2ExpTime-complete in GF2 = GF ∩ FO2

plus: implications for uniform interpolation But then: Why are conservative extensions in modal and description logics decidable and how far does the decidability extend?

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 22

SLIDE 89

Conclusions

1 Why are conservative extensions in modal and description

logics decidable?

CEs in GF and FO2 behave very different from satisfiability!
Tree model property seems necessary for decidability of CEs

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 23

SLIDE 90

Conclusions

1 Why are conservative extensions in modal and description

logics decidable?

CEs in GF and FO2 behave very different from satisfiability!
Tree model property seems necessary for decidability of CEs

2 How far does the decidability extend?

GF2 extended with guarded counting quantifiers?
GF2 extended with least/greatest fixed points?
GF2 extended with transitive/equivalence relations?

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 23

SLIDE 91

Conclusions

1 Why are conservative extensions in modal and description

logics decidable?

CEs in GF and FO2 behave very different from satisfiability!
Tree model property seems necessary for decidability of CEs

2 How far does the decidability extend?

GF2 extended with guarded counting quantifiers?
GF2 extended with least/greatest fixed points?
GF2 extended with transitive/equivalence relations?
Are CEs decidable in FO2 when the consequences ψ are

formulated in modal logic or something weaker?

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 23

SLIDE 92

Conclusions

1 Why are conservative extensions in modal and description

logics decidable?

CEs in GF and FO2 behave very different from satisfiability!
Tree model property seems necessary for decidability of CEs

2 How far does the decidability extend?

GF2 extended with guarded counting quantifiers?
GF2 extended with least/greatest fixed points?
GF2 extended with transitive/equivalence relations?
Are CEs decidable in FO2 when the consequences ψ are

formulated in modal logic or something weaker?

Are CEs decidable in GF when the second formula ϕ2 can only

use fresh relations that are at most binary?

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 23

SLIDE 93

Conclusions

1 Why are conservative extensions in modal and description

logics decidable?

CEs in GF and FO2 behave very different from satisfiability!
Tree model property seems necessary for decidability of CEs

2 How far does the decidability extend?

GF2 extended with guarded counting quantifiers?
GF2 extended with least/greatest fixed points?
GF2 extended with transitive/equivalence relations?
Are CEs decidable in FO2 when the consequences ψ are

formulated in modal logic or something weaker?

Are CEs decidable in GF when the second formula ϕ2 can only

use fresh relations that are at most binary?

Thanks!

Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 23