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

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

Conservative Extensions Fundamental notion in mathematical logic to relate different theories Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

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

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

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 FO 2 ? Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 1

Overview It is known that conservative extensions are • undecidable in FO • decidable in many modal and description logics Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 2

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 FO 2 • decidable and 2ExpTime -complete in GF 2 = GF ∩ FO 2 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 2

Decidable Fragments of FO FO 2 is the two-variable fragment of FO Example: ∀ xyRxy Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 3

Decidable Fragments of FO FO 2 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 ) Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 3

Decidable Fragments of FO FO 2 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 ) GF 2 = GF ∩ FO 2 Example: ∀ x ( Rxy → Pxy ) Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 3

Conservative Extensions A theory T 2 is said to be a (deductive) conservative extension of a theory T 1 if the language of T 2 extends the language of T 1 : • Any consequence of T 1 is also a consequence of T 2 ; and • Any consequence of T 2 , which uses only symbols from T 1 , is also a consequence of T 1 . Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 4

Conservative Extensions A theory T 2 is said to be a (deductive) conservative extension of a theory T 1 if the language of T 2 extends the language of T 1 : • Any consequence of T 1 is also a consequence of T 2 ; and • Any consequence of T 2 , which uses only symbols from T 1 , is also a consequence of T 1 . 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 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 4

Conservative Extensions A theory T 2 is said to be a (deductive) conservative extension of a theory T 1 if the language of T 2 extends the language of T 1 : • Any consequence of T 1 is also a consequence of T 2 ; and • Any consequence of T 2 , which uses only symbols from T 1 , is also a consequence of T 1 . 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 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 4

Conservative Extensions in Fragments of FO Sentence ϕ 1 extended to ϕ 1 ∧ ϕ 2 , and ϕ 2 may use fresh symbols • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 5

Conservative Extensions in Fragments of FO Sentence ϕ 1 extended to ϕ 1 ∧ ϕ 2 , and ϕ 2 may use fresh symbols • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 • Equivalently, ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if, ϕ 1 ∧ ϕ 2 ∧ ¬ ψ is UNSAT ⇒ ϕ 1 ∧ ¬ ψ is UNSAT Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 5

Conservative Extensions in Fragments of FO Sentence ϕ 1 extended to ϕ 1 ∧ ϕ 2 , and ϕ 2 may use fresh symbols • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 • Equivalently, ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if, ϕ 1 ∧ ϕ 2 ∧ ¬ ψ is UNSAT ⇒ ϕ 1 ∧ ¬ ψ is UNSAT ϕ 1 ∧ ψ is SAT ⇒ ϕ 1 ∧ ϕ 2 ∧ ψ is SAT Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 5

Conservative Extensions in Fragments of FO Sentence ϕ 1 extended to ϕ 1 ∧ ϕ 2 , and ϕ 2 may use fresh symbols • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 • Equivalently, ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if, ϕ 1 ∧ ϕ 2 ∧ ¬ ψ is UNSAT ⇒ ϕ 1 ∧ ¬ ψ is UNSAT ϕ 1 ∧ ψ is SAT ⇒ ϕ 1 ∧ ϕ 2 ∧ ψ is SAT • Deciding conservative extensions mean, given two sentences ϕ 1 , ϕ 2 , to decide whether ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 5

Σ-Entailment In applications, it is useful to consider a vocabulary Σ of interest Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 6

Σ-Entailment In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form: • ϕ 1 | = Σ ϕ 2 (“Σ-entails”), if ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from Σ Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 6

Σ-Entailment In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form: • ϕ 1 | = Σ ϕ 2 (“Σ-entails”), if ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from Σ • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 6

Σ-Entailment In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form: • ϕ 1 | = Σ ϕ 2 (“Σ-entails”), if ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from Σ • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 6

Σ-Entailment In applications, it is useful to consider a vocabulary Σ of interest We study conservative extensions in a slightly more general form: • ϕ 1 | = Σ ϕ 2 (“Σ-entails”), if ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from Σ • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 if ϕ 1 ∧ ϕ 2 | = ψ ⇒ ϕ 1 | = ψ for all sentences ψ that use symbols from ϕ 1 Then • ϕ 1 ∧ ϕ 2 is a conservative extension of ϕ 1 iff ϕ 1 | = Σ=sig( ϕ 1 ) ϕ 1 ∧ ϕ 2 Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 6

Examples Consider sentences ϕ 1 , ϕ 2 in GF 2 and Σ = { R } Mauricio Martel Conservative Extensions in Guarded and Two-Variable Fragments 7