Kripke completeness of strictly positive modal logics Michael - PowerPoint PPT Presentation

Kripke completeness of strictly positive modal logics Michael Zakharyaschev Department of Computer Science and Information Systems Birkbeck, University of London Joint work with Stanislav Kikot, Agi Kurucz, Yoshihito Tanaka and Frank Wolter supported by UK EPSRC grant iTract EP/M012670

Based on the (submitted) paper S. Kikot, A. Kurucz, Y . Tanaka, F . Wolter & M. Zakharyaschev Kripke Completeness of Strictly Positive Modal Logics over Meet-semilattices with Operators https://arxiv.org/abs/1708.03403

SP -terms, equations, and theories Strictly positive terms (or SP -terms ) are defined by the grammar σ ∧ σ ′ σ ::= p i | ⊤ | j σ | ✸ where p i are propositional variables SP -equation takes the form e = ( σ ≤ τ ) σ , τ are SP -terms ( SP -implication) NB SP -equations are always Sahlqvist formulas in Modal Logic ( SP -sequent) SP -theory (or logic ) is a set, E , of SP -equations Wormshop, Moscow 2017 1

SP -terms, equations, and theories Strictly positive terms (or SP -terms ) are defined by the grammar σ ∧ σ ′ σ ::= p i | ⊤ | j σ | ✸ where p i are propositional variables SP -equation takes the form e = ( σ ≤ τ ) σ , τ are SP -terms ( SP -implication) NB SP -equations are always Sahlqvist formulas in Modal Logic ( SP -sequent) SP -theory (or logic ) is a set, E , of SP -equations Edith Hemaspaandra (2001) called terms with p i , ¬ p i , ∧ , ✸ i , ✷ i poor man’s formulas Who needs the pauper’s SP -terms, equations, and theories? Wormshop, Moscow 2017 1

SP -theories in Knowledge Representation Description logic EL and the OWL 2 EL profile of the Web Ontology Language OWL 2 SNOMED CT EntireFemur ⊑ StructureOfFemur FemurPart ⊑ StructureOfFemur ⊓ ∃ partOf . EntireFemur ⊑ BoneStructureOfDistalFemur FemurPart ⊑ EntireDistalFemur BoneStructureOfDistalFemur DistalFemurPart ⊑ BoneStructureOfDistalFemur ⊓ ∃ partOf . EntireDistalFemur Comprehensive healthcare terminology with approximately 400 000 definitions (400 000 concept names and 60 binary relations) OWL 2 is undecidable , OWL 2 DL ( SROIQ ) is 2N EXP T IME -complete validity of quasi-equations EL is tractable w.r.t. first-order semantics, i.e., Kripke models consequence relation 1: E | = Kr e Wormshop, Moscow 2017 2

SP -theories in Knowledge Representation Description logic EL and the OWL 2 EL profile of the Web Ontology Language OWL 2 SNOMED CT EntireFemur ⊑ StructureOfFemur FemurPart ⊑ StructureOfFemur ⊓ ∃ partOf . EntireFemur ⊑ BoneStructureOfDistalFemur FemurPart ⊑ EntireDistalFemur BoneStructureOfDistalFemur DistalFemurPart ⊑ BoneStructureOfDistalFemur ⊓ ∃ partOf . EntireDistalFemur Comprehensive healthcare terminology with approximately 400 000 definitions (400 000 concept names and 60 binary relations) OWL 2 is undecidable , OWL 2 DL ( SROIQ ) is 2N EXP T IME -complete validity of quasi-equations EL is tractable w.r.t. first-order semantics, i.e., Kripke models E | = Kr e consequence relation 1: E | = Kr e Wormshop, Moscow 2017 2

SP -definable first-order properties first-order property SP -equations notation reflexivity p ≤ ✸ p e refl transitivity ✸✸ p ≤ ✸ p e trans symmetry q ∧ ✸ p ≤ ✸ ( p ∧ ✸ q ) e sym � � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → R ( y, z ) ✸ p ∧ ✸ q ≤ ✸ ( p ∧ ✸ q ) e eucl Euclideanness quasi-order { e refl , e trans } E S4 equivalence { e refl , e trans , e sym } E S5 E ′ { e refl , e trans , e eucl } S5 � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → ✸ ( p ∧ q ) ∧ ✸ ( p ∧ r ) ≤ e wcon � � � �� R ( y, y ) ∧ R ( y, z ) ∨ R ( z, z ) ∧ R ( z, y ) ✸ ( p ∧ ✸ q ∧ ✸ r ) linear quasi-order { e refl , e trans , e wcon } E S4 . 3 � � �� ∀ x, y R ( x, y ) → ∃ z R ( x, z ) ∧ R ( z, y ) ✸ p ≤ ✸✸ p e dense density � � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → ( y = z ) ✸ p ∧ ✸ q ≤ ✸ ( p ∧ q ) e fun functionality Wormshop, Moscow 2017 3

SP -undefinable first-order properties by a general necessary condition for SP -definability first-order property modal formula(s) notation � ∀ x, y, z R ( x, y ) ∧ R ( y, z ) → � R ( x, z ) ∨ ( x = z ) ✸✸ p ≤ p ∨ ✸ p pseudo-transitivity ϕ ptrans pseudo-equivalence e sym , ϕ ptrans Diff � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → ✸ p ∧ ✸ q ≤ ✸ ( p ∧ q ) ∨ ϕ wcon � R ( y, z ) ∨ R ( z, y ) ∨ ( y = z ) ✸ ( p ∧ ✸ q ) ∨ ✸ ( q ∧ ✸ q ) weak connectedness transitivity and weak connectedness e trans , ϕ wcon K4 . 3 � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → � �� confluence ∃ u R ( y, u ) ∧ R ( z, u ) ✸✷ p ≤ ✷✸ p ϕ conf transitivity and confluence e trans , ϕ conf K4 . 2 transitivity and e trans , ✷✸ p ≤ ✸✷ p K4 . 1 ∀ x ∃ y ( R ( x, y ) ∧ ∀ z ( R ( y, z ) → ( y = z ))) Wormshop, Moscow 2017 4

SP -undefinable first-order properties by a general necessary condition for SP -definability first-order property modal formula(s) notation � ∀ x, y, z R ( x, y ) ∧ R ( y, z ) → � R ( x, z ) ∨ ( x = z ) ✸✸ p ≤ p ∨ ✸ p pseudo-transitivity ϕ ptrans pseudo-equivalence e sym , ϕ ptrans Diff � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → ✸ p ∧ ✸ q ≤ ✸ ( p ∧ q ) ∨ ϕ wcon � R ( y, z ) ∨ R ( z, y ) ∨ ( y = z ) ✸ ( p ∧ ✸ q ) ∨ ✸ ( q ∧ ✸ q ) weak connectedness transitivity and weak connectedness e trans , ϕ wcon K4 . 3 � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → � �� confluence ∃ u R ( y, u ) ∧ R ( z, u ) ✸✷ p ≤ ✷✸ p ϕ conf transitivity and confluence e trans , ϕ conf K4 . 2 transitivity and e trans , ✷✸ p ≤ ✸✷ p K4 . 1 ∀ x ∃ y ( R ( x, y ) ∧ ∀ z ( R ( y, z ) → ( y = z ))) For any E ⊇ E S4 , Kr E is closed under subframes S4 . 1 -frames and S4 . 2 -frames are not SP -definable But � � e refl , e trans , e wcon defines S4 . 3 -frames Wormshop, Moscow 2017 4

SP -undefinable first-order properties by a general necessary condition for SP -definability first-order property modal formula(s) notation � ∀ x, y, z R ( x, y ) ∧ R ( y, z ) → � R ( x, z ) ∨ ( x = z ) ✸✸ p ≤ p ∨ ✸ p pseudo-transitivity ϕ ptrans pseudo-equivalence e sym , ϕ ptrans Diff � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → ✸ p ∧ ✸ q ≤ ✸ ( p ∧ q ) ∨ ϕ wcon � R ( y, z ) ∨ R ( z, y ) ∨ ( y = z ) ✸ ( p ∧ ✸ q ) ∨ ✸ ( q ∧ ✸ q ) weak connectedness transitivity and weak connectedness e trans , ϕ wcon K4 . 3 � ∀ x, y, z R ( x, y ) ∧ R ( x, z ) → Svyatlovsky’s talk � �� confluence ∃ u R ( y, u ) ∧ R ( z, u ) ✸✷ p ≤ ✷✸ p ϕ conf transitivity and confluence e trans , ϕ conf K4 . 2 transitivity and e trans , ✷✸ p ≤ ✸✷ p K4 . 1 ∀ x ∃ y ( R ( x, y ) ∧ ∀ z ( R ( y, z ) → ( y = z ))) For any E ⊇ E S4 , Kr E is closed under subframes S4 . 1 -frames and S4 . 2 -frames are not SP -definable But � � e refl , e trans , e wcon defines S4 . 3 -frames Wormshop, Moscow 2017 4

SP -theories: algebraic view Bounded meet-semilattices with normal monotone operators (or SLOs ) A = ( A, ∧ , ⊤ , ✸ i ) ( σ ≤ τ is a shorthand for σ ∧ τ = σ ) ( σ = τ is a shorthand for σ ≤ τ and τ ≤ σ ) – p ∧ p = p – p ∧ q = q ∧ p – p ∧ ( q ∧ r ) = ( p ∧ q ) ∧ r – p ≤ ⊤ – ✸ i ( p ∧ q ) ≤ ✸ i q ( monotonicity ) Wormshop, Moscow 2017 5

SP -theories: algebraic view Bounded meet-semilattices with normal monotone operators (or SLOs ) A = ( A, ∧ , ⊤ , ✸ i ) ( σ ≤ τ is a shorthand for σ ∧ τ = σ ) ( σ = τ is a shorthand for σ ≤ τ and τ ≤ σ ) – p ∧ p = p Birkhoff’s equational calculus – p ∧ q = q ∧ p ϕ = ϕ E ⊢ SLO e – p ∧ ( q ∧ r ) = ( p ∧ q ) ∧ r ϕ = ψ/ψ = ϕ – p ≤ ⊤ ϕ = ψ, ψ = χ/ϕ = χ – ✸ i ( p ∧ q ) ≤ ✸ i q ( monotonicity ) ϕ = ψ, α = β/ϕ ( α/p ) = ψ ( β/p ) consequence relation 2: E | = SLO e ⇐ ⇒ E ⊢ SLO e ∀ A ( A | = E = ⇒ A | = e ) Wormshop, Moscow 2017 5

SP -theories in Provability Logic Reflection Calculus RC (Beklemishev 2012, Dashkov 2012) Birkhoff’s equational calculus for SLOs + ✸ n ✸ n σ ≤ ✸ n σ , ✸ n σ ≤ ✸ m σ , ✸ n σ ∧ ✸ m σ ≤ ✸ n ( σ ∧ ✸ m σ ) n > m axiomatises the SP -fragment of G. Japaridze’s provability logic GLP – RC is tractable , while GLP is PSpace -complete – RC is complete w.r.t. finite Kripke frames while GLP is Kripke incomplete – RC preserves main proof-theoretic applications of GLP – RC allows more general arithmetical interpretations Wormshop, Moscow 2017 6

SP -theories in Provability Logic ? s e i r o Reflection Calculus RC (Beklemishev 2012, Dashkov 2012) e h t - P S Birkhoff’s equational calculus for SLOs e s + U ! s c ✸ n ✸ n σ ≤ ✸ n σ , ✸ n σ ≤ ✸ m σ , ✸ n σ ∧ ✸ m σ ≤ ✸ n ( σ ∧ ✸ m σ ) n > m i g o l l a axiomatises the SP -fragment of G. Japaridze’s provability logic GLP d o m n – RC is tractable , while GLP is PSpace -complete a e l o o – RC is complete w.r.t. finite Kripke frames B h c while GLP is Kripke incomplete t i D – RC preserves main proof-theoretic applications of GLP – RC allows more general arithmetical interpretations Wormshop, Moscow 2017 6

The problem Kripke completeness: is a given SP -theory E complete w.r.t. its Kripke frames? ? for all SP -equations e , E | ⇐ ⇒ E | = SLO e = Kr e Wormshop, Moscow 2017 7