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
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
j σ
where pi are propositional variables
σ, τ are SP-terms (SP-implication)
NB SP-equations are always Sahlqvist formulas in Modal Logic (SP-sequent)
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j σ
where pi are propositional variables
σ, τ are SP-terms (SP-implication)
NB SP-equations are always Sahlqvist formulas in Modal Logic (SP-sequent)
Edith Hemaspaandra (2001) called terms with pi, ¬pi, ∧, ✸i, ✷i
Who needs the pauper’s SP-terms, equations, and theories?
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the Web Ontology Language OWL 2
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 2NEXPTIME-complete
w.r.t. first-order semantics, i.e., Kripke models validity of quasi-equations
consequence relation 1: E |
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the Web Ontology Language OWL 2
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 2NEXPTIME-complete
w.r.t. first-order semantics, i.e., Kripke models validity of quasi-equations
consequence relation 1: E |
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first-order property SP-equations notation reflexivity p ≤ ✸p erefl transitivity ✸✸p ≤ ✸p etrans symmetry q ∧ ✸p ≤ ✸(p ∧ ✸q) esym ∀x, y, z
eeucl Euclideanness quasi-order {erefl, etrans} ES4 equivalence {erefl, etrans, esym} ES5 {erefl, etrans, eeucl} E′
S5
∀x, y, z
✸(p ∧ q) ∧ ✸(p ∧ r) ≤ ewcon
linear quasi-order {erefl, etrans, ewcon} ES4.3 ∀x, y
edense density ∀x, y, z
efun functionality
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by a general necessary condition for SP-definability
first-order property modal formula(s) notation ∀x, y, z
pseudo-transitivity R(x, z) ∨ (x = z)
ϕptrans pseudo-equivalence esym, ϕptrans Diff ∀x, y, z
✸p ∧ ✸q ≤ ✸(p ∧ q) ∨ ϕwcon R(y, z) ∨ R(z, y) ∨ (y = z)
weak connectedness transitivity and weak connectedness etrans, ϕwcon K4.3 ∀x, y, z
confluence ∃u
ϕconf transitivity and confluence etrans, ϕconf K4.2 transitivity and etrans, ✷✸p ≤ ✸✷p K4.1 ∀x∃y (R(x, y) ∧ ∀z (R(y, z) → (y = z)))
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by a general necessary condition for SP-definability
first-order property modal formula(s) notation ∀x, y, z
pseudo-transitivity R(x, z) ∨ (x = z)
ϕptrans pseudo-equivalence esym, ϕptrans Diff ∀x, y, z
✸p ∧ ✸q ≤ ✸(p ∧ q) ∨ ϕwcon R(y, z) ∨ R(z, y) ∨ (y = z)
weak connectedness transitivity and weak connectedness etrans, ϕwcon K4.3 ∀x, y, z
confluence ∃u
ϕconf transitivity and confluence etrans, ϕconf K4.2 transitivity and etrans, ✷✸p ≤ ✸✷p K4.1 ∀x∃y (R(x, y) ∧ ∀z (R(y, z) → (y = z)))
S4.1-frames and S4.2-frames are not SP-definable But
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by a general necessary condition for SP-definability
first-order property modal formula(s) notation ∀x, y, z
pseudo-transitivity R(x, z) ∨ (x = z)
ϕptrans pseudo-equivalence esym, ϕptrans Diff ∀x, y, z
✸p ∧ ✸q ≤ ✸(p ∧ q) ∨ ϕwcon R(y, z) ∨ R(z, y) ∨ (y = z)
weak connectedness transitivity and weak connectedness etrans, ϕwcon K4.3 ∀x, y, z
confluence ∃u
ϕconf transitivity and confluence etrans, ϕconf K4.2 transitivity and etrans, ✷✸p ≤ ✸✷p K4.1 ∀x∃y (R(x, y) ∧ ∀z (R(y, z) → (y = z)))
S4.1-frames and S4.2-frames are not SP-definable But
Svyatlovsky’s talk
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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) ≤ ✸iq (monotonicity)
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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) ≤ ✸iq (monotonicity)
ϕ = ϕ E ⊢SLO e ϕ = ψ/ψ = ϕ ϕ = ψ, ψ = χ/ϕ = χ ϕ = ψ, α = β/ϕ(α/p) = ψ(β/p)
∀A (A | = E = ⇒ A | = e)
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(Beklemishev 2012, Dashkov 2012)
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
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(Beklemishev 2012, Dashkov 2012)
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
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Kripke completeness: is a given SP-theory E complete w.r.t. its Kripke frames?
?
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Kripke completeness: is a given SP-theory E complete w.r.t. its Kripke frames?
?
by Sahlqvist completeness, E | =Kr e ⇐ ⇒ E | =BAO e ⇐ ⇒ E ⊢K e
?
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Kripke completeness: is a given SP-theory E complete w.r.t. its Kripke frames?
?
by Sahlqvist completeness, E | =Kr e ⇐ ⇒ E | =BAO e ⇐ ⇒ E ⊢K e
?
modal logic LE = K ⊕ E?
?
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(Kurucz, Tanaka, Wolter & Z, 2010)
with FO-correspondent
∀x, y
E1 | =Kr p ∧ ✸⊤ ≤ ✸p but E1 | =SLO p ∧ ✸⊤ ≤ ✸p ⊥ a ⊤
p
SLO ‘general’ frame
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(Kurucz, Tanaka, Wolter & Z, 2010)
with FO-correspondent
∀x, y
E1 | =Kr p ∧ ✸⊤ ≤ ✸p but E1 | =SLO p ∧ ✸⊤ ≤ ✸p ⊥ a ⊤
p
SLO ‘general’ frame
with FO-correspondent
in modal logic, Kripke incomplete logics are ‘rare’ and ‘complex’
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(Kurucz, Tanaka, Wolter & Z, 2010)
with FO-correspondent
∀x, y
E1 | =Kr p ∧ ✸⊤ ≤ ✸p but E1 | =SLO p ∧ ✸⊤ ≤ ✸p ⊥ a ⊤
p
SLO ‘general’ frame
with FO-correspondent
in modal logic, Kripke incomplete logics are ‘rare’ and ‘complex’
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Kripke frame F = (W, Ri) full complex BAO
i )
✸+
i X = {w ∈ W | ∃v ∈ X Ri(w, v)}
A
A |
Can we do something similar for SP-theories and SLOs? no canonical models
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Kripke frame F = (W, Ri) SLO-type reduct of full complex BAO
i )
✸+
i X = {w ∈ W | ∃v ∈ X Ri(w, v)}
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Kripke frame F = (W, Ri) SLO-type reduct of full complex BAO
i )
✸+
i X = {w ∈ W | ∃v ∈ X Ri(w, v)}
An SP-theory E is complex if every SLO A | = E is embeddable into F⋆ for some Kripke frame F | = E
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Kripke frame F = (W, Ri) SLO-type reduct of full complex BAO
i )
✸+
i X = {w ∈ W | ∃v ∈ X Ri(w, v)}
An SP-theory E is complex if every SLO A | = E is embeddable into F⋆ for some Kripke frame F | = E
Theorem Every SLO is embeddable into F⋆, for some Kripke frame F
(via elements of SLOs or via filters)
The empty SP-theory is complex, and so complete: | =Kr e implies | =SLO e, for every SP-equation e
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Kripke frame F = (W, Ri) SLO-type reduct of full complex BAO
i )
✸+
i X = {w ∈ W | ∃v ∈ X Ri(w, v)}
An SP-theory E is complex if every SLO A | = E is embeddable into F⋆ for some Kripke frame F | = E
Theorem Every SLO is embeddable into F⋆, for some Kripke frame F
(via elements of SLOs or via filters)
The empty SP-theory is complex, and so complete: | =Kr e implies | =SLO e, for every SP-equation e
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SP-terms as Kripke models σ = ✸(r ∧ ✸q ∧ ✸p) Mσ = (Wσ, Rσ, vσ) Tσ r p q
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SP-terms as Kripke models σ = ✸(r ∧ ✸q ∧ ✸p) Mσ = (Wσ, Rσ, vσ) Tσ r p q M, w | = σ ⇐ ⇒ ∃h: Mσ → M with h(rσ) = w
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SP-terms as Kripke models σ = ✸(r ∧ ✸q ∧ ✸p) Mσ = (Wσ, Rσ, vσ) Tσ r p q M, w | = σ ⇐ ⇒ ∃h: Mσ → M with h(rσ) = w
Sahlqvist’s correspondence: every SP-equation e = (σ ≤ τ) has the FO-correspondent
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based on the form of their FO-correspondents
∀x, y, z
∀x, y, z
∀x, y, z
∀x, y
closed under subframes
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based on the form of their FO-correspondents
∀x, y, z
∀x, y, z
∀x, y, z
∀x, y
closed under subframes
polynomial model property, and so is decidable in CONP if E is finite
is decidable in PTIME
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rooted profile π
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rooted profile π
‘standard’ equations eπ = ✸✸✸p ≤ ✸p type 1 e′
π = p1 ∧ ✸(p2 ∧ ✸(p3 ∧ ✸p4)) ≤ p1 ∧ ✸p4
type 2
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rooted profile π
‘standard’ equations eπ = ✸✸✸p ≤ ✸p type 1 e′
π = p1 ∧ ✸(p2 ∧ ✸(p3 ∧ ✸p4)) ≤ p1 ∧ ✸p4
type 2
π of type 2) for rooted π
e.g., ✸1 . . . ✸np ≤ ✸0p
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rooted profile π
‘standard’ equations eπ = ✸✸✸p ≤ ✸p type 1 e′
π = p1 ∧ ✸(p2 ∧ ✸(p3 ∧ ✸p4)) ≤ p1 ∧ ✸p4
type 2
π of type 2) for rooted π
e.g., ✸1 . . . ✸np ≤ ✸0p
‘non-standard’ equations p ≤ ✸✸(p ∧ ✸p) for π = eπ = e′
π = (p ≤ ✸p)
✸✸p ∧ ✸✸✸p ≤ ✸p for
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rooted profile π
‘standard’ equations eπ = ✸✸✸p ≤ ✸p type 1 e′
π = p1 ∧ ✸(p2 ∧ ✸(p3 ∧ ✸p4)) ≤ p1 ∧ ✸p4
type 2
π of type 2) for rooted π
e.g., ✸1 . . . ✸np ≤ ✸0p
‘non-standard’ equations p ≤ ✸✸(p ∧ ✸p) for π = eπ = e′
π = (p ≤ ✸p)
✸✸p ∧ ✸✸✸p ≤ ✸p for
Normal modal logics axiomatisable by SP-equations can be
✸
R ✸ P ✸ R p ≤ ✸ P p,
✸
Q ✸ R p ≤ ✸ Q p,
✸
Q ✸ P p ≤ ✸ P p
however, the corresponding SP-theory is tractable
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Theorem: Any EL-theory E consisting of equations e = (σ ≤ τ) such that – every variable in σ occurs in it only once, – τ corresponds to the tree Tτ = (Wτ, Rτ, Vτ) with – |Wτ| ≥ 2 and all points in any Vτ(p) are leaves of Tτ, – Vτ(p) ∩ Vτ(q) = ∅ whenever p = q
Example: density axiom edense = ✸p ≤ ✸✸p with
p q p q
✸p ∧ ✸q ≤ ✸(✸p ∧ ✸q)
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For P = {p0, . . . , pn}, n ≥ 1,
fun}
fun = Q⊆P |Q|=n
fun =
EAltn, {erefl, en
fun}, {etrans, en fun}, ES4 ∪ {en fun}, En S5 = ES5 ∪ {en fun}
for n ≥ 2 a0 a1 an ⊥ ⊤
fun} but not embeddable into F⋆, for any n-functional F
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For P = {p0, . . . , pn}, n ≥ 1,
fun}
fun = Q⊆P |Q|=n
fun =
EAltn, {erefl, en
fun}, {etrans, en fun}, ES4 ∪ {en fun}, En S5 = ES5 ∪ {en fun}
for n ≥ 2 a0 a1 an ⊥ ⊤
fun} but not embeddable into F⋆, for any n-functional F
{ewcon}, {erefl, ewcon}, ES4.3
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E is complete if, for any e = (σ ≤ τ),
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E is complete if, for any e = (σ ≤ τ),
(1) E-normal form E ⊢SLO (τ =
reflecting Kripke frames for E (σ ≤
̺∈Nτ ̺) is the
syntactic proxy of e
(2) for any ̺ ∈ Nτ,
for some complete E− ⊆ E
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E is complete if, for any e = (σ ≤ τ),
(1) E-normal form E ⊢SLO (τ =
reflecting Kripke frames for E (σ ≤
̺∈Nτ ̺) is the
syntactic proxy of e
(2) for any ̺ ∈ Nτ,
for some complete E− ⊆ E
– EAltn Nτ = {≤ n-functional full subtree of Tτ} E− = ∅ – ES4.3 Nτ = {full branches of Tτ} E− = ES4
tractable
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(M. Jackson 2004)
ES5 E3
S5
E2
S5
E1
S5 ES5 + (✸p ≤ p) ES5 + (✸p ≤ ✸q) Triv
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