PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9....
Transcript of PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9....
![Page 1: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/1.jpg)
PSPACE-decidability of Japaridze’s Polymodal Logic
Ilya Shapirovsky
Institute for Information Transmission ProblemsMoscow, Russia
![Page 2: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/2.jpg)
Satisfiability on ordinal sums of transitive frames
Ilya Shapirovsky
Institute for Information Transmission ProblemsMoscow, Russia
![Page 3: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/3.jpg)
Japaridze’s Polymodal Logic
GLP is the normal propositional modal logic with countably manymodalities:
�i(�ip → p) → �ip for all i
�ip → �jp for all i < j
♦ip → �j♦jp for all i < j
[Japaridze, Ignatiev, Boolos, Beklemishev,...]
![Page 4: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/4.jpg)
Japaridze’s Polymodal Logic[Beklemishev, 2007] Hereditary orders.
![Page 5: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/5.jpg)
Japaridze’s Polymodal Logic[Beklemishev, 2007] Hereditary orders.
![Page 6: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/6.jpg)
Japaridze’s Polymodal Logic[Beklemishev, 2007] Hereditary orders.
![Page 7: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/7.jpg)
Japaridze’s Polymodal Logic[Beklemishev, 2007] Hereditary orders.
![Page 8: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/8.jpg)
Main idea
If a class of frames can be represented as a class of ordinal sums of”simple” frames, then it is also ”simple”.
![Page 9: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/9.jpg)
Decision procedures for ordinal sums of frames
![Page 10: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/10.jpg)
Decision procedures for ordinal sums of frames
1. Monomodal case
![Page 11: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/11.jpg)
Decision procedures for ordinal sums of frames
1. Monomodal caseI Ordinal sums of transitive frames
![Page 12: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/12.jpg)
Decision procedures for ordinal sums of frames
1. Monomodal caseI Ordinal sums of transitive framesI Truth-preserving transformations for ordinal sums of frames
![Page 13: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/13.jpg)
Decision procedures for ordinal sums of frames
1. Monomodal caseI Ordinal sums of transitive framesI Truth-preserving transformations for ordinal sums of framesI Conditional satisfiability and moderate classes of frames
![Page 14: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/14.jpg)
Decision procedures for ordinal sums of frames
1. Monomodal caseI Ordinal sums of transitive framesI Truth-preserving transformations for ordinal sums of framesI Conditional satisfiability and moderate classes of frames
2. Polymodal case
![Page 15: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/15.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of frames
![Page 16: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/16.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of frames
I = (W ,R) is a finite partial order, W = {w1, . . . ,wn}F1 = (V1,S1), . . . ,Fn = (Vn,Sn) are transitive frames
![Page 17: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/17.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of frames
I = (W ,R) is a finite partial order, W = {w1, . . . ,wn}F1 = (V1,S1), . . . ,Fn = (Vn,Sn) are transitive frames
I[(F1, . . . ,Fm)/(w1, . . . ,wm)] = (W ,R):
W = ({w1} × V1) ∪ · · · ∪ ({wm} × Vm)
(w ′, v ′)R(w ′′, v ′′) ⇔ (w ′ 6= w ′′ & w ′Rw ′′) or (w ′ = w ′′ = wi & v ′Siv′′)
![Page 18: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/18.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of frames
I = (W ,R) is a finite partial order, W = {w1, . . . ,wn}F1 = (V1,S1), . . . ,Fn = (Vn,Sn) are transitive frames
For a class F of frames,
I[F ] = {I[(F1, . . . ,Fm)/(w1, . . . ,wm)] | F1, . . . ,Fm ∈ F}
![Page 19: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/19.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of frames
I = (W ,R) is a finite partial order, W = {w1, . . . ,wn}F1 = (V1,S1), . . . ,Fn = (Vn,Sn) are transitive frames
For a class F of frames,
I[F ] = {I[(F1, . . . ,Fm)/(w1, . . . ,wm)] | F1, . . . ,Fm ∈ F}
For a class I of finite partial orders,
I[F ] =⋃{I[F ] | I ∈ I}
![Page 20: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/20.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of framesExample: skeleton
![Page 21: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/21.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Ordinal sums of frames
Ordinal sums of framesExample: skeleton
Every transitive frame can be considered as an ordinal sum of its clusters.
![Page 22: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/22.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Truth-preserving transformations for ordinal sums of frames
Ordinal sums of framesExample: transitive frames
PO denotes the class of all finite (strict or non-strict) partial orders.
For n ≥ 1, Cn = (Wn,Wn × Wn), where Wn = {1, . . . , n};C0 denotes the irreflexive singleton ({0},∅).
Let F = {C0,C1,C2 . . . }, G = {C1,C2, . . . }.Then (up to isomorphisms):PO[F ] is the class of all finite K4-frames,PO[G] is the class of all finite S4-frames.
![Page 23: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/23.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Truth-preserving transformations for ordinal sums of frames
Truth-preserving transformations for ordinal sums of framesTreelike frames
T denotes the class of all finite transitive trees.
LemmaLet F be a class of frames, I be a finite rooted partial order. Then for any
H ∈ I[F ] there exists a tree T ∈ T such that for some H′ ∈ T [F ] we have
H′ � H.
Corollary
ϕ is PO[F ]-satisfiable ⇒ ϕ is T [F ]-satisfiable
![Page 24: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/24.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Truth-preserving transformations for ordinal sums of frames
Truth-preserving transformations for ordinal sums of framesRestricting height and branching
〈ϕ〉 denotes the cardinality of Sub(ϕ).
Well-known fact: Any K4-satisfiable formula ϕ is satisfiable in some finiteframe with the height and the branching of its skeleton not more then 〈ϕ〉.
![Page 25: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/25.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Truth-preserving transformations for ordinal sums of frames
Truth-preserving transformations for ordinal sums of framesRestricting height and branching
Let Tn,b denotes the class of transitive trees with the height not more thenh and the branching not more then b:
Th,b = {T ∈ T | Ht(T) ≤ h, Br(T) ≤ b}.
LemmaLet F be a class of transitive frames. If a formula ϕ is PO[F ]-satisfiable,
then ϕ is T〈ϕ〉,〈ϕ〉[F ]-satisfiable.
![Page 26: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/26.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Truth-preserving transformations for ordinal sums of frames
Selective filtration
A model ((W ′,R ′), θ′) is a weak submodel of ((W ,R), θ), ifW ′ ⊆ W , R ′ ⊆ R ,θ(p) = θ′(p) ∩ W ′ for any propositional variable p.
DefinitionLet M be a model, Ψ a set of formulas closed under subformulas. A weaksubmodel M′ of M is called a selective filtration of M through Ψ , if forany w ∈ M′, for any formula ψ, we have
♦ψ ∈ Ψ & M,w � ♦ψ ⇒ ∃u ∈ R ′(x) M, u � ψ,
where R ′ is the accessability relation of M′.
LemmaIf M′ is a selective filtration of M through Ψ, then for any w ∈ M′, for any
ψ ∈ Ψ, we have
M,w � ψ ⇔ M′,w � ψ.
![Page 27: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/27.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Big and small
![Page 28: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/28.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Conditional satisfiability
M is a Kripke model
M,w 6� ⊥;M,w � p ⇔ w ∈ θ(p);M,w � ϕ→ ψ ⇔ M,w 6� ϕ or M,w � ψ;M,w � ♦ϕ ⇔ ∃v(wRv & M, v � ϕ).
”ϕ is true at w in M”.
![Page 29: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/29.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Conditional satisfiability
M is a Kripke model, Ψ is a set of formulas
(M,w |Ψ) 6� ⊥(M,w |Ψ) � p ⇔ w ∈ θ(p);(M,w |Ψ) � ϕ→ ψ ⇔ (M,w |Ψ) 6� ϕ or (M,w |Ψ) � ψ(M,w |Ψ) � ♦ϕ ⇔ ∃v(wRv & M, v � ϕ)
or ϕ ∈ Ψ or ♦ϕ ∈ Ψ
“ϕ is true at w in M under the condition Ψ ”.
![Page 30: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/30.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Consider transitive models M0, M,their ordinal sum M0 + M, and aformula ϕ. Put
Ψ = {ψ ∈ Sub(ϕ) | M, v � ψ for some v}
Then for any w ∈ M0,
M0 + M,w � ϕ⇔ (M0,w |Ψ) � ϕ
More generally: for any set offormulas Φ,
(M0+M,w |Φ) � ϕ⇔ (M0,w |Ψ∪Φ) � ϕ
![Page 31: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/31.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Moderate classes of frames
For a cone F, F ϕ means that ϕ is satisfiable at a root of F;F | Ψ ϕ means (M,w |Ψ) � ϕ for some model M based on F, where w isa root of F.For a class of cones F , F | Ψ ϕ means that F | Ψ ϕ for some F ∈ F
DefinitionA sequence (Fn)n∈N
of sets of rooted frames is called d-moderate for
d ∈ N, if there exists an algorithm such that for any formula ϕ and any
Ψ,Φ ⊆ Sub(ϕ) it decides whether
F〈ϕ〉 | Ψ ∧
Φ
in space O(〈ϕ〉d ).
![Page 32: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/32.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Moderate classes of frames: examples
DefinitionA sequence (Fn)n∈N
of sets of rooted frames is called d-moderate for
d ∈ N, if there exists an algorithm such that for any formula ϕ and any
Ψ,Φ ⊆ Sub(ϕ) it decides whether
F〈ϕ〉 | Ψ ∧
Φ
in space O(〈ϕ〉d ).
![Page 33: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/33.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Moderate classes of frames: examples
DefinitionA sequence (Fn)n∈N
of sets of rooted frames is called d-moderate for
d ∈ N, if there exists an algorithm such that for any formula ϕ and any
Ψ,Φ ⊆ Sub(ϕ) it decides whether
F〈ϕ〉 | Ψ ∧
Φ
in space O(〈ϕ〉d ).
Example: (Fn)n∈Nis moderate, if:
I Fn is the set of all (non-degenerate) clusters with cardinality notmore then n:for all n Fn = {C0, . . . ,Cn} or for all n Fn = {C1, . . . ,Cn};
I Fn consists of a single frame which is a singleton:for all n Fn = {C0} or for all n Fn = {C1}.
![Page 34: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/34.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Moderate classes of frames: examples
DefinitionA sequence (Fn)n∈N
of sets of rooted frames is called d-moderate for
d ∈ N, if there exists an algorithm such that for any formula ϕ and any
Ψ,Φ ⊆ Sub(ϕ) it decides whether
F〈ϕ〉 | Ψ ∧
Φ
in space O(〈ϕ〉d ).
For classes F , G, put
F + G = {F + G | F ∈ F , G ∈ G}
LemmaIf (Fn)n∈N
, (Gn)n∈Nare moderate, then (Fn + Gn)n∈N
are moderate.
![Page 35: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/35.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Main lemma
If (Fn)n∈Nis d -moderate sequence of sets of cones, and P is a polynomial
of degree d ′, then the sequence (TP(n),P(n)[Fn])n∈Nis
max{2 + d ′, d}-moderate.
![Page 36: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/36.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
AlgorithmLet SatSimple(ϕ,Φ,Ψ) decide whether F | Ψ
∧Φ for any ϕ,
Φ,Ψ ⊆ Sub(ϕ) in space f (〈ϕ〉).
Then the following algorithm decides whether Th,b[F ] | Ψ ∧
Φ for anyϕ, Φ,Ψ ⊆ Sub(ϕ), h, b > 0 in space O (f (〈ϕ〉) + 〈ϕ〉bh):
Function SatTree(ϕ; Φ,Ψ; h, b) returns boolean;Begin
if SatSimple(ϕ,Φ,Ψ) then return(true);if h > 1 then
for every integer b′ such that 1 ≤ b′ ≤ b
for every Φ1, . . . ,Φb′ ⊆ Sub(ϕ)if
∧1≤j≤b′
SatTree(ϕ,Ψj ,Ψ, h − 1, b) then
if SatSimple(ϕ,Φ,Ψ ∪ Ψ1 · · · ∪ Ψb′) thenreturn(true);
return(false);End.
![Page 37: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/37.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Algorithm
LemmaLet F be a class of frames, and let G ∈ Th+1,b[F ] for some h, b ≥ 1. Then
G is either isomorphic to a frame F ∈ F or isomorphic to a frame
F + (G1 t · · · t Gb′), where 1 ≤ b′ ≤ b, F ∈ F , G1, . . .Gb′ ∈ Th,b[F ].
![Page 38: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/38.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Semantic condition
TheoremSuppose that a logic L is characterized by PO[F ] for some class F . If
there exists a moderate sequence (Fn)n∈Nsuch that Fn ⊆ F for all n ∈ N,
and any L-satisfiable formula ϕ is PO[F〈ϕ〉]-satisfiable, then L is in
PSPACE.
Corollary
If L = L(PO[F ]) for some finite class of finite cones F , then L is in
PSPACE.
![Page 39: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/39.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
ExamplesConsider the logics K4,S4 , Godel-Lob logic GL, and Grzegorczyk logicGRZ. They are well-known to be PSPACE-decidable. Let us illustrate,how this fact follows from the above theorem.
![Page 40: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/40.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
ExamplesConsider the logics K4,S4 , Godel-Lob logic GL, and Grzegorczyk logicGRZ. They are well-known to be PSPACE-decidable. Let us illustrate,how this fact follows from the above theorem.
GRZ (GL) is the logic of all finite non-strict (strict) partial orders:GRZ = L(PO[{C1}]), GL = L(PO[{C0}]) .By the above corollary, GRZ and GL are in PSPACE.
![Page 41: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/41.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
ExamplesConsider the logics K4,S4 , Godel-Lob logic GL, and Grzegorczyk logicGRZ. They are well-known to be PSPACE-decidable. Let us illustrate,how this fact follows from the above theorem.
Any K4-satisfiable formula is satisfiable at some finite transitive frame Fsuch that the cardinality of any cluster in F does not exceed 〈ϕ〉.Put
FK4n = {C0, . . . ,Cn}, FS4
n = {C1, . . . ,Cn}.
Then for any ϕ we have:
ϕ is K4-satisfiable iff ϕ is PO[FK4
〈ϕ〉]-satisfiable,
ϕ is S4-satisfiable iff ϕ is PO[FS4
〈ϕ〉]-satisfiable.
Since the sequences (FK4n )n∈N
and (FS4n )n∈N
are moderate, then K4 andS4 are in PSPACE.
![Page 42: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/42.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Example: logic LM
LM = K4+♦>+♦p1∧♦p2 → ♦(♦p1∧♦p2) (the logic of interval inclusion,compact inclusion, chronological future)
![Page 43: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/43.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Example: logic LMFor a transitive finite frame F,F � LM ⇔ every its degenerate cluster has a unique successor C, and C isnon-degenerate.
![Page 44: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/44.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Example: logic LMFor a transitive finite frame F,F � LM ⇔ every its degenerate cluster has a unique successor C, and C isnon-degenerate.
For F = {C0 + C | C is a finite cluster},PO[F ] is the class of all finite LM-frames (up to isomorphisms).
![Page 45: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/45.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Example: logic LMFor a transitive finite frame F,F � LM ⇔ every its degenerate cluster has a unique successor C, and C isnon-degenerate.
For F = {C0 + C | C is a finite cluster},PO[F ] is the class of all finite LM-frames (up to isomorphisms).
LM has the FMP, thus: ϕ is LM-satisfiable ⇒ ϕ is PO[F〈ϕ〉]-satisfiable,where Fn = {C0 + C1, . . . ,C0 + Cn}
![Page 46: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/46.jpg)
Decision procedures for ordinal sums of frames. Monomodal case Conditional satisfiability and moderate classes of frames
Example: logic LMFor a transitive finite frame F,F � LM ⇔ every its degenerate cluster has a unique successor C, and C isnon-degenerate.
For F = {C0 + C | C is a finite cluster},PO[F ] is the class of all finite LM-frames (up to isomorphisms).
LM has the FMP, thus: ϕ is LM-satisfiable ⇒ ϕ is PO[F〈ϕ〉]-satisfiable,where Fn = {C0 + C1, . . . ,C0 + Cn}
The sequence (Fn)n∈Nis moderate, thus LM is PSPACE-decidable.
![Page 47: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/47.jpg)
Multimodal case
![Page 48: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/48.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
Ordinal sums of multimodal framesI = (W ,R) is a finite partial order, W = {w1, . . . ,wn}F1 = (W1,R
11 , . . . ,R
1N), . . . ,Fn = (Wn,R
n1 , . . . ,R
nN) are N-frames;
1 ≤ k ≤ N. I[k ; (F1, . . . ,Fm)/(w1, . . . ,wm)]:
For a class F of N-frames, put
G[k ;F ] = {I[k ; (F1, . . . ,Fm)/(w1, . . . ,wm)] | F1, . . . ,Fn ∈ F},
for a class I of finite partial orders, put
I[k ;F ] =⋃
{I[k ;F ] | I ∈ G}.
![Page 49: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/49.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
LemmaLet F be a class of N-frames, 1 ≤ k ≤ N. If an N-formula ϕ is
PO[k ;F ]-satisfiable, then ϕ is T〈ϕ〉,〈ϕ〉[k ;F ]-satisfiable.
![Page 50: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/50.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
DefinitionLet M be an N-model. A condition for M is a tuple Ψ = (Ψ1, . . . ,ΨN) ofsets of N−formulas. For an N-formula ϕ and a point w ∈ M, we definethe truth-relation (M,w |Ψ) � ϕ (“ϕ is true at w in M under the condition
Ψ”):
(M,w |Ψ) � p ⇔ M,w � p
(M,w |Ψ) 6� ⊥
(M,w |Ψ) � ϕ→ ψ ⇔ (M,w |Ψ) 6� ϕ or (M,w |Ψ) � ψ
(M,w |Ψ) � ♦kϕ ⇔ ϕ ∈ Ψk or ♦kϕ ∈ Ψk or
for some v ∈ Rk(w) we have (M,w |Ψk) � ϕ,
where R1, . . . ,RN are the accessability relations in M.
![Page 51: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/51.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
An N-frame G = (W ,R1, . . . ,RN) is called rooted, if for some w ∈ W wehave {w} ∪ R1(w) ∪ · · · ∪ RN(w) = W ; w is called a root of G.For an N-formula ϕ, put
Sub∗(ϕ) = Sub(ϕ) ∪ {♦iψ | 1 ≤ i , j ≤ N, ♦jψ ∈ Sub(ϕ)}.
Definition. A sequence (Fn)n∈N
of sets of rooted N-frames is called d-moderate, ifthere exists an algorithm such that for any N-formula ϕ and anyΦ,Ψ1, . . . ,Ψn ⊆ Sub∗(ϕ), it decides whether
F|Sub∗(ϕ)| | (Ψ1, . . . ,Ψn) ∧
Φ
in polynomial space.
![Page 52: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/52.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
TheoremIf (Fn)n∈N
is d-moderate sequence of sets of rooted N-frames, 1 ≤ k ≤ N,
P is a polynomial of degree d ′, then the sequence
(TP(n),P(n)[k ;Fn])n∈N
is max{2 + d ′, d}-moderate.
![Page 53: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/53.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
Japaridze’s Polymodal Logic
For an N-frame F = (W ,R1, . . . ,RN) let
F+ denote the (N+1)-frame (W ,∅,R1, . . . ,RN),
F∞ denote the frame (W ,R1, . . . ,RN ,∅,∅, . . . )Hereditary strict orders:F (1) is the class of all finite strict partial orders;F (N+1) = PO[1;G(N)], where GN = {F+ | F ∈ F (N)}.FJ={F∞ | F ∈ F (N) for some N}.
Theorem (Beklemishev, 2007)
There exists a polynomial-time translation f such that for any formula ϕwe have
GLP ` ϕ⇔ FJ � f (ϕ).
![Page 54: PSPACE-decidability of Japaridze's Polymodal Logicaiml08.loria.fr/talks/shapirovsky.pdf · 2008. 9. 10. · Japaridze’s Polymodal Logic [Beklemishev, 2007] Hereditary orders. Main](https://reader036.fdocuments.in/reader036/viewer/2022081615/5fdeba11d0a9a62b6730b7e3/html5/thumbnails/54.jpg)
Decision procedures for ordinal sums of frames. Multimodal case
Let T(1)h,b = Th,b[{C0}], i.e., T
(1)h,b is the class (up to isomorphisms) of all
finite transitive irreflexive trees with the height not more then h and thebranching not more then b. Put
T(N+1)h,b = Th,b[1; {F+ | F ∈ T
(N)h,b }].
Corollary
If an N-modal formula ϕ is FJ-satisfiable, then ϕ is T(N)N,N -satisfiable.
TheoremThe satisfiability problem for F J is in PSPACE.
TheoremJaparidze’s Polymodal Logic is PSPACE-decidable.