A Semantic PSPACE Criterion for Coalgebraic Modal Logic Lutz Schr - - PowerPoint PPT Presentation

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A Semantic PSPACE Criterion for Coalgebraic Modal Logic Lutz Schr oder IFIP WG 1.3 Meeting, Braga, Mar 2007 2 Introduction Modal logics (ideally) combine the right (taylored) level of expressivity relative computational

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A Semantic PSPACE Criterion for Coalgebraic Modal Logic

Lutz Schr¨

  • der

IFIP WG 1.3 Meeting, Braga, Mar 2007

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2

Introduction

  • Modal logics (ideally) combine
  • the right (taylored) level of expressivity
  • relative computational tractability
  • Large zoo of non-normal modal logics
  • Upper complexity bounds frequently non-trivial and ad-hoc
  • Coalgebraic modal logic provides unifying framework
  • Here: semantic PSPACE algorithm for rank-0-1 logics
  • Applications:
  • modal logics of quantitative uncertainty
  • conditional logics
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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3

Complexity of Modal Logics

  • ‘Next-step’ modal logics are typically PSPACE, e.g.
  • K (KB, S4, . . . ): witness algorithm for shallow Kripke models

(Ladner 77, Halpern/Moses 92)

  • Graded modal logic (GML): constraint set algorithm (Tobies 01)
  • Logic of knowledge and probability: shallow model method based on

local small model property (Fagin/Halpern 94)

  • Epistemic logic (Vardi 89), coalition logic (Pauly 02): shallow

neighbourhood models.

  • Conditional logic: sequent calculus (Olivetti/Schwindt 01)
  • Presburger and regular modal logic: check local models on the fly

(Demri/Lugiez IJCAR 06).

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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4

Coalgebraic Modal Logic

Coalgebraic modal logic unifies all these different logics

  • Prove meta-theoretic results in the general framework:
  • Hennessy-Milner [Pattinson NDJFL 04, Schr¨
  • der FOSSACS 05]
  • Completeness [Pattinson TCS 03, Schr¨
  • der FOSSACS 06]
  • Duality, ultrafilter extensions [Kupke/Kurz/Pattinson CALCO 05]
  • Finite model property [Schr¨
  • der FOSSACS 06]
  • Generic algorithms and upper complexity bounds:
  • Obtain PSPACE bounds by uniform methods

[Schr¨

  • der/Pattinson LICS 06 and here]
  • clarity, reusability, extendability
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples

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Examples 6

Quantitative Uncertainty

(Fagin, Halpern, Megiddo, Pucella) φ ::= ⊥ | p | ¬φ | φ1 ∧ φ2 | n

i=1 aiw(φi) ≥ b

(ai, b ∈ Q) Weights w(φ) ∈ R: ‘likelihood’ of φ in the next step, e.g.

  • Probability (FHM IC 1994)
  • Upper probability (HP JAI 2002)
  • Dempster-Shafer belief (HP UAI 2002)
  • Dubois-Prade possibility (HP UAI 2002)

Extension: Expectations e( ciφi) (HP UAI 02)

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 7

Examples 7

Quantitative Uncertainty: Semantics

  • Set X of states
  • For x ∈ X, probability distribution (belief function, . . . )

Px on X describing uncertain transitions

  • 0.2

◮ ◭

0.5

  • 0.1
  • 0.2
  • 0.9

◮ ◭

1 0.8

◮

  • 0.3

◮

  • w(φ) is interpreted in state x as Px{y | y |

= φ}

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 8

Examples 8

Conditional Logics

φ ::= ⊥ | p | ¬φ | φ1 ∧ φ2 | φ1 ⇒ φ2

  • φ ⇒ ψ reads ‘ψ holds under condition φ’
  • Basic axioms ❀ conditional logic CK:
  • Replacement of equivalents on the left
  • Commutation with ∧ on the right
  • Relevance logics: a ⇒ a, (a ⇒ b) → a → b
  • Default logics, e.g. Burgess’ System C (generalizing KLM)

(REF) a ⇒ a (OR) (a ⇒ c) → (b ⇒ c) → (a ∨ b ⇒ c) (CM) (a ⇒ c) → (a ⇒ b) → ((a ∧ b) ⇒ c)

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 9

Conditional Logic CK: Semantics

Conditional Frames:

  • Set X of states
  • For x ∈ X, function fx : P(X) → P(X).
  • x |

= φ ⇒ ψ iff fx[ [φ] ] ⊆ [ [ψ] ] (where [ [φ] ] = {y | y | = φ})

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 10

Coalgebra

T : Set → Set T-Coalgebra (X, ξ) = map ξ : X → TX X: set of states ξ: transition map ξ(x): structured collection of observations/successor states

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 11

Coalgebraic Modal Logic

[Pattinson NDJFL 04; Schr¨

  • der FOSSACS 05]

Interpret n-ary modal operator L by predicate lifting [ [L] ]X : Qn → Q ◦ T op (Q contravariant powerset). Semantics of L in T-coalgebra (X, ξ): x | =(X,ξ) L (φ1, . . . , φn) ⇐ ⇒ ξ(x) ∈ [ [L] ]X([ [φ1] ], . . . , [ [φn] ]) (where [ [φ] ] = {y | y | = φ})

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 12

Examples

  • Quantitative Uncertainty:
  • TX = likelihood measures over X
  • operators L(a1, . . . , an; b)(φ1, . . . , φn) ≡ aiw(φi) ≥ b
  • [

[L(a1, . . . , an; b)] ]X(A1, . . . , An) = {P ∈ TX | aiP(Ai) ≥ b}

  • Conditional Logic CK:
  • TX = Q(X) → P(X)
  • [

[ ⇒ ] ]X(A, B) = {f ∈ TX | f(A) ⊆ B}.

  • Coalition Logic, majority logic, . . .
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 13

The One-Step Logic

No nesting, no transitions Formally: equivalent to one-step pairs (φ, ψ) over V , where φ ∈ Prop(V ) ψ conjunctive clause over atoms L(a1, . . . , an), ai ∈ V For set X, P(X)-valuation τ:

  • [

[φ] ]τ ⊆ X

  • X, τ |

= φ ⇐ ⇒ [ [φ] ]τ = X

  • [

[ψ] ]τ ⊆ TX, with [ [L(a1, . . . , an)] ] = [ [L] ](τ(a1), . . . , τ(an))

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 14

Examples 14

The One-Step Polysize Model Property

One-step model (X, τ, t) of (φ, ψ):

  • X, τ |

= φ

  • t ∈ [

[ψ] ]τ ⊆ TX OSPMP: (φ, ψ) one-step satisfiable = ⇒ has one-step model of polynomial size in ψ (Size refers to |X| and size(t))

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 15

PSPACE semantically

One-step model checking: t ∈ [ [L] ](A1, . . . , An)? Theorem If L has the OSPMP, then L has the polynomially branching shallow model property Corollary If L has the OSPMP and one-step model-checking is in P, then

  • L is in PSPACE
  • Lk (nesting of modal operators bounded by k) is in NP.
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 16

Example: Quantitative Uncertainty

  • [FHM IC 1994; HP JAI 02; HP UAI 02] prove

small model properties for one-step logics (→ NP)

  • From these proofs, extract OSPMP
  • Obtain polynomially branching shallow models,

PSPACE/NP bounds

  • Known for probability [FHM 94]
  • Novel for other cases (only one-step logics considered so far)
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 17

Example: CK

Representation of f : Q(X) → P(X) by lists of maplets, default: f(A) = ∅.

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 17

Example: CK

Representation of f : Q(X) → P(X) by lists of maplets, default: f(A) = ∅. (φ, ψ) one-step pair, ψ = ±i(ai ⇒ bi); (X, τ, f) one-step model of (φ, ψ):

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 19

Examples 17

Example: CK

Representation of f : Q(X) → P(X) by lists of maplets, default: f(A) = ∅. (φ, ψ) one-step pair, ψ = ±i(ai ⇒ bi); (X, τ, f) one-step model of (φ, ψ):

  • Pick yij ∈ τ(ai)∆τ(aj) if τ(ai) = τ(aj);
  • pick zi ∈ f(τ(ai)) \ τ(bi) if ±i = ¬;
  • put Y = {yij | . . . } ∪ {zi | . . . };
  • put τ ′(v) = τ(v) ∩ Y , f : τ ′(ai) → f(τ(ai)) ∩ Y
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 20

Examples 17

Example: CK

Representation of f : Q(X) → P(X) by lists of maplets, default: f(A) = ∅. (φ, ψ) one-step pair, ψ = ±i(ai ⇒ bi); (X, τ, f) one-step model of (φ, ψ):

  • Pick yij ∈ τ(ai)∆τ(aj) if τ(ai) = τ(aj);
  • pick zi ∈ f(τ(ai)) \ τ(bi) if ±i = ¬;
  • put Y = {yij | . . . } ∪ {zi | . . . };
  • put τ ′(v) = τ(v) ∩ Y , f : τ ′(ai) → f(τ(ai)) ∩ Y

→ have polysize one-step model (Y, τ ′, f ′), DONE!

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 17

Example: CK

Representation of f : Q(X) → P(X) by lists of maplets, default: f(A) = ∅. (φ, ψ) one-step pair, ψ = ±i(ai ⇒ bi); (X, τ, f) one-step model of (φ, ψ):

  • Pick yij ∈ τ(ai)∆τ(aj) if τ(ai) = τ(aj);
  • pick zi ∈ f(τ(ai)) \ τ(bi) if ±i = ¬;
  • put Y = {yij | . . . } ∪ {zi | . . . };
  • put τ ′(v) = τ(v) ∩ Y , f : τ ′(ai) → f(τ(ai)) ∩ Y

→ have polysize one-step model (Y, τ ′, f ′), DONE! (Olivetti et al. ACM TOCL: sequent calculus, 51 pages)

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 18

Rank-0-1 Logics

  • Rank-0-1 axioms: clauses over atoms a or L(a1, . . . , an).
  • Models: coalgebras satisfying the axioms for all valuations
  • Examples:
  • KT: ✷a → a, models: reflexive frames
  • Almost all conditional logics, e.g. CK+MP:

(a ⇒ b) → a → b, models: x ∈ A = ⇒ x ∈ fx(A).

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 23

Examples 19

One-Step Models in Rank-0-1

. . . are quadruples (X, τ, t, x0), where x0 ∈ X present state Require that (X, t, x0) satisfies the axioms for all valuations:

  • (X, κ, t, x0) |

= L(a1, . . . , an) defined as before

  • (X, κ, t, x0) |

= a iff x ∈ κ(a) OSPMP:

  • Require preservation of τ-theory of x0 in small model.
  • Represent pairs (t, x) ∈ TX × X.

For PSPACE, need ‘Is (X, τ, t, x0) one-step model?’∈ P.

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 24

Examples 20

Example: CK+MP

Everything as for CK, except

  • Default f(A) = A ∩ {x0}
  • retain x0 in small one-step model
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 25

Examples 21

Conclusion

  • Coalgebraic modal logic subsumes ‘all’ modal logics
  • Generic algorithms reproduce tight PSPACE bounds

and prove new ones

  • Compact, reusable, and clear proofs
  • Bounding the nesting depth reduces complexity to NP
  • Can now handle all axioms without nested modalities
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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Examples 22

Future Work

  • Coalgebraic CTL
  • Coalgebra automata (Kupke/Venema)? :-(
  • Pseudomodels (Emerson/Halpern) :-|
  • How do we tackle rank n?
  • Optimized automatic reasoners?
  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007
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SLIDE 27

Examples 23

Coalition Logic

(Pauly 2002) TX = ∃

sets of strategies

  • Σ1 . . . Σn

. Σi → X

  • utcome function

, where N = {1, . . . , n} set of agents. For coalition C ⊂ N, [C]φ = ‘C can force φ’. [ [[C]] ]XA = {f : Σi → X | ∃σC. ∀σN−C. f(σC, σN−C) ∈ A}.

  • L. Schr¨
  • der: A Semantic PSPACE Criterion for Coalgebraic Modal Logic; IFIP WG 1.3 Meeting, Braga, Mar 2007