A Few Pearls in the Theory of Quasi-Metric Spaces Jean - PowerPoint PPT Presentation

Quasi-Metric Spaces A Few Pearls in the Theory of Quasi-Metric Spaces Jean Goubault-Larrecq ANR Blanc CPP TACL — July 26–30, 2011

Quasi-Metric Spaces Outline 1 Introduction 2 The Basic Theory 3 Transition Systems 4 The Theory of Quasi-Metric Spaces 5 Completeness 6 Formal Balls 7 The Quasi-Metric Space of Formal Balls 8 Notions of Completion 9 Conclusion

Quasi-Metric Spaces Introduction Outline 1 Introduction 2 The Basic Theory 3 Transition Systems 4 The Theory of Quasi-Metric Spaces 5 Completeness 6 Formal Balls 7 The Quasi-Metric Space of Formal Balls 8 Notions of Completion 9 Conclusion

Quasi-Metric Spaces Introduction Metric Spaces Definition (Metric) x = y ⇔ d ( x , y ) = 0 Center d ( x , y ) = d ( y , x ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) Radius

Quasi-Metric Spaces Introduction Quasi-Metric Spaces Definition (Quasi-Metric) x = y ⇔ d ( x , y ) = 0 Center d ( x , y ) = d ( y , x ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) Radius

Quasi-Metric Spaces Introduction Hemi-Metric Spaces Definition (Hemi-Metric) x = y ⇒ d ( x , y ) = 0 Center d ( x , y ) = d ( y , x ) d ( x , y ) ≤ d ( x , z ) + d ( z , y ) Radius

Quasi-Metric Spaces Introduction Goals of this Talk 1 Quasi-, Hemi-Metrics a Natural Extension of Metrics 2 Most Classical Theorems Adapt . . . proved very recently. 3 Non-Determinism and Probabilistic Choice 4 Simulation Hemi-Metrics

Quasi-Metric Spaces Introduction Quasi-Metrics are Natural [Lawvere73]

Quasi-Metric Spaces Introduction Quasi-Metrics are Natural [Lawvere73] y d ( x , y ) = 100 x

Quasi-Metric Spaces Introduction Quasi-Metrics are Natural [Lawvere73] y d ( y , x ) = 100 ? x

Quasi-Metric Spaces Introduction Quasi-Metrics are Natural [Lawvere73] y d ( y , x ) = 100 = 0 . x

Quasi-Metric Spaces The Basic Theory Outline 1 Introduction 2 The Basic Theory 3 Transition Systems 4 The Theory of Quasi-Metric Spaces 5 Completeness 6 Formal Balls 7 The Quasi-Metric Space of Formal Balls 8 Notions of Completion 9 Conclusion

Quasi-Metric Spaces The Basic Theory The Open Ball Topology As in the symmetric case, define: Definition (Open Ball Topology) An open U is a union of open balls. U

Quasi-Metric Spaces The Basic Theory The Open Ball Topology As in the symmetric case, define: Definition (Open Ball Topology) An open U is a union of open balls. . . . but open balls are stranger. Note: there are more relevant topolo- U gies, generalizing the Scott topology [Rutten96,BvBR98], but I’ll try to re- main simple as long as I can. . .

Quasi-Metric Spaces The Basic Theory The Specialization Quasi-Ordering Definition ( ≤ ) Let x ≤ y iff (equivalently): every open containing x also contains y d ( x , y ) = 0. This would be trivial in the symmetric case. Example: d ℝ ( x , y ) = max( x − y , 0) on ℝ . Then ≤ is the usual ordering.

Quasi-Metric Spaces The Basic Theory Excuse Me for Turning Everything Upside-Down. . . . . . but I’m a computer scientist. To me, trees look like this: 1 C : faux vrai 2 3 B : faux vrai faux vrai 4 5 6 7 A : faux vrai faux vrai faux vrai faux vrai 8 9 10 11 12 13 14 15 B A A, B, C B A A C B A A with the root on top, and the leaves at the bottom.

Quasi-Metric Spaces The Basic Theory Excuse Me for Turning Everything Upside-Down. . . . . . but you should really look at hills this way: y d ( x , y ) = 0 (indeed x ≤ y ) x

Quasi-Metric Spaces The Basic Theory Symmetrization Definition ( d sym ) If d is a quasi-metric, then d sym ( x , y ) = max( d ( x , y ) , d ( y , x ) ) � �� � d op ( x , y ) is a metric. Example: d sym ( x , y ) = ∣ x − y ∣ on ℝ . ℝ Motto: A quasi-metric d describes a metric d sym a partial ordering ≤ ( x ≤ y ⇔ d ( x , y ) = 0) and possibly more.

Quasi-Metric Spaces The Basic Theory My Initial Impetus Consider two transition systems T 1 , T 2 . Does T 1 simulate T 2 ? ( T 1 ≤ T 2 ) ( d sym ( T 1 , T 2 ) ≤ 휖 ) Is T 1 close in behaviour to T 2 ? . . . notions of bisimulation metrics [DGJP04,vBW04] These questions are subsumed by computing simulation hemi-metrics between T 1 and T 2 [JGL08].

Quasi-Metric Spaces Transition Systems Outline 1 Introduction 2 The Basic Theory 3 Transition Systems 4 The Theory of Quasi-Metric Spaces 5 Completeness 6 Formal Balls 7 The Quasi-Metric Space of Formal Balls 8 Notions of Completion 9 Conclusion

Quasi-Metric Spaces Transition Systems Non-Deterministic Transition Systems Definition init State space X Transition map 훿 : X → ℙ ( X ) insert-coin Wait I’ll assume: cash-in 훿 ( x ) ∕ = ∅ (no deadlock) cancel M 1 Served 훿 continuous cancel (mathematically serve-coffee insert-coin practical) press-button Serving M 2 훿 ( x ) closed (does not restrict generality) Lower Vietoris topology on ℙ ( X ), generated by ♢ U = { A ∣ A ∩ U ∕ = ∅} , U open

Quasi-Metric Spaces Transition Systems The Hausdorff-Hoare Hemi-Metric Under these conditions, 훿 is a continuous map from X to the Hoare powerdomain ℋ ( X ) = { F closed, non-empty } with lower Vietoris topology.

Quasi-Metric Spaces Transition Systems The Hausdorff-Hoare Hemi-Metric Under these conditions, 훿 is a continuous map from X to the Hoare powerdomain ℋ ( X ) = { F closed, non-empty } with lower Vietoris topology. When X , d is quasi-metric: Definition (“One Half of the Hausdorff Metric”) d ℋ ( F , F ′ ) = sup x ′ ∈ F ′ d ( x , x ′ ) inf x ∈ F Theorem (JGL08) If X op is compact (more generally, precompact), then lower Vietoris = open ball topology of d ℋ on ℋ ( X )

Quasi-Metric Spaces Transition Systems Probabilistic Transition Systems Definition Halt State space X 0.7 0.4 0.1 0.2 Biased Good Transition map 0.2 0.4 훿 : X → V 1 ( X ) 0.3 0.6 0.5 0.5 0.7 0.4 0.5 0.5 I’ll assume: 0.6 0.3 0.5 0.5 V 1 ( X ) space of 0.4 0.7 0.5 0.5 probabilities Probabilistic 0.5 0.5 0.5 0.5 choice 훿 continuous Flip Flip 2 1 0.5 0.5 (mathematically Start practical) Weak topology on V 1 ( X ), generated by ∫ f lsc, r ∈ ℝ + [ f > r ] = { 휈 ∈ V 1 ( X ) ∣ f ( x ) d 휈 > r } , x

Quasi-Metric Spaces Transition Systems The Hutchinson Hemi-Metric Call f : X → ℝ c -Lipschitz iff d ℝ ( f ( x ) , f ( y )) ≤ c × d ( x , y ) (i.e., f ( x ) − f ( y ) ≤ d ( x , y )) When X , d is quasi-metric: Definition (` a la Kantorovich-Hutchinson) ∫ ∫ d H ( 휈, 휈 ′ ) = sup f 1-Lipschitz d ℝ ( x f ( x ) d 휈 ′ ) x f ( x ) d 휈, Theorem (JGL08) If X is totally bounded (e.g. X sym compact) Weak = open ball topology of d H on V 1 ( X ) in the symmetric case, replace d ℝ by d sym ℝ . . . replace total boundedness by separability+completeness?

Quasi-Metric Spaces Transition Systems A Unifiying View Represent spaces of non-det./prob. choice as previsions [JGL07], i.e., certain functionals ⟨ X → ℝ + ⟩ → ℝ + � �� � lsc ∫ 휈 ∈ V 1 ( X ) by 휆 h ∈ ⟨ X → ℝ + ⟩ ⋅ x h ( x ) d 휈 (Markov) F ∈ ℋ ( X ) by 휆 h ∈ ⟨ X → ℝ + ⟩ ⋅ sup x ∈ F h ( x ) Theorem V 1 ( X ) ∼ (F ( h + h ′ ) = F ( h ) + F ( h ′ ) ) = linear previsions ℋ ( X ) ∼ = sup-preserving previsions Leads to natural generalization. . .

Quasi-Metric Spaces Transition Systems A Unifiying View Represent spaces of non-det./prob. choice as previsions [JGL07], i.e., certain functionals ⟨ X → ℝ + ⟩ → ℝ + � �� � lsc ∫ 휈 ∈ V 1 ( X ) by 휆 h ∈ ⟨ X → ℝ + ⟩ ⋅ x h ( x ) d 휈 (Markov) F ∈ ℋ ( X ) by 휆 h ∈ ⟨ X → ℝ + ⟩ ⋅ sup x ∈ F h ( x ) Theorem V 1 ( X ) ∼ (F ( h + h ′ ) = F ( h ) + F ( h ′ ) ) = linear previsions ℋ ( X ) ∼ = sup-preserving previsions Leads to natural generalization. . . Definition and Theorem (Hoare Prevision) � P ( X ) = sublinear previsions ( F ( h + h ′ ) ≤ F ( h ) + F ( h ′ )) encode both ℋ and V 1 , their sequential compositions, and no more.

Quasi-Metric Spaces Transition Systems Mixed Non-Det./Prob. Transition Systems Definition Halt State space X 0.7 0.4 0.1 0.2 Biased Good Transition map 0.2 0.4 훿 : X → � P ( X ) 0.3 0.6 0.5 0.5 0.7 0.4 0.5 0.5 I’ll assume: Non−determ− � P ( X ) space of Hoare inistic choice previsions Probabilistic 0.5 0.5 0.5 0.5 choice 훿 continuous Flip Flip 2 1 (mathematically Start practical) Weak topology on � P ( X ), generated by � f lsc, r ∈ ℝ + [ f > r ] = { F ∈ P ( X ) ∣ F ( f ) > r } ,

Quasi-Metric Spaces Transition Systems Pearl 1: The Hutchinson Hemi-Metric . . . on Previsions ∫ Motto: replace x f ( x ) d 휈 by F ( f ) (“generalized average”) When X , d is quasi-metric: Definition (` a la Kantorovich-Hutchinson) d H ( F , F ′ ) = sup f 1-Lipschitz d ℝ ( F ( f ) , F ′ ( f )) Theorem (JGL08) If X is totally bounded (e.g. X sym compact) Weak = open ball topology of d H on � P ( X ) Also, we retrieve the usual hemi-metrics/topologies on ℋ ( X ) , V 1 ( X ) through the encoding as previsions