Bisimilarities Induced by Relations on Home Page Actions Title - PowerPoint PPT Presentation

Bisimilarities Induced by Relations on Home Page Actions Title Page ◭◭ ◮◮ S. Arun-Kumar sak@cse.iitd.ernet.in ◭ ◮ Department of Computer Science and Engineering Page 1 of 47 I. I. T. Delhi, Hauz Khas, New Delhi 110 016. Go Back http://www.cse.iitd.ernet.in/ ∼ sak Full Screen September 14, 2006 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Outline • Vanilla Bisimulations • Example Home Page • The basic framework Title Page • Bisimulations: Generalisation ◭◭ ◮◮ • Inheritance ◭ ◮ • The Proxy Revisited Page 2 of 47 • An On-the-fly Algorithm Go Back • Conclusions Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Outline • Vanilla Bisimulations • Example Home Page • The basic framework Title Page • Bisimulations: Generalisation ◭◭ ◮◮ • Inheritance ◭ ◮ • The Proxy Revisited Page 3 of 47 • An On-the-fly Algorithm Go Back • Conclusions Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Bisimulations: Vanilla flavoured Let P be the set of processes defined on a set Act of actions. Definition 1 A binary relation R ⊆ P × P is a (strong) Home Page bisimulation if pRq implies the following conditions for all Title Page a ∈ Act . → p ′ ⇒ ∃ q ′ : q → q ′ ∧ p ′ Rq ′ a a ◭◭ ◮◮ p − − (1) ◭ ◮ and → q ′ ⇒ ∃ p ′ : p → p ′ ∧ p ′ Rq ′ a a q − − (2) Page 4 of 47 The largest bisimulation is bisimilarity and is an equivalence, Go Back (denoted ∼ ). Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Bisimulations and Bisimilarity • Simple and intuitively appealing theory • Very nice algebraic properties Home Page • Bisimilarity is the smallest equivalence relation which re- Title Page spects branching behaviour ◭◭ ◮◮ • Park’s induction principle ◭ ◮ • Very efficient algorithms for proving bisimilarity of systems • Has a nice game theoretic interpretation Page 5 of 47 • Algorithms for verification of other equivalences of con- Go Back current systems use bisimulation Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Properties of Bisimulations • The identity relation on processes is a bisimulation • Arbitrary unions of bisimulations are bisimulations. • The converse of each bisimulation is also a bisimulation • The relational composition of bisimulations is a bisimula- Home Page tion. Title Page • Let B be a function on binary relations on P s.t. � p, q � ∈ B ( R ) if p and q satisfy the conditions of definition 1. ◭◭ ◮◮ Then ◭ ◮ – B is monotonic i.e. R ⊆ S ⇒ B ( R ) ⊆ B ( S ) . Page 6 of 47 – R is a strong bisimulation iff R ⊆ B ( R ) . – If R is a strong bisimulation then so is B ( R ) . Go Back – ∼ = � { R | R ⊆ B ( R ) } is the largest fixpoint of B . Full Screen • ∼ is the largest bisimulation and an equivalence relation Close on processes Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Outline • Vanilla Bisimulations • Example Home Page • The basic framework Title Page • Bisimulations: Generalisation ◭◭ ◮◮ • Inheritance ◭ ◮ • The Proxy Revisited Page 7 of 47 • An On-the-fly Algorithm Go Back • Conclusions Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Browser Web Server Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 47 Go Back Browser Browser Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Browser Web Server Home Page 1 t @ Title Page p : w q ◭◭ ◮◮ e r ◭ ◮ Page 9 of 47 Go Back Browser Browser Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Browser Web Server Home Page 2 t 1 @ t @ Title Page p p : w : w q ◭◭ ◮◮ e r ◭ ◮ Page 10 of 47 Go Back Browser Browser Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Browser Web Server Home Page 2 t 1 @ t @ Title Page p p : w : w q ◭◭ ◮◮ e ’ r 1 t @ p ◭ ◮ : w q e r Page 11 of 47 Go Back Browser Browser Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Browser Web Server Home Page 2 t 1 2 @ t t @ @ Title Page p p : p w : : w w q ◭◭ ◮◮ e ’ r 1 t @ p ◭ ◮ : w q e r Page 12 of 47 Go Back Browser Browser Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 47 Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page Title Page ◭◭ ◮◮ req @w:p ◭ ◮ Page 15 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page Title Page t2 ◭◭ ◮◮ req @w:p ◭ ◮ Page 16 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page t1=t2 Title Page t2 ◭◭ ◮◮ req @w:p ◭ ◮ Page 17 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page t1 <> t2 Title Page t2 req w:p ◭◭ ◮◮ req @w:p ◭ ◮ Page 18 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page t1 <> t2 Title Page t2 req w:p ◭◭ ◮◮ req @w:p ◭ ◮ w:p@t3 Page 19 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t1 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page t1 <> t2 Title Page t2 req w:p ◭◭ ◮◮ req @w:p ◭ ◮ w:p@t3 Page 20 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t3 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Example: Proxy Server Web Server Home Page t1 <> t2 Title Page t2 req w:p ◭◭ ◮◮ req @w:p ◭ ◮ w:p@t3 Page 21 of 47 req w:p Go Back Proxy Browser Server Full Screen w:p@t3 w:p@t3 Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Modelling in CCS: Actions The action set. Home Page gp () – g et p age op ( a ) – o utput p age a on screen Title Page drp () – d irect r equest for p age ◭◭ ◮◮ dsp ( h, a ) – d irectly s erve p age irp () – i ndirect r equest for p age ◭ ◮ isp ( h, a ) – i ndirectly s erve p age Page 22 of 47 drh () – d irect r equest for h eader dsh ( h ) – d irectly s erve h eader Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Modelling in CCS A typical client DClient , which accesses the web-server di- rectly , has the following definition. Home Page △ DClient = gp () .drp () .dsp ( h, a ) .op ( a ) . DClient Title Page With the introduction of a proxy server, the clients commu- ◭◭ ◮◮ nicate only with the proxy. The actions involving communications of the clients with ◭ ◮ proxy server are irp and isp . Page 23 of 47 △ = gp () .irp () .isp ( h, a ) .op ( a ) . IClient IClient Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

Modelling in CCS: Proxy • Assume it serves only one request at a time • Initial undefined content ( ⊥ , ⊥ ) in cache • On the first request it obtains the full page from the web- server. Home Page • For each subsequent request it merely sends a request Title Page with the header h 0 as parameter and waits in the state ◭◭ ◮◮ PrWait ( h 0 , a 0 ) , where ( h 0 , a 0 ) is the current content in its cache. ◭ ◮ △ Page 24 of 47 Proxy 0( ⊥ , ⊥ ) = irp () . ReqPage ( ⊥ , ⊥ ) △ Proxy ( h 0 , a 0 ) = irp () . ClWait ( h 0 , a 0 ) Go Back △ ReqPage ( h 0 , a 0 ) = drp () . ReqSent ( h 0 , a 0 ) Full Screen △ ClWait ( h 0 , a 0 ) = drh ( h 0 ) . PrWait ( h 0 , a 0 ) Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit