Equivalence between transition systems. Modal logic and first order logic. In pictures: forth condition. Bisimilation. In pictures: back condition

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1 odal logic and first order logic odal logic: local ie of the strctre ( here can I get by folloing links from here ). First order logic: global ie of the strctre (can see eerything, qantifiers do not follo edges). eaningfl eqialence beteen strctres: first order logic: (artial) isomorhism modal logic: to strctres shold hae the same edge-folloing behaior Eqialence beteen transition systems Trace eqialence? A a a b c B a choice beteen b and c is made at different oints b c GS odal Logic: lectre 3 1 GS odal Logic: lectre 3 2 Bisimilation In ictres: forth condition Let = (W,R,V) and = (W,R,V ) be to Krike strctres. A non-emty binary relation is called a bisimlation beteen and if the folloing conditions are satisfied: if (, ) then and satisfy the same roositional letters if (, ) and R(,) in, then there exists in sch that R (, ) and (, ) (the forth condition) if (, ) and R (, ) in, then there exists in sch that R(,) and (, ) (the back condition) R = {, q} = {,q} = {,q} GS odal Logic: lectre 3 3 GS odal Logic: lectre 3 4 In ictres: forth condition In ictres: back condition = {, q} = {, q} = {, q} R R R = {,q} = {,q} = {,q} = {,q} GS odal Logic: lectre 3 5 GS odal Logic: lectre 3 6 1

2 In ictres: back condition Bisimilar strctres = {, q} = {,q} R R = {, q} = {,q} We shall call and bisimilar if there is a non-emty bisimlation relation beteen them. Sometimes e consider rooted strctres: strctres here there is a distingished root/initial state of the system. For rooted strctres, and are bisimilar if there is a bisimlation sch that (root of, root of ) are in. Let s look at rooted examles. GS odal Logic: lectre 3 7 GS odal Logic: lectre 3 8 Examles: are and bisimilar? Examles: are and bisimilar? = {, q} = {, q} = {, q} y = {,q} y = {,q} x = {,q} = {,q} x = {,q} root = {,q} = {,q} GS odal Logic: lectre 3 9 easier if e color orlds satisfying same roositional ariables the same color! GS odal Logic: lectre 3 10 Are and bisimilar? Are and bisimilar? GS odal Logic: lectre 3 11 GS odal Logic: lectre

3 Are and bisimilar? Bisimlation and modal logic Sose and are to Krike strctres and is a bisimlation beteen and sch that (, ). Theorem: for eery formla φ of basic modal logic,, = φ iff, = φ In other ords, modal logic cannot distingish bisimilar strctres (can describe them to bisimlation eqialence). GS odal Logic: lectre 3 13 GS odal Logic: lectre 3 14 Proof, = φ iff, = φ if (, ) Proof contined, = φ iff, = φ if (, ) Proof by indction on φ. φ is a roositional ariable: from (, ) and satisfy the same roositional ariables φ is ψ:, = ψ iff, ψiff (by the indctie hyothesis), ψiff, = ψ φ is ψ χ:, = ψ χ iff, ψor, = χ iff (by the indctie hyothesis), ψor, = χ iff, = ψ χ. Proof by indction on φ. φ is []ψ:, = []ψ iff for all sch that R(,),, = ψ. Sose, = []ψ and, []ψ. Then there is a orld in sch that R (, ) and, ψ. By the back condition, there is a in sch that R(,) and (, ). By the indctie hyothesis,, ψ. So, []ψ: a contradiction. Similarly for, = []ψ and, []ψ. GS odal Logic: lectre 3 15 GS odal Logic: lectre 3 16 Reerse? Conterexamle for the infinite case odal logic does not distigish bisimilar strctres. Is the reerse tre: if to strctres are indistingishable by modal formlas, they are bisimilar? This is tre for finite strctres. If to finite strctres satisfy the same modal formlas, then they are bisimilar GS odal Logic: lectre 3 17 a branch of length n for eery n a branch of length n for eery n, ls an infinite branch GS odal Logic: lectre

4 Some other illminating roerties Examle Define modal deth of φ as the maximal deth of nesting of modalities in φ. For examle, ( [] ) ( [] ) [](( [] ) ( [] )) has modal deth 2. A formla of modal deth n can t see frther than n stes from the crrent orld. If e change something abot the orlds accessible in more than n stes, the trth ale of formla ill not change. ( [] ) ( []) [](( [] ) ( [])) GS odal Logic: lectre 3 19 GS odal Logic: lectre 3 20 Examle Examle ( [] ) ( []) [](( [] ) ( [])) ( [] ) ( []) [](( [] ) ( [])) GS odal Logic: lectre 3 21 GS odal Logic: lectre 3 22 Illminating roerties contined Examle: nraelling Tree model: eery satisfiable formla has a tree model. We obtain it by nraelling the original model into a tree (and roing that it is bisimilar to the original model). By or reios reslt, e already kno that eery satisfiable formla has a finite model. If e nrael a finite cyclic model, e get an infinite tree. Hoeer, e kno that the formla only cares abot orlds accessible in n stes, here n is the formla s modal deth. So e can cho the infinite tree after n leels. Each satisfiable formla has a finite tree model. GS odal Logic: lectre 3 23 GS odal Logic: lectre

5 Examle: nraelling Original strctre Unraelled strctre Examle: nraelling Each orld is bisimilar to all coies of itself in the tree GS odal Logic: lectre 3 25 GS odal Logic: lectre 3 26 Bisimilarity checking algorithm Sose e are gien to rooted grahs (W,R, ) and (W,R, ) (ignore V for the moment). We ant to check if they are bisimilar ith resect to back and forth conditions, that is if there is a bisimlation beteen them sch that (, ). This is basically the same as saying: is there a bisimlation-indced eqialence relation on the set of ertices of the grah (W,R) = (W W, R R ) sch that and are in the same eqialence class? Relational coarsest artition Let V, V W Define R -1 (V) as { in W: for some in V, R(,)} V is stable ith resect to V if either V R -1 (V ) or V R -1 (V ) =. Intitiely, V is `a kind of ertices and to be an eqialence class, V shold either only see ertices in V or not see any ertices in V. Otherise ertices in V are not eqialent ith resect to seeing ertices in V. If V is not stable ith resect to V, say V slits V. We need to find a coarsest stable artition of W, i.e. sch that none of the eqialence classes slits another. GS odal Logic: lectre 3 27 GS odal Logic: lectre 3 28 Relational coarsest artition algorithm Examle Start ith any artition of Q of W (for examle, Q={W, }). Reeat ntil the reslting artition is stable: find a set S hich is a nion of some of the eqialence classes in Q, sch that S slits some of the classes in Q relace Q = {Q 1,,Q n } ith {Q 1 R -1 (S), Q 1 -R -1 (S),, Q n R -1 (S), Q n -R -1 (S)} Q = { W, } slitter: W; R -1 (W) ={,x} Q = {{,x}, {,,y} } no stable x y GS odal Logic: lectre 3 29 GS odal Logic: lectre

6 Exercise The algorithm aboe ignores roositional ariables. Write a ersion hich does not ignore ariables (checks for bisimlation satisfying all three conditions). GS odal Logic: lectre