6 I R Relations and Posets 2 Model o Distibuted systems events beinnin o pocedue oo temination o ba send o a messae eceive o a messae temination o a p

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1 Relations and Posets 1 Goals o the lectue Relations Posets A un o a distibuted computation Happened-beoe elation cvijay K. Ga Distibuted Systems Fall 94

2 6 I R Relations and Posets 2 Model o Distibuted systems events beinnin o pocedue oo temination o ba send o a messae eceive o a messae temination o a pocess happened-beoe elation Time 12:01 San ose Withdaw $ 10 Time 12:04 - Austin Deposit $ 20 Communication Netwok Time 11:58 New Yok Tanse $ 10 cvijay K. Ga Distibuted Systems Fall 94

3 Relations and Posets 3 Relation X = any set binay elation R is a subset o X X. a Example: X = a; b; c, and = (a; c); (a; a); (b; c); (c; a). R a b h h h c cvijay K. Ga Distibuted Systems Fall 94

4 Relations and Posets 4 Relation [Contd.] Reexive: I o each x 2 X; (x; x) 2 R: Example: X is the set o natual numbes, and R = (x; y) j x divides y: Ieexive: Fo each x 2 X; (x; x) 62 R: Example: X is the set o natual numbes, and R = (x; y) j x less than y: - h o ieexive Reexive h h - h 6 6 h - h h - h cvijay K. Ga Distibuted Systems Fall 94

5 Relations and Posets 5 Relation [Contd.] Symmetic: (x; y) 2 R implies (y; x) 2 R. Examples: is siblin o, x mod k = y mod k: Anti-symmetic: (x; y) 2 R; (y; x) 2 R inplies x = y. Examples:, divides. Asymmetic: (x; y) 2 R implies (y; x) 62 R. Examples: is child o, <. cvijay K. Ga Distibuted Systems Fall 94

6 Relations and Posets 6 Relation [Contd.] Tansitive: (x; y); (y; z) 2 R implies (x; z) 2 R. Examples: is eachable om, <, divides. Example o a symmetic and tansitive but Puzzle: eexive elation. not cvijay K. Ga Distibuted Systems Fall 94

7 R Relations and Posets 7 Patially Odeed Sets [Posets] Patial Ode Reexive Ieexive Tansitive Tansitive Anti-symmetic Anti-symmetic Example: Example: < Examples: X: Gound Set, (2 X ; ) is a ieexive patial ode (N ; divides) is a eexive patial ode (R; ) is a eexive patial ode (also a total ode) causality in a distibuted system (late..) cvijay K. Ga Distibuted Systems Fall 94

8 8 >< >: b); (a; c); (b; d); (a; ); (c; e); (d; e) (c; 9 >= >; QQk 7 Relations and Posets 8 Posets [Contd.] Let Y X, whee (X; ) is a poset. Inmum: m = in(y ) i 8y 2 Y : m y 8x 2 X : (8y 2 Y : x y) ) x m m is also called lb o the set Y. Supemum: s = sup(y ) i (s is also called lub) 8y 2 Y : y s 8x 2 X : (8y 2 Y : y s) ) s x We denote the lb o a; b by a u b, and lub by a t b. e I 6 d X = a; b; c; d; e; c 6 b R = Q Q a cvijay K. Ga Distibuted Systems Fall 94

9 H HHHH H HHj C CO 6 C C C C 7 7 I S S Relations and Posets 9 Lattices sups and ins o nite sets * Lattices Poset Let S be any set, and 2 S be its powe set. The poset (2 S ; ) is a lattice. Set o ationals with usual. Set o lobal states A lattice is an alebaic system (L; t; u) whee t and u commutative, associative and absoption laws. satisy e b So d e d 6 6 I 6 C c b a b c I a a cvijay K. Ga Distibuted Systems Fall 94

10 Relations and Posets 10 Monotone unctions A unction : X! Y is monotone i 8 x; y 2 X : x y ) (x) (y): Examples union, intesection addition, multiplication with positive numbe clocks in distibuted systems y y (y) (x) (y) (x) x x cvijay K. Ga Distibuted Systems Fall 94

11 Relations and Posets 11 Down-Sets and Up-Sets Let (X; <) be any poset. We call a subset Y X a down-set (altenatively, ode ideal) i 2 Y ^ e < ) e 2 Y: Similaly, we call Y X an up-set (altenatively, ode lte) i e 2 Y ^ e < ) 2 Y: We use O(X) to denote the set o all down-sets o X. now show a simple but impotant lemma. We Lemma 1 Let (X; <) be any poset. Then, (O(X); ) is a lattice. cvijay K. Ga Distibuted Systems Fall 94

12 Relations and Posets 12 Run 0; 1 1; 3 2; 3 3; 2 (pc; x) [1] x = x 1 send (x) x = x 1 y = y + 3 eceive (y) y = 2 y [2] 0; 1 1; 4 2; 3 3; 6 (pc; y) Each P in un an tace pocess a execution i eneates e s : 1s, which is a nite sequence o local : states : l i; l 1 i; 0s i; 0e i; i; and events in the pocess P i. state = values o all vaiables, poam counte event = intenal, send, eceive A un is a vecto o taces with [i] as the tace o the P i. pocess cvijay K. Ga Distibuted Systems Fall 94

13 Relations and Posets 13 Relations 0; 1 1; 3 2; 3 3; 2 (pc; x) [1] x = x 1 send (x) x = x 1 y = y + 3 eceive (y) y = 2 y [2] 0; 1 1; 4 2; 3 3; 6 (pc; y) s 1 t i and only i s immediately pecedes t in the tace [i]. s:next = t o t:pev = s wheneve s 1 t. = ieexive tansitive closue o 1. = eexive tansitive closue o 1. event e in the tace [i] ; event in the tace [j] i e is send o a messae and is the eceive event o the same the messae. cvijay K. Ga Distibuted Systems Fall 94

14 pecedes elation the tansitive closue o union o causally 1 and ;. That is, s! t i 2. 9u : (s! u) ^ (u! t) Relations and Posets 14 Relations [Contd.] 0; 1 1; 3 2; 3 3; 2 (pc; x) [1] x = x 1 send (x) x = x 1 y = y + 3 eceive (y) y = 2 y [2] 0; 1 1; 4 2; 3 3; 6 (pc; y) 1. (s 1 t) _ (s ; t), o s and t ae concuent (denoted by sjjt) i :(s! t)^:(t! s). cvijay K. Ga Distibuted Systems Fall 94