D EPARTMENT OF C OMPUTER S CIENCES Semantics of Orc William Cook and Jayadev Misra f cook,misra g @cs.utexas.edu Department of Computer Science University of Texas at Austin Email: web: http://www.cs.utexas.edu/users/psp U NIVERSITY OF T EXAS AT A USTIN 0
D EPARTMENT OF C OMPUTER S CIENCES Outline � Define an asynchronous semantics, using labeled transitions. � Refine asynchronous semantics to a synchronous semantics. An expression transits to another expression, causing an event. Labels are events. U NIVERSITY OF T EXAS AT A USTIN 1
D EPARTMENT OF C OMPUTER S CIENCES f ; g ; h 2 E xpr M ( P ) Formal Syntax j j E ( P ) j j f > x > g j j f j g j j f where x : 2 g ::= Site call Expression call p 2 A tual x j j j j M Sequential composition Symmetric composition q 2 F or mal x j j M Asymmetric composition ::= ::= U NIVERSITY OF T EXAS AT A USTIN 2
D EPARTMENT OF C OMPUTER S CIENCES ? u and l et ( ) as two possible expressions. Enhanced Syntax f ; g ; h 2 E xpr M ( P ) j j E ( P ) j j f > x > g Add j j f j g j j f where x : 2 g ::= Site call j j ? u Call to definition j j let ( ) Sequential composition Symmetric composition Asymmetric composition Waiting for response Ready to Publish U NIVERSITY OF T EXAS AT A USTIN 3
D EPARTMENT OF C OMPUTER S CIENCES Events l 2 E v ent M h ; u i u j j u ? j j y There are 4 kinds of events. j j � ::= Site call with handle Response publish silent transition Response is outside the control of Orc. U NIVERSITY OF T EXAS AT A USTIN 4
D EPARTMENT OF C OMPUTER S CIENCES u fresh Rules for Site Call M h ;u i M ( ) , ! ? u u ? (S ITE C ALL ) ? u , ! let ( ) y let ( ) , ! 0 (S ITE R ET ) (L ET ) U NIVERSITY OF T EXAS AT A USTIN 5
D EPARTMENT OF C OMPUTER S CIENCES l 0 f , ! f Symmetric Composition l 0 f j g , ! f j g l 0 g , ! g (S YM 1) l 0 f j g , ! f j g (S YM 2) U NIVERSITY OF T EXAS AT A USTIN 6
D EPARTMENT OF C OMPUTER S CIENCES l 0 f , ! f l 6 = y Sequencing l 0 f > x > g , ! f > x > g y 0 f , ! f (S EQ 1N) � 0 f > x > g , ! ( f > x > g ) j [ =x ℄ g (S EQ 1V) U NIVERSITY OF T EXAS AT A USTIN 7
D EPARTMENT OF C OMPUTER S CIENCES l 0 f , ! f l 6 = y Asymmetric Composition l 0 g where x : 2 f , ! g where x : 2 f y 0 f , ! f (A SYM 1N) � g where x : 2 f , ! [ =x ℄ g l 0 g , ! g (A SYM 1V) l 0 g where x : 2 f , ! g where x : 2 f (A SYM 2) U NIVERSITY OF T EXAS AT A USTIN 8
D EPARTMENT OF C OMPUTER S CIENCES [[ E ( q ) � f ℄℄ 2 D Expression Call � E ( p ) , ! [ p=q ℄ f (D EF ) U NIVERSITY OF T EXAS AT A USTIN 9
D EPARTMENT OF C OMPUTER S CIENCES Rules U NIVERSITY OF T EXAS AT A USTIN 10
l u fresh 0 f , ! f l 6 = y M h ;u i l 0 M ( ) , ! ? u f > x > g , ! f > x > g D EPARTMENT OF C OMPUTER S CIENCES u ? y 0 ? u , ! let ( ) f , ! f � 0 y f > x > g , ! ( f > x > g ) j [ =x ℄ g let ( ) , ! 0 l 0 l f , ! f l 6 = y 0 f , ! f l 0 l g where x : 2 f , ! g where x : 2 f 0 f j g , ! f j g y 0 l 0 f , ! f g , ! g � l g where x : 2 f , ! [ =x ℄ g 0 f j g , ! f j g l 0 g , ! g [[ E ( q ) � f ℄℄ 2 D � l 0 E ( p ) , ! [ p=q ℄ f g where x : 2 f , ! g where x : 2 f U NIVERSITY OF T EXAS AT A USTIN 11
D EPARTMENT OF C OMPUTER S CIENCES Pending Event � transform the expression, At any moment, there is a set of pending events. Processing a pending event may � change the set of pending events. U NIVERSITY OF T EXAS AT A USTIN 12
D EPARTMENT OF C OMPUTER S CIENCES M ( x ) where x : 2 N j R , both N h u i and R h v i are pending. Example of Event Processing N h u i M ( x ) where x : 2 N j R , ! M ( x ) where x : 2 ? u j R R h v i is still pending. So are u ? x , for all x . In u ? M ( x ) where x : 2 ? u j R , ! M ( ) R h v i and u ? x are no longer pending. After U NIVERSITY OF T EXAS AT A USTIN 13
D EPARTMENT OF C OMPUTER S CIENCES Rules for Event Processing � (Fairness) If there is an internal pending event, some pending event is � (Asynchrony) Order and timing of event processings are arbitrary. processed eventually. U NIVERSITY OF T EXAS AT A USTIN 14
D EPARTMENT OF C OMPUTER S CIENCES Notes on Event Processing � Fairness is minimal progress. It does not say that: � Only internal events are under client’s control. � If there are only external pending events, no event may be processed. an event which remains pending is eventually processed. U NIVERSITY OF T EXAS AT A USTIN 15
D EPARTMENT OF C OMPUTER S CIENCES Examples � let ( x ) where x : 2 let (0) j R timer (1) � R R ’s response, if any, is never fully processed. � (Fairness) Metr onome � ( l et (0) j l et (1)) If it publishes 0, � (Asynchrony) let ( x ) where x : 2 let (0) j R timer (1) � l et (1) May publish 0 forever. may publish 0 or 1. U NIVERSITY OF T EXAS AT A USTIN 16
D EPARTMENT OF C OMPUTER S CIENCES Synchronous Semantics l 2 A tion M h ; u i u Specify time (and order) of event processing. j j y l et ( ) is internal j j � Internal event (action): u ? ::= Site call with handle publish; silent transition External Event (response): Rule 1: Process a response only if there is no action. Order among internal events is arbitrary. Order among external responses is arbitrary. U NIVERSITY OF T EXAS AT A USTIN 17
D EPARTMENT OF C OMPUTER S CIENCES let ( x ) where x : 2 let (0) j R timer (1) � l et (1) Examples let ( x ) where x : 2 let (0) j let (2) j R timer (1) � l et (1) let ( x ) where x : 2 if ( true ) � let (0) j R timer (1) � l et (1) let ( x ) where x : 2 R timer (1) � let (0) j R timer (2) � l et (1) (1) (2) (3) (4) (1) publishes 0. (2) publishes 0 or 2. (3) publishes 0 or 1. (4) publishes 0 or 1. U NIVERSITY OF T EXAS AT A USTIN 18
D EPARTMENT OF C OMPUTER S CIENCES Handle Time let ( x ) where x : 2 R timer (1) � let (0) j R timer (2) � l et (1) Rule 2: Process events as soon as possible. Assume event processing is instantaneous. (4) publishes 0. U NIVERSITY OF T EXAS AT A USTIN 19
D EPARTMENT OF C OMPUTER S CIENCES Immediate/deferred Sites let , if , add , or , � � � � Designate certain sites as immediate, rest as deferred. R timer , CNN � � � An immediate site has to respond instantaneously. Immediate Sites: let ( x ) where x : 2 if ( true ) � let (0) j R timer (1) � l et (1) Deferred Sites: Rule 3: A response from an immediate site is an internal event. (3) publishes 0. U NIVERSITY OF T EXAS AT A USTIN 20
D EPARTMENT OF C OMPUTER S CIENCES Positive/Negative Response � positive response: a result value, written as u ? , or An immediate site responds immediately with � negative response: that it will be silent, written as u ?? u ?? ? u , ! 0 (S ITE R ET ) U NIVERSITY OF T EXAS AT A USTIN 21
D EPARTMENT OF C OMPUTER S CIENCES Summary of Rules for Synchronous Semantics � Process a response only if there is no action. � Process events as soon as possible. � A response from an immediate site is an internal event. U NIVERSITY OF T EXAS AT A USTIN 22
D EPARTMENT OF C OMPUTER S CIENCES Formal definition of Synchronous Semantics � Define quiescent expression, QE xpr , one that has no internal event. � Define non-quiescent expression, NExpr , its complement. , ! : Expr � Event � Expr f Defined earlier g r r , ! R : QExpr � R esp onse � Expr = f ( q ; r ; e ) j q , ! e g a a , ! A : NExpr � A tion � Expr = f ( q ^ ; a; e ) j q ^ , ! e g l l l , ! S : Expr � Event � Expr = , ! R [ , ! A U NIVERSITY OF T EXAS AT A USTIN 23
D EPARTMENT OF C OMPUTER S CIENCES Round-based Execution � A round consists of processing internal events. This includes calls to � A round ends when no more internal events can be processed. � First round starts at the beginning of evaluation. and responses from immediate sites. � Subsequent rounds start by processing a response from a deferred site. U NIVERSITY OF T EXAS AT A USTIN 24
D EPARTMENT OF C OMPUTER S CIENCES Laws of Kleene Algebra j ) f j 0 = f j ) f j g = g j f j ) ( f j g ) j h = f j ( g j h ) j ) f j f = f � ) ( f � g ) � h = f � ( g � h ) (Zero and � ) 0 � f = 0 (Commutativity of � ) f � 0 = 0 (Associativity of � ) S ig nal � f = f (Idempotence of � ) f > x > let ( x ) = f (Associativity of � over j ) f � ( g j h ) = ( f � g ) j ( f � h ) (Left zero of � over j ) ( f j g ) � h = ( f � h j g � h ) (Right zero of (Left unit of (Right unit of (Left Distributivity of (Right Distributivity of U NIVERSITY OF T EXAS AT A USTIN 25
D EPARTMENT OF C OMPUTER S CIENCES j ) f j f = f Laws which do not hold � ) f � 0 = 0 � over j ) f � ( g j h ) = ( f � g ) j ( f � h ) (Idempotence of (Right zero of (Left Distributivity of U NIVERSITY OF T EXAS AT A USTIN 26
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