D EPARTMENT OF C OMPUTER S CIENCES Distributed Execution 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 Centralized Execution Model � semantics (and execution) described for one machine. � Then, � We develop a theory to permit distributed execution. – immediate sites respond instantaneously. – Rtimer is exact. U NIVERSITY OF T EXAS AT A USTIN 1
D EPARTMENT OF C OMPUTER S CIENCES � Assign subexpressions to different machines. Distributed Execution Model � Execution starts on a goal machine. � A machine requests another machine to start evaluation. � The calling machine supplies the context and values of variables as � The calling machine orders all tokens to be killed, and asks for ack. � All messages among these machines are delivered after arbitrary but they become defined. finite delay. U NIVERSITY OF T EXAS AT A USTIN 2
D EPARTMENT OF C OMPUTER S CIENCES � R timer (1) � l et (0) j R timer (2) � l et (1) When can we distribute? � P ( ; e ) � :g et > x > Compute ( x ) > y > e:put ( y ) � P ( ; e ) N � P ( ; e ) j P ( d; e ) Neither branch can be executed on another machine. :g et and e:put are on different machines. P ( ; e ) and P ( d; e ) can be evaluated on different machines. � R timer (1) � M j R timer (2) � N The two branches can be executed on different machines. The theory identifies subexpressions which can be distributed. U NIVERSITY OF T EXAS AT A USTIN 3
D EPARTMENT OF C OMPUTER S CIENCES Distributed Execution has to be faithful � A distributed execution is valid if it mimics a centralized execution. � The sites can not distinguish between the two execution styles. U NIVERSITY OF T EXAS AT A USTIN 4
D EPARTMENT OF C OMPUTER S CIENCES An example � Site k ey boar d responds at arbitrary times, or never. � Site s r een responds immediately and displays a message. � � � k ey boar d � s r een � � � Its response is received immediately. can not be distributed. Video-game: A human responds sometime on the keyboard. Expects to see an echo immediately. U NIVERSITY OF T EXAS AT A USTIN 5
D EPARTMENT OF C OMPUTER S CIENCES � A site is punctual if communication delay with it is zero. Punctual Site � A site is unpunctual if communication delay with it is arbitrary. � Simplification: Assume all sites are either punctual or unpunctual. � A punctual site has to be implemented on the caller’s machine. � An unpunctual site may be implemented on a remote machine. R timer , l et , if , � � � , all immediate sites, � � � , k ey boar d , s r een 0 , CNN , PostWeb , � � � M Punctual: Unpunctual: Generic symbol U NIVERSITY OF T EXAS AT A USTIN 6
D EPARTMENT OF C OMPUTER S CIENCES Punctual Expression � Any delay in starting it or processing its response can be detected. � Some timing information for site calls or publications is known. � A punctual expression has to be implemented on the caller’s machine. � An unpunctual expression may be implemented on another machine. � A punctual expression may have unpunctual subexpressions. The subexpression may be implemented on another machine. U NIVERSITY OF T EXAS AT A USTIN 7
D EPARTMENT OF C OMPUTER S CIENCES Definition of Punctual Expression � S ( x ) : S is punctual, A punctual expression is of the form � f j g : either f or g is punctual, � f > x > g : both f and g are punctual, � f where x : 2 g : both f and g are punctual, or f n x is punctual. f n x from f : x by 0 . Obtain Replace all site calls which have parameter U NIVERSITY OF T EXAS AT A USTIN 8
D EPARTMENT OF C OMPUTER S CIENCES Examples: Punctual Expressions l et ( x ) l et is punctual (immediate) l et ( x ) > x > R timer ( x ) R timer (1) where x : 2 g f n x = R timer (1) is punctual l et ( x ) where x : 2 R timer (1) l et ( x ) and R timer (1) are both punctual both sites are punctual R timer (1) j h where x : 2 g f n x = R timer (1) j h n x is punctual l et ( x ) > x > R timer (1) l et ( x ) > x > R timer (1) and R timer (1) where x : 2 R timer (1) are both punctual U NIVERSITY OF T EXAS AT A USTIN 9
D EPARTMENT OF C OMPUTER S CIENCES M � R timer (1) Examples: Unpunctual Expressions R timer (1) � M N ( x ) > x > R timer ( x ) R timer ( x ) where x : 2 M N ( x ) where x : 2 R timer (1) l et ( x ) where x : 2 M U NIVERSITY OF T EXAS AT A USTIN 10
D EPARTMENT OF C OMPUTER S CIENCES k ey boar d � s r een j JoySti k � ( M j N ) , where punctual within unpunctual, and vice versa k ey boar d , s r een , JoySti k are punctual M and N are unpunctual. k ey boar d � s r een j JoySti k � ( M j N ) | {z } | {z } | {z } pun tual pun tual unpun tual | {z } unpun tual | {z } pun tual U NIVERSITY OF T EXAS AT A USTIN 11
D EPARTMENT OF C OMPUTER S CIENCES k ey boar d � s r een j JoySti k � ( M j N ) Example: Distributed Execution | {z } | {z } pun tual unpun tual � k ey boar d � s r een j f f � JoySti k � ( M j N ) | {z } unpun tual � k ey boar d � s r een j f f � JoySti k � g g � M j N U NIVERSITY OF T EXAS AT A USTIN 12
D EPARTMENT OF C OMPUTER S CIENCES � If all sites in an expression are punctual, the expression is punctual. Some properties � If all sites in an expression are unpunctual, the expression is � Monotonicity: An unpunctual expression remains unpunctual if you unpunctual. M punctual, f ( M ) unpunctual, N unpunctual ) f ( N ) unpunctual. replace any site by an unpunctual site. M v N ) f ( M ) v f ( N ) , < unpunctual where punctual Proofs by structural induction. U NIVERSITY OF T EXAS AT A USTIN 13
D EPARTMENT OF C OMPUTER S CIENCES � l et ( true ) > b > ( if ( b ) � g � g Punctuality is conservative j if ( : b ) � R timer (1) ) � R andom > x > R timer ( x ) R andom publishes a random natural number within some bounds. Though the execution can be distributed, the expression is punctual. � Nat > x > R timer ( x ) N at publishes any natural number. The expression is punctual, and it can not be distributed. Though the execution can be distributed, the expression is punctual. U NIVERSITY OF T EXAS AT A USTIN 14
D EPARTMENT OF C OMPUTER S CIENCES M etr onome � S ig nal j R timer (1) � M etr onome , i.e. Recursively defined Expressions Metr onome � if ( true ) j R timer (1) � Metr onome S ig nal is punctual, so the rhs is punctual. Metr onome is punctual. Met � S ig nal j M � Met | {z } unpun tual Met is punctual. Its subexpression M � Met may be distributed. S ig nal j M � Met � S ig nal j f f � M � Met U NIVERSITY OF T EXAS AT A USTIN 15
D EPARTMENT OF C OMPUTER S CIENCES f � M j R timer (1) � f Least fixed point f punctual. f � M j R timer (1) � f |{z} |{z} | {z } unpun tual pun tual pun tual | {z } pun tual | {z } Assume pun tual f unpunctual. j f � M R timer (1) � f |{z} |{z} | {z } unpun tual pun tual pun tual | {z } unpun tual | {z } Assume unpun tual f is punctual. punctual < unpunctual. The least fixed point is U NIVERSITY OF T EXAS AT A USTIN 16
D EPARTMENT OF C OMPUTER S CIENCES f . Existence of least fixed point f is punctual. � definition of f is punctual: f is punctual. Least fixed point exists for any defined expression Assume � definition of f is unpunctual: f is unpunctual. f unpunctual, definition of f is unpunctual. Monotonicity: assuming U NIVERSITY OF T EXAS AT A USTIN 17
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