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Tracelets: a Model for the Laws of Concurrent Programming Tony Hoare OxfordFeb 2012

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Our Universe E is a set of events that can occur in or around a computer during execution of a program – drawn as boxes. D is a set of dependencies between the events – drawn as arrows between boxes – e --> f means f depends on e source, target: D --> E source(d) --> target(d)

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Labels P are sets of properties of D or E – e.g., the type of command executed – the objects involved in the event – the value of the data transmitted on the arrow labels:E + D --> P – labels(e) are drawn in the box, – labels(d) on top of the arrow

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A tracelet is a subset of E, denoted by p, q, r. For example: a trace of a single execution of a program or of a single execution of a command or of a single object used in the execution I = { }, the tracelet that contains no event

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An object is used by a program to store or transmit data – e.g., a variable, a semaphore, a channel, … Its behaviour is modelled by a tracelet – containing some or all events in which it has engaged A trace of a complete program execution – is the union of all the tracelets for every resource that it has used

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Pictorially νδ ν labels the allocation of an object. δ labels its disposal. All other events of its tracelet lie in between

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A variable νδ :=4labels an assignment of value 4 =4 labels a fetch of value 4 : =3 :=2 : =4 =4 =3 =2

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Object names νxνxδxδx may be added to the labels x: =3 x:=2 x: =4 x= 4 x=3 x=2

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A variable νδ The arrows from each fetch to the next assignment ensures prompt overwriting : =3 :=2 : =4 =4 =3 =2

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Weak memory νδ … which does not occur in modern weak memories. : =3 :=2 : =4 =4 =3 =2

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A Semaphore ν δ P is an acquisition of the semaphore V is a release (by the same owner) V P V P

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A buffered channel ν δ !4labels an output of value 4 ?4 labels an input of value 4 ! =3 !2 !4!4 ?4 ?3 ?2

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A single-buffered channel δ !3!3 !2 !4!4 ?4 ?3 ?2 Each output depends on prior input of the previous message ν

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A complete program trace is the union of the tracelets for every command that it contains. The tracelet of a command can be analysed into sub-tracelets for each of its immediate sub-commands. The analysis determines whether the trace is a valid trace for the program.

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Concurrent Composition p|q = p q, provided that p q = { } – otherwise the analysis is invalid, because no event is an execution of two distinct commands. Theorem: | is associative and commutative with unit

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Definitions p --> q = e є p, f є q. e --> f p => q = p = q or p is undefined.

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Sequential Composition p ; q= p|q provided not q --> p – otherwise the analysis is invalid, because no event in execution of the first command can depend on any event in the execution of the second. Theorem: ; is associative with unit

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Example x := 3 x := 4 If x is a shared variable x := 3 ; x:=4=

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Or, x := 3 x := 4 if blue arrows are equal, x := 3 ; x:=4=

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Theorems p;q => p|q – Proof: they are equal whenever not q --> p – otherwise, lhs is undefined (p|p’);q=> p|(p’;q) – they are equal when they are both defined – if rhs is undefined, then q --> p’ – which implies that q --> (p’|q), – therefore the lhs is also undefined.

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Exchange laws p;(q|q’) => (p;q)|q’ – proof similar (p|p’);(q|q’) => (p;q)|(p’;q’) – proof similar All exchange laws are derivable from the last, – by substituting for p’, or q’, – or for both q and q’

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A command is modelled by the set of tracelets of all its possible executions in all its possible environments. = { { } } x := 3 = { p | ‘x :=3’ є labels(p)} x := y = { p | n. {‘x:= n’, ‘y = n’} labels(p)}

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Let P, Q, R, be commands P | Q = { (p|q)| p є P & q є Q } P ; Q = { (p;q) | p є P & q є Q } P \/ Q = { r | r є P or r є Q } P Q = r. r є P => r є Q All our theorems p => q also hold for P Q, – because every variable appears exactly once on each side of the inequation.

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Separation Let L be the set of red arrows – they must not cross thread boundaries Blue arrows must cross thread boundaries Black arrows may cross either boundaries. Definitions of ; and | must be changed to ensure these rules

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Dependency ordering Let e < f mean that there is a path of arrows from e to f. e

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Interfaces of a tracelet p arrows(p) = s(p) u t(p) ins(p) =t(p) - s(p) outs(p) =s(p) - t(p) – where s(p) = {d| source(d) є p} t(p) = {d| target(d) є p}

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In pictures… outs ins

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Separating ; p;q= p|q provided that not q < p & outs(p) L = ins(q) L – outs(p;q) L = outs(q) L – ins(p;q) L = ins(p) L Theorem: ; is associative and has unit

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Ok is a set of tracelets that are always preferred – e.g., no overflow, no races, no divergence, etc. p => qmeansp = q orp is not defined ornot q є ok є ok p;q є ok=p є ok & q є ok

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Separating concurrency e < f= there is a path of red dependencies from e to f p*q є ok=p є ok & q є ok ¬ p < q & not q < p p*q = p|qif p*q є ok – because that’s the way it will be implemented!

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Lift to sets P => Q means p є P. p є ok => p є Q otherwise q є Q. not q є ok & ins(q) = ins(p)

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