Unifying Theories Execution History
Tony Hoare
In honour of Manfred Broy30 October 2009
Manfred Broy and Tony Hoare at Marktoberdorf.
Unifying…
• Memory– shared/private, weakly/strongly consistent
• Communication– synchronised/buffered, reliable/unreliable
• Resource management– dynamic/nested, disposed/collected
Unifying…
• Sequential programming– C, Java, C#, ...
• Process algebras and calculi– stream processing functions– CCS, CSP, pi
• Shared memory, threads– fine-grained, coarse-grained, transactions– weakly consistent memory
Labelled graphs
• Trace semantics (Mazurkiewicz)• Regular expressions (Kleene)• Causal nets (Petri)• Event structure configurations (Winskel)• Message Sequence Charts (UML)
INSIGHT! They are all labeled graphs
A labeled Graph
• E: a set of nodes (events)• A: a set of arrows (denoting data flow)• L: a set of labels (to be determined)• source, target: A -> E• label: A -> L (labelling the arrow)• label: E -> L (labelling the events)
Program Execution is recorded as a trace of• all events that have occurred– drawn as boxes
• all dependencies between them– drawn as arrows
source target– the target could not occur before source
Program Execution is recorded as a trace of:• all events that have occurred– drawn as boxes– with labels naming the executed command
• all dependencies between them– drawn as arrows
source target– with labels naming resource.value, etc.
x := 3
x.3x := 3 x = 3
A Sequential Resource
begin endnextnextnext next
Implementation
begin endnextnextnext next
•allocated globally/on stack/in heap/…
•disposed from stack/by command/by collector/by OS
Fork
fanout
all arrows of the graphhave the same source
Join
fanin
all arrows of the graph have the same target
Shared Resource
faninfanout
begin end
Publication
faninfanout
publish publish next
Assignment
faninfanout
:= 3 := 7next
=3
=3
=3
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
A variable
begin end
nextnextnext next:= := :=
fanin faninfanoutfanout =
=
= =
Unassigned fetch
begin end
nextnextnext next:= := :=
fanin
fanin
fanout fanout faninfanout
Communication
! ! !
? ? ?
send send send
Ordering
nextnext
nextnext
! ! !
? ? ?
send send send
Channel
nextnextnext
nextnextnext
! ! !
? ? ?
endbegin
next
next
send send send
Single-buffered Channel
nextnextnext
nextnextnext
! ! !
? ? ?
endbegin
next
nextsend send send
sync sync
Synchronised
nextnextnext
nextnextnext
! ! !
? ? ?
endbegin
next
nextsend send send
sync sync sync
Lossy channel
nextnext
next
! ! !
? ?
send send
Stuttering channel
next
nextnext
! !
? ? ?
send send send
Fraudulent channel
next
nextnext
! !
? ? ?
send send
Overtaking
next
next
! !
? ?
sendsend
Reliable channel
• reliable = loss & fraud & stutter & merge & overtaking
Threads
forkbegin endnextnextnext next
beginnextnext
joinbeginnextnextnext next
end
fork
join
An Atomic Assignment
x := x + y
x = 3
y= 4
x := 7
An Atomic Assignment
x := x + y
x = 3
y= 4
x := 7
x.fanout.3 x.fanin
y.fanout.4 y.fanin
x.next x.next
x.fanout.7x.fanin
t.next t.next
An Atomic Assignment
x := x + y
x = 3
y= 4
x := 7
x.fanout.3 x.fanin
y.fanout.4 y.fanin
x.next x.next
x.fanout.7x.fanin
t.next t.next
Events and atomic actions
• Each occurrence of an event in the trace of program execution belongs to the trace of exactly one resource (thread, variable, channel,…)
• Atomic actions are groups of synchronised events, including exactly one from the thread which invoked the action, and one (or more) from every resource used by it.
v.fanout
v:= 4
v := 3 v:= 6
v.next
v.faninv.fanout
v.fanin
=3 = 4 = 6
v.next
v.next
v.next
A variable (fully labelled)
v.fanout
t.nextt.next
v:= 4
v := 3 v:= 6 thread t
interfering thread
v.next
=3 = 4 = 6
v.next
v.next
A shared variable
v.next
Weakly consistent memory
as implemented in multi-core architecture,is even more complicated to define… and more still to use!
A common architecture is TSO
Total Store Ordering
:=4
sync
:=3 :=6 global memory
sync
Local memory
nextnext
val
next
:=4
sync
:=3 :=6
:= 4:= 3 := 6 local memory
global memory
sync
Local memory access
nextnext
val
next
:=4
sync
:=3 :=6
:= 4:= 3 := 6 local memory
global memory
= 3 = 4 = 6local memory
sync
nextnext
val
next
:=4
sync
:=3 :=6
:= 4:= 3 := 6 local memory
global memory
other thread:=4
= 3 = 4 = 6local memory
sync
sync
:=3
Memory Barrier
nextnext
val
next
sync
:=4 :=3 :=6
:= 4:= 3 := 6 bar localmemory
globalmemory
Summary
• Dependency and data flow are a primitive concepts adequate to describe the dynamic behaviour of many kinds of computing resource
• Labelled graphs provide a general frameworkadequate for a unifying theory of dependency
