Denotational event structure for relaxed memory
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1 Denotational event structure for relaxed memory Jade Alglave 1, Simon Castellan 2, Jean-Marie Madiot 3 1 ARM, and University College London, UK 2 Imperial College London, UK 3 INRIA 7th July, 2018 LOLA 2018
2 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data}
3 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer:
4 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer: Wdata:=17
5 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer: Wdata:=17 W flag:=1
6 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer: Wdata:=17 W flag:=1 R flag=1
7 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer: Wdata:=17 W flag:=1 R flag=1 R data=17
8 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 2 / 20 Message-passing on my computer Consider the program mp: data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my computer: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1
9 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1
10 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1 Wflag:=1
11 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1 Wflag:=1 R flag=1
12 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1 Wflag:=1 R flag=1 R data=0
13 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1 Wflag:=1 R flag=1 R data=0 W data:=17
14 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 3 / 20 Message-passing on my phone data = flag = 0 data := 17; r flag; flag := 1 if(r == 1){v data} Possible execution traces on my phone: Wdata:=17 W flag:=1 R flag=1 R data=17 Wdata:=17 R flag=0 W flag:=1 Rflag=0 W data:=17 W flag:=1 Wflag:=1 R flag=1 R data=0 W data:=17 Wflag:=1 R flag=1 W data:=17 R data=17 Wflag:=1 W data:=17 R flag=1 R data=17 Rflag=0 W flag:=1 W data:=17 A dierent architecture, much harder to reason about...
15 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 4 / 20 Structure behind traces W flag:=1 W data:=17 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 W data:=17 W flag:=1 R flag=1 R data=17 { W flag:=1 R flag=1 R data=0 W data:=17 R flag=0 W data:=17 W flag:=1 W data:=17 R flag=0 W flag:=1 R flag=0 W flag:=1 W data:=17
16 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 4 / 20 Structure behind traces W flag:=1 W data:=17 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 W data:=17 W flag:=1 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 { W flag:=1 R flag=1 R data=0 W data:=17 R flag=0 W data:=17 W flag:=1 W data:=17 R flag=0 W flag:=1 R flag=0 W flag:=1 W data:=17
17 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 4 / 20 Structure behind traces W flag:=1 W data:=17 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 W data:=17 W flag:=1 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 { W flag:=1 R flag=1 R data=0 W data:=17 W flag:=1 R data=0 R flag=1 W data:=17 R flag=0 W data:=17 W flag:=1 W data:=17 R flag=0 W flag:=1 R flag=0 W flag:=1 W data:=17
18 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 4 / 20 Structure behind traces W flag:=1 W data:=17 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 W data:=17 W flag:=1 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 { W flag:=1 R flag=1 R data=0 W data:=17 W flag:=1 R data=0 R flag=1 W data:=17 R flag=0 W data:=17 W flag:=1 W data:=17 R flag=0 W flag:=1 R flag=0 W flag:=1 W data:=17 R flag=0 W flag:=1 W data:=17
19 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 4 / 20 Structure behind traces W flag:=1 W data:=17 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 W data:=17 W flag:=1 R flag=1 R data=17 W flag:=1 R flag=1 W data:=17 R data=17 { W flag:=1 R flag=1 R data=0 W data:=17 W flag:=1 R data=0 R flag=1 W data:=17 R flag=0 W data:=17 W flag:=1 W data:=17 R flag=0 W flag:=1 R flag=0 W flag:=1 W data:=17 R flag=0 W flag:=1 W data:=17
20 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 5 / 20 Sets of partial orders and event structures The set of partial orders describes the semantics of mp: W flag:=1 W flag:=1 W data:=17 R flag=1 R data=17, R flag=1 R data=0, R flag=0 W data:=1 W data:=17 W data:=17
21 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 5 / 20 Sets of partial orders and event structures The set of partial orders describes the semantics of mp: W flag:=1 W flag:=1 W data:=17 R flag=1 R data=17, R flag=1 R data=0, R flag=0 W data:=1 W data:=17 W data:=17 This set of partial orders can be summed by an event structure: W flag:=1 R flag=0 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 W data:=17
22 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 5 / 20 Sets of partial orders and event structures The set of partial orders describes the semantics of mp: W flag:=1 W flag:=1 W data:=17 R flag=1 R data=17, R flag=1 R data=0, R flag=0 W data:=1 W data:=17 W data:=17 This set of partial orders can be summed by an event structure: W flag:=1 R flag=0 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 W data:=17
23 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 6 / 20 This talk 1. From programs to event structures: in a denotational style. Combine a semantics for the threads and for the memory 2. Using the model to reason about programs For race-free programs, optimisations preserve behaviour. 3. Using the model to explore the behaviour of programs Herd investigations: How to avoid a linear history per-variable?
24 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 7 / 20 I. From programs to event structures Modelling MiniRMO ( MiniARM): Syntax. e ::= r e + e... t ::= fence; t x := e; t r x; t p ::= t... t Two kinds of idents: thread-local registers and global variables.
25 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 8 / 20 Operational semantics Operational semantics is formulated as a LTS over the labels Σ ::= W x:=k R x=k fence. The states of the LST are pairs (p, µ : V N). t@µ l l fence var(l) x x := k; t@µ l x := k;
26 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 8 / 20 Operational semantics Operational semantics is formulated as a LTS over the labels Σ ::= W x:=k R x=k fence. The states of the LST are pairs (p, µ : V N). t@µ l l fence var(l) x x := k; t@µ l x := k; Our goal: a mapping from states to event structures s.t.: (p, µ) bisimilar to (p, µ).
27 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 9 / 20 Labeled event structures Denition A (Σ-labeled) event structure is a tuple (E, E, E, l : E Σ) where (E, E ) is a partial order and E is a symmetric relation on E, satisfying nite causes and conict inheritance. a c e b d
28 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 9 / 20 Labeled event structures Denition A (Σ-labeled) event structure is a tuple (E, E, E, l : E Σ) where (E, E ) is a partial order and E is a symmetric relation on E, satisfying nite causes and conict inheritance. a c e b d Congurations are downclosed, conict-free subsets of E. C (E) is the set of congurations of E.
29 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 9 / 20 Labeled event structures Denition A (Σ-labeled) event structure is a tuple (E, E, E, l : E Σ) where (E, E ) is a partial order and E is a symmetric relation on E, satisfying nite causes and conict inheritance. a c e b d Congurations are downclosed, conict-free subsets of E. C (E) is the set of congurations of E. Congurations form naturally a LTS: x a y when y = x {e} & lbl(e) = a
30 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 10 / 20 An overview of the semantics 1. Thread semantics: context is left open (and unknown) W flag:=1 W data:=17 R flag=0 R flag=1 R flag=2... R data=0 R data= Final semantics: context is assumed empty Compute interactions with memory: W flag:=1 R flag=0 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 W data:=17
31 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 11 / 20 Thread semantics By induction on threads. For instance x := k; t = W x:=k ; t W x:=k t The partial order is given by: ( l;e = E {(l, e) e E e is a fence or an operation on x} Program. No interaction: t 1... t n = t 1... t n.
32 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 12 / 20 How to interpret memory? Memory histories (eg. [W x:=2 R x=2 W x:=3]) form an innite e.s. C µ. (p, µ) = p C µ W flag:=1 R flag=0 W flag:=1 W data:=17 W flag:=1 = R R data=0 flag=0 R flag=1 R flag=1 W data:=17 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 R data=1 R flag=0 W flag:=1 R data=0 W data:=17 R data=17 W data:=17
33 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 12 / 20 How to interpret memory? Memory histories (eg. [W x:=2 R x=2 W x:=3]) form an innite e.s. C µ. (p, µ) = p C µ W flag:=1 R flag=0 W flag:=1 W data:=17 W flag:=1 = R R data=0 flag=0 R flag=1 R flag=1 W data:=17 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 R data=1 R flag=0 W flag:=1 R data=0 W data:=17 R data=17 W data:=17 The product is obtained by a coreection: C ( ) Event Structures Stable Families
34 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 12 / 20 How to interpret memory? Memory histories (eg. [W x:=2 R x=2 W x:=3]) form an innite e.s. C µ. (p, µ) = p C µ W flag:=1 R flag=0 W flag:=1 W data:=17 W flag:=1 = R R data=0 flag=0 R flag=1 R flag=1 W data:=17 W data:=17 R flag=1 W flag:=1 R data=17 R data=0 R data=1 R flag=0 W flag:=1 R data=0 W data:=17 R data=17 W data:=17 The product is obtained by a coreection: C ( ) Event Structures Stable Families Pr( )
35 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 13 / 20 Reasoning with the model Sequential consistency: the standard model for shared memory. Via the same technique, we can build (p, µ) SC (no reordering). Can we relate (p, µ) MiniRMO and (p, µ) SC if p is well-bahved?
36 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 14 / 20 Races and sizes A race: two co-located concurrent accesses (among which a write). data := 0xdeadbeef r data assert (data {0, 0xdeadbeef}) If data is two words, we might see: data = 0xdead0000.
37 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 14 / 20 Races and sizes A race: two co-located concurrent accesses (among which a write). data := 0xdeadbeef r data assert (data {0, 0xdeadbeef}) If data is two words, we might see: data = 0xdead0000. But, mp should be ok: data := 17; flag := 1 r flag; if(r == 1){v data} To model this, we split variables into two groups: atomic and non-atomic. Races on atomic variables are ok (necessary for eg. locks). Atomic variables should have release/acquire semantics.
38 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 15 / 20 Race-free programs Denition A race of a program p is a a trace w (N Σ) of the form: with i j and x is non-atomic. w =... (i, R x=k ) (j, W x:=k ) Denition A program is race-free when none of its traces on SC are races.
39 Proof. By studying properties of the simulation p SC p MiniRMO. Denotational event structure for relaxed memory Alglave, Castellan, Madiot 15 / 20 Race-free programs Denition A race of a program p is a a trace w (N Σ) of the form: with i j and x is non-atomic. w =... (i, R x=k ) (j, W x:=k ) Denition A program is race-free when none of its traces on SC are races. Theorem (Strong Data Racefreedom (DRF)) For a race-free program p, p SC weakly bisimilar to p MiniRMO. (Where operations on small variables are considered internal.)
40 Smaller memory models? Denotational event structure for relaxed memory Alglave, Castellan, Madiot 16 / 20
41 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 17 / 20 Not seen, not caught! We choose a particular memory implementation, C µ. x := 1 x := 2 W x:=1 W x:=2 W x:=2 W x:=1 p However no one is observing the order between the writes... Can we change to have a less sequentializing implementation?
42 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 18 / 20 When do we need to sequentialize? W x:=1 W x:=2 R x=1 R x=2 R x=2 R x=1
43 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 18 / 20 When do we need to sequentialize? W x:=1 W x:=2 R x=1 R x=2 R x=2 R x=1
44 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 18 / 20 When do we need to sequentialize? W x:=1 W x:=2 R x=1 R x=2 R x=2 R x=1
45 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 18 / 20 When do we need to sequentialize? W x:=1 W x:=2 R x=1 R x=2 R x=2 R x=1
46 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 18 / 20 When do we need to sequentialize? W x:=1 W x:=2 R x=1 R x=2 R x=2 R x=1 We nd a notion of lazily consistent partial-ordered history: W x:=1 W x:=2 R x=1 W x:=1 W x:=2 R x=1 not consistent R x=2 consistent R x=2 Theorem (Weaker correctness) Any trace of a lazy consistent history can be reordered without permuting writes on the same variable to a consistent trace. MemStates( p lazy ) = MemStates(p).
47 Denotational event structure for relaxed memory Alglave, Castellan, Madiot 19 / 20 A demo P = x := 1 x := 3 Q = x := 1 x := 3 r x s x
48 Related work / Extensions Related work. Brookes & Kavanagh's model of TSO with pomsets. Pichon & Sewell's operational semantics on event structures Jerey & Riely's axiomatic model using event structures Extensions. Extend this to: Real ARM, Linux-C, etc. More complicated C11 models. Denotational event structure for relaxed memory Alglave, Castellan, Madiot 20 / 20
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