1. Programming Language SemanticsRely/Guarantee Reasoning Parallel Programs Tal Lev-Ami Viktor Vafeiadis Mooly Sagiv

2. An Axiomatic Proof Technique for Parallel Programs Owicky & Gries Verification of Sequential and Concurrent Programs Apt & Oldrog Chapters 4-6

3. Interference • A command T with a precondition pre(T) does not interfere with the proof of {p} c {q} if: • {q  pre(T) } T {q} • For any command c’ inside c with a precondition pre(c’) • {pre(c’)  pre(T) } T {pre(c’)} • {p1} c1 {q1} {p2} c2 {q2}.. {pk} ck {qk} are interference free if for every i j and for every assignment in T in ci does not interfere with {pj} cj {qj}

4. Parallel Composition Rule {p1} c1 {q1} {p2} c2 {q2}.. {pk} ck {qk} {p1p2 …pk} cobegin c1 || c2 || …|| ck coend {q1q2…  qk} {p1}q1 {q1} {p2} c2 {q2}.. {pk} ck {qk} are interference free

5. Limitations of the Owicky & Griesproof rules • Checking interference can be hard • Non-compositionality • Until you finished the local proofs cannot check interference • A non-standard meaning of Hoare triples • Depends on the interference of other threads with the proof • Proofs need to be saved • Hard to handle libraries and missing code • Soundness is non-trivial • Completeness depends on auxiliary variables

6. Commands as relations • It is convenient to view the meaning of commands as relations between pre-states and post-states • In {p} C {Q} • p is a one state predicate • Q is a two-state predicate • Example • {true} x := x + 1 {x= x + 1}

7. Global Reasoning {x 0} x := x + 1; y := 1 {x >0} {x 0} x := x + 2; y := 2 {x >0} {x 0} x := x + 200; y := 200 {x >0} …  

8. Rely/Guarantee Reasoning {x0} • x0 0 •  • x1 x0 • x2 = x1+7  y2 = y1 • x3 x2 •  • x4 = x3  y4=7 •  • x5 x4  x := x + 7; y:=7 {x  x} {x>0} x5 >0 

9. Rely/Guarantee Reasoning(2) {x0} x := x + 7; y:=7 {x>0} {x  x}  x =x+7 y=y  x  x x =x y=7  x  x

10. {Q} {Q} Hoare vs. Rely/Guarantee Reasoning {P} c {Q} {P} c {P} G G G c R* R* R*

11. The rest of the lecture • Operations on relations • (Informal) Semantics of Rely/Guarantee • A sound Rely/Guarantee inference rules

12. p(, ) =p() p(, ) =p() A single state predicate p is preserved by a two-state relation R if p R p , : p() R(, ) p() From one- to two-state relations

13. Operations on Relations • (P;Q)(, )=:P(, ) Q(, ) • ID(, )= (=) • R*=IDR (R;R) (R;R;R) … 

14. Formulas • ID(x) = (x = x) • ID(p) =(pp) • Preserve (p)= pp

15. Informal Semantics • c(p, R, G, Q) • For every state  such that p: • Every execution of c on state  with (potential) interventions which satisfy R results in a state  such that (, )  Q • The execution of every atomic sub-command of c on any possible intermediate state satisfies G • c[p, R, G, Q] • For every state  such that p: • Every execution of c on state  with (potential) interventions which satisfy R must terminate in a state  such that (, )  Q • The execution of every atomic sub-command of c on any possible intermediate state satisfies G

16. A Formal Semantics • Let cR denotes the set of quadruples <1, 2, 3, 4 > s.t. that when c executes on 1 with potential interferences by R it yields an intermediate state 2 followed by an intermediate state 3 and a final state 4 • as usual 4= when c does not terminate • cR = {<1, 2, 3, 4> :  : <1, >  R  ( <com, >* 2  2 =3=4   ’, c’: <c, >* <c’, ’>  ( ((2= 12 = )  (3 =  3=’)  4= ) <’, 2, 3, 4 >  c’R) • c(p, R, G, Q) • For every <1, 2, 3 , 4 >  cR such that 1p • < 2, 3>  G • If 4 : <1, 4 >  Q

17. Simple Examples • X := X + 1  (true, X=X, X =X+1X=X, X =X+1) • X := X + 1  (X 0, X X, X>0 X=X, X>0) • X := X + 1 ; Y := Y + 1  (X 0Y 0, X X  YY, G, X>0 Y>0)

18. A Realistic Example ES=eventopM+1  even(i)  l: (even(l)  0<l<i) x(l)0  eventopM x(eventop)>0 Findpos: begin initialize: i :=2 ; j :=1; eventop = M+1 ; oddtop := M+1; search: cobegin Evensearch: while i < min(oddtop, eventop) do if (x(i) > 0) then eventop := i else i := i + 2; || Oddsearch: while j < min(oddtop, eventop) do if (x(j) > 0) then oddtop := j else j := j + 2; coend k := min(eventop, oddtop) end{k M+1 (l: 1 l <k  x(l) 0)  (k Mx(k)>0)} OS=oddtopM+1  odd(j)  l: (odd(l)  0<l<j) x(l)0  oddtopM x(oddtop)>0

19. Inference Rules • Define c  (p, R, G, Q) by structural induction on c • Soundness • If c  (p, R, G, Q) then c  (p, R, G, Q)

20. (Atomic) atomic {c}  (p, preserve(p), QID, Q) Atomic Command {p} c {Q}

21. (Critical) await b then c  (p, preserve(p), QID, Q) Conditional Critical Section {pb} c {Q}

22. (SEQ) c1 ; c2 (p1, R, G, (Q1; R*; Q2)) Sequential Composition c1(p1, R, G, Q1) c2(p2, R, G, Q2) Q1 p2

23. (IF) if atomic {b} then c1 else c2 (p, R, G, Q) Conditionals c1(p b1, R, G, Q) p b  R* b1 c2(p b2, R, G, Q) p b  R* b2

24. (WHILE) while atomic {b} do c  (j, R, G, b j) Loops c (j b1, R, G, j) j b  R* b1 R  Preserve(j)

25. (REFINE) c  (p’, R’, G’, Q’) Refinement c (p, R, G, Q) p’  p Q Q’ R’  R G  G’

26. (PAR) c1 || c2 (p1 p1, (R1R2), (G1G2), Q) where Q= (Q1 ; (R1R2)*; Q2) (Q2 ; (R1R2)*; Q1) Parallel Composition c1(p1, R1, G1, Q1) c2(p2, R2, G2, Q2) G1 R2 G2 R1

27. A Realistic Example ES=eventopM+1  even(i)  l: (even(l)  0<l<i) x(l)0  eventopM x(eventop)>0 Findpos: begin initialize: i :=2 ; j :=1; eventop = M+1 ; oddtop := M+1; search: cobegin Evensearch: while i < min(oddtop, eventop) do if (x(i) > 0) then eventop := i else i := i + 2; || Oddsearch: while j < min(oddtop, eventop) do if (x(j) > 0) then oddtop := j else j := j + 2; coend k := min(eventop, oddtop) end{k M+1 (l: 1 l <k  x(l) 0)  (k Mx(k)>0)} OS=oddtopM+1  odd(j)  l: (odd(l)  0<l<j) x(l)0  oddtopM x(oddtop)>0

28. EvenSearch||OddSearch (ES OS, RE RO, GEGO, ES OS imin(et, ot) jmin(et, ot)) (PAR) EvenSearch (ES, RE , GE, ES imin(et, ot)) (WHILE) (IF) if (x(i) >0) … (ESi<et), RE , GE, ES) OddSearch (OS, RO , GO, OSjmin(et, ot)) ES i<min(et, ot)RE* i<et REPreserve(ES) ES i<et)) x(i) >0 RE* x(i) >0 ES i < et  x(i) 0 RE* x(i) 0 et :=i (ESi <et) x(i) >0, RE , GE, ES) i :=i+2 (ESi< et) x(i) 0, RE , GE, ES) RE=i =iototet =et GE=j =jetet  ot =ot RO=GE Go=RE

29. Auxiliary in Owicky-Gries • {X = Y} X := X + 1 || Y := Y+1 • {X=Y}

30. Issues in R/G • Total correctness is trickier • Restrict the structure of the proofs • Sometimes global proofs are preferable • Many design choices • Transitivity and Reflexivity of Rely/Guarantee • No standard set of rules • Suitable for designs

31. Summary • Reasoning about concurrent programs is difficult • Oweeki-Gries suggest to carefully design the sequential proofs to simplify the proof procedure • The use of auxiliary variables can make proofs difficult • Can have difficulties with fine-grained concurrency • Benign dataraces • Rely/Guarantee style allows more elegant/general reasoning • Compositional • Local • Adapts to the complexity of the proof • Soundness is simple • Naturally handles libraries and missing code