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Translation Validation

Translation Validation. A.Pnueli M.Siegel E.Singerman. Motivation. Prove that high level specification is correctly implemented in low level code. Verifying compiler is not feasible. Development freezing. Solution: Translation Validation. Translation Validation.

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Translation Validation

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  1. Translation Validation A.Pnueli M.Siegel E.Singerman

  2. Motivation • Prove that high level specification is correctly implemented in low level code. • Verifying compiler is not feasible. • Development freezing. Solution: Translation Validation

  3. Translation Validation After each compiler run verify that the target code produced on this run correctly implements the source code.

  4. Necessary Ingredients • A common semantic framework. • Notion of “correct implementation”. • A proof method. • Automation of the proof method.

  5. Example logical DEC_iterate() { l0: h1C = TRUE; l1: h2C = ZNC <= 1; l2: if (h2C) l2.1: read(FBC); l3: if (h2C) l3.1: NC = FBC; else l3.2: NC = ZNC - 1; l4: write(NC); l5: ZNC = NC; return TRUE; } process DEC = ( ? integer FB ! integer N ) ( | N := FB default (ZN-1) | ZN := N $ init 1 | FB ^= when (ZN <= 1) |) where integer ZN init 1 ; end

  6. FB :  N : ZN : 1 FB : 3 N : 3 ZN : 1 FB :  N : 2 ZN : 3 FB :  N : 1 ZN : 2 FB : 5 N : 5 ZN : 1 FB :  N : 4 ZN : 5 … FB : * N : * ZN : 1 h1 : * h2 : * pc : l0 FB : * N : * ZN : 1 h1 : t h2 : t pc : l2 FB : 3 N : * ZN : 1 h1 : t h2 : t pc : l3 FB : 3 N : * ZN : 1 h1 : t h2 : t pc : l3.1 FB : 3 N : 3 ZN : 1 h1 : t h2 : t pc : l5 … FB : 3 N : 3 ZN : 3 h1 : t h2 : t pc : l0 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l2 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l3 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l3.2 FB : 3 N : 2 ZN : 3 h1 : t h2 : f pc : l5 …

  7. Common Semantic Framework Synchronous Transition System S = (V,O,Θ, ρ) • V a set of state variables • O  V a set of observable variables • Θ an initial condition characterizing the initial states of the system • ρ a transition relation, relating a state to its possible successors

  8. process DEC = ( ? integer FB ! integer N ) ( | N := FB default (ZN-1) | ZN := N $ 1 | FB ^= when (ZN <= 1) |) where integer ZN init 1 ; end V = {FB,N,ZN,m.ZN} Θ = (FB =  N =  ZN =  m.ZN = 1) N’ = if FB’  then FB’ else ZN’ -1  m.ZN’ = if N’  then N’ else m.ZN  ZN’ = if N’  then m.ZN else   ZN’  1  FB’   =

  9. V = {FBC,NC,ZNC,h1C,h2C} Θ = (ZNC = 1  pc = l0) (pc=l0 h1’C=T  pc’=l1  pres_but(pc.h1c))  (pc=l1  h2’C=(ZNC 1)  pc’=l2 pres_but(pc,h2C))  (pc=l2 h2C  pc’=l2.1 pres_but(pc))  (pc=l2h2C pc’=l3 pres_but(pc))  (pc=l2.1 pc’=l3 pres_but(pc,FBC))  (pc=l3 h2C pc’=l3.1 pres_but(pc))  (pc=l3h2C pc’=l3.2 pres_but(pc))  (pc=l3.1 N’C=FBC pc’=l4 pres_but(pc,NC))  (pc=l3.2 N’C=ZNC–1  pc’=l4 pres_but(pc,NC))  (pc=l4 pc’=l5 pres_but(pc))  (pc=l5 ZN’C=NC pc’=l0 pres_but(pc,ZNC)) C= logical DEC_iterate() { l0: h1 = TRUE; l1: h2 = ZN <= 1; l2: if (h2) l2.1: read(FB); l3: if (h2) l3.1: N = FB; else l3.2: N = ZN - 1; l4: write(N); l5: ZN = N; return TRUE; }

  10. STS computation Let A = (V,O,Θ, ρ) • s[v] – a value state s assigns to each variable vV. • σ: s0,s1… - A computation s0|= Θ (si,si+1) |= ρiN • ||A|| - the set of computations of A.

  11. Defining Refinement OAOC A = (VA,OA,ΘA, ρA) C = (VC,OC,ΘC, ρC) Clocked interface mapping: I: C OA xOA, sC. I(s)[x]=s[x] or I(s)[x]= Definition: C refines A if there exists a clocked interface mapping I from C to A such that I(||C||)||A||O.

  12. Proving Refinement Clocked refinement mapping from C to A: f: C A xOA, sC. f(s)[x]=s[x] or f(s)[x]= Theorem: C refines A if there exists a clocked refinement mapping f: C A such that • sC . s|= ΘC f(s) |= ΘA • s,s’ Cr . (s,s’)|= ρC (f(s),f(s’))|= ρA Such f called inductive.

  13. Proof Rule • : VA (VC) sA  ā(sC) For  - state formula over VA: ā(sC)|=  iff sC|= [] For assertion inv and substitution  : VA E(VC) R1. ΘC inv inv holds initially R2. inv  ρC inv` inv is propagated R3. ΘC  ΘA[] Initiation R4. inv  ρC  ρA[] Propagation R5. inv (v[] = v  v[] = ) vOA C refines A

  14. Translation Validation: from Signal to C A.Pnueli O.Shtrichman M.Siegel

  15. Observation Functions and Correct Implementation A = (VA,ΘA,A,OA) C = (VC,ΘC,C,OC) • OA, OC – observation functions • Given  : s0, s1, …, - O(s0),O(s1), …, is observation of STS. • Obs(A) is the set of A observations. Definition: C refines A if Obs(C)  Obs(A)

  16. Adaptation to Signal compilation

  17. Choosing Observation process MUX = ( ? integer FB ! integer N ) ( | N := FB default (ZN-1) | ZN := N $ init 1 | FB ^= when (ZN <= 1) |) where integer ZN init 1 ; end OCFB: if rd.FBC then FBC else  OCN : if wr.NC then NC else  • OA = (FB,N) • OC = (OCFB,OCN) logical MUX_iterate() { rd.FBC=F; wr.NC=F; l0: h1C = TRUE; l1: h2C = ZNC <= 1; l2: if (h2C){ l2.1: read(FBC); rd>FBC=T; } l3: if (h2C) l3.1: NC = FBC; else l3.2: NC = ZNC - 1; l4: write(NC); wr.NC=T; l5: ZNC = NC; return TRUE; } logical MUX_iterate() { l0: h1C = TRUE; l1: h2C = ZNC <= 1; l2: if (h2C) l2.1: read(FBC); l3: if (h2C) l3.1: NC = FBC; else l3.2: NC = ZNC - 1; l4: write(NC); l5: ZNC = NC; return TRUE; }

  18. FB :  N : ZN : 1 FB : 3 N : 3 ZN : 1 FB :  N : 2 ZN : 3 FB :  N : 1 ZN : 2 FB : 5 N : 5 ZN : 1 FB :  N : 4 ZN : 5 … FB : * N : * ZN : 1 h1 : * h2 : * pc : l0 FB : * N : * ZN : 1 h1 : t h2 : t pc : l2 FB : 3 N : * ZN : 1 h1 : t h2 : t pc : l3 FB : 3 N : * ZN : 1 h1 : t h2 : t pc : l3.1 FB : 3 N : 3 ZN : 1 h1 : t h2 : t pc : l5 FB : 3 N : 3 ZN : 3 h1 : t h2 : t pc : l0 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l2 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l3 FB : 3 N : 3 ZN : 3 h1 : t h2 : f pc : l3.2 FB : 3 N : 2 ZN : 3 h1 : t h2 : f pc : l5 …

  19. Composite STS • V : {FBC,NC,ZNC,h1C,h2C,rd.FBC,wr.NC} • Θ : ZNC = 1  pc = l0 • (h1’C=T) • (h2’C=(ZNC 1)) • (h2’C(N’C=FBC) • (h2’C(FB’C=FBC N’C=ZNC–1)) • (ZN’C=N’C) • (rd.FB’C=h2’C) • (wr.N’C=T) • OCFB: if rd.FBC then FBC else  • OCN : if wr.NC then NC else  • Compose the transition relations of the individual statements inside the loop’s body. • no nested loops C :

  20. Composite STS • V : {FBC,NC,ZNC,h1C,h2C} • Θ : ZNC = 1  pc = l0 • (h1’C=T) • (h2’C=(ZNC 1)) • (h2’C(N’C=FBC) • (h2’C(FB’C=FBC N’C=ZNC–1)) • (ZN’C=N’C) • OCFB: if h2C then FBC else  • OCN : NC C :

  21. Rule Ref. Establish by induction that, for every C:s0C,s1C,… there exists A:s0A,s1A,… such that sjA=(sjC) and their observations are equal. For an abstraction mapping VA = (VC) R1. ΘC VA = (VC) ΘA Initiation R2. VA = (VC)  C V’A = (V’C)  A Propagation R3. VA = (VC)  OA=OC Compatibility with observations C refines A

  22. Construction of the Mapping  For vVA, v(Vc) – the value of v in sA related to sC. • For v  IO, v(Vc) = OCv(VC) • For each register flow m.r = rC ’m.r = r’C • For each Register or Local variable v’ = eqv  ’v = eqv(determinate programs) W1. ΘC  rR(m.r = rC)  vIORL(v = )  ΘA W2. rR(m.r = rC  m.r’ = r’C)  C  vIO( v’ = (OCv)’ )  vRL(v’ = eqv)  A

  23. Theorem: If verification conditions W1 and W2 are valid, then C refines A.

  24. FB =   N =   ZN =   m.ZN= 1 FB =   N =   ZN =  m.ZN = ZNC  FB’ = if h2’C then FB’ else   N’ = N’C  m.ZN’ = ZN’C  ZN’ = if N’   then m.ZN else  Example U1. ZNC = 1  m.ZN = ZNc   U2. C   A

  25. m.ZN = ZNC  FB’ = if h2’C then FB’ else   N’ = N’C  m.ZN’ = ZN’C  ZN’ = if N’   then m.ZN else  m.ZN = ZNC  FB’ = if h2’C then FB’ else   N’ = N’C  m.ZN’ = ZN’C  ZN’ = ZNC Example U2. C   A U2. C   A

  26. The End

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