Hoare Calculus for SIMT Programms

Transcription

1 Introduction The extension of the Hoare Logic Proof of Equivalence Conclusion and Future Work Hoare Calculus for SIMT Programms Viktor Leonhardt Embedded Systems Group Department of Computer Science University of Kaiserslautern, Germany October 5, 2017

2 Introduction The extension of the Hoare Logic Proof of Equivalence Conclusion and Future Work 1 Introduction Hoare Logic SIMT Execution Model 2 The extension of the Hoare Logic Example Regular Programs 3 Proof of Equivalence 4 Conclusion and Future Work

3 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Axioms {ϕ} P {ψ} {ϕ[φ/x]} x := φ {ψ} {ϕ} P {χ} {χ} Q {ψ} {ϕ} P; Q {ψ} {ϕ e} P {χ} {ϕ e} Q {ψ} {ϕ} if e then P else Q {ψ} {ϕ e} P {ψ} {ϕ} while e do P {ϕ e} Hoare triple Assignment Axiom Composition Rule if-then-else Rule while Rule

4 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Example P 1 : Integer division of two natural numbers x and y a = 0 ; b = x ; w h i l e ( b >= y ) { b = b y ; a = a +1; } {x 0 y 0} P 1 {a y + b = x 0 b < y}

5 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Example Transition State σ weakest-precondition {0 y + x = x x 0} {x 0 y 0} a = 0; {a y + x = x x 0} b = x; {a y + b = x b 0} {(a + 1) y + b y = x {a y + b = x b 0 b y 0} b y} b = b y; {(a + 1) y + b = x b 0} a = a + 1; {a y + b = x b 0} while(b >= y){} {a y + b = x b 0 b < y}

6 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Sequential Program start σ 1 σ 2 a = 0 b = x σ 3 b = b y σ 4 b < y a = a + 1 σ 5 σ 6

7 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Parallel Program P 2 : Vector addition k = t i d ; w h i l e ( k < n ) { c [ k ] = a [ k ] + b [ k ] ; k = k + n t i d ; }

8 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model Hoare Logic Parallel Program States start σ 1 σ 21 σ 2 σ 22 σ 211 σ 212 σ 3 σ 222 σ 221

9 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model SIMT Execution Model start σ 1 σ 21 σ 2 σ 22 σ 211 σ 212 σ 3 σ 222 σ 221 lockstep execution branches are executed sequentially, while disabling threads

10 Introduction The extension of the Hoare Logic Proof of Equivalence Hoare Conclusion Logic SIMT and Future Execution Work Model SIMT Execution Model Parallel Program n = 3 ntid = 2 P 2 : Vector addition k = t i d ; w h i l e ( k < n ) { c [ k ] = a [ k ] + b [ k ] ; k = k + n t i d ; } a, b, c SV n i Thread 1 Thread 2 1 k = 0 k = 1 2 while(0 < 4) while(1 < 4) 3 c[0] = a[0] + b[0] c[1] = a[1] + b[1] 4 k = k = while(2 < 3) while(3 < 3) 6 c[2] = a[2] + b[2] 7 k = while(4 < 3) while(3 < 3)

11 Introduction The extension of the Hoare Logic Proof of Equivalence Example Conclusion Regular andprograms Future Work The extension of the Hoare Logic {ϕ} m P {ψ} {ϕ} m skip {ϕ} {all(m) none(m) ϕ} m sync {ϕ} {ϕ} m P {ψ} {ψ} m Q {χ} {ϕ} m P; Q {χ} Hoare quadruple (H-Skip) (H-Sync) (H-Seq)

12 Introduction The extension of the Hoare Logic Proof of Equivalence Example Conclusion Regular andprograms Future Work { x.assign(x, m, x, ē, e) ϕ[x /x]} m x[ē] := e {ϕ} {ϕ e = z} m && z P {ψ} {ψ} m &&!z Q {χ} {ϕ} m if e then P else Q {χ} {ϕ e = z} m && z P {ϕ} {ϕ} m while e do P {ϕ none(m && e)} (H-Assign) (H-If) (H-While)

13 Introduction The extension of the Hoare Logic Proof of Equivalence Example Conclusion Regular andprograms Future Work Extended Hoare Logic Parallel Program P 3 : Counterexample for the soundness w h i l e ( x [ t i d ] == t i d ) { x [ 0 ] = 1 ; x [ 1 ] = 1 ; } Execution of P 3 before loop: x[0] = 0 x[1] = 1 after first iteration: x[0] = 1 x[1] = 1

14 Introduction The extension of the Hoare Logic Proof of Equivalence Example Conclusion Regular andprograms Future Work Regular Programs Lemma 1 Let P be a program and assume that for any subprogram of the form while e do Q, e and Q satisfy the condition of lemma 2. Then P is regular. Lemma 2 Let P be a program and e an expression. Suppose that any shared variable occurring in e does not occur on the left-hand side of any assignment in P. Then e is stable under P.

15 Introduction The extension of the Hoare Logic Proof of Equivalence Conclusion and Future Work Proof of Equivalence [..] the lockstep and interleaving semantics are equivalent for race-free programs [..] 1 Lemma 3 Let P be a program and µ a mask and suppose that (P, ε i µ), σ is race-free. Then, P, µ, σ σ if and only if (P, ε i µ), σ I (, ε)i, σ 1 Kojima, Kensuke, and Atsushi Igarashi. A Hoare Logic for GPU Kernels. ACM Transactions on Computational Logic (TOCL) 18.1 (2017): 3.

16 Introduction The extension of the Hoare Logic Proof of Equivalence Conclusion and Future Work Conclusion and Future Work Closer look into the work of Kojima, Kensuke, and Atsushi Igarashi. A Hoare Logic for GPU Kernels. Analyse OpenCL and Hoare Logic for GPU Kernels to fit together Test the analysis tool of Kojima et al., which was developed in 2016