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Model Checking C-Programs – An Example: Integer Square Root

Model Checking C-Programs – An Example: Integer Square Root. Wenhui Zhang http://lcs.ios.ac.cn/~zwh. Contents. Integer Square Root. Model Checking. Compositional Reasoning. Summary. A Concrete Example of such a Software. start. initialize. s0. Take a number n. in(). s1.

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Model Checking C-Programs – An Example: Integer Square Root

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  1. Model Checking C-Programs – An Example: Integer Square Root Wenhui Zhang http://lcs.ios.ac.cn/~zwh

  2. Contents Integer Square Root Model Checking Compositional Reasoning Summary

  3. A Concrete Example of such a Software start initialize s0 Take a number n in() s1 Transformation isr(n,k), isk(n,k) s2 Correctness Claim: The ouput is the integer square root of the input Correctness Claim: (m*m)<=n; (m*m)+2*m+1>n Output a number m

  4. Example: Function main() #include <stdio.h> /********************************************/ int in(); int isr(int x,int k); int isk(int n,int k); /********************************************/ int main() { int n=0, m=0; int k=1; printf("INFO: system is now active\n"); while (1) { n=in(); m=isr(n,k); k=isk(n,k); printf("RESULT: %i\n\n",m); } }

  5. Example: Function in() int in() { char c=0; int k=0; while (1) { k=0; putc('N',stdout); putc(':',stdout); putc(9,stdout); c=getc(stdin); if (c=='\n') { printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } if (c<'0'||c>'9') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } k=c-'0'; c=getc(stdin); if (c=='\n') { return k; } if (c<'0'||c>'9') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } if (k<2) k=k*10+(c-'0'); else if (k==2&&c=='0') k=20; else { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input number must be in {0,...,20}\n\n"); continue; } c=getc(stdin); if (c!='\n') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } return k; } }

  6. Example: isr() and isk() int isr(int x,int k) { int y1=0; int y2=0; int y3=0; y1=0; y2=1; y3=1; if (x==2||(x>2&&k==20)) x=x-1; while (y3<=x) { y1=y1+1; y2=y2+2; y3=y3+y2; } return y1; } intisk(intn,int k) { if (k!=20) { if (k!=n) k=21; else if (k==19) k=0; else k=k+2; } else { k=21; } return k; }

  7. Execution and Interaction #include <stdio.h> /**************************************************/ int in(); int isr(int x,int k); int isk(int n,int k); /**************************************************/ main(int argc, char **argv ) { int n=0,m=0; int k=1; printf("system is now active\n"); while (1) { n=in(); m=isr(n,k); k=isk(n,k); printf("RESULT: %i\n\n",m); } } /**************************************************/ int isr(int y,int k) { int y1=0; int y2=0; int y3=0; int z=0; int x=y; y1=0; y2=1; y3=1; if (x==2||(x>2&&k==20)) x=x-1; while (y3<=x) { y1=y1+1; y2=y2+2; y3=y3+y2; } z=y1; return z; } /**************************************************/ int isk(int n,int k) { if (k!=20) { if (k!=n) k=21; else { if (k==19) k=0; else k=k+2; } } else { k=21; } return k; } /**************************************************/ int in() { char c; int k=0; while (1) { k=0; putc('N',stdout); putc(':',stdout); putc(9,stdout); c=getc(stdin); /* printf("%i\n",c); */ if (c=='\n') { printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } if (c<'0'||c>'9') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } k=c-'0'; c=getc(stdin); if (c=='\n') { return k; } if (c<'0'||c>'9') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } k=k*10+(c-'0'); if (k>20) { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input number must be in {0,...,20}\n\n"); continue; } c=getc(stdin); if (c!='\n') { while (1) { c=getc(stdin); if (c=='\n') break; } printf("INFO: the input must be 1 or 2 digits\n\n"); continue; } return k; } } /**************************************************/ [zwh@panda 2013cp]$ ./isr1a INFO: system is now active N: 1 RESULT: 1 N: 23 INFO: the input number must be in {0,...,20} N: 19 RESULT: 4 N: ad INFO: the input must be 1 or 2 digits N: 9 RESULT: 3 N:

  8. Program Correctness It looks that the correctness claim holds, according to the sample executions. Question: Does the claim holds for all input sequences? In fact, there is an error when the input sequence is: 1 3 5 7 9 11 13 15 17 19 0 2 4 6 8 10 12 14 16 18 4 The input history may affect the behavior of the software, and it may cause errors in certain cases

  9. Program Correctness Question: Is the claim correct? Use model checking!

  10. Contents Integer Square Root Model Checking Compositional Reasoning Summary

  11. Modeling and Model Checking • Model Checking with VERDS • http://lcs.ios.ac.cn/~zwh/verds • Input to VERDS • VVM (VERDS verification model) • Modeling Language • VML (VERDS modeling langauge)

  12. Verification Process Automatic Translator C Program Model Properties VERDS Model Checker

  13. Correctness Claim (isr1a.sp) (at line 17): ((m*m)<=n)&&((m*m)+2*m+1>n)

  14. Model Checking [zwh@panda 2013cp]$ ../verds -c isr1a.c -sp isr1a.sp VERSION: verds 1.43 - JAN 2013 FILE: isr1a.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 326 ---------- time = 326 bound = 1 time = 326 ---------- time = 326 . . . . bound =105 time = 675 ---------- time = 675 The property is false, preparing files ... CONCLUSION: FALSE (time=3985 sec) ./verds –c isr.c –sp isr.sp VERSION: verds 1.42 - DEC 2012 FILE: isr.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 667 ---------- time = 667 . . bound =105 time = 1068 ---------- time = 1068 The property is false, preparing files ... CONCLUSION: FALSE (time=3156)

  15. Verification Process Automatic Translator C Program Model Properties VERDS Model Checker Negative Conclusion Error Trace

  16. Error Trace (isr1a.cex) --- STATE 0 --- n =0 m =0 k =1 pc =0 --- TRANS 1 --- --- STATE 1 --- n =0 m =0 k =1 pc =1 --- TRANS 2 --- --- STATE 2 --- n =0 m =0 k =1 pc =2 --- TRANS 3 --- --- STATE 3 --- n =1 m =0 k =1 pc =3 … … … --- STATE 102 --- n =18 m =4 k =20 pc =2 --- TRANS 3 --- --- STATE 103 --- n =4 m =4 k =20 pc =3 --- TRANS 4 --- --- STATE 104 --- n =4 m =1 k =20 pc =4 --- TRANS 5 --- --- STATE 105 --- n =4 m =1 k =21 pc =5

  17. Analysis of the Error Trace • An error occurred with the following sequence of inputs: 1 3 5 7 9 11 13 15 17 19 0 2 4 6 8 10 12 14 16 18 4 • Confirmation of the error by testing • Correction to the program is made

  18. Correction: Function main() #include <stdio.h> /********************************************/ int in(); int isr(int x,int k); int isk(int n,int k); /********************************************/ int main() { int n=0, m=0; int k=1; printf("INFO: system is now active\n"); while (1) { n=in(); k=isk(n,k); m=isr(n,k); printf("RESULT: %i\n\n",m); } }

  19. Verification of the Corrected One [zwh@panda 2013cp]$ ../verds -c isr2a.c -sp isr1a.sp VERSION: verds 1.43 - JAN 2013 FILE: isr2a.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 276 ---------- time = 276 bound = 1 time = 276 ---------- time = 276 . . . . bound =105 time = 607 ---------- time = 607 CONCLUSION: TRUE (time=607 sec) ./verds –c isr.c –sp isr.sp VERSION: verds 1.42 - DEC 2012 FILE: isr.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 667 ---------- time = 667 . . bound =105 time = 1068 ---------- time = 1068 The property is false, preparing files ... CONCLUSION: FALSE (time=3156)

  20. Verification of the Corrected One Automatic Translator C Program Model Properties VERDS Model Checker Positive Conclusion

  21. Contents Integer Square Root Model Checking Compositional Reasoning Summary

  22. Verification Times

  23. Use of Assumption-Guarantee • Each function is augmented with a pair of assumption-guarantee as follows (isr2a.fsp): FUNCTION z=in() ASSUMPTION TRUE; GUARANTEE 0<=z&&z<=20; FUNCTION z=isk(x,y) ASSUMPTION 0<=x&&x<=20; GUARANTEE (!(z==20)||x=18);

  24. Verification Process (A/G) Automatic Translator C Program Model Properties VERDS Model Checker A/G Specification

  25. Verification Subgoals 102 verification subgoals, one for main(), and one for each of the functions. ../verds -c isr2a.c -sp isr1a.sp -fsp isr2a.fsp ../verds -ck in -Dint=i5 isr2a.vvm ../verds -ck isk -Dint=i5 isr2a.vvm

  26. Verification of the Corrected One [zwh@panda 2013cp]$ ../verds -c isr2a.c -sp isr1a.sp -fsp isr2a.fsp VERSION: verds 1.43 - JAN 2013 FILE: isr2a.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 75 ---------- time = 75 bound = 1 time = 75 ---------- time = 75 bound = 2 time = 75 ---------- time = 75 . . . . bound = 11 time = 82 ---------- time = 82 CONCLUSION: TRUE (time=82 sec) ./verds –c isr.c –sp isr.sp VERSION: verds 1.42 - DEC 2012 FILE: isr.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 667 ---------- time = 667 . . bound =105 time = 1068 ---------- time = 1068 The property is false, preparing files ... CONCLUSION: FALSE (time=3156)

  27. Verification of the Corrected One [zwh@panda 2013cp]$ ../verds -Dint=i5 -ck in isr2a.vvm VERSION: verds 1.43 - JAN 2013 FILE: isr2a.vvm bound = 0 time = 31 ---------- time = 31 bound = 1 time = 31 ---------- time = 31 INFO: A/G=1 CONCLUSION: TRUE (time=34 sec) ./verds –c isr.c –sp isr.sp VERSION: verds 1.42 - DEC 2012 FILE: isr.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 667 ---------- time = 667 . . bound =105 time = 1068 ---------- time = 1068 The property is false, preparing files ... CONCLUSION: FALSE (time=3156)

  28. Verification of the Corrected One [zwh@panda 2013cp]$ ../verds -Dint=i5 -ck isk isr2a.vvm VERSION: verds 1.43 - JAN 2013 FILE: isr2a.vvm bound = 0 time = 32 ---------- time = 32 bound = 1 time = 32 ---------- time = 32 INFO: A/G=1 CONCLUSION: TRUE (time=37 sec) ./verds –c isr.c –sp isr.sp VERSION: verds 1.42 - DEC 2012 FILE: isr.vvm PROPERTY: A G (! (pc = 5 )| (((m * m ){ n )& (((m * m )+ ((2 * m )+ 1 ))> n ))) bound = 0 time = 667 ---------- time = 667 . . bound =105 time = 1068 ---------- time = 1068 The property is false, preparing files ... CONCLUSION: FALSE (time=3156)

  29. Verification Times • Original task with model checking time = 607 seconds • 3 tasks with model checking time < 100 for each

  30. Contents Integer Square Root Model Checking Compositional Reasoning Summary

  31. Questions?

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