Constraint Systems used in Worst-Case Execution Time Analysis Andreas Ermedahl Dept. of Information Technology Uppsala University

September 2000ASTEC/WCET2 Definition of WCET  WCET = Worst possible execution time for a program running on target hardware  One program in isolation  No interrupts or context switches  Other estimates:  Best Case ET = Best case, Inverse of WCET, hard  Average Case ET = Soft real-time, not hard 0tighter safe BCET estimates safe WCET estimates actual BCET actual WCET possible execution times

September 2000ASTEC/WCET3 Uses of WCET Estimates  Hard real-time systems  Scheduling  Schedulability analysis  System dimensioning  Formal verification  Program performance tuning void foo(int j, int a[]) { int i; for(i=100, i>0; i--) { if(j>50) a[i] = 2 * i; else a[i] = i; }  ?

September 2000ASTEC/WCET4 Static WCET Analysis  Don’t run the program - analyze it!  Guaranteed safe WCET  Trying to be as tight as possible  Provided all input is correct 0tighter safe BCET estimates safe WCET estimates actual BCET actual WCET possible execution times Static analysis will give results in the safe area Measurement will give results in the unsafe area

September 2000ASTEC/WCET5 Static WCET Analysis  Flow analysis  Determine the dynamic behavior of program  Low level analysis  Determine execution time for program parts on the hardware  Calculation  Combine flow and low-level times to give a WCET estimate. Compiler Object Code Target Hardware Reality program Actual WCET Low level analysis Calculation Flow analysis Analysis WCET Estimate Actual WCET WCET Estimate

September 2000ASTEC/WCET6 WCET Analysis Architecture Compiler Program Flow Analysis Intermediate Code Cache Analysis Cache Info Simulator Pipeline Analysis WCET Calculation Constraint Problem Flow Info Object Code  Modularization of WCET analysis  Several separate analysis steps  Make the analysis more retargetable Flow Fact Conversion  All analysis results are converted to constraints

September 2000ASTEC/WCET7 Example program Flow Info Characteristics do { if(...) do { if(...)... else... if(...)... else... } while(...) else... } while(...)... Basic finiteness Statically allowed Actual feasible paths // A // B // C // D // E // F // G // H // I // J Structurally possible flows (infinite) Relation between possible executions and flow info max = 10 max = 20 samepath(D,G) WCET found here = desired result A B C D F G H E Basic block graph J I WCET found here = overestimation overestimation

September 2000ASTEC/WCET8  Program model:  Basic block graph: nodes and edges  Execution count variable (x entity ) holds number of times entity gets executed Foo() C A B D E F G end Count Variables XAXAXAXA XBXBXBXB XCXCXCXC XDXDXDXD XEXEXEXE XFXFXFXF XGXGXGXG  x node for nodes X GA X AB X BC X BD X DE X EG X CE X EF X FG X fooA  x edge for edges

September 2000ASTEC/WCET9  Constraints: Foo() C A B D E F G end Constraints Generated XAXAXAXA XBXBXBXB XCXCXCXC XDXDXDXD XEXEXEXE XFXFXFXF XGXGXGXG X GA X AB X BC X BD X DE X EG X CE X EF X FG X fooA X foo =1 X end =1  Start and end condition X AB =X A X E =X CE +X DE X A =X fooA +X GA X BC +X BD =X B  Program structure X A <=100  Loop bounds X C +X F <=X A  Other flow information

September 2000ASTEC/WCET10 A B Pipeline Analysis  Time for a basic block:  Time from first instruction enters pipeline to last instruction leaves pipeline t A = 7 B IF EX M F t B = 5 IF EX M F A IF EX M F

September 2000ASTEC/WCET11 A B Pipeline Analysis  Pipeline overlap between basic blocks  Timing effect of going from A to B is negative to indicate pipeline overlap  We use a general purpose simulator to extract times for nodes and edges t A = 7 B IF EX M F t B = 5 t AB = 10  AB = -2  AB = t AB - t A - t B = = IF EX M F 10 IF EX M F A IF EX M F

September 2000ASTEC/WCET12 Pipeline Analysis Result Result: Foo() C A B D E F G end  Each edge annotated with pipeline timing effect t GA =-1 t AB =-2 t BC =-3 t BD =-5 t DE =-2 t EG =-2 t CE =-1 t EF =-3 t FG =-3 t fooA =-4  Each basic block annotated with its execution time t A =7 t D =2 t B =5 t C =12 t E =4 t F =8 t G =20

September 2000ASTEC/WCET13  WCET= max  (x entity * t entity )  Where each x entity satisfies all constraints Foo() C A B D E F G end t A =7 t D =2 t B =5 t C =12 t E =4 t F =8 t G =20 IPET Calculation XAXAXAXA XBXBXBXB XCXCXCXC XDXDXDXD XEXEXEXE XFXFXFXF XGXGXGXG X GA X AB X BC X BD X DE X EG X CE X EF X FG X fooA X foo =1 X AB =X A X E =X CE +X DE X A =X fooA +X GA X BC +X BD =X B X A <=100 X C +X F =100

September 2000ASTEC/WCET14  Solution methods:  Integer linear programming  Constraint satisfaction  Solution:  Counts for each individual node and edge  The value of the WCET Foo() C A B D E F G end Calculation methods X A =100 X B =100 X C =100 X D =0 X E =100 X F =0 X G =100 WCET=4800 X foo =1 X end =1

September 2000ASTEC/WCET15  Some flow information complicates the picture  Examples:  Local semantics: “For each entry of inner loop...”  Partially valid flow info: “During iterations 5 to 10 of inner loop it holds that...”  Loop dependencies: ”Number of iterations of inner loop depends on current iteration of outer”  Non-linear constraints: ”If D was taken then K will be taken once” B C D E G H I F J A LK Complicating Flow Information

September 2000ASTEC/WCET16 scope foo scope loop Local semantics  Create entry count variable: x entry(scope) holds number of times loop is entered  Local flow info are raised to the global level  For example: loop:[ ]:X H  10 gets converted to: gets converted to: X H  10 * X entry(loop) X H  10 * X entry(loop) B C D E G H I F J A Scope graph “For each entry of the loop block H will be executed at most 10 times” “Block H can not be executed more than 10 * the number of times the loop is entered” x entry(loop)

September 2000ASTEC/WCET17 scope loop Partial Flow Information B A C D F  Facts can (partially) overlap scope:range  Virtual scopes: scope:range let facts be valid for complete range of iterations G E Loop bound: 20 loop : 1..5 : X C =5 (f1) loop : : X C + X F  8 (f2) loop:1..2 {f1} loop:3..5 {f1,f2} loop:6..10 {f2} loop: { } X C = X C + X C + X C + X C X C + X C =5 (f1)

September 2000ASTEC/WCET18 scope loop Partial Flow Information B A C D F  Graph is unrolled according to overlapping flow info  Can generate large graphs  Flow information often local (but can stretch over loop borders)  Dependent flow info can be used to consider subpart of graph in isolation G E loop:1..2 {f1} loop:3..5 {f1,f2} loop:6..10 {f2} Fact f1 and f2 only overlaps iterations 1..10

September 2000ASTEC/WCET19  Some flow info generates non-linear constraints  ”Number of iterations of inner loop depends on current iteration of outer” outer: X B  55 or outer: X A * X A  X B  ”If D was taken then K will be taken once” if X D > 0 then X K > 0 and if X D = 0 then X K = 0  More powerful solver needed? B C D E G H I F J A LK Non-linear Constrains

September 2000ASTEC/WCET20 WCET Tool Prototype  Prototype tool implemented  Works over whole program or use flow info to work bottom up over smaller program parts  Fast solution times when testing with CPLEX or similar solver (a network flow problem?)  We use integer linear programming (ILP) and lp_solve()  Not all flow information can be handled  Rather fast calculation times  Flow information sometimes generates large graphs and solution times  Other WCET research: Expressing hardware effects using constraints generates huge constraint systems and large solution times

September 2000ASTEC/WCET21 Experimental Results  Small programs: fast calculation (even with complex structure and flow)  Larger programs: Flow information slows down but increase precision  Example program: Nsichneu  Automatically generated program with massive amount of if-statements (> 250)

September 2000ASTEC/WCET22 Questions  What constraint solver should we use?  Are the generated constraints of a certain type?  More comments....

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