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Real-Time Systems, COSC-4301-01, Lecture 5

Real-Time Systems, COSC-4301-01, Lecture 5. Stefan Andrei. Reminder of the last lecture. Multiprocessor Scheduling Available scheduling tools Available real-time operating systems. Overview of This Lecture. A Satisfiability Approach for the Scheduling Problem. Introduction.

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Real-Time Systems, COSC-4301-01, Lecture 5

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  1. Real-Time Systems, COSC-4301-01, Lecture 5 Stefan Andrei COSC-4301-01, Lecture 5

  2. Reminder of the last lecture • Multiprocessor Scheduling • Available scheduling tools • Available real-time operating systems COSC-4301-01, Lecture 5

  3. Overview of This Lecture • A Satisfiability Approach for the Scheduling Problem COSC-4301-01, Lecture 5

  4. Introduction • Most of the existing scheduling problems are NP-hard. • Researchers have been putting many efforts in finding good heuristics for solving scheduling problem in polynomial time. • However, there still exist classes of task sets that are feasible and most traditional schedulers (e.g. RM, EDF, LL) fail to solve the scheduling problem. • Existing SAT encodings could not be adopted for the given scheduling problem. COSC-4301-01, Lecture 5

  5. The main idea • We present a new satisfiability (SAT) based approach to the scheduling problem. • We transform time constraints into a satisfiability problem. • The obtained clauses are converted to the DIMACS format and passed as input to a SAT solver, such as Zchaff. • We conclude that our satisfiability-based approach is a promising technique for scheduling. COSC-4301-01, Lecture 5

  6. Related work • At first glance it seems to be an obvious translation of scheduling problems into SAT: • Create variables to represent the start times of the operations and create clauses to represent the necessary inequalities. • However the search space in SAT problems so generated turns out to be much larger than necessary. • Previous SAT encodings approaches like those expressed in [Memik, Fallah; 2002] and [Crawford, Baker; 1994] could not be applied either because they do not consider the exact time relations [Crawford, Baker; 1994] or they are too large [Memik, Fallah; 2002]. COSC-4301-01, Lecture 5

  7. References • [Memik, Fallah; 2002] S.O.Memik, F.Fallah: “Accelerated SAT-based scheduling of control/data flow graphs”, Computer Design: VLSI in Computers and processors, 2002. Proceedings, 2002 • [Crawford, Baker; 1994] J.M.Crawford, A.B.Baker: Experimental Results on the Application of Satisfiability Algorithms to Scheduling Problems, The 12th National Conference on Artificial Intelligence, 1994. COSC-4301-01, Lecture 5

  8. Scheduling • A task T is characterized by the following parameters: • S: start (also called release, ready, or arrival) time. • c: (maximum) computation time. • d: relative deadline (deadline relative to the task's start time) • p: period (how often the tasks are supposed to be executed) • D: absolute deadline (wall clock time deadline) e.g., D = S + d. COSC-4301-01, Lecture 5

  9. The propositional satisfiability problem • The SAT problem: “Given a set of clauses C (disjunction of literals – also known as CNF: Conjunctive Normal Form) on a finite set U of variables, find a truth assignment for U that satisfies all the clauses in C”. • The SAT problem is solved by the SAT solvers. • Example: Given U = {A, B, C} the set of variables and F = (A B C)  (A B C)  (A B C) a propositional formula, then a truth assignment for F is A = false, B = false, C = false. COSC-4301-01, Lecture 5

  10. Encoding Scheduling problems as SAT problems • For each operation i executing between time t and t+1 there exists a boolean variable ei,t. • We have considered this encoding for solving many types of problems which are elaborated as follows, such as: • Scheduling unit computation tasks in uniprocessor environment • Scheduling preemptive non-unit tasks for uniprocessor environment • Scheduling non-preemptive tasks in uniprocessor environment COSC-4301-01, Lecture 5

  11. Scheduling unit computation tasks in uniprocessor environment • Consider T={T1, .., Tn} a task set with Ti = (Si, ci, di, pi). • For simplicity, we assume pi=di. • If task Ti can execute one time unit between times t, t+1, …, t+ci, then the corresponding SAT clause is ei,t V ei,t+1 V... V ei,t+ci-2 V ei,t+ci-1. • Given tasks Ti and Tj for time between t and t+1, only either Ti or Tj is executing between t and t+1. • Hence, there exists a clause ei,t V ej,t (this is equivalent to: ei,tej,t and ej,tei,t). • The SAT encoding has to be done from 0 to LCM(p1, p2, …, pn). COSC-4301-01, Lecture 5

  12. Scheduling unit computation tasks in uniprocessor environment. Example • T1: s1 = 0, c1 = 1, d1 = p1 = 2 • T2: s2 = 0, c2 = 1, d2 = p2 = 3 • The SAT encoding has to be done by LCM(d1, d2), that is, 6: • e1,0 V e1,1 • e1,2 V e1,3 • e1,4 V e1,5 • e2,0 V e2,1 V e2,2 • e2,3 V e2,4 V e2,5 • e1,0 V e2,0 • e1,1 V e2,1 • e1,2 V e2,2 • e1,3 V e2,3 • e1,4 V e2,4 • e1,5 V e2,5 COSC-4301-01, Lecture 5

  13. Scheduling preemptive non-unit tasks for uniprocessor environment • Solution: • Divide each non unit-time task into tasks with unit computation time, then apply the previous SAT encoding. • The starting time and deadline for the subsequent tasks are incremented. • Precedence constraints are introduces between the sub-tasks created. • Consider sub-task j with less precedence than sub-task i then at any time t, sub-task i will not occur after sub-task j is executing, that is, ej,t Vei,z where z > t. COSC-4301-01, Lecture 5

  14. Scheduling preemptive non-unit tasks for uniprocessor environment. Example • T1: s1 = 0, c1 = 2, d1 = p1 = 4 • T2: s2 = 0, c2 = 2, d2 = p2 = 4 • The task set is converted to unit computation time: • T1=T1,1: s1,1 = 0, c1,1 = 1, d1,1 = 3, p1,1 = 4 • T2=T1,2: s1,2 = 1, c1,2 = 1, d1,2 = 4, p1,2 = 4 • T3=T2,1: s2,1 = 0, c2,1 = 1, d2,1 = 3, p2,1 = 4 • T4=T2,2: s2,2 = 1, c2,2 = 1, d2,2 = 4, p2,2 = 4 • The precedence constraints: T1 T2, T3 T4 COSC-4301-01, Lecture 5

  15. Scheduling preemptive non-unit tasks for uniprocessor environment. Example • e1,0 V e1,1 V e1,2 • e2,1 V e2,2 V e2,3 • e3,0 V e3,1 V e3,2 • e4,1 V e4,2 V e4,3 • e1,0 V e3,0 • e1,1 V e2,1 • e1,1 V e3,1 • e1,1 V e4,1 • e1,2 V e2,2 • e1,2 V e3,2 • e1,2 V e4,2 • d1 = 3  e1,3 • s2 = 1  e2,0 • d3 = 3  e3,3 • s4 = 1  e4,0 COSC-4301-01, Lecture 5

  16. The precedence constraints: • e2,1 V  e1,2 • e4,1 V e3,2 COSC-4301-01, Lecture 5

  17. Scheduling non-preemptive tasks in uniprocessor environment • A non-preemptable task is the one that once started cannot be stopped by any other process or task. • In practice tasks may contain critical sections that cannot be interrupted. • These critical sections are needed to access and modify shared variables or use shared resources. • The non-preemptable condition can be obtained by grouping together executing constraints from starting time to deadline with size equal to that of computation time. COSC-4301-01, Lecture 5

  18. Scheduling with non-preemption for uniprocessor environment - cont • So, a non-preemptable task i is allowed to execute for a computation time c starting at time t until deadline d. • Since a non-preemptive task T=(s, c, d, p) cannot not be interrupted, the conversion to the SAT problem should contain all possible solutions for task T to be scheduled in the intervals [s, c], [s + 1,c + 1], ..., [d - c, d]. COSC-4301-01, Lecture 5

  19. Example with two non-preemptive tasks • T1: s1 = 0, c1 = 2, d1 = 4 • T2: s2 = 0, c2 = 2, d2 = 4 • The SAT encoding is: • T1: (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) V • (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) V • (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) • T2: (e2,0 Λ e2,1 Λ e2,2 Λ e2,3) V • (e1,0 Λ e2,1 Λ e1,2 Λ e2,3) V • (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) COSC-4301-01, Lecture 5

  20. Example with two non-preemptive tasks • The following are the clauses corresponding to the fact that the processor is busy with at most one task at a given time: • e1,0 V e2,0 • e1,1 V e2,1 • e1,2 V e2,2 • e1,3 V e2,3 • Problem: the sub-formula from previous slide is not expressed in CNF, but in DNF (Disjunctive Normal Form). COSC-4301-01, Lecture 5

  21. DNF to CNF conversion • Some the above clauses are in Disjunctive Normal Form (DNF). • They have to be converted into Conjunctive Normal Form (CNF). • Once the whole formula is expressed in CNF, it has to be converted into DIMACS format as an input for SAT solvers. COSC-4301-01, Lecture 5

  22. DNF to CNF conversion. Example • New variables are introduced for each clause. • Example: F = (A1 A2  A3  A4)  (A5 A6  A7  A8) is converted to the following CNF formula (X1 and X2 are new propositional variables): • (X1  A1)  (X1  A2)  (X1  A3)  (X1  A4)  (A1 A2 A3 A4  X1) • (X2  A5)  (X2  A6)  (X2  A7)  (X2  A8)  (A5 A6 A7 A8  X2) • X1  X2 COSC-4301-01, Lecture 5

  23. Complexity of this translation • The traditional approach for converting a general DNF formula needs an exponential space and time complexity: • (A1,1 A1,2  …  A1,n1)  …  (Ak,1 Ak,2  …  Ak,nk) leads to a propositional formula with n1 * … * nk CNF clauses. • The approach from previous slide needs only n1 + … + nk + k + 1 CNF clauses and k new propositional clauses. COSC-4301-01, Lecture 5

  24. DNF to CNF conversion. Example • Coming back to the example from slide 18: • T1: (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) V • (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) V • (e1,0 Λ e1,1 Λ e1,2 Λ e1,3) • we get the following CNF formula: • (X1  e1,0)  (X1  e1,1)  (X1  e1,2)  (X1  e1,3)  (e1,0 e1,1 e1,2  e1,3  X1) • (X1  e1,0)  (X1  e1,1)  (X1  e1,2)  (X1  e1,3)  (e1,0 e1,1 e1,2  e1,3  X1) • (X1  e1,0)  (X1  e1,1)  (X1  e1,2)  (X1  e1,3)  (e1,0 e1,1 e1,2  e1,3  X1) • X1  X2  X3 COSC-4301-01, Lecture 5

  25. Some state-of-the-art SAT solvers • Siege - http://www.cs.sfu.ca/research/groups/CL/software/siege/ • zChaff - http://www.princeton.edu/~chaff/software.html • Cachet - http://www.cs.rochester.edu/u/kautz/Cachet/index.htm • SharpSAT - http://www2.informatik.hu-berlin.de/~thurley/sharpSAT/index.html • Others: http://www.satlive.org/bytype.jsp?reftypefrom=-2 • If the answer provided by this SAT solver or #SAT solver is ‘Unsatisfiable’, then the corresponding propositional formula is unsatisfiable. COSC-4301-01, Lecture 5

  26. The zChaff SAT Solver • It is a state-of-the-art SAT solver designed for robustness and efficiency. • It features a highly optimized deduction engine. • zChaff is designed with performance and capacity in mind. • zChaff has been tested on Solaris/Linux/Cygwin machines with g++ as the compiler. • It can also be compiled with Visual Studio Net under Windows. • zChaff can be compiled into a linkable library for integration purpose so that the users do not need to export instance into intermediate files to use zChaff. COSC-4301-01, Lecture 5

  27. DIMACS CNF Format • In the format the number of variables and the number of clauses is defined by the line “p cnf VARIABLES CLAUSES”. • The variables are assumed to be numbered from 1 up to VARIABLES. • It is not necessary that every variable appears in an instance. • Each clause will be represented by a sequence of integers, which are separated by a space, a tab, or a new line character. • The non-negated version of the variable i is represented by i; the negated version is represented by -i. • Each clause is terminated by a value 0. COSC-4301-01, Lecture 5

  28. Example • The boolean formula F = (v1 V v3) Λ (v1 V v2 V v3) has the following DIMACS format: p cnf 3 2 1 -3 0 2 3 -1 0 COSC-4301-01, Lecture 5

  29. Tool Description • The input of our tool is the number of tasks which are to be scheduled. • For each of the tasks it asks for its starting time, computation time and deadline. • It develops various clauses using the SAT encoding described in above part. • Based on the encoding it passes the DIMACS CNF format file to zChaff SAT Solver which gives whether it is either SAT solvable or not. • The output of zChaff also gives the variables which are true, using this variables we can schedule the tasks. COSC-4301-01, Lecture 5

  30. Running the Code • Compile the zChaff code obtained from the Internet using the make command in C++ compiler and keep it inside a folder zchaff. • Inside the zchaff folder create a folder called Translate. • Add all the Java source code in this folder. • Compile the java source code using javac *.java in J2SE. • Then run the Main class file from zchaff folder using java Translate/Main COSC-4301-01, Lecture 5

  31. Experimental Results COSC-4301-01, Lecture 5

  32. The experimental results - comments • The number of tasks in the original column are converted into unit computation time tasks and thus increasing the number of tasks in the for preemption column. • The increase in the number of tasks increases the size of SAT and thus affects the performance of the SAT solver. • It can be seen from the above results that the performance is faster if there are fewer tasks. COSC-4301-01, Lecture 5

  33. Conclusion • We have explored the potential of satisfiability to be used as a tool for modeling and solving scheduling problems. • We have also researched on the earlier work and proven that we need to consider computation time so as to make it suitable for real time systems. • We have presented experimental results on the performance of SAT solver zChaff. • The performance of the SAT decreases as the number of tasks increases. • We have demonstrated that a SAT solver presents a competitive alternative as a tool to find optimal solutions to the NP-complete time constrained scheduling problem. COSC-4301-01, Lecture 5

  34. Future work • We would like to consider the other categories of tasks set constraints. • Also adding resourses constraints would make the system more suitable for real time systems. • Minimize the number of preemptions (selects the solution provided by the SAT solver that has the minimum number of preemptions). • Assuming there are new tasks added to the initial specification, design an incremental conversion to SAT without repeating the time spend for the initial tasks set). COSC-4301-01, Lecture 5

  35. Summary • A Satisfiability Approach for the Scheduling Problem COSC-4301-01, Lecture 5

  36. Reading suggestions • Research papers • Ştefan Andrei: Schedulability Analysis COSC-4301-01, Lecture 5

  37. Coming up next • An Efficient Power-Aware Scheduling Algorithm for the Multiprocessor Platform COSC-4301-01, Lecture 5

  38. Thank you for your attention!Questions? COSC-4301-01, Lecture 5

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