1. On Using Linearly Priced Timed Automata for Flow Analysis
MdH – 23/11/2005 Meeting
On Using Linearly Priced Timed Automata for Flow Analysis

2. What is a LPTA?
Timed Automata with extension: cost accumulation during behavior
More formally: tuple A=(L, C, l0, E, I, P)
Cost for taking an edge is Pe, and for staying n cycles in a state is Ps*n (should be > 0)
Look for min cost ending of traces ending in a goal state
Can “guide” the state space exploration with manually entered heuristics
Use branch and bound techniques

3. Example of LPTA
Min from left to right: 14

4. LPTA and Scheduling Problems
Optimal Task Graph Scheduling (TGS): schedule a number of interdependent tasks on heterogeneous processors
Use a network of TA to simulate the multiple processes
Inter-process Interaction: synchronization channels
Time duration: Guarded edges over clock variables
Associate cost to edges and states
Uses work on priced symbolic state for solving

5. Similarity to Flow Analysis
Flow analysis: extract the dynamic behavior of the prog
Flow extraction, representation, calculation conversion
LPTA: good for extraction and representation at least (safe and tight # of iterations)
Cost is time (can also be modeled as clock guards…)
Network of TA can simulate multiple processes

6. Cont’d
Instead of trying to find the optimal path cost-wise, we try to find the *least* optimal price
Need to modify Branch and Bound algorithm
Alternatively: use negative cost for guards: the more transitions, the better=WCET (need to extend the framework a little bit)
Priced Symbolic State ↔ Symbolic Evaluation method
Reachability Analysis ↔ Infeasible Path Determination
Priced Clock Region ↔ State Space Reduction

7. Min and Max Algorithms Cost = inf Passed =empty Waiting = {(l0, C0)}
While Waiting != empty
select (l, C) from Waiting
if (l, C) ╞  and min(C) < cost then
Cost = min(C)
if  (l, C’) in Passed: C’ !subset C then
add (l, C) to Passed
 (m, D), (l, C) ~>(m, D):
add (m, D) to Waiting
Return Cost
Cost = 0
Passed =empty
Waiting = {(l0, C0)}
While Waiting != empty
select (l, C) from Waiting
if (l, C) ╞  and max(C) > cost then
Cost = max(C)
if  (l, C’) in Passed: C’ !subset C then
add (l, C) to Passed
 (m, D), (l, C) ~>(m, D):
add (m, D) to Waiting
Return Cost

8. Obvious Corollaries Problems:
Still need to derive platform specific timing to quantify actual WCET
Need to convert existing code to LPTA and extract platform dependent models
Recursion!
BUT:
Can work on a higher level than either source/intermediate/machine code
Allows for upstream formal testing
Simplicity of modeling aspect
Powerful tools for analysis exist (notably Uppaal CORA)

9. Tiny Uppaal Example Taken from Ebbe’s thesis
Note: we’re not using the CORA variant

10. Closing Thoughts Concepts still need refinement
How does LPTA compare to syntax/time trees?
Is it really useful?