1. IIT Kharagpur & Kingston Uni
6/4/2018
Verification of Scheduling in
High-level Synthesis
Presented by
Chris Reade
Authors
C. Karfa, C. Mandal, D. Sarkar, S. R. Pentakota
Indian Institute of Technology, Kharagpur, India
&
C. Reade
Kingston Business School, Kingston University, England
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2. IIT Kharagpur & Kingston Uni
6/4/2018
Overview
Formal method for checking equivalence between two descriptions of the target system, (before and after scheduling).
Based on equivalence checking of FSMDs (finite state machines with data paths).
(Roughly) Find cutpoints in one FSMD to break loops, Visualize computations as concatenation of paths from cutpoints to cutpoints Identify equivalent finite path segments in the other FSMD; Repeat with the FSMDs interchanged.
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3. IIT Kharagpur & Kingston Uni
6/4/2018
Application
Algorithm works for
Basic block based scheduling.
Path-based scheduling where path segments are merged by the scheduler. Most of the existing algorithms failed in this case.
Can handle many arithmetic transformations.
Supports simple code motions.
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4. IIT Kharagpur & Kingston Uni
6/4/2018
FSMD vs. FSM (Why FSMD ?)
The general equivalence problem of FSMDs (EPFSMD) is undecidable and not even partially decidable.
Final targeted hardware only has finite datapath. So the restricted problem could be reduced to equivalence problem of FSM models (EPFSM). EPFSM is decidable.
BUT… this gives a state explosion
An n-bit datapath results in a number of states of the order 2kn . Thus, FSM modelling faces state explosion.
Also, scheduling algorithms may never use the finiteness of the targeted hardware - so neither should their analysis.
Hence look for approximation with FSMDs…
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5. IIT Kharagpur & Kingston Uni
6/4/2018
 - Path from qm to qn
A finite transition sequence.
All intermediate qj are distinct,
qn may be equal to qm.
R - Condition of execution of  Logical expression over variables.
If R is satisfied at the beginning of , then  is executed.
r - Data transformation over  r =  S, O  where S is transformation of variables
O is the output list along the path .
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6. Computations of an FSMD
IIT Kharagpur & Kingston Uni
6/4/2018
Computations of an FSMD
A Computation of an FSMD:
A finite walk from the reset state back to itself without intermediate occurrences of the reset state.
Computation can be represented as concatenation of paths.
Two computations c1 and c2 are equivalent if Rc1 = Rc2 & rc1 = rc2.
A sample computation is shown in Fig. 2.
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7. IIT Kharagpur & Kingston Uni
6/4/2018
Normalization
Checking Rc1 = Rc2 & rc1 = rc2
could involve the whole of integer arithmetic which is an undecidable problem (validity problem of first order logic).
In this work, we use a syntactic approximation - normal forms for expressions
R and r may be calculated by forward or backward substitution method.
So normal forms for R and r as well.
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8. IIT Kharagpur & Kingston Uni
6/4/2018
Correctness Problem
Finite Path Cover :
A set P = { p0, p1, p2, … pk } of FSMD M is said to be a path cover of M iff any computation of M can be viewed as a concatenation of paths from P.
Equivalence of two FSMDs :
For any computation c0 of M0, there exists an equivalent computation c1 in M1 and vice-versa.
In other words, there exists a finite path cover of M0, such that each path has a corresponding equivalent in M1 (with consistency for corresponding endpoints),
and vice-versa.
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9. Verification Algorithm
IIT Kharagpur & Kingston Uni
6/4/2018
Verification Algorithm
For machines M0 and M1 keep track of:
pairs of corresponding states (found so far)
matched pairs of paths (found so far)
unmatched paths to be processed
Step 1: Identify initial cutpoints in M0 to break loops:
Reset state + states with more than one output transition
Step 2: (Main loop)
Process an unmatched path  from M0 (succeed when none left).
Use findEquivalent (, q) on M1 where q corresponds to start of 
may find direct corresponding path in M1, or
may require path to be extended
(creating new cutpoints and new paths to match), or
may fail with no possible extension (exit with failure)
Step 3 : Identify initial cutpoints in M1.
Step 4 : Repeat the same procedure as described in step 2 with roles of M0 and M1 interchanged.
Step 5 : If succeed for both step 2 and 4 then
M0 and M1 are computationally equivalent, otherwise report a failure.
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10. IIT Kharagpur & Kingston Uni
6/4/2018
Algorithm Complexity
The complexity of the algorithm is determined by step 2.
For n control states in M0 or M1 we have O(n) cutpoints.
Time complexity of function findEquivalent (, q) is O(kn2).
Proportional to number of paths from q examined. with k parallel edges between two states
Lower bound is (1) when equivalent path found directly
Path extensions may be required. Maximum length after extensions is O(n), so starting from a cutpoint, up to O(kn-1), if k  1 or O(n2), if k = 1 paths of M0 may need to be examined.
Thus, total number of pairs of the form (, q) that need to be examined is O(n2kn-1), if k  1 or O(n4), if k = 1.
Lower bound is (n) as all the nodes of M0 need to be examined at least once and each cutpoint of M0 paired with one cutpoint in M1.
Overall complexity of step 2 is O(n4kn) if k1 or O(n6) if k=1 and (n).
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11. IIT Kharagpur & Kingston Uni
6/4/2018
Experimental Results
Implemented in ‘C’.
Results are shown for several High-level Synthesis benchmarks.
Name
#state
#path in cover
#path extended
CPU time
In ms M0 M1
DIFFEQ 4 12 3
2.442
EWF 35 1
1.820
GCD 7 11
3.976
DCT 29
1.754
TLC 8 13 14 2
4.196
MODN 6
4.324
PERFECT 9 5
4.028
Execution time sensitive on number of paths explored, not on number of states in FSMD.
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12. IIT Kharagpur & Kingston Uni
6/4/2018
Summary
Equivalence checking of FSMDs introduced as a formal verification method for scheduling. .
Algorithm is strong enough to accommodate
path based scheduling algorithm where consecutive path segments of input behavior are merged by the scheduler.
Most of the existing verification algorithms can fail in this case. Algorithm can also handle simple code motions.
Normal form is considered for the representation of condition of execution and data transformation to handle the arithmetic transformation.
Implementation has been completed. Experimental results are for several high-level synthesis benchmarks. Efficiency reflects in the execution time for these examples.
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13. IIT Kharagpur & Kingston Uni
6/4/2018
An Example
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14. IIT Kharagpur & Kingston Uni
6/4/2018
Extend this path
NO PATH FOUND
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15. Verification Problem Formulation
IIT Kharagpur & Kingston Uni
6/4/2018
Verification Problem Formulation
Finite State machine with Data path (FSMD) :
An ordered tuple < Q, q0, I, V, O, f, h > where
Q = { q0, q1, q2, …, qn } is the finite set of control states.
q0  Q is the reset state.
I is the set of primary input signals.
V is the set of storage variables.
O is the set of output signals.
f : Q x S → Q, is the state transition function.
h : Q x S → U, is the update function of the output and storage variables. S and U is defined below.
U = { x  e | x  O  V and e  E } represents a set of storage or output assignments, where E represents a set of arithmetic expressions over the set I  V.
S is a set of arithmetic relations between two expressions from the set E.
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