1. Sequential Flexibility

2. Overview Computing flexibilities in combinational logic networks
FSM networks
Computing sequential flexibilities
Contrasting combinational and sequential flexibility computations
FSM minimization using register splitting
Examples

3. Minimizing a Node – Computing the Flexibility at a Node
Combinational
Logic Network
Definition. A flexibility at a node is a relation (between the node’s inputs and output) such that any well-defined sub-relation used at the node leads to a network that conforms to the external specification.
Definition. The complete flexibility (CF) is the maximum flexibility possible at a node.

4. Computing the CF for a node - global step

5. Computing CF - local stepYi yi

6. CF Yi yi Yi yi
Note that essentially the same computation applies for multiple-output nodes, i.e. where

7. FSM networks Network of finite state machines (FSMs)
Problem: Compute the Complete Sequential Flexibility of a node

8. Complete Sequential Flexibility
CSF is the maximum set of FSM behaviors (represented by a pseudo-non-deterministic FSM), such that implementing any sub-behavior and replacing the sub-network by the new implemented part does not violate the specification of the total network.

9. FSMs and Automata
If the distinction between inputs and outputs is taken away (i/o becomes io), then an FSM becomes an automaton.
It has the following properties as an automaton
All states are accepting
It is incomplete
can make it complete by adding one non-accepting state
Its language is
prefix closed
i -progressive
1 0
1 1
0 0
1/0
1/1
0/0
Automaton
FSM

10. FSM networks – computing complete sequential flexibility (CSF)
Specification S (i,o)
Context C (i,v,u,o)
Unknown X (u,v)
context
unknown i o u v
spec
Problem: Given S and C, find the Most
General Solution (MGS) of

11. FSM Networks The most general solution (MGS) of is
context
unknown i o u v
spec
The most general solution (MGS) of is
In general, MGS is deterministic automaton
but as an FSM it is non-deterministic (NDFSM)

12. Languages.
A Language is set of finite length strings on the symbol set
i.e. a subset of
(a b c a c d f g g g)
At this point, we don’t care how the language is
generated or represented. So initially the comments
apply to all kinds of languages
A symbol can be made up of a vector of
variable values, e.g. 1a3de0 or
These are examples of a single symbol.
Languages can be manipulated as follows:

13. Union -
Intersection -
Complement -

14. Language over cross-product of alphabets
A language over where X and Y are symbol sets consists of finite strings of pairs
Such that

15. Projection and Lifting
Given a Language
over the alphabet
projection is defined as –
Given a Language L over the alphabet X lifting
to the alphabet
is defined as -
where - stands for any symbol in Y

16. Classes of Languages A language is prefix closed if A language over
is I-progressive if

17. Composition of Languages
Given disjoint alphabets I,U,O and languages L1 over
and L2 over
their synchronous composition is .

18. Solving a language equation
Theorem A: Let A and C be languages over alphabets
and
respectively. For the equation,
the Most General Solution is A I U O X .

19. A X Proof: We prove Theorem A. Let . Then means that I U O Thus
is the largest solution of

20. Complete Sequential Flexibility (CSF)
CSF is maximum sub-behavior in MGS which is prefix closed and u-progressive.
For unknown to be an FSM, it must be progressive in its inputs
CSF u v

21. Example: Coin Game (NIM)
Context describes the state of the game and legal moves. Its input is random moves by first player and its output tells if the game is in a losing state.
In God We
Trust
Specification is a 3-state automaton, playing, won, and lost.
Players alternate turns
On each turn, player can take 1-n coins from any one pile
Player who takes last coin loses
Winning strategy: Give your opponent a pile of coins with even number of 1’s in bit columns (except at end)
6 = 1 1 0
5 = 1 0 1
3 = 0 1 1
____
2 2 2
Example: 5 3 6

22. Example of CSF computation: NDFSM represented as automaton
The CSF is a non-deterministic FSM
5 inputs, 5 outputs, 21 states, 34 transitions
Inputs p1_0 p1_1 d1_0 d1_1 d1_2 Outputs p2_0 p2_1 d2_0 d2_1 d2_2
In God We
Trust

23. Computational Procedure
Specification: S (i,o)
Context: C (i,v,u,o)
(these have been converted into automata)
Computing the transition
relation of CSF
Complement S
Raise to variables of C
Compose with C
Hide variables not in X
Complement result

24. Finite Automata
A finite automaton (FA) is
where S is a set of states,
is an input alphabet,
is a transition relation, r is the initial state, and
is the set of accepting states.

25. An input sequence
leads from r to s’ if there exists a sequence of states,
for all i = 0, ... ,n-1.
w is in the language of F ( )
such that
if and only if w leads from r to
i.e.
where
denotes the set of states that can be reached from r
under the input sequence w.

26. Theorem: A languages is regular if and only if it is the language of a finite automaton
Theorem: The set of all languages for deterministic FA is the same as for non-deterministic FA.
(this can be shown by using the so-called subset construction.)

27. Operations on FA.
projection (
): convert F over
into F’ over X by replacing each edge (xv s s’) by the edge (x s s’)
lifting (
): convert F over X into F’ over
where
by replacing each edge (x s s’) by
stands for any .

28. Operations on FA.
Product
Given FAs
both over
, the product is
where
Complementation
If F is deterministic, then .
If F is non-deterministic, the only known way for
complementation is to determinize it first. This is
done by the sub-set construction.

29. Composition Synchronous Composition. Given two automata and
over alphabets
and
their synchronous composition is
i.e. the product of the two automata when they are
made to have the same alphabet.

30. Subset Construction Given NFA
we create a DFA F’ with the same language as F:
where
and s’
Theorem: F and F’ have the same language.
Proof:

31. Finite State Machines as Automata
A FSM is
where I is the set of input symbols, O the set of output symbols, r the initial state, and T(s,i,s’,o) is the transition relation. A transition (s,i,s’,o) from state s to s’ with output o can happen on input i can if and only if If
then M is complete, otherwise partial.

32. It is deterministic if for all (s,i) there is at most one (s’,o)
such that
It is pseudo-non-deterministic if for all (s,i,o) there is at
most one s’ such that

33. Converting an FSM to an automaton
An FSM M can be converted into an automaton F by the
following:
where
Note that Q = S, i.e. all states are accepting
The resulting automaton is typically not complete, since
there are io combinations for which a next state is not
defined. We can complete it by augmenting
to include a transition to a new non-accepting state DCN.
DCN s

34. FSMs as Automata
The language of an FSM is defined to be the language of the
associated automaton
A pseudo non-deterministic FSM is one whose automaton
is deterministic.
The language of an FSM is prefix closed.
The language of an FSM is I-progressive
Conversion
is done by grouping i/o on edges to (io) and making all
states accepting.
Conversion
can be done only if the language is prefix closed and
I-progressive. In this case, delete all non-accepting states
(prefix), and change edges from (io) to i/o.

35. Computational Procedure
Specification: S (i,o)
Context: C (i,v,u,o)
(these have been converted into automata)
Computing the transition
relation of CSF
Complement S
Raise to variables of C
Compose with C
Hide variables not in X
Complement result

36. Comparison with combinational case
Sequential
context
unknown i o u v
spec
Combinational Yi yi
unknown

37. Application - splitting FSM blif filesu i v o

38. A run on s298.blif splitting 14 latches into 7 each
mvsis 07> source langi.script
Extracting STG of spec ...
The extracted STG has 218 states and 1078 transitions.
Extracting STG of fixed ...
The extracted STG has 43 states and 128 transitions.
Extracting STG of particular solution ...
The extracted STG has 20 states and 400 transitions.
Determinizing the spec ...
The automaton is deterministic; determinization is not performed.
Computing the product ...
Product: (43 st, 128 trans) x (219 st, 1297 trans) -> (966 st, 9975 trans)
Determinizing the product and making progressive...
Checking containment …
The solution composed with fixed is contained in the spec.
The particular solution is contained in the solution
mvsis 13> psa x.aut
x.aut: The automaton is incomplete (552 states) and deterministic.
mvsis 13> minimize x.aut x-min.aut
State minimization: (554 states, 5674 trans) -> (272 states, 2704 trans)

39. X-min.aut but not from splitting s298 (too big to show);
instead from splitting s27

40. The End