Sequential Circuit Synthesis - II

1. Sequential Circuit Synthesis - II Virendra Singh Indian Institute of Science Bangalore IEP on Digital System Synthesis, IIT Kanpur

2. Incompletely Specified Machine • The specified behaviour of a machine with partially specified transitions can be described by another machine whose state transitions are completely specified • The transformation is accomplished by replacing all the dashes in the next state entries by T and adding a terminal state T whose output are unspecified Sequential@iitk

3. Compatible States • State Si of M1 is said to cover, or contain, state Sj of M2 if and only if every input sequence applicable to Sj is also applicable to Si, and its application to both M1 and M2 when they are initially in Si and Sj, respectively, results in identical output sequences whenever the outputs of M2 are specified • Machine M1 is said to cover machine M2 if and only if, for every state Sj in M2, there is a corresponding state Si in M1 s.t. Si covers Sj Sequential@iitk

4. Compatible States • Two states Si and Sj of machine M are compatible, if and only if, for every input sequence applicable to to both Si and Sj, the same output sequence will be produced whenever both outputs are specified and regardless of whether Si or Sj is the initial state • Si and Sj are compatible, if and only if, their outputs are not conflicting (i.e., identical when specified) and their Ii-successor, for every Ii for which both are specified, are the same or also compatible. • A set of states (Si, Sj, Sk, ….) is called compatible if all its members are compatible Sequential@iitk

5. Compatible States • A compatible Ci is said to be larger than, or to cover, another compatible Cj, if and only if, every state in Cj is also contained in Ci • A compatible is maximal if it is not covered by any other compatible Sequential@iitk

6. Compatible States Sequential@iitk

7. Compatible States • A set of states is compatible if and only if every pair of the states in that set is compatible Sequential@iitk

8. Compatible States B+ = B” B+ = B’ Sequential@iitk

9. Merger Graph • Transforming into fully specified machine may not be optimal one • First generate the entire set of compatibles • Select an appropriate subset, which will form the basis of state reduction leading to minimum machine Sequential@iitk

10. Merger Graph • A set of states is compatible if and only if every pair of states in that set is compatible • It is sufficient to consider pair of states and use them to generate entire set • Compatible pair of states is referred as compatible pairs • Let the Ik-successors of Si and Sj be Sp and Sq, respectively; then (Sp Sq) said to be implied by (SiSj) • (SpSq) are referred as implied pair Sequential@iitk

11. Merger Graph • Merger graph is undirected graph • It consists of n-vertices, each of which corresponds to a state of M • For each pair of states (Si Sj) in M whose next state and output entries are not conflicting, an undirected arc is drawn between vertices Si and Sj • If for a pair of states (Si Sj) the corresponding outputs under all inputs are not conflicting, but successors are not the same, an interrupted arc is drawn between Si and Sj, and then implied pairs are entered in the space Sequential@iitk

12. Merger Graph A (CF) B F (CF) (AB) (CD) (EF) (BE) C E D Sequential@iitk

13. Merger Graph Merger graph is undirected graph Nine compatible pairs (AB), (AC), (AD), (BC), (BD), (BE), (CD), (CF), (EF) (AB), (AC), (BC) are compatible => (ABC) compatible Find minimal set of compatible {(ABCD), (BC), (BE), (DE)} Sequential@iitk

14. Merger Graph • A set of compatible (for machine M) is said to be closed if, for every compatible contained in the set, all its implied also contained in the set. • A closed set of compatibles which contains all the sates of M is called closed covering • A set {(AD), (BE), (CD)} is a closed covering Sequential@iitk

15. Compatible Graph • The compatible graph is a directed graph whose vertices corresponding to all compatible pairs, and arc leads from vertex (Si Sj) to vertex (Sp Sq) if and only if (Si Sj) implies (Sp Sq) • It is a tool which aids in the search for a minimal closed covering • Compatible pairs are obtained from merger graph Sequential@iitk

16. Compatibility Graph A (CF) (BE) (BC) B E (AE) (BC) (BC) (AB) (AD) D (DE) C Merger Graph Sequential@iitk

17. Compatibility Graph A AC (CF) (BE) (BC) B E AD (AE) BE (BC) (BC) (AB) (AD) BC D (DE) CD C DE Merger Graph Sequential@iitk

18. Minimization using Network Model • Behaviour of sequential circuit can be described by traces, i.e., sequence of inputs and outputs • Various approaches to optimize • Ignore registers and optimize the combinational logic • Retiming – move position of registers only Sequential@iitk

19. Retiming • Minimize cycle time or the area of synchronous circuits by changing the position of the registers • Cycle time is bounded from below by the critical path delay in the combinational circuit • Retiming aims at placing the registers in the appropriate position, so that the critical paths they embrace are as short as possible • Moving registers may increase or decrease the number of regsisters Sequential@iitk

20. Retiming + + + Host δ δ δ δ Sequential@iitk

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23. Example P0 = (ABCDEFG) P1 = (ABCDFG) (E) P2 = (AF) (BCDG) (E) P3 = (AF) (BD) (CG) (E) P4 = (A) (F) (BD) (CG) (E) P5 = (A) (F) (BD) (CG) (E) Machine M2 Sequential@iitk