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Sequential System Synthesis -- Finite State Machine

Sequential System Synthesis -- Finite State Machine. Outline: Finite State Machine. Definitions FSM Representations State Transition Graph (STG) Flow Table Cube Table State Minimization Completely Specified FSM Incompletely Specified Machine (ISM) State Encoding.

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Sequential System Synthesis -- Finite State Machine

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  1. Sequential System Synthesis-- Finite State Machine

  2. Outline: Finite State Machine • Definitions • FSM Representations • State Transition Graph (STG) • Flow Table • Cube Table • State Minimization • Completely Specified FSM • Incompletely Specified Machine (ISM) • State Encoding ENEE 644

  3. Definition: Finite State Machine • A Finite State Machine (FSM) of Mealy type is a 6 tuple <I,S,,S0,O,> • I: input alphabet, a non-empty set of input values; • S: a non-empty, finite set of states; •  : SxI  S, a function defines the next state; • S0: S, the set of initial/reset states; • O: output alphabet; •  : SxI  O, a function defines the output. • A finite state machine of Moore type is defined in the same way except that the output function : S  O does not depend on the present inputs. ENEE 644

  4. Example: Finite State Machine • I = {x,y} • S = {A,B,C} • S0= {A} • (A,x) = A, (B,x) = A, (C,x) = C (A,y) = B, (B,y) = C, (C,y) = A • O = {0,1} • (A,x) = 0, (B,x) = 0, (C,x) = 0 (A,y) = 1, (B,y) = 0, (C,y) = 1 ENEE 644

  5. FSM Representation: STG In sum, a STG is a weighted, directed graph where self loops and duplicated edges are allowed. Each node has at most |I| outgoing edges and |I|x|S| incoming edges. Total number of edges is  |I|x|S|+|S0|. • State Transition Graph: • Node  state (S) • Edge  transition ( : SxI  S,  : SxI  O, S0) • Direction: from the current state to the next state • Label: input/output information for the transition • Special edges: edges without source, their ending nodes are initial states ENEE 644

  6. x/0 A y/1 x/0 y/1 y/0 C B x/0 Example: FSM as an STG • I = {x,y} • S = {A,B,C} • S0= {A} • (A,x) = A, (A,y) = B, (B,x) = A, (B,y) = C, (C,x) = C, (C,y) = A • O = {0,1} • (A,x) = 0, (A,y) = 1, (B,x) = 0, (B,y) = 0, (C,x) = 0, (C,y) = 1 ENEE 644

  7. x/0 A y/1 x/0 y/1 y/0 B C x y A B,1 x/0 B A,0 C,0 C C,0 A,1 FSM Representation: Flow Table • The flow table of an FSM <I,S,,S0,O,> is a |S|x|I| table, where the i-th row represents state Si, the j-th column represents input value xj. The entry at (i,j) is a 2 tuple <(Si,xj), (Si,xj)>. The initial states S0 can be specified separately. • Example: A,0 ENEE 644

  8. x/0 A y/1 I PS NS O x/0 y/1 x A A 0 y/0 C B x y A y B 1 x B A 0 A A,0 B,1 x/0 y B C 0 B A,0 C,0 x C C 0 C C,0 A,1 y C A 1 FSM Representation: Cube Table • The cube table of an FSM <I,S,,S0,O,> is a (|S|x|I|)x4 table, where in each row, the first column represents input value xj, second column is the state Si, third column is the next state (Si,xj), and the last column is the output (Si,xj). The initial states S0 can be specified separately. • Example: ENEE 644

  9. FSM with Incomplete Specification • An FSM <I,S,,S0,O,> is incompletely specified if  and/or  are incompletely specified functions. (I.e., they are not defined on some combinations of inputs and present states.) Otherwise, it is completely specified. • In STG, this means there exist nodes with less than |I| outgoing edges; • In flow table, this means there exist undefined entries; • In cube table, this means there exist undefined rows. ENEE 644

  10. Make Incomplete Complete • In STG: add a dummy state called trap state. • In flow table: leave the entry empty or fill it by <don’t care, don’t care>. • In cube table: delete the undefined row or fill the last two columns by don’t cares. y/- x/0 x/0 A A D D x/- y/1 y/1 y/1 y/1 x/0 x/0 ? x/- y/0 y/0 B C B C -/- ENEE 644

  11. FSM Minimization • FSMs may contain redundant states, i.e. states whose function can be accomplished by other states. • Removing the redundant states decreases the number of states in the FSM, and in general results in a simplification in the final implementation. • State minimization is the transformation of a given FSM into an equivalent FSM with no redundant states (I.e. minimal number of states). ENEE 644

  12. compatibility relation Binary Relations • Given two sets A and B, a binary relationR between A and B is a subset of AxB={(x,y)|xA,yB}. We write xR y if (x,y)R . • Relation R BxB is • reflexive iff xRx for any xB; • symmetric iff xR y  yR x; • anti-symmetric iff xR y, yR x  x=y; • transitive iff xR y, yR z  xR z. • A binary relation R BxB is an equivalent relation if it is reflexive, symmetric, and transitive. ENEE 644

  13. Partition into Equivalent Classes • A partition of a set of B is a set of subsets BiB, such that • BiBi (ij) • iBi=B. • Given an equivalent relation R BxB, the equivalent class of xB is [x]={yB|xRy}. • x,yB, [x]=[y] or [x][y]=; • If B1,B2,…,Bn are all the different equivalent classes, then {B1,B2,…,Bn} is a partition of B. An equivalent relation gives a unique partition. ENEE 644

  14. Refinement of a Partition • Given two partitions P1={B11,B21,…,Bm1} and P2={B12,B22,…,Bn2} of a set B, P1 is a refinement of P2 if every subset (block) Bi1Bj2 for some j. • Let P1={B11,B21,…,Bm1} and P2={B12,B22,…,Bn2} be two sets of subsets of a set B, the meet of P1 and P2 is defined as the following set: P1•P2={Bi1Bj2|i=1,2,…m,j=1,2,…,n} • Theorem: If P1 and P2 are partitions, then P1•P2 is also a partition of the same set B, furthermore, it is a refinement for both P1 and P2. [Proof:] ENEE 644

  15. s t == ? s t Equivalent States of an FSM • Given two states s and t in an FSM, and a k-string x=(x0x1…xk-1), suppose zs=(zs0zs1…zsk-1) and zt=(zt0zt1…ztk-1) are the corresponding output strings when states s and t are used as starting state respectively. x is called a length-k distinguishing sequence for states s and t iff zsk-1 ztk-1. xk-1…x1x0 zsk-1…zs1zs0 xk-1…x1x0 ztk-1…zt1zt0 ENEE 644

  16. Equivalent States of an FSM • Two states s and t are k-equivalent, written as skt, iff there does not exist a distinguishing sequence for s and t of length k or less. • Two states are equivalent iff they are |S|-equivalent. • Define k={(s,t)| skt}, the set of all pairs of k-equivalent states. • k is an equivalent relation, I.e., it is • Reflexive: sks • Symmetric: skt  tks • Transitive: rks, skt  rkt ENEE 644

  17. 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Equivalent States of an FSM • 1={(A,C),(A,E),(C,E),(B,D),(B,F),(D,F), (C,A),(E,A),(E,C),(D,B),(F,B),(F,D), (A,A),…,(F,F)} • B11={A,C,E} • B21={B,D,F} • 2={(A,C),(A,E),(C,E),(B,D), (C,A),(E,A),(E,C),(D,B), (A,A),…,(F,F)} • B12={A,C,E} • B22={B,D} • B32={F} • 3={(A,C),(B,D),(C,A),(D,B),(A,A),…,(F,F)} ENEE 644

  18. Equivalent States Checking: Theory • Two states are equivalent iff they are |S|-equivalent. • Theorem 1. Let sx and tx be the x-successors of s and t in an FSM, then sk+1t  skt and xI, sxktx. • Theorem 2. Two states of a given FSM are equivalent iff they are (|S|)-equivalent. ENEE 644

  19. State Equivalence Checking: Practice • Goal: determine |S|(S), all pairs of equivalent states in an FSM S. • Partition-Refinement procedure: • Pk={B1k,B2k,…}: the partition determined by k, the k-equivalent state pairs. (P0=S={B10}) • Idea: For each block in Pk • partition it (for all xI) if its x-successors are not in the same block; • Refine the partition by taking the meet of these finer partitions; Stop when Pk+1=Pk ENEE 644

  20. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P0={(A,B,C,D,E,F)} (1-block) P1={(A,C,E),(B,D,F)} for block P12=(A,C,E): on x=0: next states: EEC blk indices: 111 Pb10={(A,C,E)}=P12 (no refinement) on x=1: next states: DBF blk indices: 222 Pb11={(A,C,E)}=P12 (no refinement) P2={(A,C,E)} level input blk no. ENEE 644

  21. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P1={(A,C,E),(B,D,F)} P2={(A,C,E)} for block P22=(B,D,F): on x=0: next states: DBB blk indices: 222 Pb20={(B,D,F)}=P22 on x=1: next states: FFC blk indices: 221 Pb21={(B,D),(F)} refine: P22=P22 •Pb21= Pb21 ={(B,D),(F)} P2={(A,C,E),(B,D),(F)} ENEE 644

  22. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P1={(A,C,E),(B,D,F)} P2={(A,C,E),(B,D),(F)} for block P13=(A,C,E): on x=0: next states: EEC blk indices: 111 Pb10={(A,C,E)}=P13 on x=1: next states: DBF blk indices: 223 Pb11={(A,C),(E)} refine: P13=P13 •Pb11= Pb13 ={(A,C),(E)} P3={(A,C),(E)} ENEE 644

  23. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P1={(A,C,E),(B,D,F)} P2={(A,C,E),(B,D),(F)} P3={(A,C),(E)} for block P23=(B,D): on x=0: next states: DB blk indices: 22 Pb10={(B,D)}=P23 on x=1: next states: FF blk indices: 33 Pb11 ={(B,D)}=P23 P3={(A,C),(E),(B,D)} ENEE 644

  24. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P1={(A,C,E),(B,D,F)} P2={(A,C,E),(B,D),(F)} P3={(A,C),(E),(B,D)} for block P33=(F): contains single state, cannot be partitioned. P3={(A,C),(E),(B,D),(F)} P3={(A,C),(E),(B,D),(F)} ENEE 644

  25. PS NS, z x=0 x=1 A E, 0 D, 1 B D, 0 F, 0 C E, 0 B, 1 D B, 0 F, 0 E C, 0 F, 1 F B, 0 C, 0 0/0 B D 0/0 1/0 0/0 1/1 1/0 A F 1/0 0/0 1/1 1/1 0/0 E C 0/0 Example: Finding Equivalent States P1={(A,C,E),(B,D,F)} P2={(A,C,E),(B,D),(F)} P3={(A,C),(E),(B,D),(F)} One can compute P4 in the same way, which gives P4={(A,C),(E),(B,D),(F)} P4=P3 so we stop Conclusion: A and C are equivalent B and D are equivalent ENEE 644

  26. 0/0 0/0 0/0 B D 0/0 0/0 B D 1/0 0/0 D 1/1 0/0 1/0 1/0 1/0 A F 0/0 1/1 1/1 0/0 1/0 1/0 0/0 0/0 A F A F 1/1 1/0 1/0 1/1 0/0 1/0 0/0 0/0 1/1 1/1 E C 1/1 0/0 0/0 0/0 0/0 E C E 0/0 FSM Minimization with Equivalent States STG: collapse states in the same equivalent class to one state; update edges. Equivalent classes: (A,C), (B,D), (E), (F) ENEE 644

  27. Definition: Finite State Machine Recall: A Finite State Machine (FSM) of Mealy type is a 6 tuple <I,S,,S0,O,> • I: input alphabet, a non-empty set of input values; • S: a non-empty, finite set of states; •  : SxI  S, a function defines the next state; • S0: S, the set of initial/reset states; • O: output alphabet; •  : SxI  O, a function defines the output. ENEE 644

  28. FSM Equivalence Checking M1=<I1,S1,1,S01,O1,1> M2=<I2,S2,2,S02,O2,2> • What do we mean by M1 and M2 are equivalent? For any input, they should produce the same output. • I1 = I2 • O1 = O2 • How to verify that M1 and M2 are equivalent? • Assuming that S01={s01} and S02={s02}, then if there is no input string can distinguish s01 and s02, we claim that M1 and M2 are equivalent. ENEE 644

  29. The Product Machine • The product machine of two FSMs, M1=<I,S1,1,S01,O,1> and M2=<I,S2,2,S02,O,2>, is defined as M12=<I,S12,12,S012,{0,1},12> • S12=S1xS2={(s1,s2)| s1S1,s2S2} • 12: S12xI S12 12(s12,x)=t12=(t1,t2) 12(s12,x)=(1(s1,x), 2(s2,x)) =(t1,t2) • S012 =S01xS02={(s01,s02)| s01S01,s02S02} • 12: S12xI {0,1}12(s12,x)=1 iff 1(s1,x)=2(s2,x) • M1 and M2 are equivalentM12always outputs 1. ENEE 644

  30. Run and Reachable State • For an FSM M1=<I1,S1,1,S01,O1,1>, an input string x0x1…xk-1produces a sequence of states s0s1…sk(called a run, where s0 is the starting state)and an output string z0z1…zk-1. • A state t is reachable from state s if there exists an input string that produces a run with s as the starting state and t as the ending state. • The reachable states of an FSM <I,S,,S0,O,> is defined as: {tS| t is reachable from s, sS0} We only need to check the reachable states for FSM equivalence. ENEE 644

  31. FSM Equivalence Checking M1=<I1,S1,1,S01,O1,1> M2=<I2,S2,2,S02,O2,2> • If I1I2or O1 O2return not equivalent; • Build the product machine M12; • Start with the initial state s012, traverse the STG of the FSM M12; • For each reachable state in M12, if it can output 0, return not equivalent; • Return equivalent; • To get a distinguishing sequence in the case of not equivalent, we need to store the predecessor information and do backtracking. ENEE 644

  32. x/1 A,D A,E x/1 x/0 x/0 y/0 y/1 x/1 B,F A D y/1 y/1 ••• C,F B,E y/1 y/1 x/0 x/0 y/1 y/1 x/1 y/0 y/0 C F B E x/0 x/0 Example: FSM Equivalence Checking • M1=<{x,y},{A,B,C},1,{A},{0,1},1> • M2 =<{x,y},{D,E,F},2,{D},{0,1},2> • M12=<{x,y},S12,12,{(A,D)},{0,1},12> • |S12| = 9, however, only 3 states are reachable: (A,D),(B,E),(C,F) • Every reachable state outputs 1 on all inputs. • So M1 and M2 are equivalent. ENEE 644

  33. x/1 A,D A,E x/1 x/0 x/0 y/0 y/1 x/1 B,F A D y/1 y/1 ••• C,F B,E y/1 y/1 x/0 x/0 y/1 y/1 x/0 y/0 y/0 C F B E x/0 x/1 Example: FSM Equivalence Checking • Now, M1 and M2 are not equivalent. • Consequently, one of the reachable state (C,F) outputs 0 on input x. • Backtracking to find the distinguishing sequence. ENEE 644

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