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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 MachineOutline: 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 644Definition: 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 644Example: 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) = 1ENEE 644FSM Representation: STGIn 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 644x/0Ay/1x/0y/1y/0CBx/0Example: 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) = 1ENEE 644x/0Ay/1x/0y/1y/0BCxyAB,1x/0BA,0C,0CC,0A,1FSM 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,0ENEE 644x/0Ay/1IPSNSOx/0y/1xAA0y/0CBxyAyB1xBA0AA,0B,1x/0yBC0BA,0C,0xCC0CC,0A,1yCA1FSM 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 644FSM 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 644Make 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/0x/0AADDx/-y/1y/1y/1y/1x/0x/0?x/-y/0y/0BCBC-/-ENEE 644FSM 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 644compatibility relationBinary 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 644Partition 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 644Refinement 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 644st==?stEquivalent 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…x1x0zsk-1…zs1zs0xk-1…x1x0ztk-1…zt1zt0ENEE 644Equivalent 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 6440/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Equivalent 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 644Equivalent 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 644State 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=PkENEE 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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)}levelinputblk no.ENEE 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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)}=P23P3={(A,C),(E),(B,D)}ENEE 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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 644PS NS, z x=0 x=1A E, 0 D, 1B D, 0 F, 0C E, 0 B, 1D B, 0 F, 0E C, 0 F, 1F B, 0 C, 00/0BD0/01/00/01/11/0AF1/00/01/11/10/0EC0/0Example: 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 givesP4={(A,C),(E),(B,D),(F)}P4=P3 so we stopConclusion: A and C are equivalent B and D are equivalentENEE 6440/00/00/0BD0/00/0BD1/00/0D1/10/01/01/01/0AF0/01/11/10/01/01/00/00/0AFAF1/11/01/01/10/01/00/00/01/11/1EC1/10/00/00/00/0ECE0/0FSM 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 644Definition: Finite State MachineRecall: 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 644FSM Equivalence CheckingM1=<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 644The 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 644Run 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 644FSM Equivalence CheckingM1=<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 644x/1A,DA,Ex/1x/0x/0y/0y/1x/1B,FADy/1y/1•••C,FB,Ey/1y/1x/0x/0y/1y/1x/1y/0y/0CFBEx/0x/0Example: 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 644x/1A,DA,Ex/1x/0x/0y/0y/1x/1B,FADy/1y/1•••C,FB,Ey/1y/1x/0x/0y/1y/1x/0y/0y/0CFBEx/0x/1Example: 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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