Sequential System Synthesis — Finite State Machine ENEE

Sequential System Synthesis -- Finite State Machine

ENEE 644 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: 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 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 FSM Representation: Flow Table The flow table of an FSM 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 FSM Representation: Cube Table The cube table of an FSM 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 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 . In cube table: delete the undefined row or fill the last two columns by don’t cares. A C B x/0 y/1 x/0 y/0 y/1 D ? -/- A C B x/0 y/1 x/0 y/0 y/1 D x/- x/- y/-

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 Binary Relations Given two sets A and B, a binary relation R 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 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 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 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)} Example: Finding Equivalent States input blk no. level

ENEE 644 21 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 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 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 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 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 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: 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= M2= 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= and M2=, is defined as M12= 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M12 always outputs 1.

ENEE 644 30 Run and Reachable State For an FSM M1=, an input string x0x1…xk-1 produces 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 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= M2= If I1  I2 or O1  O2 return 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 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 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.