Models of Sequential Systems

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1 Models of Sequential Systems Sungho Kang Yonsei University

2 Outline Introduction to Finite State Machines Synthesis of Finite State Machines FSMs : Definition, Notation, and Examples FSM Minimization of Completely Specified Machines Graph Algorithms for FSM Traversal Models of Sequential Systems FSTs: Strings, Runs, Reachability and Products FSM Equivalence Checking Reachability Analysis Symbolic FSM State Traversal 2

3 A Verbal FSM Specification The circuit receives a stream of bits, one per clock cycle, on input x and at every clock cycle it indicates, on the three outputs Z 2 Z Z 0 the difference between the number of ones and the number of zeroes in the last three bits received. The difference is positive if the number of ones exceeded the number of zeroes and it is negative otherwise. The difference is represented as a 2's complement number, with Z 2 the sign bit and Z 0 the least significant bit. At reset, the circuit is assumed to have received an arbitrarily long string of zeroes, so that the output is -3. 3

4 2 s Complement Assume -2 n < i 2 n. Then the 2 s complement representation of an integer i is the binary code for the integer j = 2 n - i, augmented by an extra bit denoting the sign of i. Examples: Sign Bit n = 3 i= 5 3 j = BINARY( 2 i ) = BINARY( 3) = 00 i= 5 3 j = BINARY( 2 i ) = BINARY( 3) = 0 4

5 A Sequential Counter Circuit and Its STG Circuit rst ck x y0 y0' Latches 0,,2, use n=2 z2 STG (State Transition Graph) y y2 z= S00 S0 z=00 0 z=00 0 S000 S00 S0 0 S 0 0 z=0 0 z= z=0 0 z= S00 S0 z=00 0 z z0 Truth Table yyy zzz

6 Systematic FSM Design Paradigm 6

7 Synthesis of Practical FSMs We have learned basic methods for minimizing, encoding, checking equivalence, and synthesizing circuits for realizing completely specified FSMs Now we must learn to deal with the more practical case of incomplete specification Our goal is thus to find a least cost circuit that satisfies a partial specification 7

8 FSM Synthesis--State Encoding ss ss K-map of f(s) S S S 2 3 S S S S S S S={S,S 2, S 9 } State set 4 = log 2 9 Code Length s=(s,s 2,s 3,s 4 ) Encoding Vector f () s = DECODE() s S Criteria: avoid s, promote adjacency Characteristic function of set S 8

9 Encoding the Transition Functions Given: FSM specified as Cube Table Find code length and encoding Replace symbolic states with binary codes For each row with a in the δ i column, add the input cube to the formula for δ i (s). 9

10 FSM Synthesis--State Encoding Encoding: S : s s, S : s s, S : s s I PS NS O 0 S S 0 S S 2 0 S S 0 2 S S S S S S 3 x s s 2 δ δ 2 λ Simplification: δ = x s s + xs s, δ = xs s = s λ, λ = xs (or 0) literals 0

11 Exploiting Unreachable State Don t Cares When there are unreachable states, the encodings of these states are don t cares These don t cares can be used to minimize the logic of the next state and output functions s in the output columns of the cube table are primary reason for fat logic--don t cares minimize this effect

12 FSM Synthesis--State Encoding Encoding(2 s) S :s s 2, S 2 :s s 2, S 3 :s s 2 ( S 4 ) δ =s s 2 x +s s 2 x, δ 2 =s s 2 x, λ=s 2 x DC =s s 2 (9 literals) δ =s s 2 x +s 2 x, δ 2 =s s 2 x, λ=s 2 x Encoding(4 s) S :s s 2, S 2 :s s 2, S 3 :s s 2 Conclusion: Avoid codes with s δ =s s 2 x+s s 2 x+s s 2 x δ 2 =s +s s 2 x, λ=s s 2 x DC =s s 2 ( literals) δ =s s 2 x +s x+s 2 x, δ 2 =s +x, λ=s s 2 x 2

13 FSM State Sequences M = < I, S, δ, S 0, O, λ> I : Input Alphabet O : Output Alphabet S : State Set S 0 : Initial State Set δ : Next State Transition Function λ : Output Function t=δ(s,x):sxi S o=λ(s,x):sxi O State Sequences Output Sequences 3

14 Formal FSM Specification STG ( S 4 ) Flow Table x = 0 x = S S, 0 S2, S2 S, 0 S3, 0 S S, 0 S, 3 3 Cube Table I PS NS O 0 S S 0 S S 2 0 S S 0 2 S S S S S S 3 M = < I, S, δ, S 0, O, λ> I={0,}, S={S 0, S, S 2 }, S 0 ={S }, O={0,} 4

15 State Minimization Two FSMs are equivalent if their initial states are equivalent. Two states are equivalent if there is no sequence of inputs, which, applied to both machines, produces an output difference The task of replacing two or more equivalent states with a single representative state is called State Minimization 5

16 Strings and Runs A string x=(x 0,x,,x k- ) is a finite sequence of inputs to an FSM M = < I, S, δ, S 0, O, λ> where x i I Examples: x produces the s 0 -run s=(s 0,s,,s k ), s i+ = δ(s i,x i ), that is, the sequence of states generated by x when M starts in state s 0 x= ( 00,,,, ), s = S, s= ( S, S, S, S, S, S ) x= ( 00,,,, ), s = S, s= ( S, S, S, S, S, S ) Are S,S 2 equivalent? 6

17 Output Sequences Given an input sequence x=(x 0,x,,x k- ), the run s=(s 0,s, s k ) leads to the corresponding output sequence z=(z 0,z,,z k- ) Input = (0,0,,) Run = (S,S,S,S 2,S 3 ) Output Sequence=(0,0,,0) Run = (S 3,S 3,S 3,S,S 2 ) Output Sequence=(0,0,,) 7

18 Distinguishing Sequences Two states s and t are distinguished by an input string x=(x 0,x,,x k- ), if and only if the corresponding runs lead to output sequences z S =(z S,z 2S,,z k-s ) and z t =(z t,z 2t,,z k-t ), z k-t z k- s which differ in the last output symbol. Distinguishes s=s and t=s 3 Input Sequence=(0,,) Run from S 0 =(S,S,S 2,S 3 ) Outputs from S 0 =(0,,0) Run from S 3 =(S 3,S 3,S,S 2 ) Outputs from S 0 =(0,,) 8

19 Indistinguishable States Two states S and S 2 are indistinguishable if and only if any input sequence x=(x 0,x, ) gives the same output sequence z=(z 0,z, ), irrespective of whether the FSM starts in state S or S 2 (no distinguishing sequence exists). Input Sequence : (0,0,,,) Outputs from S 0 =S : (0,0,,0,) Outputs from S 0 =S 2 : (0,0,,0,) 9

20 Merge Indistinguishable States Merge Input Sequence : (0,0,,,),(,,,0,0),. Outputs from S 0 =S : (0,0,,0,),(,0,,0,0),. Outputs from S 0 =S 2 : (0,0,,0,),(,0,,0,0),. 20

21 Indistinguishable States Two states are indistinguishable if (but not only if) they have the same next state and the same output for all inputs 0/ 0 0/ 0 S 2 S / 0 / 0 S 3 0, / 0 States S and S 2 are indistinguishable Hence, S S 2 State S 3 goes to State S 3 for both inputs 2

22 Length-k Distinguishing Sequences Two states s and t are k-distinguished by an input string x=(x 0,x,,x k- ), if and only if the corresponding runs lead to output sequences z S =(z S,z 2S,,z k-s ) and z t =(z t,z 2t,,z k-t ), z k-t z k- s which differ in the last output symbol. String (0,,) 3-Distinguishes s=s and t=s 3 Input Sequence=(0,,) Run from S 0 =(S,S,S 2,S 3 ) Outputs from S 0 =(0,,0) Run from S 3 =(S 3,S 3,S,S 2 ) Outputs from S 0 =(0,,) 22

23 k-equivalent States If S and S 2 are not k-distinguishable (no k-distinguishing sequence exists), they are k-equivalent (S k S 2 ). In an n-state FSM ( S =n), S and S 2 are equivalent S S 2 if and only if they are n-equivalent (S n S 2 ). 23

24 Equivalent States A C B D Because all runs from these states either go back and forth from B to D, outputting a 0, or converge at F Because all runs from these states either go to equivalent states B and D, outputting a, or converge at E 24

25 Equivalent FSMs For any string x, the output strings for runs starting from A and A,C will be identical x=(0,,0,), z xa =(0,,0,0), z x A,C =(0,,0,0) 25

26 Theorem s k+ t if and only if s k t and s x k t x, x X, where s x and t x are the x-successors of s and t s t 0 0 u v w Proof : Take x k+ as (x,x k ), where both x X and x k are arbitrary 26

27 Equivalence Classes The binary relation k ={(s,t) s k t} SxS is an equivalence relation (reflexive, symmetric, and transitive). We denote the m k equivalence classes of k by B ik, i=,,m k 27

28 Equivalence Classes λ λ ABCDEF,,,,, A, B, C, D, E, F ( 0) = ,,,,, () =, 0, 0,,, 0 ξ = {{ ACE,, },{ BDF,, }} δ δ ABCDEF,,,,, A, B, C, D, E, F ( 0) = E, D, E, B, C, B ( ) = DFBFFC,,,,, ξ 2 = {{ ACE,, },{ BD, },{ F}} 28

29 Equivalent Classes Two states of a given FSM with n states are equivalent if and only if they are (n-) equivalent 29

30 Refinement (Meet) of Partitions Let p 0 and p denote the partition into equivalence classes induced by inputs x=0 and x= P 0 P ={B i0 }{B j } = U ij {B i 0 B j } ={B i0 B j, B i0 B j,.., B i0 B j, B i0 B j,... } P 0 = {{,2},{2,3}}, P ={{,3},{2,3}} P 0 P ={{,2} {,3},{,2} {2,3},{2,3} {,3},{2,3} {2,3}} = {{},{2},{3}} Note the Cartesian Product element {2,3} is dropped (by the absorption) 30

31 Refinement (Meet) of Two Partitions REFINE(P, P 0 ) = P P 0 The Meet of two partitions is just the Cartesian Product (next slide) Example: P 0 = {A,B,C,D,E,F} P = {{A,C,E},{B,D,F}} P P 0 = {A,B,C,D,E,F} X {{A,C,E},{B,D,F}} = {{A,C,E},{B,D,F}} 3

32 Finding Equivalent States Procedure STATE_EQUIVALENCE ( X, S, δ, λ ) { P 0 = { B }; B S; P P 0 0 = = 0 for( x X){ 2 for( s S) x j o = λ ( s, x ) j j 3 P x = PARTITION( o x, s) 4 P = REFINE( P, P x) } if( P = 0 P ) return( 0 P, 0) 5 for( k = 2, K, S ){ P k = {} 6 for( B P k i ){ P k = { B } i i 7 8 for( x X){ for( s B ) x j i t = δ ( s, x ) j j 9 b x = BLOCK_INDEX( t x, P k ) 0 Pb x = PARTITION( x i b, s) P x = REFINE( k i P, Pbi x ) i } 2 P k = P k P k i 3 } if( k k k P = P ) return( P, k ) } } Find equivalence class partition of -equivalent states Extend to equivalence class partition of (k+)-equivalent states Return partition and length of longest distinguishing sequence 32

33 Equivalence Classes We will use the following example Flow Table STG x= 0 x= A E, 0 D, B D, 0 F, 0 C E, 0 B, D B, 0 F, 0 E C, 0 F, F B, 0 C, 0 33

34 Finding Equivalent States Procedure P0 = { B }; B SP ; P 0 0 = = 0 for( x X){ 2 for( s S) x j o = λ( s, x) j j 3 P x = PARTITION( ox, s) 4 P = REFINE( P, P x) } if( P = 0 P ) return( 0 P, 0) 0 0 Start with maximum partition: all states in a single block STATE_EQUIVALENCE (X, S, δ, λ){ Flow Table x= 0 x= A E, 0 D, B D, 0 F, 0 C E, 0 B, D B, 0 F, 0 E C, 0 F, F B, 0 C, 0 o = ( ,,,,, ), P = { A, B, C, D, E, F} o = (, 000,,,, ), P = {{ A, C, E},{ B, D, F}} 34

35 -Equivalence Classes ( ) λ (A,B,C,D,E,F) (0)=(0,0,0,0,0,0) λ (A,B,C,D,E,F) ()=(,0,,0,,0) P 0 ={A,B,C,D,E,F}={B 0 } P ={{A,C,E},{B,D,F}}={B,B 2 } P =P 0 P ={B 0 } {B,B 2 } ={B 0 B, B0 B 2 } P ={{A,C,E},{B,D,F}}={B,B 2 } x= 0 x= A E, 0 D, B D, 0 F, 0 C E, 0 B, D B, 0 F, 0 E C, 0 F, F B, 0 C, 0 The equivalence classes of ( ) are {A,C,E},{B,D,E} 35

36 Finding Equivalent States P ={{A,C,E},{B,D,F}}={B,B 2} 5 for( k = 2, K, S ){ P k = {} 6 for( B k i P ){ P k = { B } i i 7 8 for( x X){ for( s B ) (, ) j i t x = δ s x j j 9 b x = BLOCK_INDEX( t x, P k ) 0 Pb x = PARTITION( x i b, s) P x = REFINE( k i P, Pbi x ) i } 2 P k = P k P k i 3 } if( k k P = P ) return( k P, k ) Extend to equivalence class partition of (k+)-equivalent states x= 0 x= A E, 0 D, B D, 0 F, 0 C E, 0 B, D B, 0 F, 0 E C, 0 F, F B, 0 C, 0 P 2 ={A,C,E}, t 0 =(E,E,C) b 0 =(,,), Pb 0 =(A,C,E) 36

37 Finding Equivalent States Mealy machine PS NS NS P ={{A,D,F,G},{B,C},{E}, } ={{A,D,F,G},{B,C},{E}} A B x=0 E/0 C/0 x= C/0 A/ P 2 ={{A},{B,C},{D,G},{E},{F}} C B/0 G/ P 3 ={{A},{B},{C},{D},{E},{F},{G},{H}} D E G/0 F/ A/0 B/0 F E/0 D/0 G D/0 G/0 37

38 Graph Algorithms Subgraph Connected subgraph Fanin cone (logic cone) Maximal completely connected subgraph If G is undirected, Clique Maximal if it is not a proper subgraph of another subgraph Strongly connected subgraph maximal subgraph for which every pair of included vertices lies on a cycle 38

39 Graph Algorithms An (S,T) cutset C of a directed graph G=(V,E) is a set of nodes (edges) of G such that any path from s S to t T passes through one of the nodes (edges) in C If the graph is a DAG, it is often understood that S and T are the sets of source nodes and sink nodes, respectively 39

40 BFS For FSMs Preorder Each node is visited before any of its successors Postorder Each node is visited after all of its successors BFS effectively levelizes the graph with respect to a specified start set V 0 and then visits all the nodes at level k before visiting any nodes at level k+ 40

41 BFS For FSMs BFS is modified for use in FSM analysis: Trace-Back Method to identify shortest paths BDD data structures for representing large state sets Length of the longest shortest path from an initial state to any of the reachable states is called the sequential depth of the machine The sequential depth corresponding to the particular subset V 0 leading to the largest sequential depth is called diameter of the graph 4

42 Breadth First Search Procedure BFS( V, E, V ){ 0 0 Reached = From = New 0 = V ; k = 0 0 do { k= k+ To = Img( E, From) 2 New k = To Reached; From New k Reached Reached New k = = } while( New k 3 ) 4 return( k, New j, j= 0, K, k ) } New New 2 New 3 States in New i+ reached in one step from New i New 0 Note New 4 = 42

43 Depth-First Search Unlike BFS, DFS prioritizes depth of penetration, rather than exhausting the current locale before proceeding. DFS proceeds recursively, visiting all transitive successors of a vertex, before completing the DFS of the current active vertex. DFS is most useful in detecting cycles of a directed graph. 43

44 Depth First Search When started (preorder) When completed (postorder) Earliest in cycle (lowlink) 44

45 Depth First Search Active Vertex Scan successors Procedure DFS(u) { kpre=kpre+; preorder[u]=kpre foreach ( a u ) { 2 if(preorder [a]=0 DFS(a). } 3 kpost=kpost+; postorder[u]=kpost } Recur if unsearched When Started When Completed 45

46 Strongly Connected Components An SCC (Strongly Connected Component) of a directed graph is a subgraph which is maximal with respect to the following property: There is a path from every vertex to every other vertex, and back. That is, every pair of vertices lies on a cycle. 46

47 Finding SCC Procedure SCC_POP(u) { Procedure DFS_SCC(u) { while(scc()) { kpre = kpre+; v=pop(scc) lowlink[u]=preorder[u]=kpre 2 if(lowlink[v]lowlink[u]) print v push (SCC, u) else { foreach (a E(u,v) u ) { 3 push(scc,v) if (preorder[a]=0) { break DFS_SCC(a) } 2 lowlink[u]=min(lowlink[u],lowlink[a]) } } } elseif (preorder[a]<preorder[u]) { if (a SCC) { 3 lowlink[u]=min(lowlink[u],preorder[a]) } } } 4 if (preorder[u]=lowlink[u]) SCC_POP(u) } 47

48 Strongly Connected Components When started When completed Earliest (preorder index) in cycle D 5 D 4 D 2 D 3 D 48

49 Shortest Paths Procedure SHORTEST_PATH(V,E,L,v0) { for(v V) λ v = ; Reached = φ 2 S = {v 0 }; λ v0 =0 3 while (S φ) { 4 λ* = min s S {λ v } V* = {v V λ v = λ*} 5 v = SELECT(V*) 6 S=S-{v} Reached = Reached {v} 7 for (a ADJ(V,E,v)) { 8 if (a Reached ) { 9 S = S {a} 0 λ a = min(λ a, λ v + L v,a ) } } } return (λ) } 49

50 Models of Sequential Systems FST (Finite State Transition Structures) In a given state, FSTs receive an input symbol, and make a transition to a new state FAs (Finite Automata) are FSTs which also take notice when a favorable state (called accepting) is entered FSMs (Finite State Machine) are FSTs which emit a specified output symbol when they make a transition from one state to another Regular Languages Languages are just sets of sequences of input or output symbols (called strings) Regular languages are just the kind of sets of strings that can be accepted by FAs or generated by FSMs DFA (Deterministic Finite Automata) NFA (Nondeterministic Finite Automata) 50

51 Finite State Transition Structures An FST is defined as a 4-tuple Γ = <X,S,δ,S 0 > X is the input alphabet S is the state set or state alphabet δ : S X 2 S is the next state transition function S 0 S is a set of initial states Definition FST Γ = <X,S,δ,S 0 > is a deterministic if the image of all pairs (x,s) is a singleton next state In this case, the specified δ does not map any given state into a set of two or more states If in addition the initial state set S 0 of Γ is a singleton set, Γ is said to be strongly deterministic If the mapping δ is specified for all pairs of states and input symbols, Γ is said to be complete 5

52 Finite State Transition Structures NFAs and ε-moves ε-moves, which represent the further conceptual ability of an NFA to be in more than one state at a time The concept of ε-moves is important in the procedure for deciding whether a specified string is accepted by a given FA and for constructing a DFA which accepts the same language as a given NFA FSTs as labeled diagraphs Sometimes an FST Γ = <X,S,δ,S 0 > is specified by giving its STG : G=(S,E,L) The vertices s S correspond to the states of Γ The edge (s,t) E are directed from s to t and signify a possible state transition (s,t) E t = δ(s,x) For each edge s,t E of G L (s,t) ={x X t δ(s,x)} = {x X t = δ(s,x)} 52

53 Finite State Transition Structures Strings, Tapes and Runs A run of an FST is a sequence of states which starts in an initial state S 0 occurs in response to some possible input sequence A sequence of inputs is called a string if it is finite and a tape if it is infinite 53

54 Traversing the Product STG A state t is reachable from state s if there exists a string x which produces a run s, ending in state t A state t is simply reachable if it is reachable from any state in S 0 54

55 Traversing the Product STG Equivalence checking requires testing output pairs at every reachable state A state t is reachable from state s if there exists a string x which produces a run s, ending in state t The act of identifying the set of reachable states of an FSM is called FSM Traversal In Traversal, we systematically search the STG from the initial state. Any method can be used but the most efficient is based on BFS (Breadth First Search) 55

56 Building the Product Machine Check this condition for all reachable product states M M 2 λ λ 2 λ λ,,k 2, 2 2,K s s 2 x Common input M 2 Comparator FST (δ xδ 2 ) PRODUCT 56

57 Building the Product (STG View) Note output of product machine is for all inputs at all reachable states 57

58 Building the Product (STG View) Start by finding states (s,s 4 ) and (s 2,s 5 ) reachable from initial state (s,s 4 ) of product M 2 58

59 Building the Product (STG View) Continue by finding state (s 3,s 6 ) reachable from previous new state (s 2,s 5 ) of product M 2 59

60 Equivalence Checking Product Machine Same as BFS Specific to FSMs Same as BFS Procedure BFS _ FSM( SX,, δλ,, S ){ 0 0 Reached = From = New 0 = S ; k = 0 0 do { k = k+ To = Img(, From) New k δ 2 = To Reached for( s New k ){ forif ( x X) { 3 ( λ( sx, ) = 0){ j ( s, x) = ERROR_ TRACE( sxkx,,,, δ,{ New}) 4 return( FALSE, s, x) } } } From = New k Reached = Reached New k 5 } while( New k ) 6 return( TRUE, ε, ε ) } 60

61 Finding a Shortest Error Trace j Procedure ERROR_ TRACE( s, x, k, X, δ,{ New }){ E E E /* Error State: λ( sx, ) = 0 0 s= ( s ) ; x = ( x ) E E for( k = k ; k ; k++ ){ E 2 for( s New k ){ 3 forif ( x X) { ( s = δ( sx, )){ E 4 s = s; x = x E E 5 s = ( se,s); x = ( x, x) E } } } } 6 return( s, x) } Returned: run s, which starts at initial state and ends at error state. 6

62 BFS for Error Trace New 0 New 3 New New 2 Suppose all outputs were except on edge from s2 to s. Then a shortest error trace would be s=(s5,s4,s,s2), x=(0,,0,0) 62

63 Isomorphism Two graphs G a = (V a, E a ) and G b = (V b, E b ) are said to be isomorphic if V a = V b if the nodes of V a can be relabeled so that the two graphs are identical A relabeling function is a function ρ : V a V b such that for each node v b V b there is exactly one node v a V a such that v b = ρ(v a ) Such a function is said to be to and onto and is called an isomorphism 63

64 BDD Representation of Reached Sets FSM Traversal requires storage of various reachable states sets ( {New j }, Reached, etc.) For a circuit with 64 latches, there could be as many as (0 9 ) reachable states It is clearly infeasible to store an int to identify each individual state Thus BDDs must be used to represent sets of states 64

65 BDD Representation of State Sets S={[0,3],[8,5],[6-23]} Encoding: , 0000,, 23 0 (codes packed around origin) 0= 0* 2 + 0* 2 + 0* 2 + 0* 2 + 0* = 0* 2 + 0* 2 + 0* 2 + * 2 + * = * 2 + 0* 2 + * 2 + * 2 + *

66 BDD Representation of State Sets S(x)=x x 2 +x x 2 +x x 2 x 3 Minterm count = path count (3 paths for all n) n=3: 2+2+=5 n=5: 8+8+4=20 n=7: 6+6+8=80 Missing variables amplify minterm count x x 2 x 3 x 4 x 5 66

67 BDD Representation of State Sets n= 36; m = States m = 20( 23) = 40( 09) 40Billion n = 5: x x 2 x x2 x3 x4 x5 S( x) States Count [ 0 3] 4 x 3 0 [ 8 6] 8 x 0 [ 6 23] 8 4 n = 7: x 5 x x2 x3 x4 x5 x6 x7 S( x) States Count [ 0 5] 6 0 [ 32 63] 32 0 [ 64 95] 32 67

68 Symbolic FSM State Traversal Image The set of co-domain points Preimage Transition relations Given a deterministic transition function (s,x), T(s,x,t) = i=n i= (t i δ i (s,x)) T(s,x,t)= denotes a set of encoded triples s,x,t of state s, input x and x-successor t of s, each representing a transition in the FST if the given FSM Existential abstraction Given an m-variable Boolean function f(x,,,, xm), xi f = f xi + f xi where f xi (f xi ) is positive(negative) cofactor I(t) = Img(T,C) = x s C(s) T(s,x,t) 68

Sequential Equivalence Checking - I

Sequential Equivalence Checking - I Virendra Singh Associate Professor Computer Architecture and Dependable Systems Lab. Dept. of Electrical Engineering Indian Institute of Technology Bombay viren@ee.iitb.ac.in