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Outline

Outline. Composition, Conformance Topologies Proof of solution Node flexibilities Examples Node minimization Windowing C-progressive. Composition. Synchronous Asynchronous (parallel) Mapping Asynchronous to Synchronous. Composition (synthronous or asynchronous) – involves two steps

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  1. Outline • Composition, Conformance • Topologies • Proof of solution • Node flexibilities • Examples • Node minimization • Windowing • C-progressive

  2. Composition • Synchronous • Asynchronous (parallel) • Mapping Asynchronous to Synchronous

  3. Composition (synthronous or asynchronous) – involves two steps • Make the two machines have the same input alphabet (support) • Perform the product • Synchronous (changing the support) • Projection – Given a language L over the alphabet projection is defined as • Lifting – given a language L over the alphabet X, lifting to the alphabet is defined as • Asynchronous (changing the support) • Restriction – Given a language L over the alphabet , the restriction to X is defined as where • Expansion – given a language L over X, lifting to the alphabet is defined as

  4. Mapping Parallel into Synchronous Suppose F is a FSM on inputs i,v and outputs u and S is an FSM on inputs i and outputs o. u S F X o o i i v The semantics are that when an input comes into a module, it takes an unspecified amount of time for the module to produce an output. This will be modeled with a non-deterministic self-loop labeled with

  5. Transitions of F (as an FSM) are one of the forms (s i/u s’) or (s v/u s’). For S, its transitions are of the type (q i/o q’). For each, we convert into automata by creating new intermediate states between inputs and outputs. Thus a transition (s i/u s’) becomes two transitions (s i s’’) (s’’ u s’). Similarly for the others. For S a transition of the type (q i/o q’) becomes (q i q’’) (q’’ o q’). i/u s s’ s’’ q’’ i u i o s s’ q q’

  6. We want to lift up the language of F to include o and arbitrary delays so that iuvu becomes for example: And S to include u and v and arbitrary delays, so that the two languages can contain similar strings. io becomes for example In particular, they both can become the same thing: The common symbols act to synchronization This is done on the automaton F by the following: i1/u1 v1/u2 becomes Note the alphabet is

  7. Similarly for S: i1/o1 becomes i1 o1 i1 Note that all states are deterministic since the alphabet is . With these conversions, we can do synchronous composition and get the equivalent expanded result of parallel composition Thus we need to implement only one type of compositional method – synchronous, and simply have a mapping of each machine into its extended machine to compose in parallel. Finally, we can take the synchronous solution and map it back into an FSM.

  8. Conformance Simulation Relation ( )Let and be two automata over the same alphabet simulates ( ) if there exists a simulation relation such that implies that Note that simulates but these are not equivalent notions. If may be easier to find a simulation relation than to prove language containment.

  9. Language Containment ( ) To show that ) we typically show that (i.e. This requires complementing which may be hard if is non-deterministic (subset construction). So simulation may be easier to check. Use of a simulation relation instead of language containment can allow avoidance of computing in the construction of . Note that if S is deterministic or small, then there is no motivation to avoid computing so using language containment is fine.

  10. Language Equality ( ) Proposition: If two FSMs, F1 and F2, and F2 is deterministic, then Hence, if we are solving and S is a deterministic FSM then is the MSG. If S is ND, then what is MSG? Comment: Clearly, we need in order for there to be a solution of . This requires that supp(S) supp(F), since otherwise there is a variable v in F but not in S. Then would be too large.

  11. O1 F I1 most general U2 U1 O2 X I2 Topologies

  12. X X F F I1 I1 I1 I1 one-way cascade one-way cascade two-way cascade two-way cascade U2 U2 U1 U1 U1 U1 O2 O2 O2 O2 F X X F

  13. O1 F F I1 Engineering Change U2 U2 U1 U1 O2 rectification X X I2

  14. Controller F U2 U1=O1 X I2

  15. Communicating the internal state O1 O1 F F I1 I1 cs U2 U1 U2 U1 O2 O2 X X I2 I2 latch_expose

  16. Thus the only variables that X does not see are the O variables. In the construction, for the most general solution, is deterministic. The only variables eliminated before complementation are O. The only way it could become ND is if ivuo on a pair of arcs have the same values of ivu, but different o’s. Thus if o is a deterministic function of i,v, o = f (i,v,cs), this could not happen. Theorem: If O is a deterministic function of i,v,cs, the second complementation is easy (no subset construction). Hiding only the outputs O I F V U X

  17. Hiding the outputs only i’u’v’o1 The only way it could become non-deterministic after hiding o is if i’u’v’ = iuv. But then o1 = o2 which means that the product machine was ND. iuvo2

  18. Solving a language equation Solve and where In particular, find the largest solution X (most general solution). Theorem A: Let A and C be languages over alphabets and respectively. For the equation, the most general solution is Theorem B: Let A and C be languages over alphabets and respectively. For the equation, the most general solution is .

  19. Proof: We prove Theorem A. Let . Then means that A I U O X Thus is the largest solution of The proof of Theorem B is similar.

  20. Computing the CF for a node -global step

  21. yi Yi Yi Computing CF - local step

  22. yi Yi yi Yi Note that essentially the same computation applies for multiple-output nodes, i.e. where CF

  23. FSM FSM i1 FSM spec FSM i o context FSM FSM i2 o FSM v u unknown FSM networks – computing complete sequential flexibility (CSF) Specification S (i,o) Context C (i,v,u,o) Unknown X (u,v) Problem: Given S and C , find the Most General Solution (MGS) of

  24. spec i o context v u unknown FSM Networks The most general solution (MGS) of is In general, MGS is deterministic automaton but as an FSM it is non-deterministic(NDFSM)

  25. u v CSF Complete Sequential Flexibility (CSF) • CSF is maximum sub-behavior in MGS which is prefix closed and u-progressive. • For unknown to be an FSM, it must be progressive in its inputs

  26. spec i o context v u unknown yi Yi Comparison with combinational case Sequential Combinational unknown

  27. Extending CF FSM O I X u v Combinational sub-block Spec is IO behavior of FSM. Combinational block is treated as unknown X with inputs u and outputs v. We derive the CSF for X. It is different than the CF where the spec is taken to be the combinational behavior of the FSM, i.e. with inputs I,CS and outputs O,NS. Also, if we extract from X a maximum combinational subpart (combinational projection), it is also different that CF

  28. Algorithm Algorithm:LanguageEquationSolving Input:prefix-closed deterministicS(i.o)andC(i,v,u,o) Output:most general prefix-closed,progressiveX (FSM) begin 01 X := Complete( S,non-accepting ) 02 X := Determinize&Complement( X ) 03 X := Support (X, (i,v,u,o)) - raise 03 X := Product( C, X ) 04 X := Support( X, (u,v) ) - hide 05 X := Determinize &Complement( X, u ) 06 returnPrefix&Progressive (X ) end Convert to FSM

  29. Examples • Games • Nim • Tic-tac-toe • Toe-tac-tic • Board • Control • Wolf, goat, cabbage • Latch splitting

  30. In God We Trust In God We Trust In God We Trust Example: Coin Game (NIM) Context describes the state of the game and legal moves. Its input is random moves by first player and its output tells if the game is in a losing, winning or continuing state. Specification is a 3-state automaton, playing, won, and lost. • Players alternate turns • On each turn, player can take 1-n coins from any one pile • Player who takes last coin loses Winning strategy: Give your opponent a pile of coins with even number of 1’s in bit columns (except at end) 6 = 1 1 0 5 = 1 0 1 3 = 0 1 1 ____ 2 2 2 Example: 6 5 3

  31. .model game-piles .inputs p1 d1 p2 d2 .outputs out .mv p1,p2,p,pt,ptt 3 .mv d1,d2,d 7 .mv cs0,cs1,cs2,ns0,ns1,ns2,nh,h 7 .mv whoseturn,whoseturn1 2 1 2 .mv out 3 OK notOK done .latch ns0 cs0 .reset cs0 3 .latch ns1 cs1 .reset cs1 2 .latch ns2 cs2 .reset cs2 1 .latch whoseturn1 whoseturn .reset whoseturn 1 #set this to 2 if Player 2 goes first. .table whoseturn whoseturn1 .default 2 2 1 .table whoseturn d1 d2 d 1 - - =d1 2 - - =d2 .table whoseturn p1 p2 ptt 1 - - =p1 2 - - =p2 NIM spec.mva .model spec .inputs out .outputs Acc .mv out 3 OK notOK done .mv CS,NS 3 a b c .table CS ->Acc .default 1 b 0 .table out CS ->NS OK a a notOK a b done a c - b b - c c .latch NS CS .reset CS a .end

  32. # Map move into a legal move .table ptt cs0 cs1 cs2 pt 0 (1,2,3,4,5,6) - - 0 1 - (1,2,3,4,5,6) - 1 2 - - (1,2,3,4,5,6) 2 0 0 - - 1 1 - 0 - 2 2 - - 0 0 .table pt cs0 cs1 cs2 p 0 (1,2,3,4,5,6) - - 0 1 - (1,2,3,4,5,6) - 1 2 - - (1,2,3,4,5,6) 2 0 0 - - 1 1 - 0 - 2 2 - - 0 0 #selects the height of the pile chosen by player 1 .table p cs0 cs1 cs2 h 0 - - - =cs0 1 - - - =cs1 2 - - - =cs2 #computes h-d. If h<=d then =0 .table h d nh .default 0 6 1 5 6 2 4 6 3 3 6 4 2 6 5 1 5 1 4 5 2 3 5 3 2 5 4 1 4 3 1 4 2 2 4 1 3 3 2 1 3 1 2 2 1 1 # The next state ns is due to the move. .table p nh cs0 ns0 0 - - =nh (1,2) - - =cs0 .table p nh cs1 ns1 1 - - =nh (0,2) - - =cs1 .table p nh cs2 ns2 2 - - =nh (0,1) - - =cs2 #"out" indicates who the winner is. .table whoseturn ns0 ns1 ns2 out .default OK 1 0 0 0 done 2 0 0 0 notOK .end

  33. Lang.script (NIM) rl fixed.mv stg_extract fixed.mva echo "Synthesis ..." determinize -ci spec.mva spec_dci.mva support p2(3),d2(7),p1(3),d1(7),out(3) spec_dci.mva spec_dci_supp.mva support p2(3),d2(7),p1(3),d1(7),out(3) fixed.mva fixed_supp.mva product fixed_supp.mva spec_dci_supp.mva p.mva support p1(3),d1(7),p2(3),d2(7) p.mva p_supp.mva determinize -ci p_supp.mva p_dci.mva prefix p_dci.mva p_dci_pre.mva progressive -i 2 p_dci_pre.mva x.mva minimize x.mva x-min.mva prefix x-min.mva x-min.mva echo "Verification ..." support p2(3),d2(7),p1(3),d1(7),out(3) x.mva x_supp.mva product x_supp.mva fixed_supp.mva prod.mva support p2(3),d2(7),p1(3),d1(7),out(3) spec.mva spec_supp.mva check prod.mva spec_supp.mva

  34. mvsis 02> source lang.script The STG with 40 states and 110 transitions is written to file "fixed.mva". Synthesis ... The automaton is deterministic; determinization is not performed. Product: (40 st, 110 trans) x (3 st, 5 trans) -> (42 st, 112 trans) The automaton is deterministic; determinization is not performed. Warning: The automaton has been completed before state minimization. State minimization: (22 states, 45 trans) -> (13 states, 30 trans) Verification ... Product: (21 st, 34 trans) x (40 st, 110 trans) -> (21 st, 34 trans) Warning: Automaton "game-piles*spec*game-piles" is completed before checking. The behavior of "game-piles*spec*game-piles" is contained in the behavior of "spec". mvsis 03> psa x-min.mva "game-piles*spec": incomplete (9 st), deterministic, non-progressive (9 st), and non-M oore (9 st). 4 inputs (4 FSM inputs) 12 states (12 accepting) 19 trans Inputs = { p1(3),d1(7),p2(3),d2(7) } mvsis 03>

  35. Example of CSF computation: NDFSM represented as automaton In God We Trust In God We Trust In God We Trust Inputs p1(3),d1(7) Outputs p2(3),d2(7)

  36. spec.mva Tic-tac-toe # .model spec .inputs out .outputs Acc .mv out 3 .mv CS,NS 3 a b c .table CS ->Acc .default 1 b 0 .table out CS ->NS 0 a a 1 a b 2 a c - b b - c c .latch NS CS .reset CS a .end spec.mv .model spec .inputs m1 c1 m2 c2 .outputs out .mv m1,m2 9 .mv out 3 .table ->out 0 2 .end

  37. .model game-tic-tac-toe .inputs m1 m2 .outputs out .mv out 3 .mv m1,m2,m 9 .mv cs0,cs1,cs2,cs3,cs4,cs5,cs6,cs7,cs8 3 .mv ns0,ns1,ns2,ns3,ns4,ns5,ns6,ns7,ns8 3 .mv whoseturn,whoseturn1 2 1 2 .latch whoseturn1 whoseturn .reset whoseturn 1 .latch ns0 cs0 .reset cs0 0 .latch ns1 cs1 .reset cs1 0 .latch ns2 cs2 .reset cs2 0 .latch ns3 cs3 .reset cs3 0 .latch ns4 cs4 .reset cs4 2 # set this to 0 if player 2 makes the second move. .latch ns5 cs5 .reset cs5 0 .latch ns6 cs6 .reset cs6 0 .latch ns7 cs7 .reset cs7 0 .latch ns8 cs8 .reset cs8 0 .table illegal whoseturn whoseturn1 0 2 1 0 1 2 1 - =whoseturn .table whoseturn m1 m2 m 1 - - =m1 2 - - =m2 # Player makes a illegal move if the square indicated by mt in not empty .table m cs0 cs1 cs2 cs3 cs4 cs5 cs6 cs7 cs8 illegal .default 1 0 0 - - - - - - - - 0 1 - 0 - - - - - - - 0 2 - - 0 - - - - - - 0 3 - - - 0 - - - - - 0 4 - - - - 0 - - - - 0 5 - - - - - 0 - - - 0 6 - - - - - - 0 - - 0 7 - - - - - - - 0 - 0 8 - - - - - - - - 0 0 # out records if there is a line of 2's (then out=2) or a line of 1's (then out=1) .table whoseturn ns0 ns1 ns2 ns3 ns4 ns5 ns6 ns7 ns8 out .default 0 2 - - 2 - 2 - 2 - - 2 2 2 2 2 - - - - - - 2 2 2 - - 2 - - 2 - - 2 2 2 - - - 2 - - - 2 2 2 - - - 2 2 2 - - - 2 2 - - 2 - - 2 - - 2 2 2 - 2 - - 2 - - 2 - 2 2 - - - - - - 2 2 2 2 1 - - 1 - 1 - 1 - - 1 1 1 1 1 - - - - - - 1 1 1 - - 1 - - 1 - - 1 1 1 - - - 1 - - - 1 1 1 - - - 1 1 1 - - - 1 1 - - 1 - - 1 - - 1 1 1 - 1 - - 1 - - 1 - 1 1 - - - - - - 1 1 1 1

  38. .table illegal done cs3 m whoseturn ns3 #.default 0 0 1 - - - =cs3 0 0 - 3 1 1 0 0 - 3 2 2 0 0 - (0,1,2,4,5,6,7,8) - =cs3 1 - - - - =cs3 .table illegal done cs4 m whoseturn ns4 #.default 0 0 1 - - - =cs4 0 0 - 4 1 1 0 0 - 4 2 2 0 0 - (0,1,2,3,5,6,7,8) - =cs4 1 - - - - =cs4 .table illegal done cs5 m whoseturn ns5 #.default 0 0 1 - - - =cs5 0 0 - 5 1 1 0 0 - 5 2 2 0 0 - (0,1,2,3,4,6,7,8) - =cs5 1 - - - - =cs5 .table illegal done cs6 m whoseturn ns6 #.default 0 0 1 - - - =cs6 0 0 - 6 1 1 0 0 - 6 2 2 0 0 - (0,1,2,3,4,5,7,8) - =cs6 1 - - - - =cs6 .table illegal done cs7 m whoseturn ns7 #.default 0 0 1 - - - =cs7 0 0 - 7 1 1 0 0 - 7 2 2 0 0 - (0,1,2,3,4,5,6,8) - =cs7 1 - - - - =cs7 .table illegal done cs8 m whoseturn ns8 #.default 0 0 1 - - - =cs8 0 0 - 8 1 1 0 0 - 8 2 2 0 0 - (0,1,2,3,4,5,6,7) - =cs8 1 - - - - =cs8 .end # Once the game gets into a winning configuration, do not change the state. .table cs0 cs1 cs2 cs3 cs4 cs5 cs6 cs7 cs8 done .default 0 - - 2 - 2 - 2 - - 1 2 2 2 - - - - - - 1 2 - - 2 - - 2 - - 1 2 - - - 2 - - - 2 1 - - - 2 2 2 - - - 1 - - 2 - - 2 - - 2 1 - 2 - - 2 - - 2 - 1 - - - - - - 2 2 2 1 - - 1 - 1 - 1 - - 1 1 1 1 - - - - - - 1 1 - - 1 - - 1 - - 1 1 - - - 1 - - - 1 1 - - - 1 1 1 - - - 1 - - 1 - - 1 - - 1 1 - 1 - - 1 - - 1 - 1 - - - - - - 1 1 1 1 # If there is a winner (done=1) then the state remains unchanged. # Otherwise, if m=i and whoseturn=1, then csi=1. Similarly, if # m=i and whoseturn=2, then csi=2 .table illegal done cs0 m whoseturn ns0 #.default 0 0 1 - - - =cs0 0 0 - 0 1 1 0 0 - 0 2 2 0 0 - (1,2,3,4,5,6,7,8) - =cs0 1 - - - - =cs0 .table illegal done cs1 m whoseturn ns1 #.default 0 0 1 - - - =cs1 0 0 - 1 1 1 0 0 - 1 2 2 0 0 - (0,2,3,4,5,6,7,8) - =cs1 1 - - - - =cs1 .table illegal done cs2 m whoseturn ns2 #.default 0 0 1 - - - =cs2 0 0 - 2 1 1 0 0 - 2 2 2 0 0 - (0,1,3,4,5,6,7,8) - =cs2 1 - - - - =cs2

  39. Lang.script (tic-tac-toe) rl fixed1.mv latch_expose stg_extract fixed.mva echo "Synthesis ..." determinize -ci spec.mva spec_dci.mva support cs0(3),cs1(3),cs2(3),cs3(3),cs4(3),cs5(3),cs6(3),cs7(3),cs8(3), whoseturn(2),m1(9),m2(9),out(3) spec_dci.mva spec_dci_supp.mva support cs0(3),cs1(3),cs2(3),cs3(3),cs4(3),cs5(3),cs6(3),cs7(3),cs8(3), whoseturn(2),m1(9),m2(9),out(3) fixed.mva fixed_supp.mva product fixed_supp.mva spec_dci_supp.mva p.mva support cs0(3),cs1(3),cs2(3),cs3(3),cs4(3),cs5(3),cs6(3),cs7(3),cs8(3), whoseturn(2),m2(9) p.mva p_supp.mva determinize -ci p_supp.mva p_dci.mva prefix p_dci.mva p_dci_pre.mva progressive -i 10 p_dci_pre.mva x.mva minimize x.mva x-min.mva prefix x.mva x-min.mva

  40. .table stop bank in1 csw nsw 0 left (empty,goat,cabbage) boat left 0 left wolfe (left,boat) boat 0 right (empty,goat,cabbage) boat right 0 right wolfe (right,boat) boat 0 - (empty,goat,cabbage) (left,right) =csw 1 - - - =csw .table stop bank in1 csg nsg 0 left (empty,wolfe,cabbage) boat left 0 left goat (left,boat) boat 0 right (empty,wolfe,cabbage) boat right 0 right goat (right,boat) boat 0 - (empty,wolfe,cabbage) (left,right) =csg 1 - - - =csg .table stop bank in1 csc nsc 0 left (empty,goat,wolfe) boat left 0 left cabbage (left,boat) boat 0 right (empty,goat,wolfe) boat right 0 right cabbage (right,boat) boat 0 - (empty,goat,wolfe) (left,right) =csc 1 - - - =csc .table bank nsw nsg nsc out .default OK right left left - notOK left right right - notOK right - left left notOK left - right right notOK right (right,boat) (right,boat) (right,boat) done # map input (in) into any legal input .table bank in csw csg csc in1 .default empty right wolfe (right,boat) - - =in left wolfe (left,boat) - - =in right goat - (right,boat) - =in left goat - (left,boat) - =in right cabbage - - (right,boat) =in left cabbage - - (left,boat) =in .end Wolf, goat, cabbage .model wolfe .inputs in .outputs out .mv in,in1 4 empty wolfe goat cabbage .mv csw,csg,csc,nsw,nsg,nsc 3 left right boat .mv bank,bank1 2 left right .mv out 3 OK notOK done .latch stop1 stop .reset stop 0 .latch nsw csw .reset csw left .latch nsg csg .reset csg left .latch nsc csc .reset csc left .latch bank1 bank .reset bank left .table out stop stop1 .default 0 done - 1 - 1 1 .table stop bank bank1 .default left 0 left right 1 - =bank out in(boat)

  41. spec.mva for wolf-goat-cabbage lang.script .model spec .inputs out .outputs Acc .mv out 3 OK notOK done .mv CS,NS 3 a b c .table CS ->Acc .default 1 b 0 .table out CS ->NS OK a a notOK a b done a c - b b - c c .latch NS CS .reset CS a .end rl wolfe.mv stg_extract fixed.mva echo "Synthesis ..." determinize -lci spec.mva spec_dci.mva support in(4),out(3) spec_dci.mva spec_dci_supp.mva support in(4),out(3) fixed.mva fixed_supp.mva product -l fixed_supp.mva spec_dci_supp.mva p.mva support in(4) p.mva p_supp.mva determinize -lci p_supp.mva p_dci.mva prefix p_dci.mva p_dci_pre.mva progressive -i 0 p_dci_pre.mva x.mva minimize x.mva x-min.mva prefix x-min.mva x-min.mva echo "Verification ..." support in(4),out(3) x.mva x_supp.mva product x_supp.mva fixed_supp.mva prod.mva support in(4),out(3) spec.mva spec_supp.mva check prod.mva spec_supp.mva

  42. Wolf, goat, cabbagex.mva

  43. Minimized x-min.mva

  44. Other Games • Toe-tac-tic (solvable) • Like tic-tac-toe • Except that any player can play either X or O at any time • A player wins when he completes a line or either X’s or O’s • Board game (too many states) • 4 x 4 board • Each player has 4 pieces which initially at the top and bottom rows of the board. • Any piece can move forward, left or right • Player wins when he moves one of his pieces to the other side • 12870 reachable states – can’t do it right now

  45. FSM2 FSM1 Application - splitting FSM blif files u FSM i X v o This is just a syntactic change. Nothing has been done yet.

  46. S o o i i cs2 cs1 Latch split mvsis 05> _split -h Usage: _split [-v] <latch_list> splits the current network S into two parts: F and X generates the script to solve the equation F * X = S -v : toggles verbose [default = no] <latch_list> : the list of latches to be included in X no spaces are allowed in the latch list the numbers of latches are zero-based for example: 0,3,5-7,9 mvsis 05>

  47. S Latch_split example .model s27.bench .inputs G0 G1 G2 G3 .outputs G17 .reset G5 0 .latch G10 G5 .reset G6 0 .latch G11 G6 .reset G7 0 .latch G13 G7 .table G0 G1 G3 G5 G6 G7 G17 .default 0 1 1 - - - - 1 1 - 0 - - - 1 1 - - - - 1 1 - - - 1 - - 1 - 1 - - 0 - 1 - - 0 - 0 - 1 - - - - 0 1 1 .table G0 G1 G3 G5 G7 G10 .default 0 1 1 - - - 1 1 - 0 - - 1 1 - - 1 - 1 1 - - - 1 1 .table G0 G1 G3 G5 G6 G7 G11 .default 0 0 - - 0 1 - 1 - 0 1 0 - 0 1 .table G1 G2 G7 G13 .default 0 1 0 - 1 - 0 1 1 .end s27a (X’) s27f (F) .model s27.bench .inputs G0 G1 G2 G3 G5 G6 .outputs G17 .latch G13 G7 0 .names G0 G1 G3 G5 G6 G7 G17 11---- 1 1-0--- 1 1----1 1 ---1-- 1 -1--0- 1 --0-0- 1 ----01 1 .names G0 G1 G3 G5 G7 G10 11--- 1 1-0-- 1 1--1- 1 1---1 1 .names G0 G1 G3 G5 G6 G7 G11 0--01- 1 -010-0 1 .names G1 G2 G7 G13 10- 1 -01 1 .end .model s27.bench .inputs G0 G1 G2 G3 G7 .outputs G17 .latch G10 G5 0 .latch G11 G6 0 .names G0 G1 G3 G5 G6 G7 G17 11---- 1 1-0--- 1 1----1 1 ---1-- 1 -1--0- 1 --0-0- 1 ----01 1 .names G0 G1 G3 G5 G7 G10 11--- 1 1-0-- 1 1--1- 1 1---1 1 .names G0 G1 G3 G5 G6 G7 G11 0--01- 1 -010-0 1 .names G1 G2 G7 G13 10- 1 -01 1 .end

  48. # Language solving script generated by MVSIS # for sequential network "s27.blif" on Wed Feb 18 21:35:53 2004 # Command line was: "split 0,1". echo "Solving the language equation ... " solve s27f.blif s27.blif G0,G1,G2,G3,G7 G5,G6 s27x.aut psa s27x.aut echo "Verifying the containment of the known implementation ... " read_blif s27a.blif latch_expose stg_extract s27a.aut support G0,G1,G2,G3,G7,G5,G6 s27a.aut s27a.aut check s27a.aut s27x.aut read_blif s27.blif

  49. s27x.aut

  50. s27x-dcmin.aut s27a.blif G0, G1, G2, G3, G7, G5, G6 inputs outputs

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