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 1. Make the two machines have the same input alphabet (support) 2. Perform the product Synchronous (changing the support) 1. Projection – Given a language L over the alphabet X Y projection is defined as L X {{ Xi } | {( Xi Yi )} L } 2. Lifting – given a language L over the alphabet X, lifting to the alphabet X Y is defined as L Y {{( Xi )} | { Xi } L } Asynchronous (changing the support) 1. Restriction – Given a language L over the alphabet X Y , the restriction to X is defined as L X {{ p X ( i )} | { i } L } i if i X where p X ( i ) otherw ise 2. Expansion – given a language L over X, lifting to the alphabet X Y is defined as L Y {{ i x i i } | { x i } L i , i Y } * 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 i F v X o i S 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 o 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’). i/u s s’ Similarly for the others. For S a transition of the type (q i/o q’) becomes (q i q’’) (q’’ o q’). s’’ i s q’’ u i s’ q o q’ We want to lift up the language of F to include o and arbitrary delays so that iuvu becomes for example: i oo u o v o u o o And S to include u and v and arbitrary delays, so that the two languages can contain similar strings. io becomes for example i uvu vvvu uu o In particular, they both can become the same thing: i oo u o v o u o o i u v u o i u v u o i uvu vvvu uu o The common symbols act to synchronization This is done on the automaton F by the following: i1/u1 v1/u2 becomes ,o ,o ,o ,o ,o Note the alphabet is ivuo i1 u1 v1 u2 Similarly for S: i1/o1 becomes , u , v ,u,v i1 ,u,v o1 ,u,v i1 Note that all states alphabet is i v u o are deterministic since the With these conversions, we can do synchronous composition and get the equivalent expanded result of parallel composition S F 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. Conformance Simulation Relation ( ) Let F1 and F2 be two automata over the same alphabet F2 simulates F1 ( F1 F2 ) if there exists a simulation relation S S 1 S 2 such that s1 s 2 S [( s1 s ) s ' ( s 2 s 2 ) ( s1 s 2 ) S ] ' 1 ' ' ' 2 Note that F2 simulates F1 implies that L ( F1 ) L ( F2 ) but these are not equivalent notions. If may be easier to find a simulation relation than to prove language containment. Language Containment ( ) To show that F1 F2 (i.e. L ( F1 ) L ( F2 ) ) we typically show that F1 F2 This requires complementing F2 which may be hard if F2 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 S in the construction of F S . Note that if S is deterministic or small, then there is no motivation to avoid computing S so using language containment is fine. Language Equality ( ) L ( F1 ) L ( F2 ) Proposition: If two FSMs, F1 and F2, and F2 is deterministic, then ( L ( F1 ) L ( F2 )) ( L ( F1 ) L ( F2 )) Hence, if we are solving F X S and S is a deterministic FSM then X F S is the MSG. If S is ND, then what is MSG? Comment: Clearly, we need F S in order for there to be a solution of F X S . This requires that supp(S) supp(F), since otherwise there is a variable v in F but not in S. Then S v would be too large. Topologies F I1 U2 I2 X O1 most U1 general O2 F I1 U2 X I1 I1 two-way U1 cascade O2 F one-way I1 cascade X U2 F X U1 X two-way U1 cascade O2 one-way cascade U1 O2 F O2 O1 F I1 U2 U1 rectification X F U2 I2 Engineering Change U1 X O2 Controller F U2 I2 X U1=O 1 Communicating the internal state F I1 U2 I2 O1 U1 X F I1 U2 O2 I2 latch_expose U1 X O1 cs O2 Hiding only the outputs I O F V U X Theorem: X ( F ( S ) U V ) I U V If O is a deterministic function of i,v,cs, the second complementation is easy (no subset construction). Thus the only variables that X does not see are the O variables. In the construction, for the most general solution, X ( F ( S ) U V ) I U V F ( S ) U V 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. Hiding the outputs only i’u’v’o1 iuvo2 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. Solving a language equation Solve A X C where { , } and { , , } In particular, find the largest solution X (most general solution). Theorem A: Let A and C be languages over alphabets I U and I O respectively. For the equation, A X C the most general solution is X ( A O C U ) U O Theorem B: Let A and C be languages over alphabets I U and I O respectively. For the equation, A X C the most general solution is X ( A O C U ) U O Proof: We prove Theorem A. Let (U O ) * . Then A C means that ( A O I ) I O C ( A O I C U ) I O A O I C U I ( A O I C U ) U O A U X ( I ) U O ( A O C U ) U O O ( A O C U ) U O ( A O C U ) U O AC Thus A C is the largest solution of The proof of Theorem B is similar. A X C Computing the CF for a node global step yi X Z R( X , yi , Z ) R( X , yi ) Z [ R( X , yi , Z ) R spec ( X , Z )] Computing CF - local step Yi X Yij yi yi Z M ( X , Yi ) CF (Yi , yi ) X [ M ( X , Yi ) R( X , yi )] CF (Yi , yi ) X [ M ( X , Yi ) Z [ R( X , yi , Z ) R spec ( X , Z )]] CF yi Yi yi X Z M ( X , Yi ) M ( X , Yi ) Yi yi CF (Yi , yi ) X [ M ( X , Yi ) Z [ R( X , yi , Z ) R X , Z [ M ( X , Yi ) R( X , yi , Z )]R spec spec (X , Z) Note that essentially the same computation applies for multiple-output 1 k nodes, i.e. where yi { yi , , yi } ( X , Z )]] FSM networks – computing complete sequential flexibility (CSF) i1 FSM FSM FSM i2 i Specification S (i,o) FSM FSM FSM FSM o spec context v u unknown o Context C (i,v,u,o) Unknown X (u,v) Problem: Given S and C , find the Most General Solution (MGS) of X C S FSM Networks i The most general solution (MGS) of X C S spec context v o u unknown is M G S (C ( S ) ( i , v ,u , o ) ) ( u , v ) In general, MGS is deterministic automaton but as an FSM it is non-deterministic (NDFSM) 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 u v CSF Comparison with combinational case Sequential i CSF (u, v) i ,o (C ( S )( i ,v ,u ,o ) ) ( u ,v ) spec context u o v unknown Combinational CF (Yi , yi ) X , Z [ M ( X , Yi ) R( X , yi , Z )]R R( X , yi , Z ) X Yi yi unknown (X , Z) X , Z [ M ( X , Yi ) R ( X , yi , Z ) R M ( X , Yi ) spec spec ( X , Z )] Extending CF FSM I u O X Combinational sub-block v 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 Algorithm Algorithm: LanguageEquationSolving Input: prefix-closed deterministic S(i.o) and C(i,v,u,o) Output: most general prefix-closed, progressive X (FSM) begin 01 X := Complete ( S, non-accepting ) 02 X := Determinize&Complement ( X ) S 03 X := Support (X, (i,v,u,o)) - raise ( S )( i ,v ,u ,o ) 03 X := Product ( C, X ) (C ( S )( i ,v ,u ,o ) ) 04 X := Support ( X, (u,v) ) - hide (C ( S ) ) ( i ,v ,u ,o ) ( u ,v ) 05 X := Determinize &Complement ( X, u ) (C ( S )( i ,v ,u ,o ) )( u ,v ) 06 return Prefix&Progressive (X ) Convert to FSM end Examples • Games – – – – Nim Tic-tac-toe Toe-tac-tic Board • Control – Wolf, goat, cabbage • Latch splitting Example: Coin Game (NIM) In God We Trust In God We Trust In God We Trust 1. 2. 3. 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) Example: 6 5 3 6=110 5=101 3=011 ____ 222 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 b0 .table out CS ->NS OK a a notOK a b done a c -bb -cc .latch NS CS .reset CS a .end .model game-piles .inputs p1 d1 p2 d2 .outputs out .mv .mv .mv .mv .mv p1,p2,p,pt,ptt 3 d1,d2,d 7 cs0,cs1,cs2,ns0,ns1,ns2,nh,h 7 whoseturn,whoseturn1 2 1 2 out 3 OK notOK done .latch .reset 3 .latch .reset 2 .latch .reset 1 ns0 cs0 cs0 ns1 cs1 cs1 ns2 cs2 cs2 .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 # 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 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 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> Example of CSF computation: NDFSM represented as automaton Inputs p1(3),d1(7) Outputs p2(3),d2(7) In God We Trust In God We Trust In God We Trust Tic-tac-toe spec.mv .model spec .inputs m1 c1 m2 c2 .outputs out .mv m1,m2 9 .mv out 3 .table ->out 0 2 .end spec.mva # .model spec .inputs out .outputs Acc .mv out 3 .mv CS,NS 3 a b c .table CS ->Acc .default 1 b0 .table out CS ->NS 0aa 1ab 2ac -bb -cc .latch NS CS .reset CS a .end .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 021 012 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 00--------0 1-0-------0 2--0------0 3---0-----0 4----0----0 5-----0---0 6------0--0 7-------0-0 8--------00 # 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 2222------ 2 22--2--2-- 2 22---2---2 2 2---222--- 2 2--2--2--2 2 2-2--2--2- 2 2------222 2 1--1-1-1-- 1 1111------ 1 11--1--1-- 1 11---1---1 1 1---111--- 1 1--1--1--1 1 1-1--1--1- 1 1------111 1 # 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-- 1 222------ 1 2--2--2-- 1 2---2---2 1 ---222--- 1 --2--2--2 1 -2--2--2- 1 ------222 1 --1-1-1-- 1 111------ 1 1--1--1-- 1 1---1---1 1 ---111--- 1 --1--1--1 1 -1--1--1- 1 ------111 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 00-011 00-022 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 00-111 00-122 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 00-211 00-222 0 0 - (0,1,3,4,5,6,7,8) - =cs2 1 - - - - =cs2 .table illegal done cs3 m whoseturn ns3 #.default 0 0 1 - - - =cs3 00-311 00-322 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 00-411 00-422 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 00-511 00-522 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 00-611 00-622 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 00-711 00-722 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 00-811 00-822 0 0 - (0,1,2,3,4,5,6,7) - =cs8 1 - - - - =cs8 .end 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 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 -11 .table stop bank bank1 .default left 0 left right 1 - =bank out in(boat) .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 spec.mva for wolf-goat-cabbage .model spec .inputs out .outputs Acc .mv out 3 OK notOK done .mv CS,NS 3 a b c .table CS ->Acc .default 1 b0 .table out CS ->NS OK a a notOK a b done a c -bb -cc .latch NS CS .reset CS a .end lang.script 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 Wolf, goat, cabbage x.mva Minimized x-min.mva Other Games 1. 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 2. 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 Application - splitting FSM blif files u i FSM FSM1 v XFSM2 o This is just a syntactic change. Nothing has been done yet. Latch split i S o 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> o i cs2 cs1 S .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 11---- 1 1-0--- 1 1----1 1 ---1-- 1 -1--0- 1 --0-0- 1 ----01 1 .table G0 G1 G3 G5 G7 G10 .default 0 11--- 1 1-0-- 1 1--1- 1 1---1 1 .table G0 G1 G3 G5 G6 G7 G11 .default 0 0--01- 1 -010-0 1 .table G1 G2 G7 G13 .default 0 10- 1 -01 1 .end Latch_split example s27f (F) s27a (X’) .model s27.bench .inputs G0 G1 G2 G3 G5 G6 .outputs G17 .model s27.bench .inputs G0 G1 G2 G3 G7 .outputs G17 .latch G13 G7 0 .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 .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 # 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 s27x.aut s27x-dcmin.aut s27a.blif G0, G1, G2, G3, G7, G5, G6 inputs outputs FSM networks - Node Minimization Given a NDFSM CSF, find the “smallest” FSM Y, such that Y is well-defined and Y CSF Y is called a reduction of CSF State graph of X It generally looks like non-accepting don’t care state C-compatibility - dcmin Two states s1 and s2 are c-compatible if their care sets do not intersect, i.e. the care set of one is completely contained in the don’t care set of the other. s s states 1 2 u u-space Remaining DC Care Care set set v X (cs,u,v,ns) Y (cs, u ) v , ns X (cs, u, v, ns) Y (s1 , u) Y (s2 , u) A simple state reduction method-dcmin • Let X ( s, v, u, s ') be the relation for the incomplete CSF X, and compute Y ( s, u ) v , s ' X ( s, v, u, s ') – i.e. those states and inputs for which there exists a next state and output (the next state can be either accepting or not). • Order this BDD with the u variables first, and let pi ( s) be the unique functions below the u variables pointed to. • Two states s1 and s2 are c-compatible if and only if for all i, pi ( s1 ) pi ( s2 ) 0 i.e. they have no minterm u in common. • So pi ( s) is a clique of states that can't be merged, i.e. are not ccompatible and must have different colors. • Then the c-incompatibility graph is I (s, s ') pi (s) pi (s ') i which has to be colored. • Suppose Q( s, c) is the assignment of states s to colors c. The new automaton relation for X is then X '(c, v, u, c ') s , s ' Q( s, c) X ( s, v, u, s ')Q( s ', c ') Simple state reduction Merged states u-space Care Care Remaining set Care set DC set Note that this is a “simple” coloring problem in contrast to the compatibilities problem normally associated with state minimization for incompletely specified FSMs. In contrast, here a group of states is “c-compatible” iff they are pair-wise ccompatible. Other ideas on reduction of CSF • This problem is similar to SOP minimization when using CF to minimize the node in the combinational network. • Many cost functions are possible. If we try to minimize the number of states in CSF, it is the problem of minimizing a PNDFSM – – T. Kam et. al., DAC 1994. • We might want to look for a good implementation directly, rather than first minimizing the number of states. – Similarly, for a node in the combinational circuit, looking for a small SOP, or the minimum number of literals in FF, may be misleading. • A specialized algorithm has been developed to check whether a combinational solution (a single-state reduction) exists. – The problem is reduced to SAT with as many variables as there are states + transitions in the CSF. Solution is practical for, say, 100 states and 500 transitions. – A similar algorithm can be developed to check whether a 2 or 3 state solution exists • more variables, the SAT problem is harder Iterative language solving The problem of computing the CSF can be iterative. 1. 2. 3. 4. 5. 6. 7. 8. Given F and S Split F into F1 and F2 Solve F1 * X = S. If we can reduce X to a smaller implementation than F2, replace F2 Solve F2 * X = S If we can reduce X to a smaller implementation than F1, replace F1 Set F = F1 * F2 If either F1 or F2 has changed, go to 2 FSM Windowing FSM1 i X X1 FSM2 X2 FSM3 X3 X = X1 * X2 * X3 Compositionally Progressive spec i o F v u X • X should be compositionally progressive (c-progressive) with F – i.e. for every product state cs of X * F, the next state ns and output o should be defined for all i. • Roland Jiang has proposed a way to use this to additionally trim the solution X during the subset construction. But he is not 100% sure it is right. • Nina and Tiziano have another method for trimming and have proved that the largest c-progressive solution can contain well-defined FSM sub-behaviors that are not c-progressive. • Roland has demonstrated that the above paper is wrong. • Being c-progressive does not necessarily imply no combinational loops • To hear a more detailed discussion, attend MVSIS weekly meeting Friday, 111pm in DOP center library (fishbowl) • There might be a connection here with omega-automata. Future developments • Objective is to push to the limit, the size of application that can be done – Keep multi-level MV structure, given in MVSIS, as long as possible (lecture on this later) – Use SAT in subset construction • The bottleneck looks to be extracting good subbehavior of CSF (reduction) – A sub-graph of the CSF usually not good enough – “Simplified” (dcmin) state minimization of CSF may be good first step? • Try for a good sub-behavior more directly without constructing CSF • Try hierarchy and windowing applied to FSM network

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