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ENGG3190 Logic Synthesis “Sequential Circuit Synthesis” Winter 2014 S. Areibi School of Engineering University of Guelph Outline • Modeling synchronous circuits – State-based models. – Structural models. • State-based optimization methods – State minimization. • State minimization for completely specified machines • State minimization for incompletely specified machines. – State encoding • State encoding for two-level logic • State encoding for multiple-level logic • Structural-based optimization methods – Retiming 2 Combinational vs. Sequential Circuits 1. Combinational logic are very interesting and useful for designing arithmetic circuits (adders, multipliers) or in other words the Data Path of a computer. Combinational circuits cannot remember what happened in the past (i.e. outputs are a function of current inputs). 2. In certain cases we might need to store some info before we proceed with our computation or take action based on a certain state that happened in the past. Sequential circuits are capable of storing information between operations. They are useful in designing registers, counters, and CONTROL Circuits, … 3 Types of Sequential Circuits Two main types and their classification depends on the times at which their inputs are observed and their internal state changes. Synchronous: State changes synchronized by one or more clocks Asynchronous: Changes occur independently 4 What are Sequential Circuits? Some sequential circuits have memory elements. – Synchronous circuits have clocked latches. – Asynchronous circuits may or may not have latches (e.g. C-elements), but these are not clocked. Feedback (cyclic) is a necessary, but not sufficient condition for a circuit to be sequential. Synthesis of sequential circuits is not as well developed as combinational. (only small circuits) Sequential synthesis techniques are not really used in commercial software (except maybe retiming). Sequential verification is a problem. Synchronous/Asynchronous Time Analog Digital Continuous in value & time Asynchronous Discrete in value & continuous in time Synchronous Discrete in value & time 6 Comparison Synchronous Easier to analyze because can factor out gate delays Speed of the system is determined by the clock (maybe slowed!) Asynchronous Potentially faster Harder to analyze We will only look at Synchronous Circuits 7 Example in1 in2 Latch Present State Next State Registers and Latches (Netlist) ----/1 in4 (1010, 0110)/1 in3 0 (--00, 11-0)/0 out 1 primary output primary inputs ---1/1 1 State Transition Graph (STG) The above circuit is sequential since primary output depends on the state and primary inputs. Analysis of Sequential Circuits The behavior of a sequential circuit is determined from: Inputs, Outputs, Present state of the circuit. The analysis of a sequential circuit consists of: Obtaining a suitable description that demonstrates the time sequence of inputs, outputs and states (STATE DIAGRAM). 9 Derive Input Equations Can describe inputs to FF with logic equations J A ( XB YC ) K A (YB C) 10 State Table Similar to truth table with state added A sequential circuit with `m’ FFs and `n’ inputs needs 2m+n rows in state table. DA ( AX BX ) DB A X Y ( A B) X 11 State Diagram Input Output • An alternative representation to State Table Input/Output “Mealy Model” 12 State Diagram: Moore Alternative representation for state table Inputs State/Output 13 Sequential Circuit Types Moore model – outputs depend on states only. Mealy model – outputs depend on inputs & states 14 Moore vs. Mealy Machine Moore Machine: Easy to understand and easy to code. Might requires more states (thus more hardware). Mealy Machine: They are more general than Moore Machines More complex since outputs are a function of both the state and input. Requires less states in most cases, therefore less components. Choice of a model depends on the application and personal preference. You can transform a Mealy Machine to a Moore Machine and vice versa. 15 Design Procedure Design starts from a specification and results in a logic diagram or a list of Boolean functions. The steps to be followed are: 1. Derive a state diagram 2. Reduce the number of states (Optimization) 3. Assign binary values to the states (encoding) (Optimization) 4. Obtain the binary coded state table 5. Choose the type of flip flops to be used 6. Derive the simplified flip flop input equations and output equations 7. Draw the logic diagram 16 Synthesis Example • Implement simple count sequence: 000, 010, 011, 101, 110 • Derive the state transition table from the state transition diagram 010 100 101 110 011 Present State C B A 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 Next State C+ B+ A+ x x x x x x 1 0 1 1 1 0 0 1 0 0 1 1 1 0 0 x x x note the don't care conditions that arise from the unused state codes 17 Don’t cares in FSMs (cont’d) • Synthesize logic for next state functions derive input equations for flip-flops C+ A B+ C X 1 1 0 X 1 X 0 A A+ C X 0 0 1 X 1 X 1 B B A C X 1 0 0 X 0 X 1 B C+ = B B+ = A + B’ C A+ = A’ C’ + AC • Some states are not reachable!! Since they are don’t cares. • Can we fix this problem? 18 Self-starting FSMs • Deriving state transition table from don't care assignment C+ A B+ C 0 1 1 0 0 1 1 0 A A+ C 1 0 0 1 1 1 1 1 B B Present State C B A 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 A C 1 1 0 0 0 0 1 1 B Next State C+ B+ A+ 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 0 0 1 1 x 19 Self-starting FSMs • Start-up states – at power-up, FSM may be in an used or invalid state – design must guarantee that it (eventually) enters a valid state • Self-starting solution – design FSM so that all the invalid states eventually transition to a valid state may limit exploitation of don't cares 010 001 111 100 101 000 110 011 20 Modeling Synchronous Circuits • State-based model – Model circuits as finite-state machines. – Represented by state tables/diagrams. – Lacks a direct relation between state manipulation and corresponding area and delay variations. – You can Apply exact/heuristic algorithms for • State minimization. • State encoding. • Structural-based models – Represent circuit by synchronous logic network. – You can Apply • Retiming. • Logic transformations (recall Multi Level Synthesis Transformations!!) • State transition diagrams can be: – Transformed into synchronous logic networks by state encoding. – Recovered from synchronous logic networks by state extraction. 21 General Logic Structure Combinational logic (CL) Sequential elements • Combinational optimization – keep latches/registers at current positions, keep their function – optimize combinational logic in between • Sequential optimization – change latch position/function (retiming) 22 Overview of FSM Optimization Specification State Encoding Logic/Timing Optimization Verification/Testing State Minimization State-Based Models: Optimization 24 Overview of FSM Optimization Initial: FSM description 1. provided by the designer as a state table 2. extracted from netlist 3. derived from HDL description • • obtained as a by-product of high-level synthesis translate to netlist, extract from netlist State minimization: Combine equivalent states to reduce the number of states. For most cases, minimizing the states results in smaller logic, though this is not always true. State assignment: Assign a unique binary code to each state. The logic structure depends on the assignment, thus this should be done optimally. Minimization of a node: in an FSM network Decomposition/factoring: of FSMs, collapsing/elimination Sequential redundancy removal: using ATPG techniques Formal Finite-State Machine Model • • • • • Defined by the quintuple (, , S, , ). A set of primary inputs patterns . A set of primary outputs patterns . A set of states S. A state transition function – : S S. • An output function – : S for Mealy models – : S for Moore models. 26 State Minimization • Definition: Derive a FSM with similar behavior and minimum number of states. – Aims at reducing the number of machine states – reduces the size of transition table. • State reduction may reduce? – the number of storage elements. – the combinational logic due to reduction in transitions • Types: 1. Completely specified finite-state machines • No don't care conditions. • Easy to solve. 2. Incompletely specified finite-state machines • Unspecified transitions and/or outputs. • Intractable problem. 27 Redundant States: Minimization 0/0 S1 0/0 S2 S1 0/0 0/0 1/1 S3 0/0 1/1 1/1 1/0 0/0 1/1 S4 S3 1/1 1/0 S4 0/0 • Can you distinguish between State S1 and S2? • States S1, S2 seem to be equivalent! • Hence we can reduce the machine accordingly 28 State Minimization for Completely-Specified FSMs • Def: Equivalent states – Given any input sequence the corresponding output sequences match. • Theorem: Two states are equivalent iff I. they lead to identical outputs and II. their next-states are equivalent. • Since equivalence is symmetric and transitive – States can be partitioned into equivalence classes. – Such a partition is unique. 29 Equivalent States • Two states of an FSM are: – equivalent (or indistinguishable) – if for each input they produce the same output and their next states are identical. Si and Sj are equivalent and merged into a single state. 1/0 Si Sm 1/0 Sm 0/0 1/0 Si,j 0/0 Sj 0/0 Sn Sn 30 Algorithmic State Minimization • Goal – identify and combine states that have equivalent behavior – Reduced machine is smaller, faster, consumes less power. • Algorithm Sketch 1. Place all states in one set 2. Initially partition set based on output behavior 3. Successively partition resulting subsets based on next state transitions 4. Repeat (3) until no further partitioning is required • states left in the same set are equivalent Polynomial time procedure Equivalent States … Cont • Example: States A . . . I, Inputs I1, I2, Output, Z Next state, output (Z) Present state A and D are equivalent A and E produce same output Q: Can they be equivalent? A: Yes, if B and D were equivalent and C and G were equivalent. Input I1 I2 A D/0 C/1 B E/1 A /1 C H/1 D/1 D D/0 C/1 E B/0 G/1 F H/1 D /1 G A/0 F/1 H C/0 A/1 I G/1 H/1 32 Minimization Algorithm • Stepwise partition refinement. • Let i , i= 1, 2, …., n denote the partitions. • Initially – 1 = States belong to the same block when outputs are the same for any input. • Refine partition blocks: While further splitting is possible – k+1 = States belong to the same block if they were previously in the same block and their next-states are in the same block of k for any input. • At convergence i+1 = i – Blocks identify equivalent states. 33 Example … • 1 = {(s1, s2), (s3, s4), (s5)}. • Split s3, s4 • 2 = {(s1, s2), (s3), (s4), (s5)}. • 2 = is a partition into equivalence classes – States (s1, s2) are equivalent. 34 … Cont .. Example … Original FSM Minimal FSM 35 … Example Original FSM {OUT_0} = IN_0 LatchOut_v1' + IN_0 LatchOut_v3' + IN_0' LatchOut_v2' v4.0 = IN_0 LatchOut_v1' + LatchOut_v1' LatchOut_v2' v4.1 = IN_0' LatchOut_v2 LatchOut_v3 + IN_0' LatchOut_v2' v4.2 = IN_0 LatchOut_v1' + IN_0' LatchOut_v1 + IN_0' LatchOut_v2 LatchOut_v3 sis> print_stats pi= 1 po= 1 nodes= 4 latches= 3 lits(sop)= 22 #states(STG)= 5 Minimal FSM {OUT_0} = IN_0 LatchOut_v1' + IN_0 LatchOut_v2 + IN_0' LatchOut_v2' v3.0 = IN_0 LatchOut_v1' + LatchOut_v1' LatchOut_v2‘ v3.1 = IN_0' LatchOut_v1' + IN_0' LatchOut_v2' sis> print_stats pi= 1 po= 1 nodes= 3 latches= 2 lits(sop)= 14 #states(STG)= 4 36 State Minimization Example • Sequence Detector for 010 or 110 0/0 0/0 S3 0/0 S1 1/0 S0 0/0 S4 S5 0/1 Next State Present State X=0 X=1 Output X=0 X=1 Reset 0 1 00 01 10 11 S0 S1 S2 S3 S4 S5 S6 0 0 0 0 1 0 1 1/0 1/0 1/0 Input Sequence 0/0 S2 1/0 1/0 S6 0/1 1/0 S1 S3 S5 S0 S0 S0 S0 S2 S4 S6 S0 S0 S0 S0 0 0 0 0 0 0 0 Method of Successive Partitions Input Sequence Next State Present State X=0 X=1 Output X=0 X=1 Reset 0 1 00 01 10 11 S0 S1 S2 S3 S4 S5 S6 0 0 0 0 1 0 1 S1 S3 S5 S0 S0 S0 S0 ( S0 S1 S2 S3 S4 S5 S6 ) ( S0 S1 S2 S3 S5 ) ( S4 S6 ) ( S0 S1 S2 ) ( S3 S5 ) ( S4 S6 ) ( S0 ) ( S1 S2 ) ( S3 S5 ) ( S4 S6 ) S2 S4 S6 S0 S0 S0 S0 0 0 0 0 0 0 0 S1 is equivalent to S2 S3 is equivalent to S5 S4 is equivalent to S6 Minimized FSM State minimized sequence detector for 010 or 110 Input Sequence Next State Present State X=0 X=1 Output X=0 X=1 Reset 0+1 X0 X1 S0 S1' S3' S4' 0 0 0 1 S0 X/0 0/0 S1’ 1/0 S4’ S3’ X/0 0/1 1/0 S1' S3' S0 S0 S1' S4' S0 S0 0 0 0 0 Computational Complexity • Polynomially-bound algorithm. • There can be at most |S| partition refinements. • Each refinement requires considering each state – Complexity O(|S|2). • Actual time may depend upon – Data-structures. – Implementation details. 40 Implication Table Method B Present state C D E √ BD CG EH AD F G H EH AD √ AD CF CD AC I A BD CG AD CF CD AC EG AH GH DH B C AB FG BC AG AC AF Next state, output (Z) Input I1 I2 A D/0 C/1 B E/1 A/1 C H/1 D/1 D D/0 C/1 E B/0 G/1 F H/1 D/1 G A/0 F /1 H C/0 A/1 I G/1 H/1 GH DH D E F G H 41 Implication Table Method (Cont.) B Equivalent states: C D √ E BD CG EH AD F G H EH AD √ AD AB CF FG CD BC AC AG AD CF CD AC I A BD CG EG AH GH DH B C S1: A, D, G S2: B, C, F S3: E, H S4: I AC AF GH DH D E F G H 42 Minimized State Table Original Present state Minimized Next state, output (Z) Present state Input I1 Next state, output (Z) I2 Input I1 I2 A D/0 C/1 S1 = (A, D, G) S1 / 0 S2 / 1 B E/1 A/1 S2 = (B, C, F) S3 / 1 S1 / 1 C H/1 D/1 S3 = (E, H) S2 / 0 S1 / 1 D D/0 C/1 S4 = I S1 / 1 S3 / 1 E B/0 G/1 F H/1 D/1 G A/0 F/1 H C/0 A/1 I G/1 H/1 Number of flip-flops is reduced from 4 to 2. 43 Incompletely Specified Machines • Next state and output functions have don’t cares. • However, for an implementation, and are functions, – thus they are uniquely defined for each input and state combination. • Don’t cares arise when some combinations are of no interest: – they will not occur or – their outputs will not be observed • For these, the next state or output may not be specified. – (In this case, and are relations, but of special type. We should make sure we want these as don’t cares.) • Such machines are called incompletely specified. … State Minimization for Incompletely Specified FSMs • Minimum finite-state machine is not unique. • • Implication relations make problem intractable. Example – Replace * by 1. • {(s1, s2), (s3), (s4), (s5)}. Minimized to 4 states 45 … State Minimization for Incompletely Specified FSMs • Minimum finite-state machine is not unique. • Example – Replace * by 0. • {(s1, s5), (s2, s3, s4)}. 0 0 It is now completely specified Unfortunately, there is an exponential number of completely specified FSMs in correspondence to the choice of the don’t care values!! 46 Example 1/1 s1 s2 1/1 0/0 s1 1/- 0/- added transitions to all states and 0/- s1 output any value 1/1 0/- d s2 0/0 s2 1/-/- 0/0 added dummy non-accepting state 1/- By adding a dummy state this can be converted to a machine with only the output incompletely specified. Could also specify “error” as the output when transitioning to the dummy state. Alternatively (better for optimization), can interpret undefined next state as allowing any next state. State Encoding • Binary and Gray encoding use the minimum number of bits for state register • Gray and Johnson code: – Two adjacent codes differ by only one bit • Reduce simultaneous switching – Reduce crosstalk – Reduce glitch # Binary Gray Johnson One-hot 0 000 000 0000 00000001 1 001 001 0001 00000010 2 010 011 0011 00000100 3 011 010 0111 00001000 4 100 110 1111 00010000 5 101 111 1110 00100000 6 110 101 1100 01000000 7 111 100 1000 10000000 48 State Encoding • The cost & delay of FSM implementation depends on encoding of symbolic states. – e.g., 4 states can be encoded in 4! = 24 different ways • There are more than n! different encodings for n states. – exploration of all encodings is impossible, therefore heuristics are used • Heuristics Used: 1. One-hot encoding 2. minimum-bit change 3. prioritized adjacency 49 One-hot Encoding • Uses redundant encoding in which one flip-flop is assigned to each state. • Each state is distinguishable by its own flip-flop having a value of 1 while all others have a value of 0. A S=0 A 0 Z=1 S=1 B 00 1 S C B Z=0 01 10 C 0 50 Z 1 Minimum-bit Change • Assigns codes to states so that the total number of bit changes for all state transitions is minimized. • In other words, if every arc in the state diagram has a weight that is equal to the number of bits by which the source and destination encoding differ, this strategy would select the one that minimizes the sum of all these weights. Encoding with 6 bit changes Encoding with 4 bit changes 51 The Idea of Adjacency • • • • Inputs are A and B State variables are Y1 and Y2 An output is F(A, B, Y1, Y2) A next state function is G(A, B, Y1, Y2) A Karnaugh map of output function or next state function 1 1 1 1 1 1 Y2 1 1 1 1 Y1 1 Larger clusters produce smaller logic function. Clustered minterms differ in one variable. B 52 Size of an Implementation • Number of product terms determines number of gates. • Number of literals in a product term determines number of gate inputs, which is proportional to number of transistors. • Hardware α (total number of literals) • Examples of four minterm functions: • F1 = ABCD +ABCD +ABCD +ABCD has 16 literals • F2 = ABC +ACD has 6 literals 53 Rule 1 (Priority #1), Common Dest States that have the same next state for some fixed input should be assigned logically adjacent codes. Fixed Inputs Outputs Combinational logic Si Sk Sj Present state Flipflops 0/0 S0 1/0 0/1, 1/1 S3 0/1 S1 0/0 S2 Next state Clock Clear Rule #1: (S1, S2) The input value of 0 will move both states into the same state S3 54 Rule 2 (Priority #2), A Common Source States that are the next states of the same state under logically adjacent inputs, should be assigned logically adjacent codes. Adjacent Inputs I1 I2 Outputs Combinational logic Fixed present state Si Sk Next Sm state Flipflops 0/0 S0 1/0 0/1, 1/1 S3 0/1 S1 0/0 S2 Clock Clear Rule #2: (S1, S2) They are both next states of the state S0 55 Rule 3 (Priority #3), A Common Output States that have the same output value for the same input value, should be assigned logically adjacent codes. Adjacent Inputs I1 I2 Outputs Combinational logic Fixed present state Si Sk Next Sm state Flipflops Clock Clear 0/0 S0 1/0 0/1, 1/1 S3 0/1 S1 0/0 S2 0/0 Rule #3: (S0, S1), and (S2, S3), states S0 and S1 have the same output value 0 for the same input value 0 01 11 1/0 0/1, 1/1 00 0/1 0/0 10 56 Example of State Assignment Next state, output (Z) Present state Input, X 0 1 A C, 0 D, 0 B C, 0 A, 0 C B, 0 D, 0 D A, 1 B, 1 0 1 0 A B 1 C D A adj C (Rule 1) A adj B (Rule 1) A 0/1 1/0 1/0 D 0/0 B 1/1 0/0 1/0 C adj D (Rule 2) Verify that BC and AD are not adjacent. 0/0 C B adj D (Rule 2) 57 State Assignment #1 A = 00, B = 01, C = 10, D = 11 Present state Next state, output Y1*Y2*, Z Y1, Y2 Input, X 0 1 A = 00 10 / 0 11 / 0 B = 01 10 / 0 00 / 0 C = 10 01 / 0 11 / 0 D = 11 00 / 1 01 / 1 Input Present state Output Next state X Y1 Y2 Z Y1* Y2* 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 58 Logic Minimization for Optimum State Assignment X Z Y2 1 1 X Y1* Y2 1 1 1 1 Y1 Y1 X Y2* 1 1 1 1 Result: 5 products, 10 literals. Y2 Y1 59 Circuit for Optimum State Assignment Z Combinational logic X Y1* Y2* 32 transistors Y1 Y1 CLEAR Y2 Y2 CK 60 Using an Arbitrary State Assignment: A = 00, B = 01, C = 11, D = 10 Present state Next state, output Y1*Y2*, Z Y1, Y2 Input, X 0 1 A = 00 11 / 0 10 / 0 B = 01 11 / 0 00 / 0 C = 11 01 / 0 10 / 0 D = 10 00 / 1 01 / 1 Input Present state Output Next state X Y1 Y2 Z Y1* Y2* 0 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 61 Logic Minimization for Arbitrary State Assignment X Z 1 1 Y2 X Y1* 1 Y2 1 1 1 Y1 Y1 X Y2* 1 1 Current Result: 6 products, 14 literals. Y2 Previous Result: 5 products, 10 literals. 1 1 Y1 62 Circuit for Arbitrary State Assignment Comb. logic Z X Y1* Y2* 42 transistors Y1 Y1 CLEAR Y2 Y2 CK 63 Best Encoding Strategy • 1. 2. 3. 4. 5. To determine the encoding with the minimum cost and delay, we need to: Generate K-Maps for the next-state and output functions Derive excitation equations from the next-state map. Derive output equations from the output function map. Implement above equations using twolevel NAND gates. Calculate cost and delay 64 Optimizing Sequential Circuits by Retiming Netlist of Gates Netlist of gates and registers: Inputs Various Goals: Outputs – Reduce clock cycle time – Reduce area • Reduce number of latches 65 Retiming Problem – Pure combinational optimization can be myopic since relations across register boundaries are disregarded Solutions – Retiming: Move register(s) so that • clock cycle decreases, or number of registers decreases and • input-output behavior is preserved – RnR: Combine retiming with combinational optimization techniques • Move latches out of the way temporarily • optimize larger blocks of combinational 66 Synchronous Logic Network … • Synchronous Logic Network – Variables. – Boolean equations. – Synchronous delay annotation. • Synchronous network graph – Vertices equations I/O , gates. – Edges dependencies nets. – Weights synch. delays registers. Circuit Representation Circuit representation: G(V,E,d,w) – – – – V set of gates E set of wires d(v) = delay of gate/vertex v, (d(v)0) w(e) = number of registers on edge e, (w(e)0) 68 … Synchronous Logic Network Circuit Representation Example: Correlator (from Leiserson and Saxe) (simplified) + 0 7 Host 0 0 (x, y) = 1 if x=y 0 otherwise 0 2 3 3 0 Retiming Graph (Directed) a b Circuit Operation delay Every cycle in Graph has at least one register i.e. no combinational loops. 70 3 + 7 Preliminaries e0 For a path p : ek 1 e1 v0 v1 vk 1 vk k d ( p ) d (vi ) Path Delay (includes endpoints) i 0 k 1 w( p ) w(ei ) i 0 Path weight Clock cycle 0 c max {d ( p)} 71 w(p)=0 0 2 For correlator c = 13 Path with 0 0 p: w ( p ) 0 7 3 0 3 Basic Operation • • Movement of registers from input to output of a gate or vice versa A positive value corresponds to shifting registers from the outputs to inputs Retime by -1 Retime by +1 • • • This should not affect gate functionality's Mathematical formulation: A retiming of a network G(V,E,W) is: – r: V Z, an integer vertex labeling, that transforms G(V,E,W) into G’(V,E,W’), where for each edge e=(u,v), the weight after retiming wr(e) is given by: – wr(e) = w(e) + r(v) - r(u) for edge e = (u,v) 72 Summary Sequential Logic Synthesis is an important phase of the Front End Tool for VLSI Circuits. Optimization of Sequential Circuits involves: State Minimization State Encoding Retiming State Minimization may be applied to completely specified Machines or Incompletely specified Machines. State Encoding utilizes different heuristics to further minimize the logic Retiming plays an important role in reducing the latency of the circuit. 73 74