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A series of reading assignments from the ece 152a - digital design principles course at the university of california, berkeley, focusing on state minimization and equivalence. The texts cover topics such as partitioning minimization procedure, elimination of redundant states, determination of state equivalence using an implication table, and moore reduction procedure. The documents also include examples and explanations of concepts.
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8.6.1 Partitioning Minimization Procedure 8.6.2 Incompletely Specified FSMs
Reading Assignment Roth 15 Reduction of State Tables / State Assignment 15.1 Elimination of Redundant States 15.2 Equivalent States 15.3 Determination of State Equivalence Using an Implication Table 15.4 Equivalent Sequential Circuits 15.5 Incompletely Specified State Tables
Elimination of Redundant States Row Matching Recall CD player controller Mealy implementation contained two sets of rows with same next state and output Eliminate redundant states Row matching doesn’t identify “equivalent states” Row matching identifies “same state” Equivalent states are the more general case
Determination of State Equivalence using an Implication Table (1) Construct Implication Table for Pairwise Comparison (2) First Pass Compare outputs For states to be equivalent, next state and output must be the same Put “X’s” where outputs differ
Implication Table (first pass) B C D E F G H A B C D E (^) F G X X X X X X X X X X X X X X X X
PS x=0 x=1 z
Determination of State Equivalence using an Implication Table (3) One column (or row) at a time, find implied pairs
Implication Table (second pass) B C D E F G H A B C D E (^) F G D-F C-H X A-D C-E B-D C-H A-F E-H B-F H-H C-E A-D E-F B-D C-E D-G A-B E-H C-F A-B C-C A-G C-F B-G X X X X X X X X X X X X X X X
PS x=0 x=1 z
Implication Table (second pass) B C D E F G H A B C D E (^) F G D-F C-H X C-E B-D C-H A-F E-H B-F A-D E-F B-D C-E D-G A-B E-H C-F A-B A-G C-FB-G X X X X X X X X X X X X X X X Self-implied pairs Same state pairs
Determination of State Equivalence using an Implication Table (4) One column (or row) at a time, eliminate implied pairs
Implication Table (third pass) B C D E F G H A B C D E (^) F G D-F C-H X C-E B-D C-H A-F E-H B-F A-D E-F B-D C-E D-G A-B E-H C-F A-B A-G C-F B-G X X X X X X X X X X X X X X X X X X X X X X
PS x=0 x=1 z
Determination of State Equivalence using an Implication Table (5) Next pass, one column (or row) at a time, eliminate implied pairs (6) Continue until pass results in no further elimination of implied pairs
Determination of State Equivalence using an Implication Table Reduced State Table * indicates change from original state table H C G 1 G B H 0 F F B 1 C C* A* 1 B F H 0 A A* C 0 PS x=0 x=1 z NS
Determination of State Equivalence using an Implication Table Row Matching on an Implication Table Mealy Machine outputs Recall 101 sequence detector (direct Mealy conversion) D C,0 B, C A,0 D, B C,0 B, A A,0 B, PS x=0 x= NS,z
Implication Table Same state pairs Eliminate implied pairs Matching rows No implied pairs B and D are “same state” B C D A-C B-B X A-B B-B B-B C-C C X X A B X X √ D C,0 B,
PS x=0 x= NS,z
Moore Reduction Procedure States S (^) i and S (^) j of machine M are said to be equivalent If and only if, for every possible input sequence, the same output sequence will be produced regardless of whether S (^) i or S (^) j is the initial state Zvi Kohavi, Switching and Finite Automata Theory
Moore Reduction Procedure The result sought is a partition of the states of M such that two states are in the same block if and only if they are equivalent P 0 corresponds to 0-distinguishablity (includes all states of machine M) P 1 is obtained simply by inspecting the table and placing those states having the same outputs, under all inputs, in the same block P 1 establishes the sets of states which are 1-equivalent
Moore Reduction Procedure Obtain partition P 2 This step is carried out by splitting blocks of P 1 , whenever their successors are not contained in a common block of P (^1) Iterate process of splitting blocks If for some k, P (^) k+1 = P (^) k, the process terminates and P (^) k defines the sets of equivalent states of the machine P (^) k is thus called the equivalence partition The equivalence partition is unique
Moore Reduction Procedure Recall state table from earlier example H C G 1 G B H 0 F F B 1 E C A 1 D A E 0 C E D 1 B F H 0 A D C 0 PS x=0 x=1 z NS
Moore Reduction Procedure P 0 = (ABCDEFGH) P 1 is obtained by splitting states having different outputs P 1 =(ABDG)(CEFH) Block 1 = ABDG, Block 2 = CEFH H C G 1
PS x=0 x=1 z
Moore Reduction Procedure Split B out of block 1 B is “2 distinguishable” from A, D and G No states of block 2 are “2 distinguishable” P 2 = (ADG)(B)(CEFH) Block 1 = ADG Block 2 = B Block 3 = CEFH
Moore Reduction Procedure Obtain P (^3) P 2 = (ADG)(B)(CEFH) A D (1) C (3) D A (1) E (3) G B (2) H (3) C E (3) D (1) E C (3) A (1) F F (3) B (2) H C (3) G (1)
PS x=0 x=1 z
Moore Reduction Procedure Obtain P 3 (cont) Split G from block 1 G is 3-distinguishable from A and D Split F from block 3 F is 3-distinguishable from C, E and H P 3 = (AD)(G)(B)(CEH)(F) Block 1 = AD, block 2 = G, block 3 = B, block 4 = CEH and block 5 = F
Moore Reduction Procedure Obtain P (^4) P 3 = (AD)(G)(B)(CEH)(F) A D (1) C (4) D A (1) E (4) C E (4) D (1) E C (4) A (1) H C (4) G (2)
PS x=0 x=1 z
Moore Reduction Procedure Obtain P 5 (cont) No blocks split from P (^5) P 5 = P 4 = (AD)(G)(B)(CE)(H)(F) P 5 = P 4 = equivalence partition Same result as implication table
Reduction of Incompletely Specified State Tables Use “modified row matching” to combine states D A - 1 - C A - 0 - B C D - - A - B - - PS x=0 x=1 x=0 x= NS Z A and C can be combined A and D can be combined C and D cannot (outputs differ)
Reduction of Incompletely Specified State Tables Using an Implication Table State pairs are compatible, not equivalent States must be “pairwise” compatible ABC requires A-B, B-C and A-C Compatible relationship is not transitive like equality Compatible pairs must be grouped and included in reduced machine
Reduction of Incompletely Specified State Tables √ indicates “compatible pair” B C D B-D C X A B √ √ A-C A-C A-C and A-D are compatible pairs C-D are not compatible pairs A-B implies B-D; B-D implies A-C → requires ABCD grouping B-C implies A-C; A-B implies B-D → requires ABCD grouping B-D implies A-C → √
