h.-j.mathony ieee proceedings 1989
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Universal logic design algorithm and its application to the synthesis of two-level switching circuits. H.-J.Mathony IEEE Proceedings 1989. Outline. Thelen’s prime implicant algorithm Two-level logic minimisation procedures Complementation Expansion of implicants - PowerPoint PPT PresentationTRANSCRIPT
Universal logic design algorithm and its application to the synthesis
of two-level switching circuits
H.-J.MathonyIEEE Proceedings 1989
Outline
Thelen’s prime implicant algorithmTwo-level logic minimisation procedures
Complementation Expansion of implicants Detection of essential primes Computation of a mnimal cover Reduction of prime implicants
Conclusions
Thelen’s prime implicant algorithm (1)
Problem definition: Given a conjunctive normal form of F
Convert F into the sum of its all prime implicants
Time-consuming and requires large memory capacity if multiplied out straightforwardly:
• Cannot decide whether an implicant is prime or not until all products are computed
)(......)(...( 2121 nm bbbaaaF
Thelen’s prime implicant algorithm (2)
Thelen’s algorithm based on method of Nelson: All prime implicants of a function f are
obtained when an arbitrary conjunctive form F of f, i.e., a product of sums representation, is expanded into a disjunctive form by multiplying out the disjunctions of F and deleting products that subsumes others.
Thelen’s prime implicant algorithm (3)
Method of Nelson: Drop contra-valid clauses
If an occurrence of a literal is repeated within a clause, drop all occurrences and save one
Drop subsuming clauses
0aa
abcabac
abababc
Thelen’s prime implicant algorithm (4)Depth-first-search multiplication
Search tree for
b c
ae
))()(( gfedcbaF
a b c
d
f f
ae
gad
adf adg adf
Thelen’s prime implicant algorithm (5)
Pruning rules R1: An arc is pruned, if its predecessor node
conjunction contains the complement of the arc-literal (corresponds to R1 of Method of Nelson)
R2: A disjunction is discarded, if it contains a literal which appears also in the predecessor node-conjunction (corresponds to R2 of Method of Nelson)
R3: An arc is pruned, if another non-expanded arc on a higher level still exists which has the same arc-literal (corresponds to R3 of Method of Nelson)
Thelen’s prime implicant algorithm (6)
b c
))()()(( cbdcabacbaF
ba
ab c
a
aac
b
cb
cdbdbacba
Linear space complexity
ba=R1
ab
ab
=R1
dca
R1=a
R4=
dcR1=
b
R1=
cR3=
b c
dba
b c
cba
R2 R2 b cac
R2 b c
cdb
R2
ab
a c dR2
a c dac R2
Thelen’s prime implicant algorithm (7)
R1 and R2 are obviousProof of R3
Theorem: suppose arc j and (on a higher level) arc k have the same arc-literal xp, then all implicants, which result from traversing down arc j, will be adsorbed by the implicants computed by traversing down arc k
Thelen’s prime implicant algorithm (8)
)......()( rpij xxxXW :
disjunction related to the level of arc j)......()( spjk xxxXW
disjunction related to the level of arc k
)()](............[
)()()......(
)()()(
XGXWxxxxxxxx
XGXWxxx
XGXWXWF
jsprjpjij
jspj
jk
since pjp xXWx )(
pj xx Corresponds to arc j and is absorbed by xp => arc j can be pruned
Thelen’s prime implicant algorithm (9)
Further pruning rule developed by the author R4: An arc j is pruned, if another already
expanded arc k with the same arc-literal exists on a higher level i and if Rule R3 was not applied in the subtree of arc k with respect to arc p on level i which leads to arc j
Reduction of the search tree up to 25%
Applications of Thelen’s theorem in two-level logic minimisation procedures
ComplementationExpansion of implicantsDetection of essential primesComputation of a minimal coverReduction of prime implicants
Complementation(1)Complementation:
Disjoint sharp operation Complementing by recursive use of the ‘Shannon
expansion’ and the ‘unate paradigm’Sharp operation:
let A=U, the universe:Disjoint sharp operation: with the resultant cubes mutually disjoint
BABA #
BBUBA #BABA #
A B
Complementation(2)
nxxxcw ...)( 21Cube c
nxxxcwcW ...)()( 21
)()...()()(...)()(
21
21
p
p
cWcWcWFcwcwcwF
F
•Thelen’s procedure is related to the non-disjoint sharp operation, i.e., the straight forward multiplication algorithm is in a one-to-one relation to the sharp product
•Want to avoid the computation of all prime cubes of
Complementation(3)R5: Let be an arbitrary disjunction
of F; if there exists a non-expanded arc with literal on a higher level, then only arc of D must be expanded.
R6: Let be an arbitrary disjunction of F; if there exists an expanded arc k with literal on a higher level, and if neither R3 nor R5 was applied in the sub-tree of arc k with respect to arc p on level i which leads to arc j, then only arc of D must be expanded.
Rule R6 is related to rule R5 in the same way as rule R4 is related to rule R3
)......( kji xxxD
ix ix
)......( kji xxxD
ix
ix
Expansion of implicants(1)
Expand a cube ci of the ON-cover C to a prime cube ci
+ so that as many literals in ci are removed as possible
Method: ON cube ci expanded against the given OFF-cover
• Petzold, ‘An algorithm for the minimisation of Boolean functions’, Techn. Report, 1999 (in German)
• Zander and Wagner, ‘A method for the computation of prime implicants for incompletely specified Boolean functions’, Elektron. Inform. Kybern, 1972 (in German)
Expansion of implicants(2) Boolean function AF(ci) in conjunctive form:
prime implicants in a one-to-one relation to all prime cubes ci
+ which cover cube ci: derived by Zander
An algebraic representation of the blocking matrix B:
else
rcx
rcx
x
xcAF
jiii
jiii
i
ji
q
j
,0
)0()1(,
)1()0(,
))()(1
Expansion of implicants(3)Example:
1413142
1413121321
43214432421
11109
8765
4321
321
))(())()()((
)()()()(
)1,1,1,1(),0,1,1,1(),1,0,1,1()1,1,0,1(),0,1,0,1(),1,0,0,1(),0,0,0,1()0,1,1,0(),0,0,1,0(),1,1,0,0(),0,1,0,0(
:
)1,1,1,0(),0,0,1,0(),1,1,1,0(:
xxxxxxxxxxxxxxxxxxxxxxxxxxxcAF
rrrrrrrrrrr
C
cccC
off
on
Expansion of implicants(4)
Guide: choose a leave that covers the largest number of cubes Thelen’s tree pruned by additional rule R5: an
arc is pruned if it cannot lead to a prime cube which covers more cubes than the best prime cube found so far
Detection of essential primes(1)
Miller, R.: ‘Switching theory’, Vol. I: ‘Combinational circuits’, 1965 Given a prime cube ci; if the consensus of ci
with all other on-cubes cj Con and DC-cubes dk Cdc completely covers ci, then ci is not essential, otherwise ci is essential.
Detection of essential primes(2)
00
10 011
0 00 0
0 00
1
0
p: the prime to be examinedR’: OFF cubes that are distance 1 from pp is essential iff there exists minterm m such thatm is completely surrounded by R’ p
•Bahnsen, ‘Essential prime implicant tester’, IBM Technic. Disclosure Bulletin, 1981
Detection of essential primes(3)method:
for each fixed component j of cube c, compute characteristic product terms against each neighbored off cube, OR these product terms to form disjunctive form EDFj
• characteristic product term of an off cube:
– substitute the fixed values of c with the jth fixed value inverted into the off cube
form conjunctive form ECF of all these disjunctive forms EDFj. ECF describes the essential vertices covered by cube c.
c is essential iff ECF has a solution
Detection of essential primes(4)
4324
3213
4322
3211
421
)1,0,1,(
)1,1,,1(
)1,1,0,(
),1,0,1(
)0,,1,(
xxxr
xxxr
xxxr
xxxr
xxc
Example:
r1, r3 and r4 are distance 1. •substitute (c with x2 inverted), or x2= 0, x4= 0, in r1, r3 and r4 => EDF2 = x1x3. •substitute (c with x4 inverted), or x2= 1, x4= 1, in r1, r3 and r4 => EDF4 =
=>c is essential
42xx
42xx
331 xxx 03142 xxEDFEDFECF
Detection of essential primes(5)
Another Thelen’s expansion tree problem ECF is converted into a disjunctive form by the
use of Thelen’s algorithm expansion terminates when the first leaf node is
arrived or if no arc leads to a leaf node
Computation of a minimal cover(1)
Petrick function Petrick, S.: ‘A direct determination of the
irredundant forms of a Boolean function from the set of prime implicants’. Air Force Cambridge Res. Center, 1956
Computation of a minimal cover(2)
disjunction Dj of PF correspond to vertices of Con
which can be covered alternatively by the prime cubes ci represented by the literals vi which form
the disjunction Dj.
prime implicants of PF are in a one-to-one relation to the irredundant sums of the function f:
the minimal cover Cmin corresponds to the shortest prime implicant of PF
)(......)(...( 2121 nm bbbaaaPF
Computation of a minimal cover(3)
Branch and bound:• rule R3 guarantees that the first implicant which is
found is prime
• The first leaf node always represents an irredundant subcover of Con,
• the number of literals of the first prime implicant is an upper bound for the depth of the resulting search tree
Reduction of prime implicants(1)Given a prime cube ci, the maximal reduced cube
equals the supercube of The function
represents all on-vertices which are only covered by cube ci
ic
)(# dconi CCc
dckionj
k
q
jj
p
jii
CdcCc
dWcWcwcRF
},{\
)()()()(11
nxxxcwcW ...)()( 21
Reduction of prime implicants(2)
Another Thelen’s expansion tree problem: apply R1 to R6 R7: form an intermediate supercube with each
cube of a new leaf and terminate the search if this intermediate supercube equals the cube to be reduced -> ci is not reducible
Conclusion
Thelen’s theorem on finding all primes of a conjunctive form function
Universal solution of two-level minisation procedures by applying Thelen’s theorem
complementation expansion of implicants detection of essential primes computation of a minimal cover reduction of prime implicants