1. Universal logic design algorithm and its application to the synthesis of two-level switching circuits §H.-J.Mathony §IEEE Proceedings 1989

2. Outline §Thelen’s prime implicant algorithm §Two-level logic minimisation procedures l Complementation l Expansion of implicants l Detection of essential primes l Computation of a mnimal cover l Reduction of prime implicants §Conclusions

3. Thelen’s prime implicant algorithm (1) §Problem definition: l Given a conjunctive normal form of F l Convert F into the sum of its all prime implicants l 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

4. Thelen’s prime implicant algorithm (2) §Thelen’s algorithm based on method of Nelson: l 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.

5. Thelen’s prime implicant algorithm (3) §Method of Nelson: l Drop contra-valid clauses l If an occurrence of a literal is repeated within a clause, drop all occurrences and save one l Drop subsuming clauses

6. Thelen’s prime implicant algorithm (4) §Depth-first-search multiplication l Search tree for bc ae bc d ff e g ad adfadgadf

7. Thelen’s prime implicant algorithm (5) §Pruning rules l R1: An arc is pruned, if its predecessor node conjunction contains the complement of the arc-literal (corresponds to R1 of Method of Nelson) l 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) l 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)

8. Thelen’s prime implicant algorithm (6) bc b c a Linear space complexity a =R1 a d a R1= a R4= d R1= c R3= cc R2 c c a d ad

9. Thelen’s prime implicant algorithm (7) §R1 and R2 are obvious §Proof of R3 l Theorem: suppose arc j and (on a higher level) arc k have the same arc-literal x p, then all implicants, which result from traversing down arc j, will be adsorbed by the implicants computed by traversing down arc k

10. Thelen’s prime implicant algorithm (8) : disjunction related to the level of arc j disjunction related to the level of arc k since Corresponds to arc j and is absorbed by x p => arc j can be pruned

11. Thelen’s prime implicant algorithm (9) §Further pruning rule developed by the author l 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 l Reduction of the search tree up to 25%

12. Applications of Thelen’s theorem in two-level logic minimisation procedures §Complementation §Expansion of implicants §Detection of essential primes §Computation of a minimal cover §Reduction of prime implicants

13. Complementation(1) §Complementation: l Disjoint sharp operation l 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 A B

14. Complementation(2) Cube c 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

15. 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

16. Expansion of implicants(1) §Expand a cube c i of the ON-cover C to a prime cube c i + so that as many literals in c i are removed as possible §Method: l ON cube c i 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)

17. Expansion of implicants(2) §Boolean function AF(c i ) in conjunctive form: l prime implicants in a one-to-one relation to all prime cubes c i + which cover cube c i: derived by Zander l An algebraic representation of the blocking matrix B:

18. Expansion of implicants(3) §Example:

19. Expansion of implicants(4) §Guide: choose a leave that covers the largest number of cubes l 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

20. Detection of essential primes(1) §Miller, R.: ‘Switching theory’, Vol. I: ‘Combinational circuits’, 1965 l Given a prime cube c i ; if the consensus of c i with all other on-cubes c j C on and DC-cubes d k C dc completely covers c i, then c i is not essential, otherwise c i is essential.

21. Detection of essential primes(2) 0 0 1 00 1 1 00 00 00 0 1 0 p: the prime to be examined R’: OFF cubes that are distance 1 from p p is essential iff there exists minterm m such that m is completely surrounded by R’ Bahnsen, ‘Essential prime implicant tester’, IBM Technic. Disclosure Bulletin, 1981

22. Detection of essential primes(3) §method: l 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 EDF j characteristic product term of an off cube: –substitute the fixed values of c with the j th fixed value inverted into the off cube l form conjunctive form ECF of all these disjunctive forms EDF j. ECF describes the essential vertices covered by cube c. l c is essential iff ECF has a solution

23. Detection of essential primes(4) Example: r 1, r 3 and r 4 are distance 1. substitute (c with x 2 inverted), or x 2 = 0, x 4 = 0, in r 1, r 3 and r 4 => EDF 2 = x 1 x 3. substitute (c with x 4 inverted), or x 2 = 1, x 4 = 1, in r 1, r 3 and r 4 => EDF 4 = =>c is essential

24. Detection of essential primes(5) §Another Thelen’s expansion tree problem l ECF is converted into a disjunctive form by the use of Thelen’s algorithm l expansion terminates when the first leaf node is arrived or if no arc leads to a leaf node

25. Computation of a minimal cover(1) §Petrick function l 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

26. Computation of a minimal cover(2) §disjunction D j of PF correspond to vertices of C on which can be covered alternatively by the prime cubes c i represented by the literals v i which form the disjunction D j. §prime implicants of PF are in a one-to-one relation to the irredundant sums of the function f: l the minimal cover Cmin corresponds to the shortest prime implicant of PF

27. Computation of a minimal cover(3) l 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 C on, the number of literals of the first prime implicant is an upper bound for the depth of the resulting search tree

28. Reduction of prime implicants(1) §Given a prime cube c i, the maximal reduced cube equals the supercube of §The function §represents all on-vertices which are only covered by cube c i

29. Reduction of prime implicants(2) §Another Thelen’s expansion tree problem: l apply R1 to R6 l 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 -> c i is not reducible

30. 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 l complementation l expansion of implicants l detection of essential primes l computation of a minimal cover l reduction of prime implicants