1. Regular Structures

2. Levelized Structures

3. Standard Lattice Diagrams for continuous, multiple-valued and binary logic

4. Patented by Pierzchala and Perkowski 1994/1999 Lattice Structure for Multivalued and Binary Logic Realizes every binary symmetric function Realizes every non- symmetric function by repeating variables Realizes piece-wise linear multivalued functions

5. Lattice Structure for Multivalued and Binary Logic Cell has three inputs and two outputs Both outputs have the same function 01 red nose beard red eyes PerkowskiJeske Zakrevskij Al-Rabadi Multivalued variables Binary input variables Multi-valued output variable

6. Lattice Structure for Multivalued and Binary Logic 01 Redness of nose in interval [3,4] Length of beard an odd number Redness of eyes in intervals [2,4] or [7,9] PerkowskiJeske Zakrevskij Al-Rabadi Multivalued variables Multi-valued output variable binary

7. Lattice Structure for Multivalued and Binary Logic 01 A>B C<D E=G PerkowskiJeske Zakrevskij Al-Rabadi Multivalued input variables Multi-valued output variable binary Multivalued input variables

8. Lattice Structure for Multivalued and Binary Logic 01 A>B C<D E=G PerkowskiJeske Zakrevskij Al-Rabadi Multivalued input variables Multi-valued output variable Multivalued input variables A BA B C>0 E 0 E<G and G<0 C DC D E GE G Cell has 4 inputs and 2 outputs Can we make the cell reversible?

9. Control left right output 0 value - value 1 - value value Control left right 0 0 1 1 0 1 1 0 Control right left We want to make this cell reversible output Values not separated

10. Control left right 00 00 10 10 01 11 11 01 Controlright left Let us try to repeat control variable in output output output1 Still not separated

11. Control left right 000 001 101 100 010 110 111 010 Controlright left Repeating variables will not help output output1 Now it works! output2

12. This means that we added another MUX left Control right output output1 Control right 000 001 101 100 010 110 111 010 left output output1 output2

13. …. And we reinvented the Fredkin Gate ….!!! But how to use it in a Lattice?

14. Lattice Structure for Binary Logic 01 A B C 01 0 1 F = S 1,3 (A,B,C) S0S0 S1S1 S2S2 S3S3

15. 0 1 2 A PS RQ B CD A P Q R S D C B D C B Q S R A P (a) (b) (c)

16. x1x1 x2x2 x3x3 1 243 8 7 65

17. D R S Q B C A P x1x1 x2x2 x3x3 x1x1 x2x2 x3x3 1 2 3 4 7 8 6 5

18. Notation for Fredkin Gates 01 A 0101 CB PQR A P BC QR (a) (b)

19. Three Types of General Expansions 01 A f f0f0 f1f1 f and A f 0 and f 1 g,h and A g 1 A+h 0 A’ 10 1 0 g1g1 h A g 1 A+h 0 A’ A g h 10 g 0 A’+h 1 A g, h, and A g 0 A’+h 1 A and g 1 A+h 0 A’ Forward Shannon Reverse Shannon Reversible Shannon (a) (b) (c) hoho g

20. * + * + * + * + …... cici f1f1 f2f2 f3f3 f4f4 k1k1 k2k2 k3k3 k4k4 k5k5 k6k6 Previous levels next levels Other same level

21. * + * + * + * + …... f1f1 f2f2 f3f3 k1k1 k2k2 k3k3 k4k4 k5k5 k6k6 Previous levels next levels Other same level f4f4 cici

23. 01 X Y Z 101 f garbage 11 1 1 00 0 1 - - 1010 0 1 1 1 1001 1 0 g garbage fggf hi hfgfgh YZ X X X X 00 01 11 10 0101

24. 1 0 0 1 1 1 - - 0 1 1 1 - - 1 0 - - X Y Z - - 1 1 - - - - 1 0 - - 1 - - - - 1 - - - 0 - - 0 - - - =1 1 0 - - - - - - - - 0 1 - - =0=1=0 - - 1 - - - - - - 0 - - X YZ

25. Reversible Lattice Structure for Binary Logic 01 A B C 01 0 1 F = S 1,3 (A,B,C) S0S0 S1S1 S2S2 S3S3 G waste F waste

26. Two-Dimensional Lattice Diagrams for reversible logic

27. Three Types of General Expansions f 01 A f0f0 f1f1 f and A f 0 and f 1 Forward Shannon

28. Three Types of General Expansions g,h and A g 1 A+h 0 A’ 10 1 0 g1g1 h A g 1 A+h 0 A’ Reverse Shannon (b) hoho g

29. Three Types of General Expansions g 1 A+h 0 A’ A g h 10 g 0 A’+h 1 A g, h, and A g 0 A’+h 1 A and g 1 A+h 0 A’ Reversible Shannon

30. 0 0 1 1 a + + a c b x y b c  ab c   ab a a Third stage of decompositio n: Feynman gate Second stage of decomposition: Reversible Expansion for Fredkin gate First stage of decomposition: Feynman gate Realization of Toffoli Gate from Fredkin and Feynman Gates

31. 0 0 1 1 a + + a c b x y b c  ab c   ab a a First stage of composition: Feynman gate Second stage of composition: Reversible Expansion for Fredkin gate Third stage of composition: Feynman gate Realization of Toffoli Gate from Fredkin and Feynman Gates

32. 01 1010 - - 0 1 1 1 1001 f garbage g 1010 Z fg X garbage gf h i Y 1 garbage hfgfgh YZ X - - X YZ X 11 1 1 00 0 1 X 00 01 11 10 0101

33. cofactor permuter To distinguish this new general decomposition from the well- known decompositions of Ashenhurst, Curtis or Shannon, we call it the Multi-purpose Portland Decomposition, the MP-decomposition for short.

34. Generalization We mapped the logic function to a lattice structure of geometrical connections there is nothing in our method to map to only this kind of structure we can map to any selected regular structure we can also map to a irregular structure with arbitrary connections

35. Generalizations of Fredkin gate Observe, that this definition of the gate does not specify the type of signals. Thus they can be binary, multi-valued, fuzzy, continuous or complex. The only requirement is that the relation of order (<) is defined on them It is interesting and important that a single reversible gate in binary logic has many generalizations in multiple-valued logic.

36. Generalizations of Fredkin gate Because it has been shown in [1] that there are many multiple-valued and multi-output (k>3) generalizations of Fredkin gate, the name “modified” assigned by Picton is not correct. The generalization invented by him we will call the Picton Gate, while generalization of Fredkin- like gates we call “new gates”.

37. Generalizations of Fredkin gate The exhaustive list of families of all such permutative multi-valued gates (both Shannon- like and Davio-like) has been presented in [1] and even more families in [18]. These of the “new gates” that use multiplexers only are similar to the original Fredkin gate but they use multiple-valued multiplexers. Such multiplexers have been already realized in many technologies, including super-pass transistors [9], so building these new gates should be also possible.

38. Generalizations of Fredkin gate We believe therefore that they are good candidates for future reversible multiple-valued nano-technologies. The new generalization of Fredkin gate using multi-valued logic has additional advantages and is simpler. Let us observe, that equations for the binary 4 * 4 binary Fredkin gate can be rewritten as follows: P = A, Q = if A=1 then C else if A=0 then B, R = if A=1 then B else if A=0 then D, S = if A=1 then D else if A=0 then C Now, it can be easily generalized to a 4 * 4 ternary gate as follows: P = A, Q = if A=2 then B else if A=1 then C else if A=0 then D, R = if A=2 then C else if A=1 then D else if A=0 then B, S = if A=2 then D else if A=1 then B else if A=0 then C

39. Reversible Lattice Structure for Binary Logic Advantages regular structure binary Fredkin Gate planar structure (good for Quantum Logic) Easy algorithmic creation Reasonable waste Disadvantages Variable ordering? Symmetrization? Waste still exist Should be patented!

40. Do you remember that there are other binary expansions? All Binary Expansions Shannon - S Flipped Shannon - fS Positive Davio - pD Negative Davio - nD Flipped Positive Davio - fpD Flipped Negative Davio - fnD Ideas Fredkin = what about these? …. I checked some of them to work

41. Do you remember that there are other component functions of reversible gates All Binary Balanced Expansions: ….. Linear functions - L Negations - N Majorities - M Ideas Fredkin = what about these? …. I checked some of them to work

42. As you see, this opens a very broad area of research that will lead to invention of new reversible gates and regular structures that use them Easy way to become a pioneer: Investigate all combinations Use genetic programming or other search methods to build structures and map functions to them There is a place for many researchers Nobody does this research But this was only for binary What about multivalued, fuzzy, arithmetic or other logics?

43. …. And we reinvented the Fredkin Gate ….!!! But what about the variant with two control signals?

44. Multi-valued Fredkin Gate MVFG is described by equations: P = A Q = B R = C if A < B else R = D S = D if A < B else S = C ABCDABCD PQ RSPQ RS >= ABCDABCD PQ RSPQ RS < A < B

45. Lattice Structure for Multivalued and Binary Logic 01 PerkowskiJeske Zakrevskij Al-Rabadi Multivalued input variables Multi-valued output variable Multivalued input variables A BA B C DC D E GE G Cell has 4 inputs and 4 outputs Cell is reversible! MV and Generalized MV Fredkin waste

46. Multi-valued logic generates less signals Hence it generates less waste Of course, it generates also less power, less connections and is easier to test

47. The real-life functions are multi-output. Thus, there exists an opportunity to re-use some waste functions in other output functions This is a tough problem. I do not know now how to solve it! The main open research problem We need some group creativity

48. Generalized Multi-valued Fredkin Gate ABCDABCD PQ RSPQ RS < A < B Select other pairs of MUX-type functions Select other pairs of VAR-type and NOT- type functions Select other function of two variables

49. Generalized Multi-valued Fredkin Gate The number of these gates is astronomical We need both computer generation and some intelligence, simply generating them all would be a nonsense Very wide area of research It will give hints to gate designers what to look for

51. Let us go back to our fundamental invention….. What if we resign from oblique buses?

52. Buses are removed and each cell is programmed individually….. Some regularity is lost!

53. D’ C B A 3 3*A*B*C*D’ X Y Z V 2 2*Z*V B B 1 A 1 * A’ * B’

54. The general levelized method can assume any structure of the layout, thus any order and choice of input signals of successive Reversible Shannon expansions. Assuming other type of structure, cascade or non-planar lattice with intersecting signals, this other type of structure would be created. For arbitrary structures, however, the method requires small modification: if the structure is too constrained, the structural equations have no solutions or the algorithm loops.

55. This happens, for instance, when a Maitra Cascade structure is assumed for a function that is not Maitra- realizable. It happens also when we assume a levelized circuit of too narrow a bandwidth Thus the algorithm must be modified to deal with these special cases. Finally, our general approach will work also for irregular structures. In such case, any pair of signals can be the inputs to the Reversible Shannon Expansion, regardless of their order. The signals are paired to give the smallest evaluated total complexity for the level.

56. Arbitrary symmetric function can be realized in a lattice without repeated variables. Arbitrary (non-symmetric) function can be realized in a lattice with repeated variables (so-called symmetrization). Similar property exist for the presented method. This method terminates for arbitrary function, assuming that the variables are repeated in levels. Thus, if the leafs of the lattice are not constants after expanding for all input variables, some of these variables are used again in new levels of expansions, which we call variable repetition. Interestingly, the functions that do not require variable repetition in the Reversible Shannon Lattices are not symmetric functions. We work on the characterization of the functions realizable in these structures without repetitions and respective synthesis algorithms.

57. We can impose during joining the structure of the three dimensional lattice. Such lattice is typical for some crystals. There are also several other three-dimensional structures corresponding to other types of bonds or constraints that exist in Nature (for example, quantum dot computers). This leads to very many new circuit types, which are reversible and multi-valued generalizations of Shannon Lattices, Kronecker Lattices, Fat Trees, and many other structures introduced in the past.

58. Future Work Several realizations of reversible and quantum logic, such as for instance quantum dots, involve a geometrical space. –For instance, in the quantum dot model this space is two- dimensional. Here we propose to create three-dimensional regular structures, because our physical world is three dimensional. Layout-driven synthesis We plan to design these structures in CMOS and Optical technologies. Software