Fredkin gates for finite-valued reversible and conservative logics

The basic principles and results of conservative logic introduced by Fredkin and Toffoli in 1982, on the basis of a seminal paper of Landauer, are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Łukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental connectives. Two no-go theorems are also proved.

[1]  Kozo Kinoshita,et al.  On Magnetic Bubble Logic Circuits , 1976, IEEE Transactions on Computers.

[2]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[3]  Andrew F. Rex,et al.  Maxwell's Demon, Entropy, Information, Computing , 1990 .

[4]  J. von Neumann,et al.  Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components , 1956 .

[5]  C. Chang,et al.  Algebraic analysis of many valued logics , 1958 .

[6]  C. Chang,et al.  A new proof of the completeness of the Łukasiewicz axioms , 1959 .

[7]  Anton Zeilinger,et al.  The physics of quantum information: basic concepts , 2001 .

[8]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[9]  Charles H. Bennett.  Notes on the history of reversible computation , 2000, IBM J. Res. Dev..

[10]  Brian F. Chellas.  Modal Logic: Normal systems of modal logic , 1980 .

[11]  Gianpiero Cattaneo,et al.  Brouwer-Zadeh posets and three-valued Ł ukasiewicz posets , 1989 .

[12]  Takao Hinamoto,et al.  The Polish Academy of Sciences , 1961, Nature.

[13]  Gianpiero Cattaneo,et al.  BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations) , 1999, Fuzzy Sets Syst..

[14]  Robert E. Clay.  A simple proof of functional completeness in many-valued logics based on Łukasiewicz's C and N , 1962, Notre Dame J. Formal Log..

[15]  Robert McNaughton,et al.  A Theorem About Infinite-Valued Sentential Logic , 1951, J. Symb. Log..

[16]  A. R. Turquette,et al.  On the Many-Valued Logics , 1941 .

[17]  Tommaso Toffoli,et al.  Reversible Computing , 1980, ICALP.

[18]  Alasdair Urquhart.  Many-valued Logic , 1986 .

[19]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[20]  R. Feynman.  Quantum mechanical computers , 1986 .