rumer's transformation, in biology, as the negation, in classical logic

Rumer’s Transformation, in Biology, asthe Negation, in Classical Logic

TIDJANI NEGADIDepartement de Physique, Faculte des Sciences, Universite d’Oran, Es-Senia, 31100, Oran, Algerie

Received 7 October 2002; accepted 13 March 2003

DOI 10.1002/qua.10602

ABSTRACT: In this article, we make a connection between the Rumertransformation, used in the study of the genetic code-doublets, and the negation ofclassic logic. A unified classification is given, relying on two Klein’s 4-groups describingthe symmetries of the 16 doublets of nitrogenous bases and those of the 16 binaryconnectives of classic logic, both groups being subgroups of a larger noncommutativegroup with eight elements we identify as the dihedral group D4. Also, someconnections with other works are briefly considered. © 2003 Wiley Periodicals, Inc. Int JQuantum Chem 94: 65–74, 2003

Key words: biology base-doublets; logic binary connectives; Rumer symmetry;negation; dihedral group

1. Introduction

F inding a connection between logic and biol-ogy is indeed an interesting question. Cooper,

in his recent book [1], brings forward the boldhypothesis that logic is reductible to evolutionarybiology. More precisely, it states that “the laws oflogic, or at least of classical logic and certain gen-eralizations of it, are reducible to evolutionary bi-ology in a standard sense: The terms of the logicaltheory are definable in evolutionary terms and log-ical assertions are deducible from evolutionary as-sertions.” In a pioneering endeavor, Fontana andBuss [2] devised a project to use �-calculus (a theoryof proofs in logic) in the study of the biologic orga-nization, where one imagines that “the synthesis ofa molecule is like proving a theorem by using lem-

mas (other molecules) according to the rules of asymbolic logic yet to be formalized.” As a matter offact, what is needed is an analog (if any) of theCurry–Howard isomorphism, which connects logicand computer science, to link logic with biology.Leaving this particular point for the future, onefinds that, today, Boolean logic is already inti-mately connected to some research programs onDNA computing. The Warwick group (www.bio.warwick.ac.uk/hodgson/dnacomp/boolean/genetic.html), for example, describes the transcrip-tional factors in gene expression (like the lacoperon) or the metabolic pathways in enzyme in-teractions as Boolean circuits (see Section 4).

In this work, we shall be more modest in ourgoal but nevertheless adhere to these guiding ideasand report on a connection we found between theset of the 16 genetic code-doublets and the set of the

International Journal of Quantum Chemistry, Vol 94, 65–74 (2003)© 2003 Wiley Periodicals, Inc.

16 binary connectives of the classic propositionalcalculus, at least for the classification-and-symme-try aspect. We present a parallel classification forthe two sets and extend it to symmetry consider-ations. In Section 2, we summarize recent results onthe matrix classification of the 16 doublets and theirsymmetries, together with an important clarifica-tion concerning the use of two types of matrixproducts (see below and the Appendix). We definethree transformations, among them the Rumertransformation, describing their symmetries andconstituting a Klein’s 4-group, KB, after adding theidentity transformation. Next, in Section 3, we con-sider the 16 binary connectives of the (two-valued)classic logic and show explicitly how to link themwith the 16 doublets of Section 2 by resorting toRosen’s “size index” of the four nitrogenous bases,U, C, A, and G, expressing their relative atomiccomposition. The corresponding symmetries (dual-ities) are described, in this case, by a second Klein’s4-group, KL, sharing with KB the Rumer transforma-tion (R1) which is, now, associated to the negation(or the NOT operator). Section 4 is devoted to theunification of KB and KL into a larger group that, weshall show, is the dihedral group D4 and to theconnection with other works. Finally, in an impor-tant appendix, we introduce for classification pur-pose a concatenation matrix product, traced on thewell-known Kronecker product, which happens tobe useful in a two-fold way: (1) It embodies the(intrinsic) noncommutativity of the bases and (2) it

helps us to make a consistent construction of 16Boolean functions’s matrix in Section 3.

2. Symmetries of the 16 GeneticCode-Doublets

In Ref. [3], we started from the constituents of thefour nitrogenous bases, that is, the hydrogen, car-bon, oxygen, and nitrogen atoms, written as 2 � 2matrices [see Eq. (7)], and were led, after construct-ing the molecules U, C, A, and G using ordinarymatrix multiplication, to the following base matrix:

� :� �U CA G�. (1)

In this article, we improve the formalism by using anew concatenated “Kronecker”-like product, de-fined in the Appendix, to take into account thenoncommutativity of the bases (e.g., UA is not thesame as AU). Applying this product to �, we have

�1 � � � � � �UU UC CU CCUA UG CA CGAU AC GU GCAA AG GA GG

� . (2)

�1 is a C-type matrix, grouping the 16 possibledoublets, which are believed to be all different fromeach other.* Only to visualize concretely things, wegive its numeric form

�40642564064256 40642563556224 35562244064256 35562243556224

4064256130691232 40642561045529856 355622130691232 355622410455298561306912324064256 130691232355622 10455298564064256 10455298563556224

130691232130691232 1306912321045529856 1045529856130691232 10455298561045529856� ,

which shows clearly that this matrix product war-rants that the 16 objects are all different. One couldsee immediately the regularities in the matrix �1:Doublets sharing the same first base form four com-pact clusters of four members each (the four quad-rants, or quartets). Note these quartets “U1,” “C1,”“A1,” and “G1.” One could also consider the other(interesting) form

�2 � X1��1��X1��1 � �

UU CU UC CCAU GU AC GCUA CA UG CGAA GA AG GG

� ,

(3)

where X1 is a permutation matrix [4] [see Eq. (13)]so that, now, it is the doublets sharing the samesecond base that form compact clusters noted “U2,”“C2,” “A2,” and “G2.” The relation between �1 and�2 is also reflected at the level of the definition ofthe Kronecker product (see the Appendix): To gofrom �1 to �2 (or vice versa), it suffices to make theexchanges j 7 s, k 7 t. This representation dou-bling is interesting because it is well suited to fit thetwo known trends in the classification of thecodons: Second-position bases connect amino acids

*Using the same techniques, one could construct and classifythe codons, for example, in the product �1 � � or the L-oligo-nucleotides, in the product � � � � . . . � � (L times).

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that have similar properties while first-base positionbases connect amino acids from the same biosyn-thetic pathway [5]. In [4], we considered the sym-metries and introduced the following three trans-formations (permutation matrices):

R1:�� 0 0 0 10 0 1 00 1 0 01 0 0 0

�,

R2:� � � � � �0 1 0 01 0 0 00 0 0 10 0 1 0

�,

R3:� R1R2 � �0 0 1 00 0 0 11 0 0 00 1 0 0

�; �:� �0 11 0�,

� � �� 1, (4)

where R and Q are, respectively, the (ordinary)direct product and direct sum of matrices. Onecould immediately verify that the set {I4�4, R1, R2,R3} closes under matrix multiplication and consti-tutes a commutative group with four elementsknown as a Klein’s 4-group. This group, denotedKB, is different from (and also less formal than) theone in [6] but they share the same important trans-formation, which is the Rumer transformation [7],called � in [6] and R1 in [4]. The action of R1, R2, andR3 on �2 [Eq. (3)], for example, is the following:

R1�2�R1��1 � �

GG AG GA AACG UG CA UAGC AC GU AUCC UC CU UU

� , (5.1)

R2�2�R2��1 � �

GU AU GC ACCU UU CC UCGA AA GG AGCA UA CG UG

� , (5.2)

R3�2�R3��1 � �

UG CG UA CAAG GG AA GAUC CC UU CUAC GC AU GU

� . (5.3)

First, consider R1. Its action reveals the existence oftwo sets: M1, the elements of which are underlined,and M2, for the rest [see Eq. (3)]. These two sets areexchanged by R1 and this transformation exchange

M1 and M2 also in �1, as could be easily verified inEq. (2). It is well known that M1 contains doubletsfor which the third base in the triplet (codon) isirrelevant (for coding an amino acid) while in M2

the knowledge of the third base is necessary (ex-cept, of course, for the singlets). Comparison of (3)and (5.1) shows that the transformation R1 effec-tively implements the Rumer symmetry (CUGA 7AGUC, see [3]). It is transparent from Eq. (5.1) thatto each doublet B1B2 corresponds a dual doublet,B1B2, symmetrical with respect to the middle point;for example, CU7 AG. There is a second manner tolook at the set of 16 doublets, which has been con-sidered by Jimenez Montano and his collaborators[8]. When one considers the 64 codons (symbolizedby 3�-�1�2�3-5�) of the genetic table, from the�2�3 perspective, that is, the second and the thirdbases of a codon, this then leads to the existence oftwo other sets M�1, which contains doublets endingin a strong base S (C/G), and M�2, which containsdoublets ending in a weak base W (U/A). M�1 cor-responds to the third and fourth columns while M�2corresponds to the first and second columns in thematrix �2. Note that in �1 Eq. (2) columns two andthree are exchanged. (It is evident that �1 and �2

could be used either for the �1�2 perspective or the�2�3 perspective; it suffices to rename the bases.)The above partitioning is also interesting because itleads to the more symmetrical mitochondrial code,as has been shown in [8], and involves also a Klein’s4-group (see [8] and [4]). To put things simply, thisseparation into M�1 and M�2 corresponds to the sep-aration of the “eukaryotes” from the “prokaryotes,”as the former use frequently codons with �2�3 inM�1 and the latter use frequently codons with �2�3

in M�2. Returning to R1, we have the result that itperforms a three-fold exchange at the same time:M1 7 M2, M�1 7 M�2, and “U2” 7 “G2,” “C2” 7“A2,” with an overall Rumer-type exchange, U7 Gand C7 A, for the two bases in all the doublets. Forthe remaining two transformations, the results areas follows [the underlining in (5.2) and (5.3) isdropped as, M1 and M2 break out]: R2 leaves “U2,”“C2,” “A2,” and “G2” as well as M�1 and M�2 globallyinvariant, with an overall Rumer-type exchange, U7 G and C7 A, for the first base in all the doublets.R3 exchanges at the same time “U2” and “G2” and“A2” and “C2” on the one hand, and M�1 7 M�2 onthe other, with an overall Rumer-type exchange, U7 G and C 7 A for the second base in all thedoublets. Similar results hold for �1.

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 67

3. Symmetries of the 16 BinaryConnectives

In classic (two-valued or Boolean) logic, if aconnective connects two propositions, or vari-ables P and Q, and each one of these can be true(T) or false (F), there are four combinations oftruth values for each connective: TT, FT, TF, andFF. For each of these, a given logical connectiveshould yield independently either true or false sothere are exactly 24 � 16 binary functions, orbinary connectives as we call them here, whichare explicitly constructed in Eqs. (12). To see theconnection with the biologic objects of Section 2,let us, first, recall the four nitrogenous bases inmatrix form [3]:

U � �U 00 0� , C � �0 C

0 0� ,

A � �0 A0 0 � , G � �G 0

0 0� , (6)

with

U � �CC�2N2H4OO, C � �CC�2N3H5O,

A � �CC�2CN5H5, G � �CC�2CN5H5O. (7)

The capital letters representing the atoms are them-selves given by 2 � 2 matrices (see [3]), and the basematrix � [Eq. (1)] is constructed as U � C � A � G.The dual form X � �X��1 [� is defined in Eq. (4)]for a matrix X is necessary, for example, to ensurethat some matrix products do not vanish: CC � 0while CC � 0, as examples. Now, the connectionwith the bases could be seen as follows. Let usinvoke the atomic content of the four bases andtheir Rosen’s “size index” [9], in an extended form,[3], to include thymine [T: (CC)2CN2H6OO]:

n:� 2�nN � nC� � nO � 2��nH, 6� � 2. (8)

This number captures the monotonic size gradationof the bases and is a consequence of their relativeatomic content [see Eq. (7)], where the powers givealso the relative number of each atom type. Wehave

n�U/T� � 0, n�C� � 1, n� A� � 2, n�G� � 3,

(9)

which is the same as

n�U/T� � 00, n�C� � 01, n� A� � 10,

n�G� � 11, (9.1)

in two-bit binary notation. It is well known that thetwo Boolean constants F and T are associated withthe symbols 0 and 1, respectively (which is theassignment made by G. Boole in 1854; the oppositechoice is used, sometimes, but much less frequent-ly), so that the four combinations FF, FT, TF, and TTare represented in binary notation by

FF � 00, FT � 01, TF � 10, TT � 11. (10)

Comparing (9.1) and (10), we have the following(formal) correspondence: U 7 FF, C 7 FT, A 7TF, and G 7 TT. In Ref. [8], the authors also useda two-bit binary representation of the baseswhere the first bit corresponds to the chemicaltype [purine (R):0/pyrimidine (Y):1] and the sec-ond to the H-bonding character [weak (W):0/strong (S):1]: U:10, C:11, A:00, G:01. Our choicein Eq. (9.1) differs from the one in [8] only in thefirst bit. Our choice is natural in a two-fold sense:(1) It is prescribed by the atomic composition ofthe bases and (2) is in harmony with the (increas-ing) second-bit characterization because the pyri-midines have one ring and the purines have tworings. It is also in agreement with other authors’assignments in connection with genome studies(see [10]). At this point it is interesting to notethat it is indeed possible to construct a matrixhaving, as matrix elements, the above numbers 0,1, 2, and 3. It suffices to add a parameter, a, in thefirst matrix element in the atomic matrices (seethe appendix of [3]): �11

(i, j) � a[Zi � 2( j � 1)]. Inthis way, a � 1 corresponds to the ordinary caseof the atomic matrices with their atomic numberswhile the case a � 0 (and i � 1) leads to thefollowing simple matrix, written also in binarynotation:

�0 12 3� or �00 01

10 11� � 1. (11)

This matrix could represent either the symbols in(9) and (9.1), codifying the size indices of the bases,or those in (10), codifying the building blocks oflogic. Taking the second form in (11), as a starting

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point, and using our modified “Kronecker” prod-uct, we have†‡

2 � �00 0110 11� � �00 01

10 11�

� �0000 0001 0100 01010010 0011 0110 01111000 1001 1100 11011010 1011 1110 1111

� (12)

or

2 � �FF FTTF TT� � �FF FT

TF TT�

� �FFFF FFFT FTFF FTFTFFTF FFTT FTTF FTTTTFFF TFFT TTFF TTFTTFTF TFTT TTTF TTTT

� . (12.1)

These are the 16 binary connectives. Translatingthese into words, we used the usual connectives not(�), or (�), and (�), if . . . then (3, conditionalimplication), iff or 7 (biconditional implication),and xor [Q, exclusive or; please do not confuse withthe same sign used in Eq. (4) for the direct sum ofmatrices], and rewrote 2 as Table I. The correspon-dences could be obtained either directly by usingtruth tables for the connectives in Table I or resort-ing to known useful equivalences like P 3 Q NnotP or QN not(P and notQ). Take an example thatwill also fix our writing conventions. The truthtable for and is

P Q P and QT T TT F FF T FF F F

and the tables for the other connectives could beeasily found. In fact, the knowledge of only half thetable is necessary as the other half is obtained by(negation) symmetry, as we shall see now. Analo-gously to what we have done in Section 2 for the 16doublets, let us introduce the following three trans-formations:

S1 � R1, S2 � X1 � �1 0 0 00 0 1 00 1 0 00 0 0 1

� ,

S3 � R1X1 � �0 0 0 10 1 0 00 0 1 01 0 0 0

� , (13)

where we have drawn in the (biology) store ofSection 2, R1, Eq. (4) and X1. From (13), it is straight-forward to show that the set {I4�4, S1, S2, S3} closesunder matrix multiplication and constitutes an-other Klein’s 4-group, KL. (Note that KB in Section 2and KL share two elements: the identity I4�4 and theRumer transformation R1.) For the ease of reading,we represent the action of these transformations, on2, in Table II.

Let us now see what we learn from Table II. First,consider the action of the simplest transformation,S2. In this case, comparing (T) with (TS1

), 4 binaryconnectives are not transformed (the ones that areat the corners) and the 12 others are transformedinto each other, in pairs, by negating only the first

†Note that the concatenation process works also for the literalsymbols. For example, the concatenation of FF and FT is justFFFT.

‡The matrices (12) owe their (consistent) existence thanks tothe concatenation product defined in the Appendix.

TABLE I ______________________________________________________________________________________________

Contradiction notP and notQ P and notQ notQFFFF FFFT FTFF FTFT

not P and Q notP P xor Q notP or notQFFTF FFTT FTTF FTTT

(T)P and Q P 7 Q P if Q then P

TFFF TFFT TTFF TTFT

Q if P then Q P or Q TautologyTFTF TFTT TTTF TTTT



atomic proposition P, for example, P and Q7S2 notP

and Q, using not(notP) � P, the double negation.All the other cases could be derived by using theabove-mentioned equivalence or using the truthtables for xor and the biconditional implication, towhich we shall return below. Note also that in thislatter case we have symbolically XYUV7

S2 UVXYfor any two binary connectives in terms of Ts andFs. Now, we examine the action of the most impor-tant transformation, S1 (or R1 in Section 2). As doesR1 in the case of the doublets, its action on 2partitions the set of 16 binary connectives into 2subsets of 8 elements each, exactly like M1 and M2for the doublets [see Eqs. (2) and (12)]. The mostsimple, and direct, way to see the dualities, inducedby S1, is by using the FT notation (01 notation): Thesymmetrical (or dual) of a binary connective is ob-tained by exchanging the roles of F and T (0 and 1).For example, TFTT 7 FTFF (1011 7 0100). As a

good pedagogical exercise, one could also use the“word” notation. In this case, all the “components”of a compound proposition like not P and Q, forexample, must be negated using not (and) � or andnot (or) � and. One has, for example, the De Mor-gan laws

not �P and Q� � notP or notQ,

not �P or Q� � notP and notQ. (14)

or else the identity we have already met:

not �P and notQ� � if P then QN notP or Q.

(15)

This is easily seen for almost all the compoundpropositions in table TS1

, but for the dual P xor Qand P 7 Q a little work is necessary. Take, for

TABLE II ______________________________________________________________________________________________

TS1� S12(S1) � 1

Tautology P or Q if P then Q QTTTT TTTF TFTT TFTF

if Q then P P P 7 Q P and QTTFT TTFF TFFT TFFF

not P or notQ P xor Q notP not P and QFTTT FTTF FFTT FFTF

notQ P and not Q notP and notQ ContradictionFTFT FTFF FFFT FFFF

TS2� S22(S2) � 1

Contradiction P and notQ notP and notQ notQFFFF FTFF FFFT FTFT

P and Q P P 7 Q if Q then PTFFF TTFF TFFT TTFT

notP and Q P xor Q notP notP or notQFFTF FTTF FFTT FTTT

Q P or Q if P then Q TautologyTFTF TTTF TFTT TTTT

TS3� S32(S3) � 1

Tautology if P then Q P or Q QTTTT TFTT TTTF TFTF

notP or notQ notP P xor Q notP and QFTTT FFTT FTTF FFTF

if Q then P P 7 Q P P and QTTFT TFFT TTFF TFFF

notQ notP and notQ P and notQ ContradictionFTFT FFFT FTFF FFFF

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example, the latter. Using well-known equiva-lences for xor, conditional implication, and in-voking Eq. (15), one has P 7 Q N (P or Q) 3 (Pand Q) N not ((P or Q) and not (P and Q)),which is nothing but not (P xorQ). Summing up,we have the result that the Rumer transformationR1, in biology, is “isomorphic” to the negation S1in classic two-valued logic. The third and lasttransformation, S3, corresponds to the joint actionof S1 and S2, as we have S3 � S1S2 � S2S1: A firstnegation of the first (atomic) proposition P isfollowed by a negation of the resulting entirecompound proposition or a negation of the com-pound proposition followed by the negation of P,as would be expected because our Klein’s4-group KL is commutative. The operation S3 ex-changes only 12 binary connectives; the 4 centralones P, notP, P 7 Q, and P xor Q are invariant.The dualities, here, could be seen exactly as for R1above. Take, for example, notP or notQ�FTTT� N

S3

notP and Q (FFTF). We have notP ornotQ � P or notQ then not(P or notQ), which isnotP andQ, all in one step. As a final remark, allour transformations are involutions (Ri

2 � Id4�4,Si

2 � Id4�4): Applying any one of them a secondtime gives the original object on which they act.As a final remark, let us stress that the aboveformalism could be used to generate higher-di-mensional matrices corresponding to Booleanfunctions in K variables. As an example, for threevariables P, Q, and R the 223

� 256 Booleanfunctions, like (P � Q) � R� , could be calculatedin the product 3 � 2 � 2. For the general case,one forms the product K � K�1 � K�1 [K � 2,3, . . . , with 1 defined in Eq. (11)].

4. Unifying KB and KL and Conclusion

In this work, we considered a joint classifica-tion of the 16 genetic code-doublets, on the onehand, and the 16 binary functions of the (two-valued) classic logic, on the other. We gave sup-port to our unified classification by showing thatit is possible to associate to the four nitrogenousbases, U, C, A, and G the Boolean numericalvalues 00, 01, 10, and 11, expressing their relativeatomic composition. Also, and importantly, thislink between biology and logic has been made(consistently) possible thanks to the introduced“Kronecker”-like product with concatenation(presented in the Appendix). We have then

shown that the symmetries of the 16 doublets aredescribed by Klein’s 4-group KB of transforma-tion (permutation) matrices. We have also shownthat the symmetries of the 16 binary connectivesare described by a second Klein’s 4-group KL oftransformations. It appeared at the end of thisstudy that the groups KB and KL themselvescould be unified into a larger group constructedas follows. Consider the two matrices

T2 � R2S2 � �0 0 1 01 0 0 00 0 0 10 1 0 0

� , T3 � R2S3

� �0 1 0 00 0 0 11 0 0 00 0 1 0

� , T24 � T3

4 � I4�4. (16)

It is straightforward to verify that the set {I4�4, R1,R2, R3, S2, S3, T2, T3} closes under matrix multipli-cation and has all the properties of a group. Makingthe substitutions Id4�43 e, R13 q, R23 s, R33 t,S2 3 u, S3 3 v, T2 3 r, T3 3 p, we obtain thefollowing Cayley table:

which we recognize as the multiplication table ofthe dihedral group with eight elements, D4, whichis known to describe, among other things, the sym-metries of the square and has many applications inphysics, computer sciences, etc. Among the 10 sub-groups of D4, 3 are of order four and correspond tothe 3 subgroups {e, q, s, t}, {e, q, u, v}, and {e, p, q, r}.The first two are, respectively, the 4-group KB, de-scribing the symmetries of the 16 genetic code dou-blets, and KL, another Klein’s 4-group, describingthe symmetries of the 16 binary functions. The thirdsubgroup is, contrary to KB and KL, not commuta-tive and is of no use here. (Note that each of ourtransformations R1, R2, R3, S2, and S3, together withthe identity transformation e, constitute a group oforder two.) We have then the result that the sym-



metries of the two fundamental (second order) “al-phabets” of biology and logic appear to be unifiedin the dihedral group D4. Let us give some wordsconcerning the link with the logical alphabet ofZellweger [11, 12], the classification of the 16 dou-blets by Duplij and Duplij [13], and our own clas-sification. We shall, also, restrict ourselves to theRumer transformation (R1), which expresses the py-rimidine–purine duality in biology (being at theroot of many theoretical investigations on the ge-netic code ([3, 6–8, 14, 15])) and the negation inlogic. (Concerning negation, we discovered, in thecourse of shaping this work, that our matrix �,defined in Eq. (4) and used to define the Rumertransformation (recall R1 � � R �), could indeed beassociated with the negation operator. For example,Cattaneo et al. [16], in their semantic characteriza-tion of logical gates in quantum computation, usethe “NOT(1)” operator, which appears to be identi-cal with our �. We shall consider, in the following,only a 2-D representation of the Zellweger’s logicalalphabet, namely, the “clock compass” representa-tion, which is a low-dimensional shadow of thelogical garnet, itself a 3-shadow of a 4-cube. It couldbe depicted as follows:

where we have used the symbols for the connec-tives [see Eq. (12.1)] and replaced the symbol fornegation by an overbar. The symbol “�” is for con-tradiction and “�” for tautology. (See the morebeautiful original depiction of Zellweger in [12]; wedo not have the necessary fonts in our ScientificWorkplace 3.0 to reproduce it.) Comparing withtable (T), we see that, in both, any pair of binaryconnectives that are the negation of each other aresymmetrical with respect to the central point (noted� above). Especially, the De Morgan laws are trans-parent. As we have seen, in Section 3 this symmetrycorresponds to the application of the (Rumer) trans-formation R1 [see Eq. (TS1

)]. Let us now close theloop and return to the 16 doublets, of Section 2, bytranslating back the symbols of the above figure toobtain a “clock compass” classification of the 16doublets

Note that the quartets “U1,” “C1,” “A1,” and “G1,”on the one hand, and “U2,” “C2,” “A2,” and “G2,” onthe other, occupy separate parallel lines and, more-over, the subsets M�1 and M�2 are also well separated.As for the binary connectives, two dual doubletsare symmetrical with respect to the central pointand conform to the Rumer symmetry. Indeed, thereis also some order in this “clock compass” classifi-cation of the genetic code-doublets. The above fig-ure reminds us a “quick” classification given byDuplij and Duplij in Ref. [13]:

where the numbers at the right are the sum of the“determinative degrees,” dX, of the bases in eachdoublet, with dC � 4, dG � 3, dU � 2, and dA � 1, inagreement with the M1 � M2 partitioning (see Sec-tion 2 and [3]). Note that, here also, symmetricaldoublets with respect to the central point (�) havetheir two bases exchanged according to the Rumersymmetry U7 G and C7 A and the same kind oforder, as for the “clock compass,” is present.

Let us now conclude this article with some re-marks. First, the idea behind the subject of thiswork, that is, a connection between biologic objects(bases, base-doublets, codons, etc.) and logic oper-ations (Boolean binary functions and in generalK-ary Boolean functions, is not new by itself. In fact,the idea of using concretely, that is, at the experi-mental level and in view of real computations, DNAand logical operations to solve hard problems hasbeen introduced by Adleman in 1994 [17] (Hamil-tonian path, or traveling salesman problem) andLipton (1995) [18] extended Adleman’s method tobreak the Data Encryption Standard (DES). This isknown today as the Adleman–Lipton paradigm;see a recent report by Sakamoto et al. [19] and thereferences therein, where the authors describe aDNA-based solution of the satisfiability problem

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(SAT, another hard problem) with, as logical imple-mentation operations, hairpin formations by single-stranded DNA molecules. Second, there is an inti-mate relation between binary coding and the studyof the many genomes that are, now, available pub-licly. For example, Martinis [10] analyzed symbolicDNA/RNA sequences by using specific assign-ments to the bases T/U, C, A, and G that led todifferent fractal dimensions for �-helices and-sheets in proteins. These specific assignments ap-pear to be the same as those in our Eq. (11.2). Third,and finally, the concatenation of matrix elements inour proposed new matrix product would not beforbidden, in principle, by any known obstruction(at least to the knowledge of the present authorwho believes in the creativity of mathematics); onthe contrary, it seems in line with the biochemicalprocesses [it is the job for the DNA/RNA polymer-ase to assemble (or concatenate) one by one thenucleotides along the DNA strands, for example]and it is also the basic operation in the C�� lan-guage in computer science. In our case, in thisarticle it ensures that the 16 dinucleotides, on theone hand, and the 16 Boolean functions, on theother, are differently assigned objects at the literalas well as at the numeric levels. In the course ofwriting these last lines, it appeared to the presentauthor that a similar representation has been usedby Tino [20] in a recent interesting work (see hisFig. 1) to visualize Hao’s frames [21], but withoutdetails. It is also interesting to note, in connectionwith our symmetry considerations, that Horne, in[22], regarding each of the 16 Boolean functions asa “self-maintaining” (homeostatic) automaton inlogical space, invokes also permutations of the 0sand 1s in these functions and random concatena-tion of operators to reveal pattern generation.

Appendix: “Kronecker”-Like MatrixProduct with Concatenation

The usual Kronecker product of an m � m matrixM and an n � n matrix N is defined by

�M � N�js,kt � MjkNst (A1)

and a simple way to express such matrices withsingle index labeling, (M R N)p,q, is given by

p � n� j � 1� � s, q � n�k � 1� � t,

1 p mn, 1 q mn. (A2)

Now, we define a (new) “Kronecker”-like productwith concatenation (in base b) by

�M � N�js,kt � Mjk � Nst � Mjk � bl�Nst� � Nst, (A1�)

so that instead of taking the product of the matrixelements as in (A1) we take the concatenation of thematrix elements; otherwise, the construction is thesame as for the ordinary Kronecker product. [Notethat the product (A�1) is defined only for matriceswith integer entries; this is the case in our applica-tions.] Recall that the concatenation of two (integer)numbers p and q is given by

p � q � pbl�q� � q, (A3)

where the length, l(q), of q in base b is given in termsof the floor function ( ): l(q) � logbq � 1. Inpractice, it suffices to join the numbers. One has 13� 31 � 1331, in decimal (base 10), and 10 � 01 � 1001,in binary (base 2), as only two elementary exam-ples. What is interesting in the product (A1�) is thatit is noncommutative because p � q � q � q, and thisis welcome [see Eq. (2)] because a doublet XY mustbe different from the doublet YX. This is not thecase when using (A1). Moreover, this product per-mits a consistent connection to logic made in Section3, contrary to (A1). This fact has been realized onlyrecently by the author. Take, for example, the def-inition (1); see also Eqs. (6) and (7). We have in thedetail [3]

� :� �U CA G� � � 4064256 3556224

130691232 1045529856�. (A4)

The numbers are the products of the nuclear chargesof all the atoms constituting a given base and so could“characterize” in a certain way said (nitrogenous)base. For example, C � (CC)2N3H5O � 64738 �3556224 (here, ordinary multiplication in the matrixordinary products is used). When, now, we take theproduct � � �, we have, for example, (b � 10) UC �40642563556224, which is different from CU �35562244064256 and similarly for the other doubletsXY with X � Y; see Eq. (2). [This concatenation func-tion exists in computer packages as the cat(.,.) func-tion.] We see therefore that both products, (A1) and(A1�), allow one to get larger matrices from smallerones. The product (A1�) takes all its importance inSection 3 but, in this work, we shall use both productswith their respective notation. In this work, classifica-tion matrices, obtained by using (A�1), are called C-



type matrices while those obtained by using (A1) arecalled T-type (transformation) matrices.

ACKNOWLEDGMENTS

The author acknowledges Shea Zellweger formany e-mail exchanges and for sending many orig-inal documents concerning his work on the logicalalphabet. It is also a pleasure to acknowledge ananonymous referee for helpful recommendations.Many thanks are extended also to Professor BailinHao for bringing to the author’s attention Ref. [20].

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1. Cooper, W. S. The Evolution of Reason: Logic as a Branche ofBiology; Cambridge University Press: Cambridge, UK, 2001.

2. Fontana, W.; Buss, L. W. Bull Math Biol 1994, 56, 164.3. Negadi, T. Int J Quantum Chem 2003, 91, 651.4. Negadi, T. Presented at the 24th meeting of Group Theoret-

ical Methods in Physics, Group 24, Paris, 2002. Submitted forpublication in 2003.

5. Taylor, F. J. R.; Coates, D. Biosystems 1989, 22, 177.

6. Danckwerts, H.-J.; Neubert, D. J Mol Evol 1975, 5, 327.7. Rumer, Y. B. Proc Acad Sci USSR 1966, 167, 1393.8. Jimenez Montano, M. A.; de la Mora Basanez, C. R.; Poschel,

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14. Shcherbak, V. I. J Theor Biol 1989, 139, 271.15. Shcherbak, V. I. J Theor Biol 1994, 166, 475.16. Cattaneo, G.; Dalla Ciara, M. L.; Giuntini, R.; Leporini, R.

ArXiv: quant-phys/0201013 v3 8 Jan 2002.17. Adleman, L. M. Science 1994, 266, 1021.18. Lipton, R. J. Science 1995, 268, 542.19. Sakamoto, K.; Gouzu, H.; Komiya, K.; Kiga, D.; Yokayama,

S.; Yokomori, T.; Hagiya, M. Science 2000, 288, 1223.20. Tino, P. Physica A 2002, 304(3/4), 480.21. Hao, B. L.; Lee, H. C.; Zhang, S. Chaos Solitons Fractals 2000,

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rumer's transformation, in biology, as the negation, in classical logic

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