Table Of ContentQuantum view on contextual logic of composite intelligent devices
∗
E. D. Vol
B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine 47, Lenin Ave., Kharkov 61103, Ukraine.
(Dated: January 24, 2013)
Basedontheideasofquantumtheoryofopensystems(QTOS)weproposetheconsistentapproach
tostudyprobabilisticmany-valuedpropositionallogicofintelligentdevicesthatarecomposedfrom
separate but interconnected logical units.In this preliminary communication we consider only the
simplest example of such systems,namely, four- valued probabilistic logical devicecomposed of two
logical subsystems.We demonstrate that similar devices can generate two classes of probabilistic
propositions :1) decomposable propositions , which in fact are equivalent to certain ordered pair
3 ofpropositions indevicesubsystemsand 2) indecomposable propositions which are connectedwith
1 inherentlogical interaction betweendeviceunits.Theindecomposable propositionsareundoubtedly
0 of greatest interest since they , as shown in the paper, provide powerful additional logical resource
2 compared to standard parallel processing in composite intelligent systems. The contextual logic
of composite devices proposed in this paper, as we believe, can be also used for analysis of highly
n
organized systems in cognitive sciences specifically in neuropsychology and linguistics.
a
J
PACSnumbers: 03.65.Ta
3
2
I. INTRODUCTION and humans, consist exactly from two hemispheres each
]
h of which is able to perfom special mental functions and
p continuously exchange with partner by cognitive infor-
- In the last years close attention of quantum physi- mation.The rest of the paper is organized as follows.In
t
n cistscommunitywasfocusedonthepossibilityofcreation Section1 for the convinience of the reader we briefly de-
a of effectively working quantum computer(QC).Although scribe the information from [2] concerning probabilistic
u
certain progress has been achieved recently in this field logicwhichisnecessaryfortheunderstandingofthegiven
q
[ ( see for example [1] ), nevertheless many essential dif- paper.Inadditionweintroduceherethevaluableconcept
ficulties yet to be overcome.Themost seriousobstacle to of distanse between two probabilistic propositions with
1 solvethisproblemisundoubtedlydecoherencethatisin- the help of which one can define the degree of closeness
v teraction of workingr part of QC with its environment of different propositions.In the Section 2 , that is the
9
that leads to the destruction of quantum states super- main in the paper, we consider the structure of logical
1
4 positions and thus to lower efficiency of quantum com- propositions in composite logical devices (CLD).In par-
5 putations.Thereby as we believe it makes sense to pay ticularthesimplestexampleoffour-valuedlogicaldevice
. attention alsoto other options to increase computing re- is studied more detail.We demonstrate that all logical
1
0 sourse ofintelligent devices which would not be so sensi- propositions in similars system can be divided into two
3 tive to destruction by inner and exterior noise.One such different classes:1)decomposible propositions (DP) each
1 optionconnectedwiththestudyofcompositeprobabilis- of which is in fact equavalent to certain ordered pair of
: tic logical devices just considered in present paper. The propositionsindevicelogicalsubsystemsand2)indecom-
v
i approachproposeduses essentially the results of preced- posible propositions (IP) which are connected with in-
X ing authors paper [2] in which on the basis of quantum herentmentalinteractionbetweendevicesubsystems.We
r theory of open systems (QTOS) were considered both believe that the existence of IP and the possibility to
a
probabilistic logical propositions and all possible logi- operate with them represents additional powerful logical
cal operations (that is logical connectivies) with them resourse in composite systems as compared with usual
.Thus it becomes possible by simple and unified method parallelinformationprocessing.Letusnowgotothe pre-
toformulatevariouslogicalcalculuses: probabilisticlogic sentation of concrete results of the paper.
(including usual Boolean logic), many- valued logic ( in-
cludingamodallogic)andsoon.Inthepresentpaperour
main goal with the help of ideas and methods of QTOS
II. PRELIMINARY INFORMATION
to state the logic of composite probabilistic devices that
composed of two or more logical units mentally inter-
acting with each other .Our study can not be consid- In this part we briefly remind main results of the pa-
ered as too much speculative.Indeed here is sufficient to per [2] which are necessary for the understanding of the
point out the well-known fact that brains, both animals following text.As well as in [2] the main subject of our
considerationwillbeasetofplausiblepropositions(PP),
that is the propositions, the truth or falsity of which
is known to recipient not exactly but only with certain
∗Electronicaddress: [email protected] probability.It is convient to represent any
2
such proposition by normalized to unity diagonal ma- D(A,B) = 1 N | p − q |(we have in mind here
trix with positive elements.For example in the simplest 2 i i
iP=1
caseoftwo-valuedlogicanyplausiblepropositionAcan that propositions A and B have representative matrices:
be represented by the matrix : diag{p1,pi,pN} and diag{q1,qi,qN} correspondently.It
p 0 is clear that above definition is satisfied to all relevant
ρ(A) = A , where p is the probability
(cid:18) 0 1−pA(cid:19) A conditions of the concept.Using this definition of close-
for A to be true.In what following everywhere where it nessandalsothedefinitionsofbasicconnectivesbetween
does not lead to confusion we will identify propositions propositionsintroducedin[2],onecanproveforexample
with their representativematrices.It turns out that var- the relation D(A=⇒B,A and B) ≧ D(B,A and B)
ious operations with PP ,that is logicalconnectives,un- andmanyothersimilarrelations.Ontheotherhandfrom
der such approachcan be realizedas special transforma- the physical point of view it is very important that one
tionsoftheirrepresentativematrices.Referringthereader canthinkofrepresentativematriceofpropositionasden-
for details to (2) we representhere only final decisive re- sitymatrixofrelevantquantumsystem,andhenceshould
sult that we will need in the form of the next theorem. torealizeallplausiblepropositionsandlogicaloperations
Theorem1. Let ρ is N ×N diagonalmatrix with posi- with them as result of certainphysical manupulations in
tiveelementswhosetraceisequalto1andleassumethat correspondent quantum system.In addition this analogy
G is some M × N matrix which posseses two defining allows one to propose concrete physical realizations of
properties:1) all elements of G are equal 0 or 1 and 2) logical devices in order to improve their effectiveness of
in each column of G only single element is equal to 1 information processing.
and all the rest are equal to zero.In this case the trans-
formation of the form ρ = GρGT maps ρ onto M ×M
diagonal matrix ρ with positive elements ρ that sat- III. THE LOGIC OF COMPOSITE LOGICAL
ii
M e DEVICE. THE SIMPLEST MODEL.
isfy to the relatioen ρii = 1.Similar apprfoach can be
iP=1
used in the general casefof N- valued probabilistic ( or Till now we consider probabilistic logic and possible
modal)logicwhereeverypropositionhasN logicalalter- logical devices in which they can be generated as cer-
natives with corresponding probabilities pi (i=1,...N) tain integral systems.However, using the analogy with
N quantum theory an essential element of which is investi-
which satisfy to normalization condition ρ = 1.We
i gationofcompositesystemsitisfullyjustifiedtoinclude
iP=1
will call the above transformations as admissible (log- in our focus also composite logic and correspondently
ical) transformations.Having in hands the above result composite logical devices (CLD) consisting of several
onecanbyunifying waydeterminealln-placelogicalop- subsystems that are logically interconnected with each
erations in N valued probabilistic logic.For example the other.Evidentlysuchapproachisnotpureacademical.To
resultofanyone-placeoperationappliedtoPPAcanbe confirmitenoughtopointoutthatbrainsofhumansand
represented in the form: animalsexactlyconsistoftwohemisphereseachofwhich
is able to perform special cognitive functions [3].In this
A=G1·A·GT1, (1) connectionitisveryessentialtoemphasizetwofacts:1)a
flow of afferent information from sensor organs to brain
e
where G1is some admissible N ×N matrix. reaches both hemispheres almost simultaneuosly and 2)
Similarly the result of any two-place logical operation left and righthemispheres can continuously exchangeby
applied to propositions B1 and B2 can be written as cognitive information with partner through special fiber
system in the brain, so called the corpus callosum.
B =G2(B1⊗B2)GT2, (2) Of course the simplest and in many respects prim-
itive model which we will study in this paper by no
where G2 is someeadmissible N2×N matrix . means can not represent to the right degree any real-
It is clear that with the help of admissible transfor- isticmodelofthebrain.Neverthelesswebelievethevery
mations one can determine logical operations for any idea about peculiar logic of CDL deserves close atten-
number of PP as well.Note that in the case of N-place tion and further study.Now let us turn to the descrip-
logical operation it is necessary to take N propositions tion of composite logic.Note that in this paper in view
A1,A2,...AN ,andchooseasinitialstatethetensorprod- of maximalsimplicity and clearnesswe restrictourselves
uct of them ,that is A1 ⊗ A2... ⊗ AN..It is clear also only the simplest case of the device in which composite
that total number of one- place logical operations in N logic should be occur.Namely we will discuss the four-
-valued logic is equal to NN, the number of two- place valuedcomposite logicaldevice consisting of two-valued
operations is equal to NN2 and so on.Note that the ap- subsystems (units).It should be noted however that al-
proachproposedalsoallowsonetointroducetheconcept most all results of our consideration can be generalized
of distance between two N valued propositions.Indeed , (with appropriate refinements) to the more compex sit-
accoding to natural reason, the distance D(A,B) be- uations.Returning to four-valued CDL remind, that ac-
tween two propositions A and B can be defined as: cording to Sect.1 any proposition in four- valued logic A
3
can be represented in the form of diagonal matrix 4×4 connectives.So , the negation of PP A that is A can be
, namely: A = diag{p1,p2,p3,p4} where probabilities written as:
of different logical outcomes p (i=1..4) satisfy to the
i
4
conditions:1) 0 ≤ pi ≤ 1 and 2) pi = 1.Our next p4
step is the statement of basic logiiP=ca1l operations (con- A= p3 , (3)
p2
nectives) with such propositions.Accordingto theorem 1 p1
fromSect.1this taskcanbe realizedwiththe helpofrel-
evant admissible matrices.Omitting intermediate and to
some extent tedious expressions for admissible matrices conjuction of two propositions A and B =
we adduce here only the required results for the basic diag{q1,q2,q3,q4} is equal to
p1q1
(A and B)= p1q2+p2q1+p2q2 , (4)
p1q3+p3q1+p3q3
p2q3+p3q2+p4+q4−p4q4
disjunctionofthesametwopropositionsAandBisequal to
p3q2+p2q3+p1+q1−p1q1
(A or B)= p4q2+p2q4+p2q2 , (5)
p4q3+p3q4+p3q3
p4q4
and so on.Clearly that in the case of four- valued logic positesystemandcorrespondentoperationsinitssubsys-
there are considerably more one-place and two- place tems. It turns out that above definitions of basic logical
connectives than in two-valued case.For example it is connectives in four- valued logic imply the next simple
easy to see that we have 44 = 256 one -place connec- rules between operations in composite system and cor-
tives in contrast of 22 similar connectives in two- val- responding operations in its subsystems.Indeed one can
ued logic.Now using the analogy with quantum theory easily verify that the simple relations hold:1) A 1 =A1
of composite systems [4] we can make the next impor- , A 2 = A2 for negation, and 2) for two -pl(cid:0)ace(cid:1)opera-
tant step.Let us consider two logical projections of some
tio(cid:0)ns(cid:1): (A and B)1 = (A1and B1 ) , (A or B)1 = (A1or
four- valued proposition on corresponding two- valued B1).
propositionsineachofthesubsystemofCLD).Forexam-
plearbitrarypropositionA=diag{p1,p2,p3,p4}hastwo Now we intend to demonstrate that there are two dis-
p1+p2 tinct classes of propositions generated in composite de-
logical projections: A1 = in the first
(cid:18) p3+p4(cid:19) vice, namely 1) decomposable propositions (DP) each
p1+p3 of which in fact is equivalent to some ordered pair of
subsystem and A2 = in the second
(cid:18) p2+p4(cid:19) propositionsindevicesubsystemsand2)indecomposable
one.On the language of the quantum theory one should propositions (IP) that do not allow such a reduction.To
say that the density matrix of composite quantum sys- prove this fact it is convinient to use the representation
tem uniquely determines correspondingdensity matrices which is valid in fact for any four- valued proposition A.
of its subsystems.By means of this mapping we can set Namelylet A=diag{p1,p2,p3,p4} thenitis easyto see
thecorrespondencebetweenalllogicaloperationsincom- that A can be represented also as
4
pq+C
p(1−q)−C
A= , (6)
q(1−p)−C
(1−p)(1−q)+C
where in Eq. (6) we use the notation: p = p1 +p2, form:
q =p1+p3 and C =p1p4−p2p3.
1 +C
4
1 −C
The collection of propositions which satisfy to con- A(C)= 4 1 −C , (7)
dition C = 0 will be called decomposable since for 4 1 +C
them the decomposition in the form of tensor product 4
:A = A1 ⊗ A2 obviously holds.Thus such ”composite” where the variable C satisfies to
propositions are in fact coincided with tensor products unequalities: −1 ≤C ≤ 1 . Evidently that if C 6= 0
oftheirlogicalprojections.Notethatinthequantumthe- 4 4
then all prop(cid:0)ositions A(C(cid:1)) are indecomposable and the
oryofcompositesystemsanaloguesofthese propositions
context variable for proposition A(C) coincides exactly
are factorable pure states and correspondentquantity C
with C.Note in passing that for any IP A its distance
(that is equal to zero in the case of factorable states)
to the nearest DP that is A1 ⊗ A2 is equal exactly to
is called concurrence.However,with respect to the case
4|C| where C is the value of context variable for A.It is
of probabilistic many-valued logic, will call the quantity
easy to see also that logical projections for proposition
C from Eq. (6) the context variable or simply as con-
A(C) from (7) are independent from context C and
text.Reasonsforsuchdesignationwillbecomeclearalit- 1
tle later.Now let us point out two nearly obvious facts equal to each other, namely A1 = A2 = 2 1 .Now
(cid:18) (cid:19)
concerningDP:1)if A is DP then Ais DP aswelland 2) 2
let us perform the concrete one-place logical operation
if A andB aretwo DP then propositions(A and B) and
:A(C) = GA(C)GT, where the admissible matrix G
(AorB)aredecomposablealso.Therebythecollectionof
has the form:
all DP is closedwith respect to basic logicaltransforma- e
tions, namely one can prove that the following theorem
0 1 0 0
has place.Theorem 2 : let A is DP in composite system
1 0 0 0
and G some one-place admissible transformation.Then G= , (8)
0 0 1 0
A = GAGT is decomposable proposition as well and 0 0 0 1
can uniquely be represented in the form: A = A1⊗A2
e
whereA1 =G1(A1⊗A2)GT1 andA2 =G2(eA1⊗fA2)GfT2 Note that above operation is isometric that is it does
where (G1,G2) is some ordered pair of two-place admis- not change the distances between propositions from col-
f f
sible transformations and A1,A2 are logical projections lection of interest .Really, it is easy to verify that:
ofDPAindevicesubsystems.Withoutstoppingtoprove D[A(C1),A(C2)] = D A^(C1),A^(C2) = 4|C1−C2|.
theorem2noteonlythattotalnumberofadmissibleone- h i
^
place transformations in CLD is equal 44 which is coin- In explicit form the matrix A(C) reads as
cideswiththenumberoforderedtwo-placepairs(G1,G2)
which is equal to 24×24. 41 −C
^ 1 +C
A(C)= 4 1 −C , (9)
Thus we come to the important conclusion: all logical 4 1 +C
operations with DP in CLD can be entirely reduced to 4
thelogicaloperationsproducedindevicesubsystemsthat
TheexpressionEq. (9)impliesthatlogicalprojectionsof
is inthis caseCLDworksexactlyaccordingto principles
1
^
of parallel processing information.On the other hand we A(C) in device subsystems has the form: A1 = 2 1
arguethatforindecomposablepropositionsthe situation (cid:18) 2(cid:19)
itshequpiotsesidbiifflietyretnot.oItpetruartneswoiutht tthheamt etxoisatelanrcgeeodfegIPreeainnd- and A2 = (cid:18)21 −2C 12 +2C(cid:19). We see thaft logical pro-
creaselogicalresoursesofCLD.This importanttopic un- jectiofn of transformed proposition in the first subsys-
doubted deserves a special and detail investigation. But tem does not change, while its projection in the sec-
in this paper which has preliminary nature we restrict ond subsystem essentially depends from context value
ourselves only to single but very good example illustrat- C.Moreover the very plausibility of proposition A2 may
ingthe abovethesis.Letus considerinCLDoneparame- bedeterminedbycontextanddependingonCcanchange
ter collectionof indecomposable propositionsofthe next its value to the contrary.Indeed, the value of Cf= −1
4
5
implies that A2 −41 = (cid:18)1 0(cid:19)and value of C = 14 im- intIenrecsotnincglusiisosnuewseheamvephraezmisaeintehdatomutasnidyeimofpoorutarnstcaonpde
(cid:0) (cid:1)
plies that A2f14 = (cid:18)0 1(cid:19).In this connection one can .1A)mthoengnetehde tmoogstenimerpaloirzteanthtetoappipcrsowacehmparyopcoaslledsuochnathse:
(cid:0) (cid:1)
say that thfe applied logical operation is connected with case of composite system consisting of arbitrary num-
recognition of context by second subsystem (agent).Of ber of subsystems with different dimensions 2) the need
course the similar transformation can be applied with to classify logical operations in CDL from the point of
respect to first subsystem (agent) as well.We are con- view of their possibility to transform DP into IP and
vincedthatapplicationofIP andlogicaloperationswith vica versa.Besides it is necessary to study in detail the
themallowsonesignificantlyexpendtheclassofdeduced problemofrealizationCDLinconcretequantumphysical
propositionsinsubsystemsofCLDcomparingwithordi- systems.In particular in the case of simplest four-valued
nary Booleanfunctions from p and q.that are realizedin CDL we should talk about two-qubit quantum system
the case of decomposable propositions. interacting with environment by special way.In part we
Thus already this preliminary analysis of simple CLD consider this issue in [2], but this question needs much
clealydemonstratestheadvantagesofsuchdevicesinlog- more detail investigation.
icalprocessingincomparancewithordinaryparallelpro- All these and some other questions we hope to discuss
cessing systems. in later publications..
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(7417), 541, (2012). and Quantum Information, Cambridge University Press,
[2] E. D. Vol, Int.J. Theor. Phys, 52, 514, (2013). Cambridge,(2001).
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