A SYSTEMS-THEORETICAL REPRESENTATION OF TECHNOLOGIES AND THEIR CONNECTIONS
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1 A SYSTEMS-THERETCAL REPRESENTATN F TECHNLGES AN THER CNNECTNS Takehiro nohara eparmen of Vale and eision Siene, Gradae Shool of Soial and eision Siene and Tehnology, Tokyo nsie of Tehnology, okayama Megro-k, Tokyo , JAPAN inohara@aldes.ieh.a.jp ABSTRACT This paper proposes a sysems-heoreial represenaion of ehnologies. A ehnology is represened as an effiien inp-op (/) sysem in he sense of mahemaial sysems heory, where he / sysem ransforms he inps proided for ihrogh he inp hannels of i ino he ops, whih are oped from ihrogh he op hannels of i. This paper also proides a definiion of onneions of / sysems as a way o onsr a bigger / sysem from smaller / sysems. f orse i is no always re ha a onneion of / sysems is a ehnology. an be erified, howeer, ha a onneion of ehnologies is always a ehnology. n his paper a mahemaial erifiaion of his fa is proided. Keywords: / sysems; ehnologies; onneions of ehnologies NTRUCTN The aim of he researh in whih his paper is inoled is o deelop a mahemaial mehod for ealaing ehnologies, in parilar, paens (Razgaiis (1999), Smih and Parr (1994)). As a firs sep of his researh, a sysems-heoreial (Mesaroi e al (1974), Mesaroi and Takahara (1975), Mesaroi and Takahara (1989)) represenaion of ehnologies will be proposed in his paper. This an be employed for frher deelopmen of he researh: for example, sh oneps as neworks of ehnologies, fngibiliy of ehnologies, ommerializabiliy of ehnologies and onneabiliy of ehnologies an be mahemaially expressed. n his paper, a ehnology is represened as an inp-op (/) sysem in he sense of mahemaial sysems heory, where he / sysem ransforms he inps proided for i hrogh he inp hannels of i ino he ops, whih are oped from ihrogh he op hannels of i. is reqired for an / sysem o be a ehnology hahe / sysem saisfies ha for eah inphere exiss an op sh hahe inp is ransformed ino he op by he / sysem, and for eah ophere exiss an inp sh hahe / sysem ransforms he inp ino he op. 1
2 n his paper, moreoer, a definiion of he onep of onneabiliy of / sysems will be proided, and a proposiion whih laims ha a onneion of ehnologies saisfies he ondiions o be a ehnology will be erified. The srre of his paper is as follows: in he nex seion, Seion 2, he mahemaial framework for reaing / sysems and heir onneions will be proided. The definiion of ehnologies will be gien in Seion 3, and sbseqenly, he proposiion menioned aboe will be erified in Seion 4. The las seion, Seion 5, is deoed o he onlsie remarks. MELS: / SYSTEMS AN THER CNNECTNS This seion gies he mahemaial framework employed in his paper. This framework is onsred based on he mahemaial sysems heory (Mesaroi e al (1974), Mesaroi and Takahara (1975), Mesaroi and Takahara (1989)). Le C be he se of all / hannels. For C, is he inp se of hannel and Y is he op se of hannel. efiniion 1 (The field of / sysems) The field F of / sysems is a ple (C, ( Y ) ). C ( ), C Tha is, he field F of / sysems onsiss of he se C of all / hannels and he inp ses and op ses Y for all C. Wihin he field F of / sysems, / sysems are defined as follows: efiniion 2 (/ sysems) An / sysem is a ple ( S,, sh ha S Y, where = and Y Π = Y. Π, ( ), ( Y ) ) For an / sysem, C is he se of all inp hannels of and C is he se of all op hannels of. Moreoer, for an / sysem, an inp hannel of and an op hannel of, ( ) is he inp se of hannel of and Y ( Y ) is he op se of hannel of. efiniion 3 (Conneabiliy of / sysems) An / sysem = ( S,,, ( ), ( Y ) ) is said o onne wih an / sysem = ( S,,, ( ), ( Y ) ) on, where φ C, if φ =. f an / sysem onnes wih an / sysem on, hen an / sysem is he onneion of and on, denoed by, if and only if = ( S,,, ( ), ( Y ) ) (see Figre ), where 1. = ( \), 2. = ( \), and 2
3 3. ( x \ S if and only if here exiss ), y ) S, and y ). y \ = y, where \ x, Y sh ha ( x =( x, x \ ) and S, (( y, y \, \ \ Figre. The onneion of and on. TECHNLGES n his paper, i is reqired for an / sysem o be a ehnology hahe / sysem is effiien in he sense ha for eah inphere exiss a leas one op sh hahe inp is ransformed ino he op by he / sysem, and for eah ophere exiss a leas one inp sh hahe / sysem ransforms he inp ino he op. The nex is a preise definiion of ehnologies. efiniion 4 (Tehnologies) A ehnology is an / sysem ( S,,, ( ( Y ) 1. for all 2. for all ) sh ha;, here exiss Y, here exiss Y sh ha ( sh ha ( S, and S. ), Anoher ype of ehnologies, whih saisfy a ondiion ha is weaker han he one ha is reqired for an / sysem o be a ehnology, an be defined as follows: efiniion 5 (Weak ehnologies) A weak ehnology is an / sysem ( S,, ( ), ( Y ) ) sh ha; 1. for eah and eah x, here exiss sh ha Y sh ha ( S, and, = x and here exiss 3
4 2. for eah and eah sh ha ( Y, here exiss S. Y sh ha = y and here exiss PRPSTNS The firs proposiion shows ha a ehnology always saisfies he ondiions o be a weak ehnology. Proposiion 1 f an / sysem = ( S,,, ( ), ( Y ) is also a weak ehnology. Proof: For and x, one an hae sh ha = arbirary. Then, here exiss is a ehnology. For and Y, one an hae sh ha Y sh ha ( Y sh ha sh ha = arbirary. Then, here exiss bease is a ehnology. ) is a ehnology, hen = x, aking, = y, aking sh ha ( for eah S, bease for eah S, The nex is he main proposiion of his paper, whih erifies ha a onneion of ehnologies is also a ehnology. Proposiion 2 (Conneion of ehnologies is a ehnology) f an / sysem is he onneion of ehnologies and on, hen is a ehnology. Proof: Take ( x = ( x, x \ ) arbirary. Then, here exiss x, z ) S, bease is a ehnology. Moreoer, for ( z, z Y sh ha (( z, x \ of he onneion of and on, Take ( y \ z x \ Y sh ha ), here exiss ), ) S, sine is a ehnology. Ths, by he definiion z \, z ) saisfies ha (, y ) Y arbirary. Then, here exiss z, y ) S, sine is a ehnology. Moreoer, for z sh ha ( z, ( definiion of he onneion ( S. y \ y \ S. z sh ha, z ), here exiss, z )) S, bease is a ehnology. Ths, by he of and on, x = ( z, z \ ) saisfies ha CNCLUSNS This paper gae a mahemaial framework for dealing wih ehnologies. The oneps of ehnologies and heir onneions were newly proided, and he faha a onneion of ehnologies saisfies he ondiions o be a ehnology was erified. The framework onsred in his paper allows s o deelop sh oneps as neworks of ehnologies, fngibiliy of ehnologies and ommerializabiliy of ehnologies. 4
5 These oneps onribe o deelop a mahemaial mehod for ealaing ehnologies, in parilar, paens (Razgaiis (1999), Smih and Parr (1994)). The nex sep of his researh ms be defining hese oneps rigorosly wihin he framework newly deeloped in his paper. REFERENCES Mesaroi, M.., Mako,. and Takahara, Y. (1974). Theory of Hierarhial, Mlileel Sysems., Aademi Press n., U.S.. Mesaroi, M.., Takahara, Y. (1975). General Sysems Theory: Mahemaial Fondaions., Aademi Press n., U.S.. Mesaroi, M.., Takahara, Y. (1989). Absra Sysems Theory, Springer-Verlag, U.S.. Razgaiis, R. (1999). Early-Sage Tehnologies: Valaion and Priing, John Wiley & Sons n., U.S.. Smih, G. V. and Parr, R. L. (1994), Valaion of nelleal Propery and nangible Asses, 2nd ed., John Wiley & Sons, n., U. S.. 5
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