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BioSystems 107 (2012) 95– 105

Contents lists available at SciVerse ScienceDirect

BioSystems

journa l h o me pa g e: www.elsev ier .com/ locate /b iosystems

ymmetrizing object and meta levels organizes thinking

atsuji Takahashia,∗, Yukio-Pegio Gunjib

School of Science and Technology, Tokyo Denki University, Hiki, Saitama 350-0394, JapanGraduate School of Science and Technology, Kobe University, Kobe, Japan

r t i c l e i n f o

rticle history:eceived 12 July 2011eceived in revised form 4 October 2011ccepted 5 October 2011

eywords:

a b s t r a c t

We present a single non-cellular finite automaton model first shown to exhibit self-organizing behaviorwith intermittency and criticality, through a self-referential process. We propose a method to make self-referential contradiction a dynamic process of interaction with the selves in first person and third persondescription. The process represents thinking as inner dialogue with the self in second person. The dynamiceffect of the rewrite shows characters proper to internal measurement, disequilibration by equilibration

nternal measurementecond personnner speechntermittencyriticalityelf-organizationinite automaton

and transfer of inconsistency to the neighborhood by local resolution of the inconsistency. As the result,the advent of contradiction is postponed by the rewrite. The duality of internal measurement subjectprevents inner dialogue in second person from lapsing into monologue. Criticality of thinking process isexpressed. A probabilistic interpretation of non-determinacy weakening oracle is the key.

© 2011 Elsevier Ireland Ltd. All rights reserved.

. Introduction

Emergence and hierarchy form one of the central problemsf science of complex systems (Kauffman, 1993; Holland, 1998).e, however, think that the problem to first cope with is forhom is it emergent and hierarchical. If we admit the omni-

cient observer, implicitly or explicitly, true emergence ceases toxist. Then a definitively specified individual observer calls a phe-omenon emergent merely because of his ignorance. Emergence is

llusory insofar as we consider internal observers (Matsuno, 1989;össler, 1998) just particular reduced reproductions of the generalranscendental existence.

We are not incomplete transcendent as a definitive part of themniscient. We do not have such a definite boundary. Well-definedoundary requires double specification simultaneously from the

nside and the outside (Wittgenstein, 1961). It is not possible for annternal observer to glance over the world encompassing it and toive itself a boundary from the outside. Its boundary is character-zed by indefinite finiteness, distinguished from definite finitenessr infiniteness. The implementation of the indefiniteness is one ofhe central methodologies of internal measurement. By the idea of

nternal measurement, many formal and experimental methodolo-ies are proposed (Gunji and Kamiura, 2004; Imai et al., 1999).

∗ Corresponding author. Tel.: +81 492965416.E-mail address: tatsujitakahashi@mac.com (T. Takahashi).

303-2647/$ – see front matter © 2011 Elsevier Ireland Ltd. All rights reserved.oi:10.1016/j.biosystems.2011.10.004

The purpose of this article is to promote the basic study ofinternal measurement. We examine the internal and dynamicrelationship of the subject with itself that internal measurementconduces. The theory of internal measurement is constructed withlinguistic instruments such as person and tense (Matsuno, 2003).In this study we concentrate on person.

Internal description entailed by internal measurement is in sec-ond person (Haruna and Gunji, 2007). It is not in third personwhere the observer is placed outside the system, looking overthe whole. It is not in first person where the observer is insidethe system, shortsighted or subjective. Second person relation-ship with the system changes the relationship with the self. Ityields not only the self-referential duality to the self (Heidegger,1996), but also the dynamic process of the duality (Gunji et al.,2004).

An example of interaction with the self is thinking with innerspeech (Bakhtin, 1986; Vygotsky, 1986). When we bring out innerspeech, we talk to ourselves. It is inner dialogue in second per-son. The dialogue is not wholly transparent, because the self as thespeaker and the self as the hearer are not identical (Derrida, 1968).If it is identical, then it is just a monologue, no dialogue (Sawa andGunji, 2007), in which one can neglect the active participation of thehearer. The opacity makes it possible for us to discover somethingnew in inner dialogue.

If that is the case, how is the relationship with the self in secondperson, such as thinking, formally represented and made possiblein well-defined ways? What does it bring about as the result? Theseare the problems to be solved in this article.

9 BioSystems 107 (2012) 95– 105

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Table 1The duality between first and third person description.

Person Object Relationship

as the label. If the finally reached state is a final state, we say thatthe FA accepts the string. Moreover, FA can convert a compositetransition to a string which is the concatenation of the transitions’labels. It is output of a word acceptable to the FA itself.

Table 2Grammar and recognizer in contrast to first and third person.

6 T. Takahashi, Y.-P. Gunji /

. “Personal difference” formalized

Self-referential sentence best presents the duality of subject thatnternal measurement brings about. With the internalist stance, theelf is one of the objects referred to by the subject itself as well.

.1. I and me in self-reference

The liar sentence is one of the representatives of self-referentialtructures. It clearly shows the duality of self. It is

“I am lying.”

f I am lying, I am telling a truth. If I am not lying, I am not telling aruth. It is a contradiction.

This sentence indicates the essential difference of the two modesf self. The sentence is definitized as

“I say “I am lying.”,”

ince for any statement there is always the subject who issues it.he self-referential paradox is caused by the identity, despite theifference in logical type, of the two “I”s, I as the subject of thepeech and the subject in the speech (de Vijver, 1996; Lacan, 2007).

e call the former “I” and the latter “me (Mead, 1934).” In the liarentence, these two selves refer to one another.

Two disciplines have their own solutions. In logic, these twospects of self are usually rigorously distinguished as typically inhe type theory (Whitehead and Russell, 1997). In scientific descrip-ion, on the other hand, “I” is made completely implicit as theonsistency of the description or the system. It does not becomen object. The former solution denies the identity of the two “I”s,hile the latter eliminates the self-referential aspect and, therefore,

he activeness of agents in the described system. Both solutions arensatisfactory for our internal stance.

We cannot neglect the duality of self derived from internaltance. We also cannot make the two irrelevant by a strictly dis-inction. Nevertheless, insofar as we consider the two selves asubstantive, contradiction is entailed. Consequently we focus onhe two modes of description where we locate the selves. First andhird person descriptions correspond to “I” and “me.” The descrip-ions have a difference in logical type. The former is at object levelnd the latter is at meta level.

.2. First and third person

Here we compare first and third person description. The unique-ess of first person is the use of indexical expression. We are in aarticular context in first person. It makes it possible to use deic-ics and demonstratives such as now, here, this and that. It dependsn the speaker and the context. We can ostensively define the ref-rent in first person. Because the demonstratives have no intentmeaning) but a unique extent (referent), the relation between thisnd that is ambiguous or undefined. In first person, we can pointo concrete and individual objects. On the other hand, the relationetween the objects is ambiguous or undefined.

In contrast, concepts are the objects in third person descrip-ion. It is because the speaker and the hearer, or the writer andhe reader, do not have a common context. It is therefore impossi-le to use indexical expressions in third person. Concepts are morebstract and ambiguous than individual objects. A concept has itswn intensional meaning. The relations among concepts can beefined by the relation among intensions of the concepts. The rela-ion between third person objects may be clear and unique in third

erson.

In summary, in first person, objects are definite but the relations indefinite, whereas the relation is definite and objects are indef-nite in third person. First and third persons are dual and they have

First (object) Tangible Ambiguous or undefinedThird (meta) Concept or collection Unique

a difference in logical type as object and meta. The comparison issummarized in Table 1.

2.3. Person in the theory of computation

There are the counterparts of the pair of first and third persons inthe theory of computation, non-determinacy and determinacy. Thesimplest yet general implementation for the pair is finite automa-ton (FA) (Hopcroft et al., 2001). The pair of a non-deterministic FA(NFA) and a deterministic FA (DFA), where the latter is constructedby determinization of the former, is the model of first and thirdperson. When we call an NFA grammar, the determinized DFA fromthe NFA is called recognizer. Their correspondence to first and thirdperson and it is shown in Table 2. The relationship parallels gener-ative and analytic grammar. We can think of FA as an instrumentto generate and analyze regular language.

2.4. Regular language

In formal language theory, language is set of words. Word isconcatenation of characters in an alphabet. Throughout this study,the alphabet is fixed to � = {0, 1}. The empty word of length 0 isdenoted by �. The set of all bit strings is denoted by �* = {�, 0, 1, 00,01, 10, 11, 000, . . . },

Regular language is set of words that can be described byregular expression. Regular expression is algebraic representa-tion of words, such as X = (0 + 1)∗01. X represents all the bitstrings satisfying the condition “ending with 01.” The regular lan-guage represented by X is L(X) = {01, 001, 101, 0001, . . .}. Regularexpression consists of characters 0 and 1, grouping operator (paren-theses, “(” and “)”) and alternation (union, “+”), concatenation (dot,“.”), and quantification (closure, “*”), with dot usually omitted. Notethat “*” means zero or more times of repetition, therefore (10)* = {�,10, 1010, 101010, . . . }.

2.5. Determinacy (DFA) and non-determinacy (NFA)

Regular language is described by FA as well. FA is a kind ofdirected graph with distinguished nodes and alphabet labels. Thenodes and edges are called states and state-transitions, respec-tively. One start state and one or more final states are defined. Alltransitions have an alphabet label. Examples are in Fig. 1. The startstate is with an arrow without the source, such as A in G0 and {A}in R0. Final states are depicted as double circles.

The function of FA is usually considered as partitioning theinput strings into two disjoint parts, acceptable and non-acceptable(rejected). Starting at the start state, the characters in the inputstring are replaced in order by transitions that have the character

State (object) State transition (relation)

Grammar (NFA) Individual objects Ambiguous or undefinedRecognizer (DFA) Set or collection of

individual objectsUnique

T. Takahashi, Y.-P. Gunji / BioSys

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ig. 1. A grammar G0 satisfies L(G0) = L(E) = L((0 + 1)∗01). It inputs or outputs bittrings ending with 01 only. For a recognizer R0, R0 = sc(G0) holds. SC0 schematizeshe calculation for subset construction sc().

There are deterministic FA (DFA) and non-deterministic FANFA). DFA R is defined by a 5-tuplet R = (Q, �, ı, q0, F), where

is the set of states, � input symbols or alphabet fixed to {0, 1}, : Q × � → Q the transition function, q0 ∈ Q the unique start state,nd F ⊆ Q the set of final states. R is called deterministic since forach pair of state and input symbol there is one and only one tran-ition and the next state.

NFA is also determined by the 5-tuplet notation. Its differencerom DFA is the type of transition function ı which is ı : Q × � →(Q ), where P(X) means the power set of X (all subsets of X, includ-

ng the empty set and X itself). DFA may be considered to be a specialase of NFA. The range of NFA’s ı being the power set of Q allows aituation in which from a state q and with an input symbol a as thenput there are multiple transitions and next states, ı(q, a) = {q ′ , q ′′ ,

. . } or |ı(q, a)| ≥ 2 (ambiguous) or there is no next state (when ı(q,) =and so |ı(q, a)| = 0) (undefined). In the example in Fig. 1, the NFA0 has ambiguity at state A with symbol 0, ı(A, 0) = {A, B}. It also hasndefinedness for 0 at B and for 0 and 1 at C, ı(B, 0) = ı(C, 0) = ı(C,) =. These are the causes of the problems, stuck and lost explained

ater, under the probabilistic interpretation of non-determinacy.

.6. Probabilistic interpretation of non-determinacy

In the theory of computation, non-determinacy is defined withacktrack and/or parallel execution. Under the interpretation, weay that an NFA accepts an input word w if only there exists a properomposite transition corresponding to w, even if other transitionsorrespond to w.

The standard interpretation of non-determinacy is too stronghat there is no actual difference between NFA and DFA in behavior.ence, in this study, We interpret non-determinacy as uniformlyrobabilistic. An FA can take only one state at once and backtracking

n the input process is not allowed. When there are multiple transi-ions for an input and a state, one is randomly chosen. Because whate model is internal measurement of thinking, that is in action

n real time, it is natural to assume that there is no time or otheresources to backtrack or to arrange all the possibilities.

.7. Transition and language

DFA A accepts a word w = w0w1· · ·wn−1 if there is a composite

ransition from the start state to a final state with the concate-ation of labels identical to w. In other words, with A’s start state0 ∈ Q and the set of final states F ⊆ Q, there exists a composite tran-ition q0→w0 q1→w1 · · · →wn−1 qn satisfying qn ∈ F. We call such a

tems 107 (2012) 95– 105 97

composite transition from the start state to a final state proper. Incontrast, under our probabilistic interpretation, NFA works differ-ently, exposed to the possibility of failure as in Proposition 3.4.

Definition 2.1. (Proper transition) A composite transition q0s0→

q1s1→ · · · sn−1→ qn (qi ∈ Q, si ∈ �, i ∈ {0, . . ., n}) on FA A = (Q, �, ı, q0, F)

is proper if qn ∈ F holds.

Definition 2.2. (Extended transition function) The extended tran-sition function ı* : Q × �* → X with the argument (state, string) isdefined from a transition function ı : Q × � → X with the argument(state, symbol) is defined as follows: when ı is deterministic, X = Q,and ı*(q0, �) : = q0 and ∀x ∈ �*, ∀ a ∈ �, ı*(q0, xa) : = ı(ı*(q0, x), a).When ı is non-deterministic, X = P(Q ), ı*(q, �) : = {q} and ∀x ∈ �*,∀ a ∈ �, ı*(q, xa) : =

⋃p∈ı(q,x) ı(p, a).

Definition 2.3. (Language expressed by FA) We denote the lan-guage expressed by FA A by L(A). L(A) is the set of words that Acan accept. If A = (Q, �, ı, q0, F) is a DFA, L(A) = {w|ı∗(q0, w) ∈ F}.If A is an NFA, L(A) = {w|ı∗(q0, w)

⋂F /= }.

Regular expression is broadly used for computational tasks suchas searching a string in a text. Given a regular expression, we canconstruct an NFA or a DFA that exclusively accepts or rejects thestrings which it expresses. There is a procedure called subset con-struction that converts any NFA G to an “equivalent” DFA R withL(G) = L(R).

2.8. Determinization by subset construction

G0 and R0 in Fig. 1 is a pair of grammar and recognizer (R0 =sc(G0)). Both accept a word 0101. In other words, 0101 ∈ L(G0) =L(R0). G0 has a proper composite transition A

0→A1→A

0→B1→C while

R0 has the unique one, {A} 0→{A, B} 1→{A, C} 0→{A, B} 1→{A, C}.Input of a string is conversion from a string to a composite tran-

sition. The conversion is unique on DFA but on NFA it may fail. Thecondition for string acceptance differs between NFA and DFA. Wesay that an NFA can accept a string if there exists a proper compos-ite transition for the string. There may be other proper transitionsand non-proper composite transitions leading to failure.

On the other hand, on DFA, the same input causes the samebehavior. When there is a proper transition corresponding to astring on a DFA, the DFA necessarily accepts the string. We can saythat sc() is an operation that converts possibility to necessity inregard to acceptance.

The basic idea of sc() is to form a collective state consisting ofthe states reachable from a state for each input. At first the startstate is the only reachable state. So sc() starts at the singleton ofthe start state, which becomes the start state of the recognizerin construction. Then, the reachable states are made a collectivestate of recognizer. For each formed collective state, the collectionis repeated. The procedure sc() stops when the transitions from allthe recognizer states are defined and the number and kind of thetransitions agree the number of alphabet. It means that the con-structed finite automaton is deterministic. Recognizer’s final statesare all the collective states that have a grammar’s final state. Thedefinition is as follows. For the justification proof, see (Hopcroft andUllman, 1969).

Definition 2.4. (Subset construction) The procedure subset con-struction sc() is defined as follows. It forms the DFA R = sc(G)= (QR, �, ıR, {q0}, FR) from an NFA G = (Q, �, ı, q0, F), equivalentin the sense that it satisfies L(R) = L(G).

1. Let {q0}, the singleton of G’s start state q0, be the start state of R.2. For each newly formed state P of R, construct the transition for

all s ∈ � as follows:

9 BioSystems 107 (2012) 95– 105

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possible transitions from qi are qi = {qi0→ q, qi

0→ q′, . . . , qi1→q ′′ ,

qi1→q ′′ ′ , . . . }, the next transition is chosen from qi with the uni-

form probability 1/|qi|. We exclude the states not reachable to any

word

self- recognition

8 T. Takahashi, Y.-P. Gunji /

(a) When P = {q1, q2, . . ., qk}, let ıR(P, s) =⋃k

i=1ı(qi, s).(b) If ıR(P, s) appears first, make it a new state and execute 2 for

it. If not, go to 3.. For R’s states which include any of G’s final states, make them

the final states of R.

It is the objectification process from first to third person. On thether hand, subjectification from third to first is ambiguous. As ayriad of subjective perspectives correspond to an objective one,

here are a myriad of NFAs equivalent to a DFA. We can also get anquivalent NFA from a DFA by just converting the type. This fact issed in the proof of Proposition 3.4.

The grammar–recognizer pair can be understood as micro-acro pair in respect to distinguishability as well. One of the

uthors has formalized a hierarchy of micro and macro perspec-ives by lattice and quotient lattice (Gunji et al., 2006). In that study,he hierarchies are connected by a sheaf-like operation (Tennison,975). In this study, it corresponds to subset construction. It islmost a closure operator since it is extensive (G ≤ sc(G)) wherehe order relation is defined to be of the number of states), idem-otent (sc(sc(G)) ∼= sc(G)), and it seems to be monotonous under anppropriate definition of a type of complexity of FA (if G ≤ G′ thenc(G) ≤ sc(G′)). It gives a description of higher order in which thenternal structure of the DFA states is neglected. The informationf the individual transitions in the original grammar is lost after its converted to the equivalent recognizer.

.9. Occam’s razor: minimizing third person description

There are three points that justify the use of non-determinacynd determinacy as the model of first and third person. First, theepresentation of definite-indefinite duality of first and third per-on descriptions (Tables 1 and 2). The second is the existence ofbjectification procedure from first to third person, subset con-truction. The third point is a reduction procedure of third personescription mentioned below. It is analogous to Occam’s razor andorresponds to the construction of the simplest objective theorymong theories with the same explanatory capability, which is theorm of science.

The reduction procedure is denoted by min(). It constructs themallest DFA MR = min(R) equivalent to a DFA R. We call MRinimal recognizer when R is called recognizer.min() ascends one more logical level as sc(). We define equiva-

ence classes of states in relation to acceptance. The classes are thenonsidered to be individual states. Two states q and q′ in a DFA Rre equivalent if they satisfy ∀w ∈ �∗, (ı(q, w) ∈ F ⇔ ı(q′, w) ∈ F).ransitions are naturally induced since the equivalent states agreen the target of transitions from them. R0 in Fig. 1 is congruent toin(R0). G1, R1 and MR1 in Fig. 2 are respectively grammar, rec-

gnizer and minimal recognizer. The states {C, D} and {B, C, D} areollected up in MR1 because they both belong to the equivalencelass ({{C, D}, {B, C, D}}).

The number of states of the minimized DFA can be useds the complexity measure of the original NFA(grammar) andFA(recognizer) (Wolfram, 1984). This measure is useful since it

s difficult to uniquely simplify regular expressions, which is a kindf intensional description of language.

efinition 2.5. (DFA minimization) We obtain the minimized DFAR = min(R) from a DFA R = (QR, �, ıR, {q0}, FR) as follows:

. Partitions R’s states into final states FR and non-final states

QR\FR.

. Partition each group’s states into ones with the same targets andthe other ones.

. Iterate (2) as far as the groups can be partitioned.

Fig. 2. An example of grammar, recognizer and minimal recognizer, G1, R1 = sc(G1)and MR1 = min(R1)

4. Make the groups the states of MR. The transitions and the spec-ified states are naturally defined. The transitions from a group isthe one that all states in the group agree. The start state and thefinal states are ones containing them of R.

3. Model of inner dialogue

Thus we have formalized first and third person. Now we defineinner dialogue process as the model of thinking with inner speech.First we review the problems proper to first and third personself-recognition, caused by our probabilistic interpretation of non-determinacy.

3.1. Self-recognition in first and third person

In our inner dialogue model, FA repeats to output a word andinput it again. Inner dialogue is repetition of utterance and recogni-tion (See Fig. 3). On an FA, self-recognition is recognition of the wordwhich was or could be uttered by the FA itself. The largest differ-ence between first and third person shows itself in self-recognition.We let A = (Q, �, ı, q0, F) be an FA in the following definitions andpropositions.

Definition 3.1. (Utterance (output)) For A, utterance of a wordw = w0w1· · ·wn−1 results from formation of a proper transition

q0w0−→ q1

w1−→ · · · wn−1−→ qn ∈ F . The proper transition is built up by akind of random walk with the uniformly assigned probability. Letus define ı|qi

:=⋃

s∈�ı(qi, s), the number of transitions out of thestate qi. When it is at a final state qi ∈ F at the ith step of the tran-sition formation, the utterance process stops with the probability1/(|ı|qi

| + 1) and the output word w is completed. Otherwise, if the

utterance

Fig. 3. The scheme of a turn in inner dialogue. The dynamics changes according toin which person is the agent, first, second or third.

BioSystems 107 (2012) 95– 105 99

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ing stuck or lost. We define a way to overcome stuck and lost, bothbeing rewrite of NFA transition structure.

First person lost resolution is simple. It is at 5 in the definitionbelow. In a self-recognition process, if the finally reached state is

T. Takahashi, Y.-P. Gunji /

f final states and the transitions to them from the FA. It is becauseith such states there are possibilities that the utterance process

ets stuck, i.e., at the ith step with the present state qi, there is noossible transition,

⋃s∈�ı(qi, s) =. The “sink state” such as the empty

tate {} of R1 or {{}} of MR1 in Fig. 2 are automatically excluded.f the start state is a final state, q0 ∈ F, the empty word � with length

can be outputted.

efinition 3.2. (Recognition (input)) The recognition of a word = w0w1· · ·wn−1 on A is a process that replaces the characters in

with a proper transition q0w0−→ q1

w1−→ · · · wn−1−→ qn ∈ F . At 0th step its at the start state q0 ∈ Q. At the ith step at the state qi, it replaces w’s

th character wi with a transition qiwi→ q. Here q is the state ı(qi, wi)

f A is a DFA and is arbitrarily chosen from ı(qi, wi) if it is an NFA.hen move to the state q ∈ ı(qi, wi) and make it the next state qi+1,o to the i + 1th step. When i = n, if qn is a final state of A, we sayhat the recognition of w on A succeeds.

s we show in Proposition 3.4, there are two cases of failure ofecognition on NFA, stuck and lost. Stuck is failure in execution ofecognition. Lost is failure that becomes apparent at the last recog-ition step.

efinition 3.3. (Self-recognition) We call recognition of the word ∈ L(A) on A itself self-recognition.

roposition 3.4. (Self-recognition incompleteness of NFA: stuck andost) If A is NFA, self-recognition on A may fail. There are two cases ofailure, stuck and lost. It is stuck if the next transition is undefined, i.e.,(qi, wi) =, at the ith step (i /= n), when trying to replace wi with aransition. It is lost if it is at the last step (i = n) but the finally reachedtate qn is not final (qn /∈ F).

ketch of Proof. By exemplification. G0 in Fig. 1 can output w =101 since it has a proper transition A

0→A1→A

0→B1→C. G0 may fail in

elf-recognition of w. It is because proper transitions A0→B

1→C0→ †

stuck) or A0→A

1→A0→A

1→A (lost) may be chosen in the randomormation of the composite transition.

roposition 3.5. (The self-recognition completeness of DFA) If A isFA, self-recognition on it does not fail, i.e., self-recognition of anyord w ∈ L(A) does not get stuck nor lost.

roof. We let the word recognized by A be w = w0w1· · ·wn−1 ∈(A). Since A is DFA, the type of the transition function is

: Q × � → Q. We “lift” it and construct an NFA A = (Q, �, ı, q0, F).ere we define ı : Q × � → P(Q ), ∀q ∈ Q, ∀s ∈ �, ı(q, s) = {ı(q, s)}.

is congruent to A and equivalent to it in the sense L(A) = L(A)s well. It is since ı∗(q0, w) ∈ F ⇔ ı∗(q0, w)

⋂F /= holds. By defini-

ion, ∀q ∈ Q, ∀s ∈ �, |ı(q, s)| = 1. Therefore ∀w ∈ L(A), |ı∗(q0, w)| = holds. It means that it does not get stuck since A can get stuck

ost if and only if ∃q ∈ Q, ∃s ∈ �, |ı(q, s)| = 0. On the other hand, itan get lost if and only if ∃w ∈ L(A), ∃q ∈ ı∗(q0, w) ⊆ Q and q /∈ F.owever, ∀w ∈ L(A), ∀q ∈ ı∗(q0, w), q ∈ F holds because L(A) =(A) = {w|ı∗(q0, w) ∈ F} by q ∈ ı∗(q0, w) ⇔ q ∈ {ı∗(q0, w)} ⇔ q =∗(q0, w).

If an FA is deterministic, it is self-recognition complete. Theonverse does not hold since there are counterexamples easilyomposable.

.2. (A)symmetry of inner dialogue in first and third person

The self-recognition incompleteness of NFA is a result ofhe context of utterance not conserved. It daily happens thate misread a sentence composed by ourselves, subjectively and

mbiguously, when we forget about the context of the composition.

Fig. 4. Inner dialogue asymmetry on NFA and symmetry on DFA. The process isreversible on DFA.

In first person inner dialogue, speaking and hearing are asymmetric(Fig. 4). Utterance is unique but recognition is ambiguous.

In contrast, speaking and hearing is completely symmetricin third person inner dialogue. Utterance is an unambiguousconversion from a proper transition to a word. Recognition is alsouniquely determined because of the determinacy. Third personinner dialogue is reversible; there is no discrepancy between speak-ing and hearing. The complete context is always given and thusshared by the selves as a speaker and a hearer. Even if we forget thecontext, it stays well-defined since the dialogue has an unaffectedstructure unambiguously arranged. Therefore, third person innerdialogue brings about no change.

In first person inner dialogue, when it self-recognizes, the infor-mation on which path the state transition in the utterance actuallyfollowed is completely lost. It is an absence of oracle in the sensethat there is no supervisor or transcendent who tells which way togo when there are multiple possible paths. Contrastingly, in thirdperson self-recognition, we need no oracle or in effect we have acomplete oracle.

3.3. Inner dialogue

We define a turn in inner dialogue process as a coupling of utter-ance and self-recognition (Fig. 5). There are three kinds of innerdialogue: first, third and second person inner dialogue hereafterdefined.

3.4. First person inner dialogue

Here we define first person inner dialogue, a combination ofutterance and first person self-recognition. First person descrip-tion/perspective, NFA, is self-rewritten through the dialogue. Asmentioned above, first person self-recognition can fail encounter-

Fig. 5. The conceptual scheme of time development of inner dialogue. Ft and wt arerespectively the FA and the uttered word at the tth turn. It is first (third) personinner dialogue if F is NFA (DFA).

1 BioSystems 107 (2012) 95– 105

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Table 3Three persons in respect to the input–output symmetry and oracle. s and h areabbreviations for speaking and hearing.

Mode of self-recognition Symmetry Oracle

First person s /= h Absent

00 T. Takahashi, Y.-P. Gunji /

nal state, it is made a final state. Another final state is made aormal state instead, conserving the number of final states.1

On the other hand, in first person, there is little guideline fortuck resolution. It is possible to search the most conservativeewrite which adds a transition from the stuck state to another stateo the NFA, so as to keep the regular expression of the NFA stayinghe most unchanged. However, it is not realistic since it requires

assive calculation. Consequently we define the stuck resolutionandom as in 3 below.

efinition 3.6. (First person self-recognition) We define firsterson self-recognition of w = w0w1 . . . wn−1 ∈ L(G) on NFA G =Q, �, ı, q0, F). i indicates step.

. (Initial setting) Make the present state q0.

. (Step, 0 ≤ i ≤ n)(a) (Termination check) If i = n go to 5.(b) (Stuck check) If ı(qi, wi) is empty, i.e. stuck, go to 3.(c) (Random transition) Otherwise go to 4 with a state q ran-

domly chosen from ı(qi, wi).. (First person stuck resolution) Randomly choose a state q from

Q. Then add the transition qiwi→ q from the present state qi to the

chosen q. Remove another transition randomly chosen.. (Preparing for the next step) Make the present state qi+1 = q, set

i forward, and go to 2 again.. (Termination processing: lost resolution) Terminate the recog-

nition process. If qn /∈ F, add qn to F. Randomly choose a state fromF \ {qn} and remove it from F.

Stuck and lost resolution in first person respectively conserveshe number of transitions and final states. This form of inner dia-ogue exhibits no interesting results (Takahashi and Gunji, 2008).f the NFA is actually deterministic, it causes no change as in thirderson below.

.5. Third person inner dialogue

Contrastively, third person self-recognition causes nothing. Nohird person rewrite takes place. It is because there is no failure,tuck and lost, as the chance for change; everything goes smoothly.t is the effect of the symmetry of speaking and hearing or of theonservation of context in third person. A memo objectively writtenn third person clearly specify all the requisite elements of the con-ext. There is no possibility for misreading. The self is transparentnd communication is ignorable (Derrida, 1968). It is monologue.

. Second person inner dialogue

Inner dialogue in first and third person produced no interest-ng result. Dynamic self-relation of onefold subject is disorderedr poor. Here we define a recognition process which resolves firsterson stuck by partially referring to third person perspective, sec-nd person self-recognition. In this process, first and third personnteracts with one another, which is the expression of dynamicaluality of first and third person.

.1. Incomplete first and third in second person: surplus andndless dialogue

We can use indexical expressions and concepts in second per-on. However, the use of both is more incomplete compared to firstnd third person. It is because second person indexical expressions

1 If it monotonously increases, the range of the acceptable words expands toouch and/or inner dialogue process easily converges to a certain state.

Second person s /= h ⇔ s = h InternalThird person s = h Unnecessary

depend on the understanding of the hearer and the use of conceptsis exposed to skepticism in dialogue (Kripke, 1982). The otherswith whom we construct second person description may misun-derstand, express skepticism, dispute and break in. The others arenot in control of the subject.

The relationship with the self sheds light on the duality of thesubject. It is as in self-referential sentences. The dual subjects arein first and third person. The former is “I,” the self referring to andthe latter is “me,” the self referred to. Conversation is based on thebilateral relationship. Conversation with the self is between “I” and“me.”

As the expression of this duality, we adopt comings and goingsbetween first and third person. The comings and goings try to thegap between first and third person. However, the new utterancetrying to infill the gap itself engenders a new gap. Therefore con-versation does not get converged. Things to say does not evaporate(Matsuno, 1989; Bakhtin, 1986).

4.2. Second person self-recognition

Since second person self-recognition is based on first person, itis exposed to the possibilities of failure, stuck and lost. It gets stuckin places because of the undefinedness even in self-recognition, dueto the ambiguity of first person description. It often gets lost as well.

Each step in second person self-recognition is a successionof generalization and specialization. It identifies or confuses thepresent state on the grammar with the present collective state onthe recognizer. It is generalization. First it traces a transition in thirdperson. Then it identifies or confuses the next collective state withthe next specific state, determining the latter. It is specialization.

The undefinedness producing stuck is resolved by a rewrite.The rewrite joins the flats by changing first person retrospectively,referring to third person. The change in first person, of course,spreads to third person which is constructed from first person.However, the local structure where first person is made consistentby the confusion with third person stays invariant. On the otherhand, in its neighborhood, new inconsistencies as discrepanciesbetween first and third person arise again (see Section 6).

In the following definition, (3) is the stuck resolution and (5) isthe lost resolution. (6) means the halt of recognition, on the occa-sion of the second stuck entailed by the change of third person inthe recognition process. Because we are modeling inner dialoguewith internal measurement, the agent has no time or computa-tional capacity to reconstruct third person from the changed firstperson in the recognition process. Second person in comparison tofirst and third persons is in Table 3, denoting speaking (utterance)and hearing (recognition) by s and h. While in first and third person,s /= h and s = h, respectively, in second person self-recognition, theprocess of making s /= h to s = h, effort of joining the flats, conducess /= h again.

Definition 4.1. (Second person self-recognition) We define sec-ond person self-recognition of w = w0w1 . . . wn−1 ∈ L(G) on NFA G =

w w

(Q, �, ı, q0, F). A proper transition P0 = {q0} 0−→ P1

1−→ · · · wn−1−→ Pn

for w on R = sc(G) is determined by Pi :=⋃

p∈Pi−1ı(p, wi). We make

use of each Pi, the set of reachable states at each step i.

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. (Initial setting) Make the present state q0 and the present possi-ble states P0 : = {q0}.

. (Step, 0 ≤ i ≤ n)(a) (Termination check) If i = n, go to 5.(b) (Stuck check or transition) If ı(qi, wi) is empty, i.e. stuck, go

to 3, or if it is the second time to get stuck in this recognitionprocess, go to 6. Otherwise go to 4 with a state q randomlydetermined from ı(qi, wi).

. (Second person stuck resolution) Randomly choose p fromPi \ {qi} satisfying |ı(p, wi)| ≥ 1 and q from ı(p, wi) ⊆ Pi. Remove

pwi→ q from G and add qi

wi→ q to it.. (Preparing for the next step) If i = n, go to 5. If not, make the

present state qi = q and the present possible states Pi+1, incrementi, and go back to 2.

. (Termination processing: lost resolution) Terminate the recog-nition process. If qn /∈ F, add qn to F. Randomly choose a state fromF \ {qn} and remove it from F.

. (Handling second stuck) When it gets stuck the second time, quitthe recognition process.

Second person self-recognition does not make any differenceith first person one unless it gets stuck and with third person onenless there is ambiguity or undefinedness. In second person it does

transition toward the next step 2 and resolves lost 5 in the sameay as in first person. However, in each step the process in seconderson comes and goes between first and third person. Since thirderson is the meta level of first person, generalization and special-

zation, going up and down the levels, are realized alternately. Thisuccession of generalization and specialization resolves stuck. Therocedure 3 is the most important, that is rewrite of the transitionaltructure for stuck resolution, proper to second person. The rewriteonserves the number of transitions as well.

.3. Succession of generalization and specialization: stuckesolution in second person

On the other hand, in second person, rewrite in stuck resolutionhanges the first person structure to conserve the local third persontructure.

Stuck resolution in second person self-recognition 3 is resolvednto five steps, as summarized in Table 4. Let us recall that the objectn first person is individual whereas in third person it is concept.he graphical scheme is given in Fig. 6. (3a) is generalization and3c) is specialization. Each step goes across the object and metaevels.

a) (Generalization) Move from an individual state qi ∈ Pi at theobject level to a collective state Pi at the meta level.

b) (Meta level inference) Trace a third person transition Piwi→ Pi+1.

(c) (Specialization) Move from Pi+1 to qi+1 ∈ Pi+1. When Pi+1 contains

multiple states, the choice of the next state, qi+1, which state tomove on, is ambiguous and hence randomly chosen.

d) (Rewrite of Relation in the Object Level) If there is no qiwi→ qi+1,

i.e. stuck at qi, then add a transition qiwi→ qi+1 and remove a

able 4ive steps of stuck resolution in second person self-recognition.

Step Logical type

(1) Conceptualization Object→meta

(2) Meta inference Meta

(3) Objectification Meta→object

(4) Rewrite (if needed) Object

(5) Object inference Object

Fig. 7. Grammar G2, recognizer R2, and the composite transitions on them corre-sponding to w = 010.

randomly chosen transition rwi→ qi+1 satisfying r ∈ Pi. It replaces

r, the source of the transition rwi→ qi+1, by qi.

(e) (Object level inference) Trace an individual transition qiwi→ qi+1.

First it transits in third person and then, according to the transition,rewrite the first person structure if needed, resolving stuck.

Any language use is in some sense generalization. Thinking ismediated by the generality of language (Vygotsky, 1986). We can-not directly use concrete objects in thinking that is conceptual. Itmay change on which relation, between concrete objects, to focusand sometimes alter the relation.

4.4. An example of second person self-recognition

As an example, we consider second person self-recognition ofw = 010 on G2 illustrated in Fig. 7.

In the second step illustrated in Fig. 8, stuck at object level isresolved by (1) generalization to meta (C

∈→{B, C}), (2) meta transi-1

tion ({B, C}→{A}), (3) specialization to object level ({A}→A), and

finally (4) “joining the flats” at object level (altering B1→A into

C1→A).

Property Form

Unique qi∈→Pi

Unique Pi

wi→ Pi+1

Choice of qi+1 ∈ Pi+1 is ambiguous Pi+1 →qi+1

Choice of r ∈ Pi is ambiguous (rwi→ qi+1) ⇒ (qi

wi→ qi+1)

Unique as a result of (3) qi

wi→ qi+1

102 T. Takahashi, Y.-P. Gunji / BioSystems 107 (2012) 95– 105

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Initial setting) First the present state is the start state q0 = A. Thefirst possible states are P0 = {A}.

i = 0) Since the first character is w0 = 0, the next possible statesare ı(q0, w0) = ı(A, 0) = {B, C}. Let q = C. Now the presentstate is q1 = q = C and the present possible states are P1 = {B,C}.

i = 1) The second input character is w1 = 1. Now it is stuckbecause ı(q1, 1) = ı(C, 1) =. So we randomly choose a stateq ∈ P2 = {A, C} and let it be q = A. Then r is randomly cho-sen from P1 \ {q1} = {B, C} \ {C} = {B}, in this case uniquely

determined as r = B. Remove the transition (B1→A) and add

(C1→A). Make the present state q2 = q = A and the present

possible states P2 = {A}. i = 2) The third, last input character is w2 = 0. Since ı(A, 0) = {B,

C}, let be the transition target q = B is chosen in the ran-dom way. Make the present state q3 = q = B and the presentpossible states P3 = {B, C}.

i = 3) In this step, termination processing is executed, sincei = 3 = n. Because it is lost at B /∈ F = {C}, add B to F, andrandomly choose another final state to remove, in thiscase uniquely C. Now F = {B} and the recognition processis terminated.

. Results

We simulated the second person inner dialogue. Our model, sec-nd person inner dialogue process, has no parameter. Initially givenrst person perspective (grammar) is the only initial setting. Nev-rtheless, the process frequently shows self-organization such asntermittency and criticality, in the time development of the gram-

ar complexity, the number of nodes of the minimized recognizer,enoted by |MRt| = |min(sc(Gt)), representing the complexity of therammar Gt.

For the time development shown here, we define the initialrammar G0 := G3 in Fig. 9 with some degree of size, because G0, G1nd G2 are all too simple to observe time development by stuck

esolution.

In the temporal change in the grammar among some types, webserve six types of MRs constructed from the grammars (Fig. 10).he initial G0 = G3 is the most complex and it gets simplified as the

Fig. 10. The six types of MRs appearing in the time development, a, b, c, d, e, and f,with the number of states 3, 3, 3, 5, 6, 7, respectively. Type f is min(sc(G3)). Type aand c are isomorphic.

results of second-person stuck resolutions. The transitional graphstructure among MRt is in Fig. 11. The nodes in the figure representthe group of MRs with the identical |MR|.

As the measure of the time development of our second-personinner dialogue process, we calculate |MRt| as the complexity foreach Gt (Wolfram, 1984). The plot along the time axis shows theintermittent behavior in Fig. 12. In Fig. 13, the length of the periodsin which |MRt| is identical is shown. There are 175 periods in the first1000 turns. As in Fig. 14, the cumulative frequency roughly obeysthe power law with the exponent −1. We can see the alternation of“laminar” periods with longer and simpler FA structure and shorterand more complicated “burst” periods.

6. Discussion

We circumstantially investigate the properties of second per-son stuck resolution. It is what drives the organized development

Fig. 11. The transitional graph structure of MRt for 10,000 turns in the time devel-opment. The number on the node is the complexity |MR| and the label on the edgeis the number of transitions.

T. Takahashi, Y.-P. Gunji / BioSystems 107 (2012) 95– 105 103

200 400 600 800 10002

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0 50 100 150

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tion is caused by the discrepancy between first and third personsthat it gets stuck in first person but it is possible to transit inthird person. The movement of correcting structural disequilib-rium or symmetrizing asymmetry causes new inconsistencies or

ig. 14. The log–log plot of the cumulative frequency for the first 10,000 turns. Theolid line is f(x) = x−1.

r inconsistency. In our second person model, organizationnd deviance from it have the same root, different in the senserom the equilibrium theory of economics where the movementonducting the equilibrium and the movement displacing the equi-ibrium are completely distinguished (Matsuno, 2000). The internalelation of organization and disorganization in our model is whatenerates our result.

.1. The properties of the second person rewrite

In second person self-recognition, stuck resolution may happeny rewriting first person perspective with local reference of thirderson perspective. The accompanied change in the third personhows the dynamics of structural equilibration = disequilibrationchematized in Fig. 15 (Matsuno, 1989). In Fig. 16, the conservedtructure and the altered structure in the example in Section 4.4 ishown.

.1.1. Coordinated and conserved structureWhat the rewrite conserves is the total number of transitions,

he number of final states and the DFA local structure referred in

discrepancy

new disc repancies

structural equilibration

structural disequilibration

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ig. 15. The succession of making consistent of an inconsistency resulting from sec-nd person stuck resolution and the regeneration of inconsistency by the resolutiontself.

n

omplexity of grammar, |MRt|, for the first 1000 turns.

the rewrite. The local structure is maintained in the rewrite as anorm. It is a transition from a collected state including both statesinvolved in the rewrite, one added and one removed. In the example

in Fig. 16 the collected state is {B, C}. The transition {B, C} 1→{A,C} is reserved through the rewrite. Consequently, if there were acollected state like {A, B, C} in R2, the unique 1-transition from itwould be conserved.

On the other hand, in the case of Fig. 16, 1-transitions {C} 1→{}and {A, C} 1→{C} with the sources {C} and {A, C} change. The transi-tions altered by a rewrite are from the collected states that includeonly one of the two states involved in the rewrite, where states ina collective state are inconsistent with one another, one added andanother removed a transition. In the example, they are B and C. Thediscrepancy may bring about a stuck at the first step for the word0010 in the next turn.

6.1.2. Equilibration = disequilibration in structureIt is extremely rare that the grammar becomes isomorphic to

the recognizer through development. The difference in logical typebetween the two perspectives never dissolves. While the locallybound change in first person keep the corresponding local structureunchanged, it generates a new inconsistency in its neighborhood.

The rewrite by stuck resolution models the basic scheme ofinternal measurement, equilibration = disequilibration (Matsuno,1989; Gunji, 1995). The rewrite of first person as the stuck resolu-

Fig. 16. The movement to pursue consistency itself brings about inconsistencies inthe example in Section 4.4. The conservation and change are depicted by the solidand broken arrows.

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iscrepancies. This is the property of second person rewrite, thats local and internal.

The rewrite resembles bad debugging in computer program-ing. A local fix causes inconsistencies with the other parts. In

rogramming, however, it is often possible to rewrite from scratchince the system can be paused and the objective is clear. In con-rast, in inner dialogue, if the system (subject) stops, everythingtops. Ad hoc self-correcting is needed. It is like a sailing shipust be repaired without docking, as for the Neurath’s boat (Quine,

960).

.2. Thinking and language, “problem” vs. “language”

In the theory of formal language, the two concepts languagend problem are treated as equivalent. A language is a set of sen-ences, while a problem is the question to determine whether aiven sentence belongs to a language or not. These are extensionalnd intensional descriptions. In a formal representation, languages of type L ⊆ �* while problem is of p : �* → 2, p = T if w ∈ L, F if

/∈ L.Given an alphabet �, there is a language L as the subset of the

et that forgets the algebraic structure of the free monoid generatedn �, and a problem of determining whether the elements in �*

elongs to L or not, p : �* → 2, p = T if w ∈ L, F if w /∈ L. The language Lnd the problem p are made equivalent. It corresponds to the equiv-lence between subsets and characteristic functions in set theory,etween the powerset P(A) and 2A where A is a set. While in setheory, there is neither discrepancy between defining intension andxtension, nor asymmetry, in theory of computation, problem has aarticular implementation, such as a finite automaton. Parallelism

s only doubtfully interpreted to seriality.The equivalence of language and problem annihilates time.

he probabilistic interpretation of non-determinacy in this studyeakens the equivalence between language and problem and it

eestablishes time. In our model, second person self-recognition,now” is the dynamic interface of a non-deterministic state and aeterministic collective state.

.3. Application for music

One of the possibilities of the application of our model is in per-ormance art. The model can handle a quite broad range of rules,ecause it is any FA that we can give to it as the initial rule. It can beasily applied to arts with a notation such as music and dance. FAs known to be eminently valid for analysing ethnic music whereetrachords play a central role (Sibata, 1978), called skeleton the-ry. We can then simulate evolutions of musical structure. We arereparing a music generation system with our model and the the-ry. In this application, defining a dialogue between us and theystem, we can also use the system as a partner of our improvisa-ion,

Solo improvisation is rightly an inner dialogue. An action isrought out, motivated by a player, and the player receives thection. It is in some sense externalized and symbolized. This justi-es the use of our model as well. Additionally, intermittency andriticality resulting from our model may be significant there.

.4. Micro, meso, macro: wave-particle duality

Second person represented in our model can be interpreted aseso between micro (first person) and macro (third person). Meso

cale is apparently important for considering biological systems

Igamberdiev, 2005). It can be defined as a scale where both of

icro and macro logics can not be neglected. If we express micros non-deterministic, probabilistic relationship among particle-likebjects and macro as deterministic and unique relationship among

tems 107 (2012) 95– 105

distributions, the pair of non-determinacy and determinacy exactlycorresponds to micro and macro, as particle-like and wave-like. Oursecond person model represents a situation where the propertiesboth as particles and waves is not neglected. It is also important tostudy the universality of criticality in meso scale in this regard.

6.5. Stuck resolution in the wild

Discrepancy between two pictures, that are at object and metalevels, drives the dynamics our model. It is found as a stuck andthe succeeding stuck resolution. Where do we find a stuck and itsresolution in nature, not merely as a theoretical construct? It isubiquitous in natural phenomena. Open physical systems in whichthe boundary is indefinite or indeterminate, components of thesystem cannot obey the system law in complete accordance or inprescribed harmony (Matsuno, 1989).

We animals move (object) and see (meta). Their consistencyis not warranted, especially with nervous systems, but they mustbe coordinated so as to postpone the advent of ruinous discrep-ancy. If we see and then move, it may be too late. We move firstand then sometimes see, changing the way to see, just as in oursecond-person stuck resolution. To realize the critical dynamicsdriven by stuck resolution, we have been practicing experiments onfish school. The twofold dynamics of an individual and the swarmand the scale-free correlation (Cavagna et al., 2010) is where wecan empirically test our theory, that local discrepancy and its res-olution is the key of our flexibility that has been missing in AI andA-Life from the outset (Matsuno, 1989; Conrad, 1983).

7. Second person at the foundation

Haruna and Gunji modeled second person with respect to theplurality of the context and the dynamical relationship betweencontext and law (Haruna and Gunji, 2007). What we have achievedhere is the formalization of second person self-relationship. It isnot static and consistent as in first and third person. It is mutualcomplement of incompleteness of first and third person.

Only in the self-relationship in first or third person, a self maybe onefold, tautologically identical. In second person, not only therelationship with the others but also the difference of dual sub-jects drives the change and development of the subject. Identity isnot static. It is absolutely an endless process in which the surplusof identification results non-identity to be resolved and identifiedagain (Vygotsky, 1986; Bakhtin, 1986; Matsuno, 1989).

Self-referential contradiction matters only when the subjectsin first and third persons are substantively established and thenthe relationship between them, identical or not, is inquired. Suchcontradiction is a contradictory state problematic when the bothsubjects are placed in an external relationship. The relationship isexternal in the sense that the both does not get essentially affectedby it.

In second person, first and third persons have an internalrelationship. It is that they change themselves by having the rela-tionship. Rather, precisely speaking, it is not that first and thirdpersons existent before everything start to have a relationshipat a moment. There is a gap between first and third persons orbetween active ad passive. The gap and the movement to infillthe gap precipitate bipolar first and third persons. Second per-son is at the foundation, not first or third. In this sense, ourmethodology to reconstruct second person from first and third per-sons is deconstructive and it does not follow the developmentalsequence.

Being in second person is the condition for all the internalobservers. Internal observers make nothing of contradiction since,primarily, self-referential contradiction is an abstraction. Whatdrives the second person movement is discrepancy as dynamic

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T. Takahashi, Y.-P. Gunji /

ontradictory process. It is not contradiction as contradictorytate.

In this study, we have shown the dynamics of second per-on self-relationship. In second person, discrepancy is not globallyesolved but locally maneuvered around. The local resolutionf discrepancy regenerates new discrepancies in the neighbor-ood. The discrepancies and the local resolution bring about theuccessive change or development. The process shows intermit-ent and critical self-organization. Thinking with inner dialogues such a process. It is the duality of the self that organizeshinking.

cknowledgments

We thank Akane Hara and Fujiko Hayashi for helpful discus-ions. Part of this work was carried out under the support of Grantumbers Q11K-02 and Q11J-08 of Research Institute for Sciencend Technology of Tokyo Denki University and of the Coopera-ive Research Project Program of the Research Institute of Electricalommunication, Tohoku University.

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