Notes on semi-Thue Systems

The main differences between symbolic and morphic formal systems: Instead of the Kleene star for the symbolic universes, morphic universes are generated by the Stirling cross. Symbolic substitution and concatenation is preserving production concatenation. Morphic substitution/concatenation is opening up a system of interactive complexions of derivations. Comparison of substitution based production systems (Thue, Post, Markov) with Hausser’s systems of “possible continuations” of Left-Associative languages is sketched. 1. Semi-Thue Systems 1.1. Production systems 1.1.1. Deconstruction remarks ”It is a gross simplification to view languages as sets of strings. The idea that they can be defined by means of formal processes did not become apparent until the 1930s. The idea of formalizing rules for transforming strings was first formulated by Axel Thue (1914). The observation that languages (in his case formal languages) could be seen as generated from semi Thue systems, is due to Emil Post. Also, he has invented independently what is now known as the Turing machine and has shown that this machine does nothing but string transformations. [...] The idea was picked up by Noam Chomsky and he defined the hierarchy which is now named after him (see for example (Chomsky, 1959), but the ideas have been circulating earlier)." Marcus Kracht 2003, The Mathematics of Language, Rewriting Systems, p. 53 "In formal language theory, languages are sets of strings over some alphabet. We assume throughout that an alphabet is a finite, nonempty set, usually called A. It has no further structure, it only defines the material of primitive letters.” (ibd, p. 16) http://www.lix.polytechnique.fr/~bournez/MPRI/formal.pdf "In formal language theory, languages are sets of strings over some alphabet. We assume throughout that an alphabet is a finite, nonempty set, usually called A. It has no further structure, it only defines the material of primitive letters.” (ibd, p. 16) http://www.lix.polytechnique.fr/~bournez/MPRI/formal.pdf A deconstruction of “sign”, “string” and “set” is necessary to understand morphogrammatics and morphogrammatic semi-Thue systems, morphic finite state machines and morphic cellular automata as introduced in recent papers. A further deconstruction has to go into the topics of finiteness and infiniteness of alphabets and strings. It also has to be seen that the term “sign” is understood as a purely syntactical mark, letter or character and is not involved in any serious semiotical distinctions. The kernel of formal language considerations is the monoid,  = (M, Î, 1) and the Kleene (star) production A*. A monoid is a triple  = (M, Î, 1) where : “Δ is a binary operation on M and 1 an element such that for all x, y, z œ M the following holds. x Î 1 = x (Idempotence) 1 Î x = x (Idempotence) (x Î y) Î z = x Î (y Î z) (Associativity). Hence, a deconstruction of a monoid  has firstly to deconstruct the binary operation (composition) “Δ and then, secondly, more or less as a consequence of it, a deconstruction of the elements of . A deconstruction of the concept of set-theoretical elements has led to the introduction of a new ‘data-type’, the kenogrammatic and morphogrammatic data patterns used in kenomic and morphic cellular automata constructions. Criticism: Just an abstraction more? At a first glance it seems that such a deconstruction which leads from the Kleene product to a Stirling distribution might simply be an abstraction over the set of values producing an equivalence class as it is well known. Hence, Stirling K* = S/eq. There are some academic publications insisting on such profound insight. Furthermore it is trivial to conclude that the same abstraction holds for the introduction of kenomic cellular automata: kenoCA = ECA/eq. In such a view the kenomic rules are just an abstraction of the CA rules. If we consider the situation for ECAH3,2L with the complete rule set 23= 8 and a complete rule range of 22 3 = 256 and the corresponding kenoCAH3,2Lwith an incomplete rule set of StirlingSn2(3, 2) = 4 and an incomplete rule range of StirlingSn2(23, 2) = 128, then results look quite trivially as an abstraction from 23 to 23í2= 4 and 22 3 to 22 /2 = 128. Unfortunately, the complete rule set for the elementary kenoCAH3,4L is StirlingSn2(4, 4) = 15 and not 8. As a consequence of this asymmetry between complete rule sets, different kinds of rules, methods and features are surpassing the classical definitions of CAs without the Stirling approach those new constellations wouldn’t be accessible. • There is not much chance to achieve such a transformation of the concept and functioning of an elementary operation like the composition "Δ in a monoid. Nevertheless, there are some still recent but well elaborated and tested approaches to recognize. The diamondization of composition has been demonstrated in my papers to a Diamond Category Theory. 2 Author Name A deconstruction of the concept of set-theoretical elements has led to the introduction of a new ‘data-type’, the kenogrammatic and morphogrammatic data patterns used in kenomic and morphic cellular automata constructions. Criticism: Just an abstraction more? At a first glance it seems that such a deconstruction which leads from the Kleene product to a Stirling distribution might simply be an abstraction over the set of values producing an equivalence class as it is well known. Hence, Stirling K* = S/eq. There are some academic publications insisting on such profound insight. Furthermore it is trivial to conclude that the same abstraction holds for the introduction of kenomic cellular automata: kenoCA = ECA/eq. In such a view the kenomic rules are just an abstraction of the CA rules. If we consider the situation for ECAH3,2L with the complete rule set 23= 8 and a complete rule range of 22 3 = 256 and the corresponding kenoCAH3,2Lwith an incomplete rule set of StirlingSn2(3, 2) = 4 and an incomplete rule range of StirlingSn2(23, 2) = 128, then results look quite trivially as an abstraction from 23 to 23í2= 4 and 22 3 to 22 /2 = 128. Unfortunately, the complete rule set for the elementary kenoCAH3,4L is StirlingSn2(4, 4) = 15 and not 8. As a consequence of this asymmetry between complete rule sets, different kinds of rules, methods and features are surpassing the classical definitions of CAs without the Stirling approach those new constellations wouldn’t be accessible. • There is not much chance to achieve such a transformation of the concept and functioning of an elementary operation like the composition "Δ in a monoid. Nevertheless, there are some still recent but well elaborated and tested approaches to recognize. The diamondization of composition has been demonstrated in my papers to a Diamond Category Theory. With “ x Î 1 = x” and “1 Î x = x” it follows that the equation “x Î 1 = x = 1 Î x” holds. This is not surprising and has its rock solid foundations in firstorder logic and category theory and their epistemologies. Does it hold for morphogrammatics? Obviously not! The equation might be interpreted as the equality of rightand left-oriented self-identity of the object “x” of a morphism. JX Î 1N x= X x JDiamond idempotenceN J1Î XN x= x X JDiamond idempotenceN f X Î f id Xœ Iter Xœ Accr http://www.thinkartlab.com/pkl/lola/Semiotics-in-Diamonds/Semiotics-in-Diamonds.html Hence, even the simplest presumbtion, namly that X = X has to be deconstructed. As a consequence, the obvious symmetry of A = B iff B = A is not obvious anymore. A deconstruction of associativity of composition follows, at first, quite automatically: The context-independent associativity "(x Î y) Î z = x Î (y Î z)" becomes the contextualized associativity "(X Î Y)|(x; y) Î Z | z = X | x Î (Y Î Z)|(y; z)". Article Title 3 http://www.thinkartlab.com/pkl/lola/Semiotics-in-Diamonds/Semiotics-in-Diamonds.html Hence, even the simplest presumbtion, namly that X = X has to be deconstructed. As a consequence, the obvious symmetry of A = B iff B = A is not obvious anymore. A deconstruction of associativity of composition follows, at first, quite automatically: The context-independent associativity "(x Î y) Î z = x Î (y Î z)" becomes the contextualized associativity "(X Î Y)|(x; y) Î Z | z = X | x Î (Y Î Z)|(y; z)". Diamondization of associativity of composition " x, y, z : Jx Î yN Î z= SEM xÎ Jy Î zN "X, Y, Z ; " x, y, z, u : JJX Î YN Jx; yN Î Z zN u= DIAM JX x Î JYÎ ZN Jy This has consequences for any introductory rule like R0: ö X. There is no simple beginning in a diamond world. Setting a beginning is always multiple, at least double: a beginning as an iterative or an accretive beginning. The act of beginning happens in a context of a beginning and has its own notion in a calculus of beginnings. Hence, R0: ö X becomes diamond R0: ö X | x. Therefore, the statement of a beginning kenogram [kg] of kenogrammatic sequences in a trito-universe TU as in TU = ([1] Tsucc) of a recursive formula is just a beginning of the process of deconstruction of the notions and terms of kenoand morphogrammatics and not an end at all. Nevertheless, diamond-theoretic thematizations had been, more or less, omitted in the proposals on kenomic and morphic cellular automata, finitestate machines and semi-Thue systems. And just the Stirling effect is in focus that is affecting the rules of the morphogrammatic game of semi-Thue systems and cellular automata deconstruction. Hence, the universe of trito-structural kenogram sequences, kgs, TU, remains defined without its diamond environment as TU = [[1] Tsucc], with x + 0 = 0 + x = x. Further deconstructions of the concept of ‘beginnings’ in formal systems at: “Quadralectic Diamonds: Four-foldness of beginnings”, http://www.thinkartlab.com/pkl/lola/Quadralectic%20Diamonds/Quadralectic%20Diamon ds.pdf 4 Author Name Therefore, the statement of a beginning kenogram [kg] of kenogrammatic sequences in a trito-universe TU as in TU = ([1] Tsucc) of a recursive formula is just a beginning of the process of deconstruction of the notions and terms of kenoand morphogrammatics and not an end at all. Nevertheless, diamond-theoretic thematizations had been, more or less, omitted in the proposals on kenomic and morphic cellular automata, finitestate machines and semi-Thue systems. And just the Stirling effect is in focus that is affecting the rules of the morphogrammatic game of semi-Thue systems and cellular automata deconstruction. Hence, the universe of trito-structural kenogram sequences, kgs, TU, remains defined without its diamond environment as TU = [[1] Tsucc], with x + 0 = 0 + x = x. Further deconstructions of the concept of ‘beginnings’ in formal systems at: “Quadralectic Diamonds: Four-foldness of beginnings”, http://www.thinkartlab.com/pkl/lola/Quadralectic%20Diamonds/Quadralectic%20Diamon ds.pdf 1.1.2. Langtonʼs rules for simple linear growth A classical example of a production system is introduced by Langton‘s Lsystem. "Here is an example of the simplest kind of L-system. The rules are context free, meaning that the context in which a particular part is situated is not considered when altering it. There must be only one rule per part if the system is to be deterministic. The rules: (the “recursive description” of a GTYPE) 1) A ö CB 2) B ö A 3) C ö DA 4) D ö C When applied to the initial seed structure “A,” the following structural history develops (each successive line is a successive time step): time structure rules applied (L to R) 1. A : start  2. CB : rule1 on 1. áä 3. DA A : rule3 on 2. C, rule2 on 2. B á  ä 4. C CB CB : rule4 on 3. D, rule1 on 3. A, rule1 on 3. A Christopher Langton, Artificial Life, 1989, p. 26 .l 1 2 3 4 5 6 7 8 9 rule= rule1 .2 .3 .4 0 Ñ Ñ Ñ Ñ A Ñ Ñ Ñ Ñ 0L initial " seed " 1 Ñ Ñ Ñ C B Ñ Ñ Ñ 1L rule 1 replaces A with CB 2 Ñ Ñ D A Ñ A Ñ Ñ 2L rule 3 : C with DA, rule 2 : B with A 3 Ñ C Ñ C B C B 3L rule 4 : D with C; rule 1 : AA with CB' s 4 Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ stop Atomic elements are substituted by unary and binary elements. Binary elements are seen as a concatenation of 2 identical unary elements. Because of this atomism or elementarism a kenomic abstraction is empty, i.e. all unary elements are kenomically equivalent. Because rewritting systems are substitutional systems the point of substitution in this case is atomistic. Article Title 5 Atomic elements are substituted by unary and binary elements. Binary elements are seen as a concatenation of 2 identical unary elements. Because of this atomism or elementarism a kenomic abstraction is empty, i.e. all unary elements are kenomically equivalent. Because rewritting systems are substitutional systems the point of substitution in this case is atomistic. kenoCAH2L rule set R1 ‡ ‡ ‡ ‡ R2 ‡ ‡ · ‡ R3 ‡ · ‡ ‡ R4 ‡ · · ‡ R6 ‡ ‡ ‡ · R7 ‡ ‡ · · R8 ‡ · ‡ · R9 ‡ · · · Example for kenoCA rules of the form: [axb] ö y rule1: [AAA] ö A rule7: [AAB] ö B rule8: [ABA] ö B rule4: [ABB] ö A Nr.l 1 2 3 4 5 6 7 8 9 rule= rule1 .7 .8 .4 0 Ñ Ñ Ñ x A x Ñ Ñ Ñ 0L A initial " seed " H0; 5L 1 Ñ Ñ x B B A x Ñ Ñ R7H0; 3, 4, 5L : BBA B, R8H0; 4, 5, 6L : ABA B, R4H0; 5, 6, 7L : ABB A 2 Ñ x A A B B A x Ñ R1H1; 2, 3, 4L, r1H1; 3, 4, 5L : BBB A, R7H1; 4, 5, 6L : BBA B, R8H1; 5, 6, 7L : ABA B, R4H1; 6, 7, 8L : ABA B kenoCA Nr.l 1 2 3 4 5 6 7 8 9 rule= rule1 .7 .8 .4 1 Ñ Ñ Ñ Ñ ‡ Ñ Ñ Ñ Ñ R7H1; 3, 4, 5L, R8H1; 4, 5, 6L, R4 H1; 5, 6, 7L 2 Ñ Ñ Ñ x x ‡ Ñ Ñ Ñ 1, 1, 7, 8, 4 3 Ñ Ñ ‡ ‡ x x ‡ Ñ Ñ 7, 4, 7, 4, 7, 8, 4 4 Ñ x ‡ x ‡ x x ‡ Ñ 1, 7, 8, 8, 8, 4, 7, 8, 4 5 ‡ x x x x ‡ x x ‡ stop 1.1.3. Introducing kenogrammatic rules Technical alphabet, standard normal form of kenograms: {A, B, C}. Rules: rule1, rule2, rule3. One trito-equivalence of the calculus, not applicable to the technical, meta-linguistic alphabet: A = KGC: PKG = [mode=KG; {A, B, C}, rule1, rule2, rule3] rule1: ö A rule2: A ö AB rule3: AB ö C A, AB, C, AB, C, AB, ...: chiastic interchange between A(2) and C(3). 6 Author Name Technical alphabet, standard normal form of kenograms: {A, B, C}. Rules: rule1, rule2, rule3. One trito-equivalence of the calculus, not applicable to the technical, meta-linguistic alphabet: A = KGC: PKG = [mode=KG; {A, B, C}, rule1, rule2, rule3] rule1: ö A rule2: A ö AB rule3: AB ö C A, AB, C, AB, C, AB, ...: chiastic interchange between A(2) and C(3). The term A(1) as operator is set, C(3) as operand of the operator AB (3) becomes the operator A(2). Hence, A(2) is involved in the chiasm of operator and operand, playing both roles at once. Considering the roles of C, the same holds for a B in place of C. Différance and memristivity Strictly speaking, we encouter with this tiny PKGsystem a situation where both, the halt and the continuation of the production, happens at once. The chiastic interplay of the situation “A” and the situation “C” is playing the différance of the difference of “A” and “C” and its defer of change from ”C” to “A”. Jaques Derrida’s différance, which is neither a word nor a term, and is phoneticlly indistinguishable from “différence”, plays on the fact that the French word différer means both "to defer" and "to differ." "Différance as temporization, différance as spacing. How are they to be joined?” (J. Derrida) Jacques Derrida, Différance, http://www.stanford.edu/class/history34q/readings/Derrida/Differance.html This mechanism of a chiastic interchange between A and C invites to interpret it as a memristive mechanism and probably as the smallest model of non-destructive self-referentiality in/of a formal system. The self-referentiality of the production scheme seems to be obvious. What isn’t obvious at a first glance is its memristivity. Memristivity is involved with the chiasm between ‘operand’ C and ‘operator’ A. The property of re-entry and sameness has to be remembered during the substitution. In a classical setting, nothing of this kind has to be reached because it is presumed and installed from the ‘outside’ by an external designer/user of the rules that the re-entry ’port’ is not missed and that the object has not changed in the process of substitution from one identity (A/C) to another identity (C/A). For the kenomic calculus, the technical alphabet is build by distinctive letters, characters, elements, but inside the kenomic game and its rules, all occurrences of monadic elements are kenomically equal. The substitution of C from rule3 to rule2 as A has the choice to decide for a kenomic or for a symbolic interpretation of the substitution. With a symbolic interpretation the calculus stops here because the application is refused. For a kenomic interpretation rule2 holds, and the game goes on. Hence, with A ≠SEM C, the semiotic rule system is terminating with C, and with A =KG C the production goes on with C =KG A. Hence, the general decision problem gets confronted with an as yet unknown situation of a rewriting system having properly a state and at once not having that state in the calculus. Consequences for the concepts and constructions of replication, cloning and self-production/production of a self (autopoiesis) have at first to deconstruct the underlying concepts of iterability in their concepts of recursion. Decidability and non-decidability, therefore, is not focussed on the identification or non-identification of an object, i.e. a state, with decidable or non-decidable properties but on the interaction between applications inside and between formal systems. Article Title 7 For the kenomic calculus, the technical alphabet is build by distinctive letters, characters, elements, but inside the kenomic game and its rules, all occurrences of monadic elements are kenomically equal. The substitution of C from rule3 to rule2 as A has the choice to decide for a kenomic or for a symbolic interpretation of the substitution. With a symbolic interpretation the calculus stops here because the application is refused. For a kenomic interpretation rule2 holds, and the game goes on. Hence, with A ≠SEM C, the semiotic rule system is terminating with C, and with A =KG C the production goes on with C =KG A. Hence, the general decision problem gets confronted with an as yet unknown situation of a rewriting system having properly a state and at once not having that state in the calculus. Consequences for the concepts and constructions of replication, cloning and self-production/production of a self (autopoiesis) have at first to deconstruct the underlying concepts of iterability in their concepts of recursion. Decidability and non-decidability, therefore, is not focussed on the identification or non-identification of an object, i.e. a state, with decidable or non-decidable properties but on the interaction between applications inside and between formal systems. Further examples PID = [mode=ID; {A, B, C}, rule1, rule2, rule3] rule1: ö A rule2: A ö AB rule3: AB ö C