Graphematics of Multisets

An application of multiset techniques to graphematical structurations

Rudolf Kaehr Dr.phil@

Copyright ThinkArt Lab ISSN 2041-4358

 

Abstract

Some applications of techniques borrowed from multiset theory to elaborate graphematical systems as ‘data’ structures with the operations of union, sum, difference and polycontextural dissemination of mixed data structures, like set, multiset, list, trito-, deutero- and protograms. The metaphor of ‘team’ observation for the study of multisets gets a polycontextural explication and application to the team observation of complexions of heterogeneous heterarchic ‘data’ structures. The elaborations remain on a ‘descriptive’ formal level. (work in progress, v.0.5)

1.  Multisets and graphematic structures

1.1.  Multisets

1.1.1.  Summary of multiset approach

Yuncheng Jiang, Description Logics over Multisets
"A naive concept of multiset was formalized by Blizard. It has the following properties:
(i) a multiset is a collection of elements in which certain elements may occur more than once;
(ii) occurrences of a particular element in a multiset are indistinguishable;
(iii) each occurrence of an element in a multiset contributes to the cardinality of the multiset;
(iv) the number of occurrences of a particular element in a multiset is a (finite) positive integer;
(v) the number of distinguishable (distinct) elements in a multiset need not be finite; and
(vi) a multiset is completely determined if we know the elements that belong to it and the number of times each element belongs to it."

"More concretely, a multiset is a collection of objects in which repetition of elements is signifcant.”
http://ceur-ws.org/Vol-654/paper1.pdf

In a multiset, repetition is not only relevant but measured by its multiplicity.

"Multisets form a generalization of sets: “identical” elements can occur a finite number of times.”
"A multiset X is a pair
(X, ρ), where X is a set and ρ an equivalence relation on X . The set X is called the field of the multiset. Elements of X in the same equivalence class will be said to be of the same sort; elements in different equivalence classes will be said to be of different sorts.”
http://obelix.ee.duth.gr/~apostolo/Articles/mset.pdf

Depending on the definition of the equivalence relation on X, different classes might be definied with ρ= equivalence relation with μ=multiplicity and λ=locus as:
sets, set = (X, ρ=⌀, λ=1, μ=1)
multisets, mset = (X, ρ, μ, λ=1)
tritoset, tset = ((X, ρ, λ, μ)
and deutero- and protosets and others.

Epistemological remarks

In the words of Wilberger:

"Thus the only possible relations between two mathematical objects are 1) they are equal, or 2) they are different.
"This leads to effectively three possible relations between any two physical objects; they are different, they are the same but separate, or they are coinciding and identical.”

This corresponds to the Geman distinctions: Selbigkeit, Gleichheit, Verschiedenheit.
Or in English: equal (identical), equivalent (same), different.
Hence, elements in multisets are equivalent. They occur in different multiplicity as the same at different places in a not ordered context. But they are nevertheless semiotically identical, i.e. a at palce i and a at place j, i!=j, of a space or a string, are semiotically identical albeit “same but separate".
In contrast, elements in tritograms are equivalent despite their semiotical difference.

1.1.2.  Recalling multisets

"For each a in A the multiplicity (that is, number of occurrences) of a is the number m(a). If a universe U in which the elements of A must live is specified, the definition can be simplified to just a multiplicity function mU : U -> N from U to the set N = {0, 1, 2, 3, ...} of natural numbers, obtained by extending m to U with values 0 outside.” (Multisets, WiKi)

"The set of all mappings ∝: ∪ -> X is denoted by ∪ X .”  

The sum or (arithmetic) addition of A and B, denoted by A + B or A ∪+ B or A ∪ B, is the mset C such that mC (x) = mA (x) + mB (x), for all x.

"For example, if A = [a, b]typeset structure and B = [a, b]typeset structure then A − B = [a, b]typeset structure ⊂ B contradicting the classical laws that (A − B) ∩ B = ∅ and (A − B) ∪ B = A.
Therefore, < p(Y ), ∪, ∩, −, ∅, Y > is only a lattice (Knuth) and not a boolean algebra. “ (Sing)

The multiplicity function mU : U -> N from U to the set N = {0, 1, 2, 3, ...} might be involved with the graphematic abstractions, defining different types of graphematic constellations (systems) in terms of multiset terminology, concepts and formalization.

∝: ∪ -> X might be parametrizised towards graphematical abstractions, hence the system of graphematical multiset shall be defined as: graphem(∪ X) =  ∪ Xtypeset structure

Conflict between Calculus of Indication (CI) and Multisets
If we accept that the CI belongs to the language of multisets, as it is supposed by some experts, it turns out that the equally proposed “boolean algebraic structure” of the CI that is characterizing the CI, is not holding properly.

Again, it becomes obvious that the CI, even if it belongs to the graphematic scriptures that are defining the languages of multisets, is of such a minimal complexity that its coincidence with boolean structures becomes arbitrary.

Sounds like: “A free Boolean algebra on no elements, namely 2."

"For example, if A = [a, b]typeset structure and B = [a, b]typeset structure then A − B = [a, b]typeset structure ⊂ B contradicting the classical laws that (A − B) ∩ B = ∅ and (A − B) ∪ B = A."

For the special case of the CI with A = [a, b]typeset structure and B = [a, b]typeset structure then A − B ∩ B = ∅:
([a, b]typeset structure - [atypeset structure) ∩ [a,b]typeset structure= ∅ and for
(A − B) ∪ B = A:
([a, b]typeset structure - [atypeset structure) ∪ [a, b]typeset structure = [a, b]typeset structure.

Properties
multiset
multiplicity of objects
cardinality of the multiset
order of objects is irrelevant

Algebraic properties of msets <br /> mset = (mset, ∩ , ∪ , ∪+ , com, ass, id ... B) ∪ (A ∩ C) . <br /> <br /> Order : ∩ x ⊆ ∪ x ⊆ ∪+ x .

1.1.3.  Polycontextural modeling of multisets

Multisets are mappings from U to N, resulting in tupels (U, N). It might be speculated that such a mapping could be represented by contextural mediation between the to distinguished domans U and N. Therefor, this kind of multisets would be represented in the mediated domains of U an N as (U, N). Both domains, U and N, are covered by a contexture, therefore, the mediation (U,N) is represented by a third contexture that is mediating the contextures for U and M.

Following the fact that multisets are answers to two different questions, a modeling in a polycontextural framework is as natural as other modelings too. One question concerns the set of elements, the other question is concerned with the multiplicity of the elements of the set. Obviously there is a kind of an order between set-theoretic and multiplicity-theoretic topics. It could even be mentioned that the multiplicity-aspect is a reflexion onto the set-aspect of the multiset construction.

On the other hand it could be argued that classical set-theoretic concepts are polycontextural too. But restricted to a mono-contextural understanding where the multiplicity of elements is always just one. This argument holds for the mono-contextural approach to sets and multiplicity too.

"Remark 1. Any ordinary set A is actually a multiset A, χ A , where χ A is its characteristic function."

There is also another interesting circularity to observe. In classical settings, multiplicity in set-theory, based on cardinality, is itself based on sets. Even if the paradoxes of the naive concept of sets are suspended by different axiomatizations, a new paradox emerges: Multiplicity of multisets is based on the cardinality of ordinary sets. Hence, multisets are ‘actually’ sets of sets. While any ordinary set is “actually a multiset".

A polycontextural thematization and formalization of the topics of multisets is replacing set- or category-theoretic mappings of different domains for a mediation of those domains. Mediation additionally opens up highly flexible and complex constellations that are not easily accessible with the concept and formalism of mappings.

As long as both domains or contextures of a mset are just separated and are not interacting and are therefore not changing during the process of manipulation, there is no need to introduce more sophisticated concepts and methods to replace or augment the well established static correlations or mappings in the sense of multiset theory. Multiset operations, like insertion, addition, subtraction, etc. are sufficient to realize change in a static context.

If it is reclaimed that msets are more directly respecting real-world and concrete life situations than their counterpart, the abstract sets of extensional set theory, the proposed claims has to be reduced to the fact of another kind of abstract notions. A separation of the two (or more) domains enables flexible concurrent interactions between otherwise stable and unified domains.

"However like other multiset theories, they are both two-sorted theories where the multiplicities are a different type of objects from the multisets they support. This would require separate axioms for multiplicity arithmetic, and in the infinite case it involves piggybacking on a predefined model of cardinal arithmetic (for example [Blizard 3] uses cardinals in a model of ZF set theory)." (Dang, 2010, p. 48)

A one-sorted approach for multiset theory is given in Dang’s thesis “Symmetric sets and graph models of set and multiset theories”.

"Therefore we will now propose a one-sorted account of multi- sets, where multiplicities and sets come from the same universe and follow the same axioms. As a result multiplicities are no longer cardinal numbers but sets themselves, with their own internal structures. The natural ordering of multiplicities will be identified with the subset relation, i.e. intuitively we consider x to be less than y as multiplicities if is a proper subset of y.” (Dang, 2010, p. 48)
http://www.dpmms.cam.ac.uk/~tf/dangthesis.pdf

1.1.4.  Bifunctorial approach to multisets

A version of a deliberation of mappings that is not yet polycontextural might be achieved with the concept and machinery of 2-categories and bifunctoriality between different types of mappings. Here, the mapping of sets and the mapping of arithmetical multiplicity, both generating the mapping of multisets. Hence, multiset mappings are based on set-theoretical mappings as definitions of multisets.

Interchangeability is a strategy to avoid unneccessary conceptual and formal hierarchies. The strategy of Ur-elements is eliminating the type difference between sets and numbers in favor of an abstract untyped concept prior to sets and numbers.

Example A = [a, b]  _ (4, 5) : (U, N) <br /> (U, N) = (mapping      ... ;                

mset: μ: U --> U,  ν: N --> N

bifunctorial:  (μ, ν):   typeset structure: (N1 typeset structureN2) o (U1 typeset structureU2) = (N1 o U1) typeset structure (N2 o U2).

     Interchangeability of    set, multiplicity and multiset <br /> <br /> ...                                                                                               mset

1.2.  Indicational structures as multisets

Indicational structures of the calculus of indication, CI, of George Spencer-Brown’s Laws of Form had been identified mathematically as multisets (Matzka).
This is no secrete. It was also pointed out by Jeffrey James’ Interpretations of Laws of Form and by others too.
http://www.lawsofform.org/interpretations.html

Supposed there exists an indicational universe, then events occur as partially ordered collections, called multisets. In CI terms that means that the events are commutative or permutative in respect of the number of observed events. Such an indicational space of events then is algebraically defined by commutativity, associativity and idempondency of its primary operation, i.e. concatenation. Distributivity is characterizing concatenation and superposition (encloser) of the CI.

Unfortunatly, no consequences had been drawn from the comparision between multisets and the CI. Therefore, there are no applications of the mathematical methods and results of multiset theory involved with the study of the CI and its possible generalizations.

On the other hand, multiset notions had been studied mathematically from the angle of set-theory and category theory but there seems no attempt to use those insights to motivate a new concept of formal reasoning.

Our concern in this paper is what the effect on logic will be if we shift from ordinary sets to multisets, i.e. collections which account not only for types but also for tokens of objects.

"Under this interpretation of formulas as extensions, a logic Λ contains exactly the syntactic rules of a calculus of extensions forming a certain kind of structure S . We express this by saying that Λ is the logic of S .

E.g. classical logic is the logic of boolean fields of sets (i.e., boolean algebras of sets), intuitionistic logic is the logic of pseudo-boolean fields (like the structure of open sets of a topological space), modal logic is the logic of topological boolean fields (that is, boolean fields equipped with a further interior operator), and so on.”
http://users.auth.gr/tzouvara/Texfiles.htm/multlog.pdf

Multiset theory is well founded in first order logic (FOL) and classical set theory. Hence, multiset theory is a new branch of mathematics, like fuzzy sets, but is not touching the fundaments of semiotics, logic and arithmetics as such.

In contrast, the ambitions of the indicational calculus, CI, are trying to develop new fundaments for formal and mathematical reasoning on the base of a restricted  “multiset” approach.

From the perspective of multisets, it turns out that the CI is a maximally restricted calculus based on minimal multisets. This is in accordance to the fact that the CI is a minimal graphematic system on the level of permutative partitions for m=2.

Brown  _ (n, m)   = ((n + m - 1)/n) = ((n/k))

1.3.  Mersenne structures

Mersenne structures are neither sets nor multisets nor strings but tuples with a Mersenne abstraction that is abstracting from the equality of homogeneous tuples. Hence a kind of restricted tuples.

Mersenne  _ (n, m)   = m ^n - (m - 1) .

2.  Graphematics of multisets

2.1.  Graphematics

2.1.1.  Little typology of graphematical systems

In contrast to the 3 graphematic systems of semiotics (identity systems, Leibniz), indicational systems (multisets, Brownian) and Mersenne systems, that are all three supporting, in different ways, the semiotic concept of identity of signs, the kenomic systems of graphematics, i.e. the trito-, deutero- and proto-systems, are involved in a subversion of the semiotic principle of identity.

The mentioned 3 graphematic systems had been studied also under the names of Stirling, Pascal and Leibniz systems or scriptural approaches of a general theory of graphematics.

There are at least two strategies to develop more reality-adequate formalisms. One is to involve parametrization over a multitude of “concrete” domains, producing a bulk of specialized ‘data types’. The other approach is to construct an even more abstract formalism to cover structures, like over-determination, interaction, mediation, etc., not accessible to concretized formalisms.

Such new abstractions are tackling with new relationships between types and tokens of semiotic and graphematic objects. The multiset account with “collections which account not only for types but also for tokens of objects” shall be continued with a dynamization of the type-token relationship of the sign-usage.

    typeset structure

m  _ U : morph (U) ->  _ type N : (systems          objects          types  ... ero                                                        Cardinal         kenomic          proto


Table of types, examples

sets : set(MG) = {aaaaaaaabbbbbbccc} = {a, b, c}  _ (1, 1, 1) trito - sets : locus dep ... aaaaaaaabbbbbbccc] : [1 ^8, 2 ^6,  3 ^3] = [a, b, c]  _ (8, 6, 3)

deutero - sets : locus independent free lexical order       &nbs ... bbbbbbccc >  _ ⌈ 17 : 3 ⌉ . <br /> pomsets : partially ordered multisets .

"The pomset type generalizes sets, bags, lists, trees, and other ordered types, and therefore provides a uniform representation for all these types. Intuitively, a pomset can be viewed as a string with a partial order instead of a total order.” (Gumbach, Milo, An algebra of pomsets, 1995)

     lists, strings: totaly ordered multisets.

"A very special case of partially ordered multisets are the strings over a given set of elements. Here the partial ordering is actually total. It is well known that strings are free monoids, meaning that they are freely generated by the signature Σstr = <ε, ·> with the following equations:
                               ε ·x         = x               (1)
                              x · ε         = x               (2)
                              (x · y) · z = x· (y · z )  (3)
ε denotes the empty string and · concatenation of strings; the equations state that concatenating the empty string to the left or right does not change a string, and that concatenation is associative.” (Resnik, Deterministic Pomsets, 1994)

Multisets. Another very special case of partially ordered multisets are the multisets (sometimes called bags) over a given set of elements. Here the elements are actually completely unordered. Multisets are known to constitute free commutative monoids; that is, they are freely generated by the signature Σmul = < ε, ∪+ > with the following equations:
                                      ε ∪+ x = x                         (4)
                             (x ∪+ y) ∪+ z = x ∪+ (y ∪+ z )        (5)
                                      x ∪+ y = y ∪+ x                  (6)
ε now denotes the empty multiset and ∪+ multiset addition; the latter is associative and commutative, whereas adding the empty multiset does not change a multiset.” (Resnik, Deterministic Pomsets, 1994)

"A labelled partially ordered set or lposet over E is a triple p = < V , <, l > where
• V is an arbitrary set of vertices ;
• < ⊆ V × V is an irreflexive and transitive ordering relation;
• l: V -> E is a labelling function.

A multiset addition is modelled by disjoint pomset union:
p ∪+ q = [Vp ∪ Vq , <p ∪ <q , lp ∪ lq ] where again the representatives p and q should be disjoint.” (Resnik, Deterministic Pomsets, 1994)

2.1.2.  Example of the general tritoset scheme

typeset structure
dec(Tritoset) = (mg1, mg2, mg3) ,
loc(dec(Tritoset)) = (loc1(mgtypeset structureloc2(mgtypeset structuremg3), loc4(mg1), loc5(mg2)).
kenom(loc1(mgtypeset structure) = [aaaa]
kenom(loc2(mgtypeset structure)) = [bb]
kenom(loc3(mg3)) = [ccc]
kenom(loc4(mg1)) = [aaaa]
kenom(loc5(mg2)) = [bbbb].

Tritoset [A]  Tritoset = (loc, mg, ken) consists of loci loc  _ i and monomorphies mg  ...  _ 2) = (bb),    ken (mg  _ 3) = (ccc) . <br /> MG : ken --> mg --> loc .

typeset structure  typeset structure     

Tritosets are mappings from an alphabet of supporting objects U to N (natural numbers): mset: U --> N defining the kenoms of distributed monomorphies. Loci are the places of different or repeated monomorphies.  And monomorphies are containing a number of kenoms as objects: Tritoset : monomorphies --> loci --> kenoms.
Repeated monomorphies might differ in the number of kenoms.

Hence, the example Tritoset[A] = [aaaabbcccaaabbbb] gets a numerical notation including the order of the monomorphies and the multiplicity of the monomorphies as the number of the kenoms over the ’support’  set [a, b, c], with:

                     typeset structure

A final explication has to inscribe the order of the loci of the monomorphies in the morphogram (tritoset), 1 -5, hence:

               &nbs ...     Tritoset[A] = [a, b, c]  _ (1 _ 1^4 2 _ 2^2 3 _ 3^3 1 _ 4^4 2 _ 5^4 ) <br />

For A =  [a, b, c]typeset structure, with [a,b,c] as support set of kenoms in trito-normal form (tnf), the indices typeset structureas numerical multiplicity of the kenoms of the monomorphies and the order of the indices, the positions (loci, 1 to 5) of the monomorphies (mg). Because of the implicit order of the indices of the loci, the notation of the positions (loci)  might be omitted.

               &nbs ...  1^4 2 _ 2^2 3 _ 3 <br /> <br /> ^3 1 _ 4^4 2 _ (5 - locus)^(4 - multiplicity)    )

Hence, a Tritoset tset is defined as a triple of [occurrence, multiplicity, locus] over a kenomic ‘support' set.
While a Multiset mset is defined as a tuple [occurrence, multiplicity] over an identitive suppoert set.

           typeset structure

Set-theoretical definitions

"Definition: Let S be a nonempty set. A multi-set M with underlying set S is a set of ordered pairs:
                       M = {(si, ni)|si∈S, ni∈Z+},

where ni is the multiplicity of the element si. A multi-set defined as, or using, a set.”
http://mathematics-diary.blogspot.co.uk/2012/03/sets-and-multisets.html

Multisets are based on 2 distinctions: elements, si, and the multiplicity of the occurrence of elements, ni. Hence: (si, ni).
Not mentioned but accepted is the identity presumption of the elements, si ∈ ID.
Therefore, the full definitions for multisets is: (elements, multiplicity; identity), i.e.
                          M = {(si, ni)|si∈S, ni∈Z+; S∈ID}.

Tritosets are based on 3 distinctions: elements, si, multiplicity, ni, location, li, in the realm (underlying set) of non-identitive kenograms, i.e., si ∈ KENO.
Therefore, the full “set"-theoretic definition for tritosets is: (elements, multiplicity, location; kenomic), i.e.

        <br />        T = {[s  ... A0; _ i ∈ Z ^+, l  _ i ∈ L ; S ∈ KENO}     <br />

2.1.3.  Definition schemes for mset, tset, dset and pset

               &nbs ... p;                

               &nbs ... p;                

               &nbs ... p;                

               &nbs ... p;                

               &nbs ... p;                

For A =  [a, b, c]typeset structure, with [a,b,c] as support set of kenoms, the indices typeset structureas multiplicity of the kenoms of the monomorphies and the order of the indices the  positions (loci, 1 to 5) of the monomorphies (mg). Because of the implicit order of the indices, the notation of the positions, loci, might be omitted.
In contrast, sets are stripped off of any additional differentiation, i.e. loci=1, multiplicity=1, ∀x∈ D: mA(x)=1. Example: [a, b, c]typeset structure = {a, b, c}.

Multiset
"The set of distinct elements of an mset is called its root or support. Formally, the root set of an mset A is the set {x|x ∈ A}. The cardinality of the root set of an mset is called its dimension.” (Sing)

2.1.4.  Summary

       GRAPHEMATICS <br />          ... p;   = min (cardA, cardB) <br />          


reduction: msets --> dpsets --> dsets:

typeset structure --> typeset structure --> typeset structure

       <br />    mset = ((n + m - 1)/n)    < ... bsp;   ]    P (n, k)    <br />      deutero


reduction: tcset --> tpsets --> tsets:
  

typeset structure -->  typeset structure -->  typeset structure   

   <br /> tcset = Underoverscript[∑, k = 1, arg3] S (n, k) ((n + m - 1)/n) &nb ... ript[∑, k = 1, arg3] S (n, k)       <br />     trito

2.1.5.  Motivations for sets, strings, multisets, pomsets, tritograms

Sets, strings, multisets and pomsets

"Now picture yourself at a bank. You withdraw ten dollars, the teller asks how you want it, you say “Two fives, please.” You have thereby specified a one-letter multiset. You have not specified an amount, in that you won’t settle for ten ones. You have not specified a set, for that would imply particular five-dollar bills. You have however specified a set up to isomorphism, meaning that any two sets of five-dollar bills in bijective correspondence will be equally acceptable. But this is what we mean by a multiset. And although you do not appear to have specified an order, this is what we mean by the discrete or empty order, in which no two elements are comparable. Thus you have specified a pomset that happens to be a multiset.

"Now suppose to this specification we add “And one at a time, please.” We may distinguish the previous specification from this one as respectively 5|5 and 5;5 or just 55, their concurrence versus their concatenation. The former is a multiset, the latter a string, but both are pomsets.”

"Linearly ordered multisets (labelled chains up to isomorphism) are strings. Pomsets as partially ordered multisets therefore constitute a generalization of strings to partial orders.”
http://homepages.inf.ed.ac.uk/gdp/publications/Teams.pdf

Trito-sets
Instead of “Two fives, please.” it would be enough to ask, independently from the amount, for “Two of the same, please.” With that, an abstraction from the occurrences of the elements happens. What ever happens, there are just two different possibilities open, both choices are equal or both choices are different. This defines a tritoset but is not yet rich enough to consider the order of the unspecified choices.

If there would be 3 choices offered, the possibilities would count up to 5: (aaa), (aab), (aba), (abb) and (abc). The new property is order of unspecified occurrences of elements. The choices (aab) and (aba) are considered as different, while choices (bab) and (aba) are seen as trito-equal.

All together, the abstraction from the occurrences, i.e. the support set, and the rule of order are defining tritosets.
Trito-sets are measured by the Stirling numbers of the second kind:typeset structure.
Example: trito([aaabbcbb]) = [a,b,c]typeset structure.
In contrast: mset([aaabbcbb]) = [13, 24, 31] = [a, b, c]typeset structure.

Deutero-sets
If we abstract in this model of tritosets from the order of the occurrences of the elements, we get a new class or type of languages, the languages based on deutero-sets. Hence, the deutero-sets (aab), (aba), (bba) and (bab), are deutero-equal, equally [aaa] and [bbb], while (aaa) and (aab) are not deutero-equal. Deutero-sets are measured by the sum of partitions:typeset structureP(n, m).
Example: deutero([aaabbcbb]) = {1typeset structure2typeset structure3typeset structure.

Proto-sets
A further abstract that is still keeping some properties of the original pattern is possible with the abstraction of the number of the occurrences of the elements. The proto-set marks the addition of the numbers of separable multiplicities and the number of the occurrence of their elements. Hence, n = Σ multiplicities and m = occurrences, written as ⌈m:n⌉.
Example: proto([aaabbcbb]) = ⌈8:3⌉, with m=8, n=3.

A reduction to the cardinality of a tritogram or a multiset counts the number of the occurrences of elements abstracting from the partition into different kinds of elements, hence proto [8:3] becomes cardinal [3], min{m, n}.

Order: multisets => trito => deutero => proto.

2.1.6.  Formal characterizations

"Remarks: Pomsets are only defined up to isomorphism to hide the identities of the elements in V, so that only the cardinality of V counts, leaving Σ as the only important set underlying a pomset.”
http://www.cs.tau.ac.il/~milo/projects/query_languages/papers/icdt95.ps

"Definition 1. A label led partial order or lpo over a set Σ is a structure (V , <=, σ, Σ) where partially orders V and σ : V -> Σ assigns to each element of V an element of Σ.
We think of Σ as an alphabet of actions and V as instances of that alphabet, or events forming a word, with the order of occurrences of letters in the word given by <=."

"Definition 3. A pomset is the isomorphism class of an lpo.
More intuitively a pomset is an lpo in which we pay no attention to the choice of the set V , other than its cardinality, but retain all other details. Thus if we replace V = {0, 1, 2} by V = {5, 6, 7} without otherwise disturbing either <= or σ the pomset does not change.” (Teams p.8)

Does that mean that two pomsets A = (0 1 2) and B = (4 5 6) are equivalent? They are by definition “defined up to isomorphism”. But on the level of representations, A and B are not equivalent, A !=pomset B.
In contrast, A and B are trito-equivalent on a “representational” level. Given V = {0, 1, 2} two tritograms A and B over V, with A = (0,1,2,2) and B = (1,2,0,0), are trito-equivalent, A =typeset structureB over their common V.

Graph representation
trees, multi-trees, graphs

List of short definitions
1.
A multi-set is a collection of objects in which repetition of elements is significant and measured by multiplicity.
2. A trito-set is a collection of objects in which identity of objects is irrelevant but distribution and permutation of elements remains significant.
3. A deutero-set is a collection of objects in which identity and permutation of objects is irrelevant but partition and repetition of elements remains significant.
4. A proto-set is a collection of objects in which just repetition of elements is significant.

2.1.7.  Combinatoris

"Sets, in which the order of elements and the number of occurrences of each element do not matter.
Multisets, in which the number of occurrences of each element is important, whereas the order of elements does not matter.”
items: number of elements typeset structure, number and order of occurrences of typeset structure elements, alphabet and support set.

    types                                                                    order             ...                                        1, 1, 2, 3, 1                                    m 

Comparision with the graphematic system

  Authors                                                                                      ...                                                                                 k = 1            n

http://memristors.memristics.com/Graphematics/Graphematics%20of%20Cellular%20Automata.pdf

Summary
There are
1. 3 kenogrammatic systems: trito, proto and deutero.
2. 3 identitive systems: semiotics, partition (Mersenne), indication (Spencer Brown)
3. 3 mixed identical-kenomic systems: trito-partitive, trito-commutative (trito-Brown), deutero-partitive.

Number of representations

Representations for tritosets
To deal with abstractions needs representations. A tritogram [abb] is written in normal form and has therefore a number of different representation that are equivalent to the abstract tritogram, written in normal form.
For the case of just 3 elements involved, the abstract tritogram has a representation of 6 concrete realizations, i.e. {[abb], [acc], [baa], [bcc], [caa], [cbb]}, all representing the tritogram [abb].

This number is calculated by the formula : <br /> <br />    card[μ]  _  ...    . <br /> Then we get the number 6 for all possible representations for m = 3, k = 2 .

Representations for deutero-sets

   <br />    card [μ]  _ deutero = n !/((1 !) ^(e &#x ...  _ n)) (m/k) k !/(e  _ 1  ! e  _ 2 !  ... e  _ n !) <br />

Repesentations for proto-sets

   <br />    card[μ]  _ proto = Underoverscript[∑, i =  ... F3A0; _ i1  ! e  _ i2 !  ... e  _ in !)     <br />

Schadach gives a full account for all classifications of the graphematic system.
http://www.ballonoffconsulting.com/PDF/1987AppendixII.pdf

Representations for multisets
What is the number, card(mset(n, k)), of representations for multisets?

Let S be a multiset that consists of n objects of which
n1 are of type 1 and indistinguishable from each other.
n2 are of type 2 and indistinguishable from each other.
...
nk are of type k and indistinguishable from each other
and suppose n1 +ntypeset structure +. . . +ntypeset structure = n.

What is the number of distinct permutations of the n objects in S?

       card[μ]  _ mset  = n !/(n  _ 1 ! n & ... xF3A0; _ k !) . <br />       (CS 171, Lecture Outline, February 03, 2010)

Example 1. How many permutations are there of the mset [abccbccbddb]?
Solution. We want to find the number of permutations of the multiset
[A] = [a,b,c,d]11243442 = {1 · a, 4 · b, 4 · c, 2 · d}.
Thus, n = 11, n1 = 1, n2 = 4, n3 = 4, n4 = 2. Then number of permutations is given by

n !/(n  _ 1 ! n  _ 2 ! ... n  _ k !) = 11 !/(1 ! 4 ! 4 ! 2 !) = 330.

Thus, the mset [A] = [a,b,c,d]11243442 has 330 identitive representations. The notation [abccbccbddb] for [A] is therefore a conventional choice and put into mset-normal form notation.

Monomorphic approach

Up to now elements of multisets had been treated as a atomic elements and their occurrences as atomic too. A morphic approach is dealing not with atomic elements but with monomorphies, i.e. with patterns of kenomic elements.
A first approach, albeit still in an identive setting, is given with the definition of multisets with repeated words instead of repeated atomic elements as supports over a common set X.

"Usually, a multiset with finite support, M, is presented as a set of pairs {x, M (x)}, for x∈supp(M)."
Paun, et al, DNA Computing, 1998

Example
{(ab, 3), (abb, 1), (aa, 2)} = {(ab), (ab), (ab), (abb), (aa), (aa)}

Application to tritograms.
1. Prolongating the “tail” of a pattern.

A =    a bb c aa =                                                                   ... nbsp;   ø            a    

prolong  _ ((bb, 1)  _ 2)  _ (/(dd, 2) ) (A ) =    a ...  _ ((bb, 1)  _ 2)  _ (/(aa, 2) ) (A ) = a bb c aa aaaa  = a bb c aa aaaa

A decomposition of A into its monomorphies is given by the table of A with monomorphies mgi distributed over the loci of their occurrence, locij.
typeset structure might be written as:
A = [(a,1)1, (bb,1)2, (c,1)3, (aa,1)4], e.i., [(a,1), (bb,1), (c,1), (aa,1)].

2. Another possible prolongation of A might be defined as prolongtypeset structuretypeset structure where the monomorphy mgtypeset structure= (bb) is repeated twice in A in form of all possible trito-occurrences of (mgtypeset structure= (bb) at locustypeset structurein A. Therefore, the context or environment of mg2 is determining the occurrences of the monomorphy mgtypeset structure.

Because the contextual iterations of the monomorphy typeset structureare independent in respect to their occurrence, all concrete prolongations might happen at once. Hence, the prolongation is involved with a contextural distribution in the mode of reflection, i.e. iteration into itself, over 4 places and the mediation (∐) of the distributed positions.

prolong  _ (bb, 2)  _  _ 2 (A) = (( [(a, 1), (aa, 2), (c, 1), (aa, 1)] ... to (aa) . <br /> The mediation sign (∐) between the loci of distribution shall be omitted .

2.2.  Trito-structures

2.2.1.  Equality for trito-sets

Mset-equality  A =typeset structure
"Equal msets. Two msets A and B are equal or the same, written as A = B, iff  for any object x ∈ D, mA(x) = mB(x) or A (x) = B (x). Equivalently, A = B if every element of A is in B and conversely.” (Sing)

With emphasis on permutation of the occurrences of the elements:

"Formally, A ∈ CAtypeset structure is multiset, denoted by A ∈ MStypeset structure , if for any permutation π of {1 . . . k}, the local function δA satisfies
                      ∀a1 , . . . , ak ∈ Qn : δA(a1 , . . . , ak ) = δA(atypeset structure , . . . , atypeset structure)."

Example: A =typeset structure
[a,b,c]typeset structure = [a,b,c]typeset structure:
∀ x ∈ D: a,b,c ∈ D
mA(x) = mB(x): mA(a) = mB(a) = 1, mA(b) = mB(b) = 2, mA(c) = mB(c) =3.

Trito-equality A =trito B based on equivalence
A =trito B iff for any objects x, y ∈ D, and all loci i,j: mA(xi) = mB(yj) with 1<=i=j <=|A| and [A] =trito[B].

A = [aabac]typeset structure =trito B = [bbcba]typeset structure
mA(x) =trito mB(y): typeset structure =trito typeset structure
       x =tritoy         :     [aabac] =trito  [bbcba].

Trito normal form tnf
A given keno-sequence might not be in a standard normal form (tnf), hence the ML function tnf, based on lexical order delivering kseq shall be applied.

Example

tnf: [cdda] --> [abbc], kseq.

- val A = [2, 2, 1, 1] : kseq
> val A = [2, 2, 1, 1] kseq
-tnf A;
>  [1, 1, 2 2] : kseq

Trito-equivalence A =trito B based on the ϵ/ν-structure
Two tritograms [A] and [B] are trito-equivalent iff their ϵ/ν-structures are equal.

[A] =trito [B] iff EN([A]) = EN([B]).

Example
typeset structure         typeset structure
typeset structuretypeset structure

typeset structure
datatype EN =E|N;
fun delta (i, j) z=
if  (pos i z) = (pos j z)
    then (i, j, E)
     else (i, j, N) ;

EN([A]):
- ENstructure ["a”, “a”, “b”, “c"];
> [[],
   [(1,2, E),
   [(1, 3, N), (2, 3, N)],
   [(1, 4, N), (2, 4, N), (3, 4, N)]]: enstruct.

EN([B]):
- ENstructure [typeset structure];
> [[],
   [(1,2, E),
   [(1, 3, N), (2, 3, N)],
   [(1, 4, N), (2, 4, N), (3, 4, N)]]: enstruct.

EN([A]) = EN([B]) <==> [A] =trito [B].

2.2.2.  Reversion of tritosets

Reversion for msets
The reversion of a mset [A] is an unchanged mset. Obviously, the reverse (inverse, dual, reflected) of a mset is obsolete because there is no order of objects that could be changed.

mset[A]:
rev([A]) = [A].
Example
mset[A] = [a,b,b] = [a, b]typeset structure
rev([A]) = rev([a,b]) rev(1,2) = [b,a]typeset structure = [a,b]typeset structure.

Reversion for tritosets
Because tritosets are ordered sets, i.e. tritograms, the possibility of reversions for tritosets follows naturally.

tset[A]:
[A] = [...]nnnn, with [...]: head, nnnn: multiplicity
rev([A]) = rev([head], multiplicity]) = (rev([head]), rev(multiplicity)])

[A] ∈ sym => rev([A]) =trito [A]
[A] ∉ sym => rev([A]) !=trito [A].

Example
[A] ∈ sym, with head, multiplicity∈Sym
[A] = [aabb]typeset structure
rev([A]) = [bbaa]typeset structure
tnf(rev([A])) = [aabb]typeset structure
Hence, [A] =trito rev([A]).

[B] ∈ sym, with head∈Sym, multiplicity∉Sym
[B] = [aabbb]typeset structure
rev([B]) = rev([aabbb]) typeset structuretypeset structure
rev([B]) = [bbbaa]) typeset structuretypeset structure
tnf(rev([B])) = [aaabb]typeset structure
Hence, [B] !=trito rev([B]).

[C] ∉ sym
C = [aabac]typeset structure
rev([C]) = [cabaa]typeset structure
tnf(rev([C])) = [abcbb]typeset structure
Hence, [aabac]typeset structure !=trito [abcbb]typeset structure.

Rules

rev(rev([A])) = [A].

rev([])     =t [],
rev([a])   =t [a],
rev([aa]) =t [aa],
rev([ab]) =t [ab],
succ([A])) =t ([A]) ∪+t [ai], i ∈ ken([A]) +1
rev(succ([A])) != succ(rev([A])).

Reflectional morphogrammatics
Reversions as basic operations for a reflector-based morphogrammatics. (cf. Morphogrammatik, 1993).

A matrix Mtypeset structure = typeset structure is easily decomposed into its 2x2-submatrices: typeset structure,   typeset structure, typeset structure.

It might sound reasonable to interprete these tritogrammatic patterns, A, B, C, by logical values, resulting into the logical pattern [typeset structuretypeset structure

rev([A]) =  typeset structure =typeset structure, logically rev([typeset structure. But rev([A]) has to be considered in the context of [ABC], hence

revA([ABC])  = typeset structure =typeset structure   typeset structure, hence logically: revtypeset structure[typeset structure

revAB([ABC]) = typeset structure =typeset structure   typeset structure, hence logically: revtypeset structure[typeset structure

revABC([ABC]) = typeset structure =typeset structure typeset structure, hence logically: revtypeset structuretypeset structure[typeset structure

http://works.bepress.com/thinkartlab/15/

2.2.3.  Insertion and prolongation

Insertion into a multisets
"The insertion of an element x into an mset A gives rise to a new mset A’ = A+x such that mtypeset structure (x) = mA(x)+1 and mtypeset structure(y) = mA(y) for all x!=y.” (Sing)

Example: multiset insertion
A = [a,a,b,b,c]typeset structure
A’  = A + x, for x = a:
mtypeset structure(a) = mA(a) + 1: mtypeset structure(a) = 3
mtypeset structure(y) = mA(y) : for y = b, c, thus
A' = [a,a,a,b,b,c]typeset structure.
A’  = A + x, for x = b:
mtypeset structure(b) = mA(b) + 1: mtypeset structure(b) = 3
mtypeset structure(y) = mA(y) : for y = a, c, thus
A' = [a,a,b,b,b,c]typeset structure.

For two equal multisets [A] and [B], [A] = [B], an insertion of the same element “x” into the msets [A] and [B] restores the equality: [A] = [B] <==> [A]/x1 = [B]/typeset structure, x1= x2. And obviously, for x1!= x2: [A] = [B] <==> [A]/x1 != [B]/typeset structure. Interestingly, this restriction isn’t leading for insertions and substitutions in tritosets.

Because multisets are sets with repetition and not ordered sets, the insertion can be applied to each element of the multiset without changing the definition of insertion.
For an ordered set, an insertion might be defined as an addition to the last element and not to any elements inside the order. This is not a necessary restriction but should be applied at first for the examples of morphogrammatic insertions.

Insertion into morphograms: prolongation

Trito-prolongation
Insertion into multisets has a correspondance to prologations in morphogrammatics. Insertion in morphograms of trito, deutero- and proto-structure has different applications. This shall be restricted to the mode of simple prolongation in morphograms.

MG' = MG +trito x,
x = kenom, x∈ MG+1
+trito is a retro-grade addition depending on the complexity of MG plus 1.
With the temporary restriction of an addition to the monomorphy of the last locus:
MG = (mg1, mg2, ..., mgn) and
MG' = (loc1(mgtypeset structureloc2(mgtypeset structuremgtypeset structure,..., loctypeset structure(mgi), locn(mgi+1)).
For i,j=1,2: loci(mgj), i=j.

MG = a bb c =                                                                                  ... lash;      prolongation (a bb c ) : MG +  _ i  x, and tables . <br />

 a bb c a  =                                                                                   ... bsp;  ø            ø    

Numerical notation <br /> num (a bb c) = [abbc] , [1 ^1 2 ^2 3 ^1] = [a, b, c]  _ (1 ^1 2 ^2 3 ^1)

<br /> Prolongation <br /> num (prolong (a bb c)) =    ( [abbca]             ...                                                  [1   2   3   4  ]

<br /> prolong    ([1 ^1 2 ^2 3 ^1] ) = (            ...                                         [1   2   3   4   ]

2.2.4.  Prolongation and induction

Induction for multisets Well - foundedness and the induction principle are well studied for mu ... 04; M ) P (M )  where the predicate P (M ) stands for M ∈ W (M (T ), <  _ 1 ) .

http://www.cs.us.es/~mjoseh/pub/Proving_termination_with_multiset_orderings_in_PVS.pdf

Induction for tritosets

Induction for tritosets is not yet studied. But the hints are well established. The succesuccessor operation for the induction is a multiple-successor operation. Therefore, the induction follows in parallel along different branches. In this sense it follows that multset conclusions are, like set- or popositional conclusions, still perceived as single-conclusion systems while tritosets and tritograms are demanding multiple-conclusion systems.

Multiple-conclusion logic might still be in its infancy (D.S. Shoesmith, T.J. Smily, Multiple-Conclusion Logic, 1978) but tritogrammatical systems are offering strong approaches to a further concretization of genuine multiple-conclusion logics.

    Induction scheme for tritosets <br /> <br />       ...  _ n | ( ∀ M ) P (M )  _ (n + 1)        <br />

The succession range, formalized with (M ∪+ {a}) is depending, not on an abstract atomic element "a" of the set {a} as a successor but on the structure of M that is determining the range of the possible successors.
Hence for [M].ken = (a1, a2, ...,an), a simple succession model, not yet based on monomorphies, is derived with the mediated parallelism of successions typeset structure||typeset structure, i!=j:

M ∪+  _ trito {a} = <br />           ... ;           [A] ∪+ a  _ (n + 1) .

2.2.5.  Trito-substitution

Substitutions are like concatenation, fusions or merging fundamental concepts for any language. Depending on the definition of the language or scripture, substitutions are involved into interesting interactions. The following gives a short glance into its spectre of differentiations.

   <br />    Context rules for tritogrammatic substitution CRS    ... _ trito Subst _ (h/m _ 2)(H _ 2)      ; modulo CRS   

Trito - substitution example Given 2 trito - equivalent tritosets   [A] and [B] with ...  delivers : <br /> Dec([A]) = ([aa], [bb], [a], [cc]), <br /> Dec([B]) = ([aa], [cc], [a], [bb]),

The objects m  _ 1 and m  _ 2  are of equal length, semiotically different but ... 0; _ 2 = [eee] shall be inserted at place of    h = [aa] of the tritograms [A] and [B] .

Conditions of substitution are given with the context rules CRS : <br /> length(m  _ 1 ... F3A0; _ 1, m  _ 2  sem(m  _ i) ∩ sem (H _ i) = ø , i = 1, 2.

Therefore, the substitution delivers : <br /> Subst(Dec([A]))  _ ([aa]/ _ [ddd ... [a], [cc]), <br /> Subst(Dec([B]))  _ ([aa]/ _ [eee]) = ([eee], [cc], [a], [bb]) .

[A] =  _ trito [B]  <==>    Subst ([A])  _ ([aa]/ _ [d ... ; _ [eee]) . <br /> [A '] = [dddbbacc], [B '] = [eeeccabb] ==> [A '] =  _ trito [B '] .

Alternative formulation The substitution examples visualized with tables

<br /> Morphograms    MG  _ 1 = aa   a  and     MG  _ ... ; -          -       dd

Subst (MG  _ 1)  _ ([aa]/ _ [ddd]) = ddd   a  ,    &nbs ...   -          -      dd

Counter - example MG _ 1  =  _ MG MG _ 2  ==>    Subst (M ...    -          -     dd

http://memristors.memristics.com/Church-Rosser%20Morphogrammatics/Church-Rosser%20in%20Morphogrammatics.html
http://memristors.memristics.com/MorphoProgramming/Morphogrammatic%20Programming.html
http://memristors.memristics.com/Dominos/Domino%20Approach%20to%20Morphogrammatics.html
http://memristors.memristics.com/semi-Thue/Notes%20on%20semi-Thue%20systems.pdf

2.2.6.  Trito-sum

A = [aab]  _ [1  ^2 2 ^1], with [a, b] as support set : A = [a ... #xF3A0; ^2 3 ^1] <br /> sum  _ trito (A, B) = A ∪+  _ t B .

[aab]  _ trito  ∪+  _ t (         B                              ...             μ 4                  μ 5                  μ 6                  μ 7

<br /> μ (B) = m !/(m - k) ! + 1 =    3 !/(3 - 3) ! + 1 = 7

With root   se t notation :             ... 3A0;                                                1   2   3   2 

Trito - Non - Commutativity for ∪+  _ trito A    ∪+  _ t ...                  2   3                         1   2   3 

<br /> Mset - commutativity for    ∪+  _ mset <br /> A    &# ... xF3A0; _ (2, 1) = [a, b]  _ (5, 1) <br />         

2.2.7.  Trito-Union

Multiset union
"Let A and B be msets. The union of A and B, denoted by A∪B, is the smallest mset C containing both A and B i.e., A ⊆ C and B ⊆ C. In other words, mC(x) = max{mA(x), mB(x)} for all objects x if such a max exists; otherwise the min is taken which always exists.” (Sing)

Example :   trito - union  _ trito (A, B) = A ∪  _ t B . <br /> A = [aab ...            μ 4                  μ 5                  μ 6                  μ 7

<br /> 2.   [a, b]  _  _  _ (1  ^2 2  ...                                                                             1   2 

               &nbs ... 3   2                               1   2   3   4 

Trito - Non - Commutativity for ∪  _ trito A    ∪  _ t B ...                  2   3                         1   2   3 

Trito-non-idempotency

A    ∪  _ t A != A, <br /> A    ∩  _ t A != A, ...   2   3                          1   2   3     

<br />             <br />     ...                                     [A "]                 [A] + [A "]    

[a, b]  _ (1 ^2 2 ^1) ∩  _ t ([a, b]             ...    2   3                          1   2   3     

<br />

            <br />     Trito  ...                          [A "]                 [A] + [c]       

[a, b]  _ (1 ^2 2 ^1) ∪+  _ t ([a, b]            ...    2   3                          1   2   3     

      <br />     Trito - idempotency for   ∪+ <br />  ...                               [A "]                 [A] ⊕ [A "]    

2.2.8.  Trito-Difference

Multiset difference " Difference and complementation . Let A and B be two msets over D an ... xF3A0; _ A (x)    - m  _ (A ∩ B) (x), for all x ∈ D . " (Sing)

Tritoset difference A, B ∈ TRITO : difference  _ trito (A, B) = A \  _ t ... F3A0;^(-1))                

 A \  _ t B = ([a, b, c]                                        &nbs ...                                                                 1   2   3 

The proposed approach of the example for the difference operation accepts negative 'occurrences' of elements. This makes sense only in the context of the whole constellation of the trito-grammatic difference operation. Negative occurrences as such, i.e. in isolation, are not (yet) considered.
Normal form: [a,b,c]typeset structure = [aabbb c=⌀  d=-1] = [aabb d=-1] = [aabbc=-1] = [a,b,c]typeset structure

2.2.9.  Trito-Complement

Multiset approach
"Let  ℑ = {A1 , A2 , ...} be a family of multisets composed of the elements of the generic set D. Then, the maximum multiset z is defined by mz(x) = typeset structuretypeset structure mA(x) for all x ∈ D and all A ∈ ℑ.
Now, the complement of an mset A, denoted by typeset structure , is defined as follows:
typeset structure = Z − A = {mtypeset structure (x) . x | mtypeset structure (x) = mz(x) − mA(x), for all x ε D}." (Sing)

"Now define the difference A − B between two multisets A and B as A − B = A + (−B)
or equivalently by the rule that for any object x mtypeset structure (x) = mA (x) − mB (x) .
(-1) A = - A
n A + m A = (n + m) A,  n (mA) = (nm) A .

"We may now derive the ‘De Morgan type’ laws
(−A) ∩ (−B) = −(A ∪ B)
(−A) ∪ (−B) = −(A ∩ B)
and their relative versions
(A − B) ∩ (A − C ) = A − (B ∪ C )
(A − B) ∪ (A − C ) = A − (B ∩ C )."
(N J Wildberger, A new look at multisets, 2003)
http://web.maths.unsw.edu.au/~norman/papers/NewMultisets5.pdf

Example: Trito-complement typeset structure (- A)

A = [a, b, c]typeset structure
B = [a, b]typeset structure

(-1) B = -B = - [a, b]  _  _ (1 ^1 2 ^1) = [a, b]  _ & ... - 2 ^3 - 3 ^1)) = [a, b, c]  _ (1 ^2 2 ^3 3 ^1) .

2.2.10.  Trito-Products (multiplication)

Multiset multiplication
"There is also a multiplicative operation for multisets. Define A × B, the direct product of the multisets A and B, to be the multiset consisting of all ordered pairs [a, b] with a ∈ A and b ∈ B. By this we mean that mtypeset structure ([a, b]) = mA (a) x mB (b)." (Wildberger)

Multiset  A × B

For example
[1 2] × [2 3 2] = [[1, 2] [1, 3] [1, 2] [2, 2] [2, 3] [2, 2]].

"A × [] = [] × A = [] .  
|A × B| = |A| |B| .
Distributive laws for finite msets
A × (B ∪ C ) = (A × B) ∪ (A × C )
A × (B ∩ C ) = (A × B) ∩ (A × C )
A × (B + C ) = (A × B) + (A × C ) .
Non-commutative and non-associative laws
A × B != B × A
(A × B) × C != A × (B × C ) .”  (Wildberger, p.9)

Additionally:
A x [1] = [1] x A = A.

Trito-multiplication

Example : trito - multipication A x B A = [a, b]  _ (1 ^1)  _ (2 ó ...                                                                                 [abb ' cdd ' abb]

<br /> A x B = mul ([a, b]  _ (1 ^1)  _ (2 ^2), [a, b]  _ (1 ^1)  _ (2 ^1 1 ^1)) = <br />

              ( [a, b] ༺ ...                             1   2   3   4   1   2 

<br /> Arithmetic <br /> | [a, b]  _ (1 ^1)  _ (2 ^2) | = 3 <b ... A0;^2), [a, b]  _ (1 ^1)  _ (2 ^1 1 ^1)) | = 9 <br /> Null

  A x B = kmul ([a, b]  _ (1 ^1)  _ (2 ^2), [a, b]  _  ... p;                

Table notation of A x B  A x B = ( [a, b]                                              ...                [1   2  ]        (3   4   ) Null

    <br />         kmul ([A], [B]) = [1 ^1 2 & ...                                                                     [ 3  ]   [4  ]

Trito-multiplication with context rule

Context - Rule CRM    <br />    ∀ i ∈ Dec(MG _ (i + 1) ... 1))) <br />    Head = first kenom of a monomorphy <br />    Body = the rest .

<br /> Decomposition <br /> kmul(MG  _ 1, MG  _ 2) : <br /> kmul([1, 2, 2], [1, 2, 3, 1]) => kmul(([1], [2, 2]), ([1], [2], [3], [1])) . <br />

MG _ 1 = [1, 2, 2] ==> [mg _ 1, mg _ 2] MG _ 2 = [1, 2, 3,  ... 3A0; _ 1.2, _]}, {Overscript[Overscript[mg _ 1.2, _], _]}, mg _ 1.2)) ∈ CRM

<br /> Order <br /> alph(MG)    --> num(MG) : <br /> [abb], [abca]    --> [1, 2, 2], [1, 2, 3, 1] .

<br /> Multiplication   table   for   kmul   ([1, 2, 2], [1, 2, 3, 1] )   <br /> kmul          ...      1 1 1 3     5 3 3 5 1     6              1  2  2

<br /> kmul (1   , 1 ) = {1            2            3            1         , 1            3    ...                                1   1   1   1   3   4   5   4   3   5   3   6                     1

               &nbs ...                       O                   O                   O                             5.6 .6

<br /> Example   <br /> [r  _ 122 r  _ 211 r  _ 233 r  _ 244 r ... 0;^2 2 ^1 4 ^2 2 ^1 3 ^2 1 ^1 2 ^2)    .

<br /> Trito - beginnings, recursion head <br /> A × [] = [] × A = [], <br /> A x  ... _ iter  [ab] = [ab] <br /> 1 x  _ accr  [ab] = [bc], hence <br /> 1 x [ab] = ([ab]/[bc]) .

2.3.  Deutero-structures

2.3.1.  Operations on deutero-sets

Multisets

Example A = [a, b]  _ (4, 5), B = [c, d]  _ (2, 3), with [a, b] != [c, d]

Deutero - sets A and B are    deutero - multisets, if their objects are deutero - eq ... > A = [a, b]  _ (4, 5), B = [c, d]  _ (2, 3), with [a, b] =  _ d [c, d] .

Example :   sum   ∪+  _ deutero A = [a, b]  _ (4, 5), B = [a, b]  ...                             μ2                    2, 3                            4, 5, 2, 3

Example :   union   ∪  _ deutero A = [a, b]  _ (4, 5), B = [a, b] ༺ ...   μ2                    2, 3                                                      4, 5, 2, 3

Example: multiset difference \mset
"Let A and B be two msets, and B ⊆ A. The (arithmetic) difference of B from A, denoted by A \ B or A − B, is the mset C ⊆ A such that mC(x) = mA(x) − mB(x), for all objects x.
In general, mC(x) = mA(x) − m(A ∩B) (x) = max{mA (x) − mB (x), 0}, for all objects x.” (Sing)

Difference \  _ mset A = [a, b]  _ (4, 5), B = [a, b]  _ (2, 3), <br / ... sp; [a, b]  _ (4, 5) \    [a, b]  _ (2, 3) = [a, b]  _ (2, 2) .

Example   :   deutero - difference   \  _ d A = [a, b, c]  _ (4, 5, 4) , B = [ ... 4, 5, 4) \    [c, d]  _ (2, 3) = [a, b, c, d]  _ (4, 5, 2, -3) . <br />

A \  _ d B = ([a, b, c]                )                                       ...                      [a, b, c, d]                                              4, 5, 2, -3

2.4.  Proto-structures

Proto - addition (n, k) +   _ proto (m, j) = {(n + m, k + j - 1), (n + m, k + j)} .

Example : addition  _ proto A = [a, b]  _ (4, 5) = [9 : 2], B = [a, b]  ... 0; _ proto B =    [9 : 2] +  _ proto   [5 : 2] = ([14 : 3]/[14 : 4])

Algebraic properties of   +  _ proto <br /> +  _ proto ∈ {distributive, commutative, associative}

3.  Polycontextural multiset theory

3.1.  Sketch of polycontexturality

3.1.1.  General framework

"A mapping : ∪ -> X, where  ∪ is a universal set and X is a numericset, is called a set if X = {0, 1}; a multiset if X = N, the set of natural numbers with 0; a signed multiset if X = Z, the set of integers.” (Sing)

A logical valuation of for a classical two-valued logic gives: val () --> {t, f}. This corresponds to a mono-contextural constellation. For a poly-contextural constellation, the mapping has to consider its dissemination in the contextural grid.

In a polycontextural setting, the mapping α is disseminated over a grid of contextures, producing a framework for disseminated sets and multisets.

Polycontextural valuation for m=3: valtypeset structure(typeset structure) --> {t, f}typeset structure:

(val ^1 (∝) --> {t, f} ∐  _ 1.2 val ^2 (∝) --> ... 01; t  _ 3, f  _ 1 ≅ t  _ 2, f  _ 2 ≡ f  _ 3 .

<br /> (t, f) ^(3) = ((t  _ 1 --> f  _ 1 ∐ t  _ 2 --& ...              ] f  _ 3)) .

<br /> val ^(3) (∝ ^(3)) --> {t, f} ^(3) = ((                 ...                                                               val (∝) --> {t, f} 

Dissemination
There are at least 5 main types of actions on disseminated multisets or modi of interaction in polycontextural systems to be studied:
1. identitive        : id: α --> α
2. permutative   : perm: α1α2 --> α2α1
3. reductive        : red: α1α2 --> α1α1
4. reflective (reflectional) :                          refl1: α1αtypeset structure --> αtypeset structuretypeset structureα2.2
5. bifurcative (transpositional, interactional) bif1:  α1αtypeset structure --> α1.1α2.1α3.1α2.2α3.3
http://crossbars.memristics.com/Poly-Crossbars/Poly-Crossbars.pdf

3.1.2.  Elements of a polycontextural quantification-theory

Multisets are first of all sets, i.e. special sets. Hence, a polycontextural framework for multisets has to consider some elements of a polycontextural set-theory. Further on, set-theory is based on first-order logic with its operators for “all” and “there exist”: ∀x (Px) and ∃x (Px).
In a two-valued setting, the tableaux for ∀x and ∃x are well defined. Hence, a distribution of the tableaux rules for quantification over different contextures follows quite naturally.

Another question is, how to define transjunctional quantifiction, i.e. quantification over discontextural universes of logics? Obviouslly, their elements are at once in at least two different contextures or discontextural domains. Therefore, they have to be taken into account as tupels of 2 elements, say a and b. The variable x of the quantification gets specialized by “a” and “b”. Syntaxtically, the variables are split to run over different contextures. In the case of Q1, (a1, b1) ∈ U1, with U1 as the universe U1 of logic Logtypeset structurein Logtypeset structure. Because of the tediuos complication of full formalisation, simplifications should support reading and understanding.

Quantification schemes

Scheme for junctional quantification (id, id, id) : Underscript[[∃ ∀] [∃  ...            3.3 x ^(i . j) ∈ U ^(i . j) type = {set, mset, tritogram, ...} .

Scheme   for   transjunctional   quantification : <br /> Example :   [   Underscript[∀ & ...                                   -                     -                                     3.3

    <br />      [bif, id, id] : [  Underscript[∀ &nbs ... ;                <br />

    <br />      (bif, id, id) : <br />    &n ...                                               ∀  (x ) P (x )

<br />     Junctional quantifiers <br /> <br />     Underscript[ ...        <br />          

 (                                                                                             ...                                                              a                  a 

        Transjunctional quantifiers <br /> <br />  &nb ... ;                

(                                                                                              ...                                                        3.3  b              3.3  b 

<br /> Duality and self - duality for quantifiers <br />      Duality <br  ... p; 1, 2, 5        ]    ∃ ∃ ∀ <br />

3.1.3.  Identitive disseminations

          <br />       Example : M ...                                                                                                  3

3.1.4.  Bifurcative disseminations

This paper is offering a sketch of some modi of interactions in a exemplary way as a logical scheme for the interactions of disseminated multiset systems, dismset.


[Graphics:HTMLFiles/Graphematics of Multisets_273.gif]

A highly explicit formalization of the basic features of the constellation <transjunction, conjunction, conjunction> as given by the logical tableau setting is proposed by the category-theoretic formalization involving bifunctoriality to model the distribution of transposition (transjunction) and mediation of the mentioned operators in the contextural grid.
A simplified scheme for (typeset structure) shows more directly the distribution for transposition, typeset structure, and mediation, typeset structure, in the 3-contextural grid (M, O).

typeset structure

 <br />           val (X <> ∧ ∧ ...                           3            3                               3.1           1           1

 <br />    false (X <> ∧ ∧ Y) :      <br /> (  ...      3.1            1           1                  3.1           1           1                   1

3.2.  Dissemination of multiset systems

3.2.1.  Junctional operations

On the base of the sketched polycontextural schemes, disseminations of multiset systems that are containing msets are naturally constructed. Junctional operations of identive mappings are modeling a kind of parallelism and concurrency without interactiviy and reflectionality between disseminated mset systems. Therefore, there are no special conflicts to consider for a consistent formalization as for a modeling of transjunctional, i.e. interactional mset systems.

Example :   union   ∪ ∪ ∪  _ deutero A = [a, b]  _ (4, 5), B ...   μ2                    2, 3                                                      4, 5, 2, 3

 (([a, b]                  &nbs ...                                                                                         4, 5, 2, 3

3.2.2.  De Morgan for 3-junctional operations

Example :   union   ∪ ∪ ∪  _ deutero      & ... p;    = [a, b, c]  _  _ (1 ^2 2 ^3 3 ^1) .

<br /> A = [a, b]  _ (4, 5), B = [a, b]  _ (2, 3) : <br /> (- _ 1 A &# ... 3 (A ^(3) ∪  _ 2 ∪  _ 1 ∪  _ 3 B ^(3)) .

3.2.3.  Transjunctional operations

Transjunctional operations on disseminated mappings are modeling a kind of interactivity and interpenetration without reflectionality between disseminated contextural systems.

Scheme   for   transjunction : Example :   Underscript[∪+  ∪  ∩, type, type, type]

(bif, id, id) : ∪+ ∪ ∩ --> ∪+ ∪ ∩ : <br /> bif  ...                                                   1.1                   2.1                    3.1

id  _ 2 : ∪    --> ∪ : type id  _ 3 : ∩  --> ...                                 -                      -                                      3.3

    <br />      [bif, id, id] : A ^(3, 3) Underscri ... p;                

3.2.4.  Reflectional operations

(refl, id, id) : ∪+ ∪ ∩ --> ∪+ ∪ ∩ : <br /> refl ༺ ...                                                   1.1                   1.2                    1.2

id  _ 2 : ∪    --> ∪ : type id  _ 3 : ∩  --> ...                                                 1.3    -                                      3.3

    <br />      [refl, id, id] : A ^(3, 3) Underscr ... p;                

3.2.5.  Dissemination of mixed abstract data types

It is supposed that mostly a complexion of processes, especially in living systems, cannot be adequately modeled with one abstract data type alone. A complexion of abstract data types that are supporting more concrete data types are necessary, all interacting at once together. Such a situation has no correct modeling and formalization in mono-contextural logics and systems theory. Earlier sketches of a general framework of complex programming and programming the complexity of living systems had been published as “Contextural Programming” at ConTxTures:
http://www.thinkartlab.com/pkl/lola/ConTeXtures.pdf

The example A ◊∪∪ Btypeset structure below shows an interpenetration of subsystem2 and subsystem3 onto subsystem1.
Specifically, the subsystem1 wit its ‘data type' "tritogram” gets an interpenetration by transjunction from subsystem2 with its ‘data type' "msets” and from subsystem3 with its ‘data type' "sets”. All this together holds simultaneously. Junctional situations that are holding or running separately in parallel are covered by subsystem2 as such with its ‘data type’ “mset” and by the subsystem3 as such with its ‘data type' "set”.

The logical constellation of this simultaneity of different abstract data types at a ‘single’ contextural locus is ruled by the logical tableaux for transjunctional quantification. The proposed formula for A ◊∪∪ Btypeset structure is not yet taking this logical constellation manifestly into account in its presentation. The logical background becomes manifest in the further use and development of the formula. A fact not unknown in set or multiset theory that are based on classical logic.

Team observation
With the dissemination of objects over the kenomic matrix it becomes obvious that an adequate observation of the events in the complexion is not realizable by a single external observer with a single or multiple observations but needs a team of observers with an adequate team structure to be observed, modeled and formalized.

Junctional dissemination

Example :   union   Underscript[∪  ∪  ∪, set, mset, mset] A  ...                                          2, 3                                                4, 6

 <br /> Example :   union   Underscript[∪  ∪  ∪, set, mset,  ...                               2, 3                                                      2, 3, 2, 3

Example :   union    Underscript[∪  ∪  ∪, set, mset, trito]  ...                                                    1   2   3   4 

Example :   union    Underscript[∪  ∪  ∪+, set, mset, trito] ...        1   2   3            

Transjunctional dissemination

Scheme   for   transjunction   ◊ :   Underscript[∪+  ∪  ∪, trito, mset, set]

(bif, id, id) : Underscript[∪+  ∪  ∪, trito, mset, set] --&g ...                                   trito           1.1    mset           2.1     mset           3.1

id  _ 2 : Underscript[∪+  ∪  ∪, trito, mset, set] -- ...                                 -                      -                        set           3.3

<br />      Underscript[(bif, id, id) : A ^(3) ∪+  & ...  (A ^3.3  Underscript[∪, set] B ^3.3)     <br />   

<br /> Example : transjunction    [bif, id, id] : <br /> A Underscript[∪+  ... A0;^2 = [b, c]  _ (2, 3)     , B ^3 = {a, b}  _ (1, 1)

<br />     [bif, id, id] : A Underscript[∪+  ∪   ...                 1   2                                       1   2 

     <br /> ([bif, id, id] : A ∪+  ∪  ∪ B = < ...                                                              3.3                              1, 1

3.2.6.  Algebraic properties of disseminated mset-operations

mset = (mset, ∩ , ∪ , ∪+ , ◊ , O) : <br /> <br /> algebr (mset ) = (ms ... , alg)) <br /> alg = {com, ass, idem, iden, distr}, <br /> sops = (id, red, iter, refl, perm, bif}

Monoform A ∪ ∪ ∪ B    <br /> Heteroform A ∪ ∪ ∩ B  A ∪ ∪+ ∩ B <br /> Bifunctorial

(A o C) ◊ (B o D) = (A ◊ B) o (C ◊ D) <br /> (A o C) O (B o D) = (A O B) o (C O D)

Duality dual              &nb ... bsp;        : O --> O      (reflection)

 <br />         Bifunctoriality for transposition <br /> &n ...                                               3                                    3.1           1

  <br />      false : log (A ◊ ∪ ∪  B  _ ...                  3.1            1                        3.1           1                         1

http://www.thinkartlab.com/pkl/lola/AFOSR-Place-Valued-Logic.pdf
http://www.thinkartlab.com/pkl/lola/From%20Ruby%20to%20Rudy.pdf

4.  Possible applications of graphematics for Membrane computing

4.1.  Membrane computing and multisets in hierarchies

[Păun 2005]: “parallelism a dream of computer science, a common sense in biology“.

"Any cell means membranes. The cell itself is defined - separated from its environment - by a membrane, the external one. Inside the cell, several membranes enclose “protected reactors”, compartments where specific biochemical processes take place. In particular, a membrane encloses the nucleus (of eukaryotic cells), where the genetic material is placed.

"We have mentioned above the notion of a multiset. The compartments of a cell contains substances (ions, small molecules, macromolecules) swimming in an aqueous solution; there is no ordering there, everything is close to everything, the concentration matters, the population, the number of copies of each molecule (of course, we are abstracting/idealizing here, departing from
the biological reality). Thus, the suggestion is immediate: to work with sets of objects whose multiplicities matter, hence with multisets. This is a data structure with peculiar characteristics, not new but not systematically investigated in computer science.”
http://psystems.disco.unimib.it/download/MembIntro2004.pdf


[Graphics:HTMLFiles/Graphematics of Multisets_310.gif]

"More mathematically stated, we look to the set of rules, and try to find a multiset of rules, by assigning multiplicities to rules, with two properties:
(i) the multiset of rules is applicable to the multiset of objects available in the respective region, that is, there are enough objects in order to apply the rules a number of times as indicated by their multiplicities, and
(ii) the multiset is maximal, no further rule can be added to it (because of the lack of available objects)."
http://www.macs.hw.ac.uk/~pier/

4.2.  Multisets in heterarchies

"Interchangeability not even a dream of computer science, a common interaction in living systems.”

Păun’s text gives an impressive plaidoyer for the use of mutisets in computer science and for the modeling of living systems. Despite the highly technical elaborations of P-systems or Membrane systems, it seems that this approach is giving just a snapshot of a living organism but is not thematizing the dynamic mechanisms of living matter as such.
Its emphasis is on hierarchical systems, described mainly by the data structure of multisets but neglecting the interchangeability of hierarchies that are establishing heterarchies between parts and wholes, or domains and system.
The interchangeability, chiasm or proemiality of the inside/outside mechanism is not in the focus and is strictly excluded by the hierarchy of the general model that is strictly mirrored in the formal apparatus.

5.  Appendix 1: Elements for programming trito-grammatics

TRITO-EQUIVALENCE: a=b, ab=ba, aa!=ab, aab!=aba ∈ ∑tnf

5.1.  Equivalence classes

Tritogram[A] = [abcddc]
Tritogram[B] = [•Oc♣♣c]
dec([A]) = ([a], [b], [c], [dd], [c])
dec([B]) = ([•], [O], [c], [♣♣], [c]).
[A] =trito [B] iff ([a] =ken[•], [b] =ken[O], [c] =typeset structure, [dd] =ken [♣♣]).

But this equivalence relation approach is, albeit correct, misleading because it is still too much relying on the identity of its signs.

5.2.  Morphogrammatics

ϵ/ν-structure
How to define the equivalence of tritograms?

The ϵ/ν-approach is checking just the equality (ϵ) or non-equality (ν) of objects (signs) of a collection or a string and not the individual atomic signs as such.

[A] = [aabc] ENstructure[A]

              ν  ... sp; a       b         c]   


datatype EN =E|N;
fun delta (i, j) z=
if  (pos i z) = (pos j z)
    then (i, j, E)
     else (i, j, N) ;

- ENstructure ["a”, “a”, “b”, “c"];
> [[],
   [(1,2, E),
   [(1, 3, N), (2, 3, N)],
   [(1, 4, N), (2, 4, N), (3, 4, N)]]: enstruct

The ϵ/ν-analysis of a constellation of objects determines the tritogram [A].trito.
For further elaborations, this ϵ/ν-result of the tritogram might be transformed into a sequential form of a keno-sequence (kseq) with the operation ENtoKS.

This is realized by the ML function: ENtoKS

- ENtoKS [[],
                [(1,2, E),
                [(1, 3, N), (2, 3, N)],
                [(1, 4, N), (2, 4, N), (3, 4, N)]]
> [1, 1, 2, 3] : kseq

- ENtoKS ENstructure ["b”, “a”, “d”, “c"];
> [1, 2, 3, 4] : kseq

Trito normal form tnf
A given keno-sequence might not be in a standard normal form (tnf), hence the ML function tnf, delivering ks shall be applied.

tnf: [c,d,d,a] --> [a,b,b,c], kseq.
- val a = [2, 2, 1, 1] : kseq
> val a = [2, 2, 1, 1] kseq
-tnf a;
>  [1, 1, 2, 2] : kseq

Trito-equivalence
Two tritograms [A] and[B] are trito-equivalent iff their ϵ/ν-structures are equal.

[A] =trito [B] iff EN([A]) = EN([B]).

typeset structuretypeset structure

In contrast: Equality for multisets
"Two multisets A and B are equal, or the same, if for any object x
                      mA (x) = mB (x)." (Sing)

Hence, from the example: [A] !=mset [B] but  [A] =tset [B].

5.3.  Monomorphies

Strings of signs consist of atomic signs form an aphabet. This is leading all the following definitions for strings, especially the definition of substitution.

Tritograms and Tritosets are patterns and not strings and consist of monomorphies.
The mein properties or operations on singular tritosets are:
primary operations ={tnf, card, lex, num, dec, ken, pos}.

          ν  _ 1.4 <br />  &nbs ... 57;  _ 3.4    <br />     <br /> [aaab]    = [C] <br />

[A] = [aabc]:
[A].dec = [mg1, mg2, mg3]
mgtypeset structure,
mg2.2.ken = [b],
mg3.3.ken = [c].

[C] = typeset structure
[C].dec = [mg1, mgtypeset structure
[mg1.1].ken = [aaa]
[mgtypeset structure

[A].dec: [A].EN = (1,2, E)
- ENstructure ["a”, “a”, “b”, “c"];
> [[],
   [(1,2, E),
   [(1, 3, N), (2, 3, N)],
   [(1, 4, N), (2, 4, N), (3, 4, N)]]: enstruct.

[A].dec = [mg1,mg2, mg3]

[C].dec: [C].EN = (1,2, E), (2,3, E)
- ENstructure ["a”, “a”, “a”, “b"];
> [[],
   [(1,2, E),
   [(1, 3, E), (2, 3, E)],
   [(1, 4, N), (2, 4, N), (3, 4, N)]]: enstruct.

Construction of monomorphies from ENstructure <br />         ... >      <br />       [aaab]    = [C] <br />

[C].dec =  [(1,2, E) ∪ (2,3, E) ∪ (1,3, E)] => mg:[mg1, mg2]
mg1 = ((1,2)+(2,3)+(1,3), E)
[C].dec = [mgtypeset structuremg2]
ken([mg1]) = [aaa],
ken[mg2] = [b].
Numeric functions
card([mg1] = 3
lex([mg1]) = 1
num([mgtypeset structure) = (lexcard([mgtypeset structure) = 13.
num([mg2]) = 21
num([MG]) = num([mgtypeset structure), num([mg2]),
num([MG]) = 1321.

     [D] = <br />      [a     &nbs ... nbsp;      ν  _ 1.6      <br />    

- ENstructure ["a”, “a”, “b”, “c”, “c"];
> [[],
   [(1,2, E),
   [(1, 3, N), (2, 3, N) ],
   [(1, 4, N), (2, 4, N), (3, 4, N)],
   [(1, 6, N), (2, 5, N), (3, 5, N), (4,5,E]: enstruct.

[D].dec = [ [(1,2, E) ∩ (4,5,E] = ∅] => mg:[mg1, mg2, mg3]

[D].dec = [mgtypeset structuremg2, mg3]
ken([mg1]) = [aa],
ken[mg2] = [b],
ken[mgtypeset structure= [cc].

ML procedures in: Morphogrammatik, p. 46, 49-52
http://works.bepress.com/thinkartlab/15/

5.4.  Math of Monomorphies

Let A and B nonempty finite sets A = {a1, a2, ..., an} and B = {b1, b2, ..., bm}
Let BA denote the set of allmappings from A to B,

BA ={μ | μ: A --> B}.

This is elaborated at: Morphogrammatik.

How to construct monomorphies mathematically?
The question: What replaces atomic signs in a kenogrammatic pattern (morphogram)? Is answered by Schadach with the introduction of monomorphies of morphograms.

From a mathematical point of view, monomorphies are partitions of mappings. This is well elaborated by [Schadach 1967]. The procedure to build monomorphies out from morphograms, as it is mathematically defined by Schadach’s approach, shall be called monomorphic decomposition.


typeset structure
typeset structure
typeset structure

(Dieter J. Schadach, BCL Report No. 4.1, August 1, 1967)