Graphematics of Multisets

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-, deuteroand 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 deuteroand 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. Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 1 of 52 24/05/2012 17:19 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] and B = [a, b] then A − B = [a, b] ⊂ 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) = ∪ X 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] and B = [a, b] then A − B = [a, b] ⊂ 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] and B = [a, b] then A − B ∩ B = ∅: ([a, b] [a ) ∩ [a,b] = ∅ and for (A − B) ∪ B = A: ([a, b] [a ) ∪ [a, b] = [a, b] . Properties multiset multiplicity of objects cardinality of the multiset order of objects is irrelevant Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 2 of 52 24/05/2012 17:19 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". Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 3 of 52 24/05/2012 17:19 A polycontextural thematization and formalization of the topics of multisets is replacing setor 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 multisets, 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. mset: μ: U --> U, ν: N --> N bifunctorial: (μ, ν): : (N1 N2) o (U1 U2) = (N1 o U1) (N2 o U2). Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 4 of 52 24/05/2012 17:19 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. Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 5 of 52 24/05/2012 17:19 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. 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-, deuteroand 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. Table of types, examples Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 6 of 52 24/05/2012 17:19 "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 , 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: A final explication has to inscribe the order of the loci of the monomorphies in the morphogram (tritoset), 1 -5, hence: For A = [a, b, c] , with [a,b,c] as support set of kenoms in trito-normal form (tnf), the indices as 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. 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. 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. Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 8 of 52 24/05/2012 17:19 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. 2.1.3. Definition schemes for mset, tset, dset and pset Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 9 of 52 24/05/2012 17:19 For A = [a, b, c] , with [a,b,c] as support set of kenoms, the indices as 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 Morphic Multisets.nb file:///Volumes/KAEHR/HD-KAE-Texte/KAE-TEXTS/Publi... 10 of 52 24/05/2012 17:19 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] = {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 reduction: msets --> dpsets --> dsets: