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# day28 - Canonical Processes Groups and Grammars Click to...

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Click to edit Master subtitle style 10/4/11 Canonical Processes, Groups and Grammars Post Canonical Systems of Varying Sorts and Their Relation to Groups and Grammars

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 22 Semi-Groups, Monoids, Groups S = (G, •) is a semi-group if G is a set, • is a binary operator, and 1. Closure: If x,y { 2. Associativity: x • (y • z) = (x • y) • z S is a monoid if 3. Identity: Pe F G Mx ° G [e • x = x • e = x] S is a group if 4. Inverse: ex ° G ± x-1 ° G [x-1 • x = x • x-1 = e]
10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 33 Finitely Presented If S is a semi-group (monoid, group) defined by a finite set of symbols h, called the alphabet or generators, and a finite set of equalities ( α i = β i), the reflexive transitive closure of which determines equivalence classes over S, then S is a finitely presented semi-group (monoid, group). Note, the set S is the closure of the generators under the semi-group’s operator. The word problem for S is the problem to determine of two elements α, β , whether or not α = β , that is, whether or not they are in the same equivalence class. If ° is commutative, then S is Abelian.

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 44 Finitely Presented Monoids Strings over an alphabet (operation is concatenation, identity is string of length zero). Natural numbers (use alphabet {1} make + the operator, identity is 0 occurrences of a 1, use shorthand that n represents n adds: 1+1+ … +1). This is actually an Abelian monoid. In above cases, we would also need rules for equivalence classes, e.g., we can get the equivalences classes dividing the even and odd numbers by 1+1 = 0 The two classes have representatives 0 and 1.
10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 55 Abelian Monoids Consider a finitely presented Abelian monoid over generators Σ = {a1,…,an}.

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day28 - Canonical Processes Groups and Grammars Click to...

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