Semiautomatic Ordinal and Ring Structures Qi Ji NUS School of - - PowerPoint PPT Presentation

Semiautomatic Ordinal and Ring Structures

Qi Ji

NUS School of Computing

13th November 2019

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Slides

https://m4th.b0ss.net/semi/semi.pdf

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Background

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Automata theory

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Finite automata

Processes input symbol by symbol with fjnite memory The fjnite memory indicates whether the word as seen so far satisfjes the condition to be checked.

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Check multiple of 3

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Automatic relations

Encode idea of process tuples in parallel Add a padding symbol # 𝑑𝑝𝑜𝑤(010, 01236) = (0

0)(1 1)(0 2)(# 3 )(# 6 )

A function is automatic ifg its graph (encoded this way) is automatic

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Verifying addition

Primary school algorithm n – correct and no carry c – carry i – wrong Correct addition Incorrect addition 2 3 5 8 3 3 3 3 # 1 1 2 # # 2 2 2 4 7 0 # 1 5 5 n n n c n i i n n n

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Automatic structures

A structure (𝐵, 𝑔1, … , 𝑔𝑜, 𝑆1, … , 𝑆𝑛) is automatic ifg 𝐵 is a regular set, 𝑔1, … , 𝑔𝑜 are automatic functions, 𝑆1, … , 𝑆𝑛 are automatic relations.

Example

(ℕ, +) is automatic but (ℕ, +, ⋅) is not automatic.

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Semiautomatic structures

(𝐵, 𝑔1, … , 𝑔𝑜, 𝑆1, … , 𝑆𝑛; 𝑕1, … , 𝑕𝑞, 𝑇1, … , 𝑇𝑟) is semiautomatic ifg 𝐵 is a regular set, 𝑔1, … , 𝑔𝑜 are automatic functions and 𝑆1, … , 𝑆𝑛 are automatic relations, 𝑕1, … , 𝑕𝑜 are semiautomatic functions and 𝑇1, … , 𝑇𝑛 are semiautomatic relations, where 𝑔 ∶ 𝐵𝑜 → 𝐵 is semiautomatic ifg fjxing 𝑜 − 1 inputs, the resultant 𝐵 → 𝐵 function is automatic.

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Semiautomatic structures

Example

(ℕ, +, <, =; ⋅) is not automatic. For any constant 𝑜, implement multiplication by 𝑜 as repeated addition.

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Set theory

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Ordinals

Equivalence classes of well-order, where a well-order (𝑇, <) obeys trichotomy, transitivity and well-foundedness (every nonempty subset has a minimum). Intuitively, measures how many times a discrete process is repeated (possibly transfjnitely many). For example, 0, 1, 2, 3, … , 𝜕, 𝜕 + 1, … , 𝜕 + 𝜕 = 𝜕 ⋅ 2, 𝜕 ⋅ 2 + 1, 𝜕 ⋅ 2 + 2, … , 𝜕 ⋅ 2 + 𝜕 = 𝜕 ⋅ 3, … , 𝜕 ⋅ 4, …, 𝜕 ⋅ 𝜕 = 𝜕2, … , 𝜕3, … , 𝜕𝜕, …. One way to generalise sum and products to infjnite structures.

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Ordinals

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Semiautomatic Ordinal Structures

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Semiautomatic ordinals with automatic addition

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Existing characterisation

Theorem (Delhommé)

For any ordinal 𝛽 the structure (𝛽, +, <) is automatic ifg 𝛽 < 𝜕𝜕.

Proof Sketch

Consider 𝛽 = 𝜕𝑜, any 𝛾 ∈ 𝛽 is of the form 𝜕𝑜−1 ⋅ 𝑑𝑜−1 + ⋯ + 𝜕 ⋅ 𝑑1 + 𝑑0 for coeffjcients 𝑑0, … , 𝑑𝑜−1 ∈ ℕ. We can defjne + on the 𝑜-ary convolution of an automatic copy of (ℕ, +, <).

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Incorporating semiautomaticity

• Observation. Using the same representation, when 𝛾 ∈ 𝛽 is fjxed, we

can defjne left and right-multiplication by 𝛾 in an automatic manner. (Addition and multiplication on ordinals are not commutative)

Theorem

For any ordinal 𝛽 < 𝜕𝜕 the structure (𝛽, +, <, =; ⋅) is semiautomatic.

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Left multiplication

Let 𝛾 = 𝜕𝑙 ⋅ 𝑐𝑙 + 𝜕𝑙−1 ⋅ 𝑐𝑙−1 + ⋯ + 𝜕 ⋅ 𝑐1 + 𝑐0 𝛿 = 𝜕𝑚 ⋅ 𝑑𝑚 + 𝜕𝑚−1 ⋅ 𝑑𝑚−1 + ⋯ + 𝜕 ⋅ 𝑑1 + 𝑑0 expanding the giant expression, we get 𝛾 ⋅ 𝛿 = 𝛾 ⋅ 𝜕𝑚 ⋅ 𝑑𝑚 + 𝛾 ⋅ 𝜕𝑚−1 ⋅ 𝑑𝑚−1 + ⋯ + 𝛾 ⋅ 𝜕 ⋅ 𝑑1 + 𝛾 ⋅ 𝑑0 … = 𝜕𝑙+𝑚 ⋅ 𝑑𝑚 + 𝜕𝑙+𝑚−1 ⋅ 𝑑𝑚−1 + ⋯ + 𝜕𝑙+1 ⋅ 𝑑1 + (𝜕𝑙 ⋅ (𝑐𝑙 ⋅ 𝑑0) + 𝜕𝑙−1 ⋅ 𝑐𝑙−1 + ⋯ + 𝜕𝑐1 + 𝑐0) ⋅ 1𝑑0≠0 where 1𝑑0≠0 is 1 is 𝑑0 ≠ 0 and 0 otherwise.

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Right multiplication

Ordinal multiplication distributes on the right, so we get a fjnite composition of right-multiplication by 𝜕, right-multiplication by fjxed constants,

• rdinal addition.

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Right multiplication by 𝜕

(𝜕3 ⋅ 𝑐3 + 𝜕2 ⋅ 𝑐2 + 𝜕 ⋅ 𝑐1 + 𝑐0) ⋅ 𝜕 = ⎧ { { { ⎨ { { { ⎩ 𝜕4 if 𝑐3 > 0 𝜕3 if 𝑐3 = 0, 𝑐2 > 0 𝜕2 if 𝑐3 = 0, 𝑐2 = 0, 𝑐1 > 0 𝜕 if 𝑐3 = 0, 𝑐2 = 0, 𝑐1 = 0, 𝑐0 > 0

• therwise

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Semiautomatic ordinals at 𝜕𝜕 and beyond

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Overview

Theorems (Jain, Khoussainov, Stephan, Teng and Zou)

Let 𝛽 be any countable ordinal, the structure (𝜕𝛽; +, <, =) is semiautomatic. The semiring of polynomials over ℕ (ℕ[𝑦]; +, ⋅, =) is semiautomatic.

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Polynomials over ℕ

Fix an semiautomatic copy 𝐵 of (ℕ, +, <; ⋅), add new “connectives” ⊕, ⊗ Represent polynomials as lists of coeffjcients from 𝐵 Represent elements as polynomials with connectives between them

1, 10, 2 ⊕ 0, 4 represents the expression (2𝑦2 + 10𝑦 + 1) ⋅ (4𝑦)

Defjne the quotient map 𝑤𝑏𝑚 sending an expression to the canonical representation. 𝑤𝑏𝑚 is not automatic, but for any polynomial 𝑞 ∈ ℕ[𝑦], there is an automatic fragment of 𝑤𝑏𝑚 that is “good enough”.

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Polynomials over 𝜕

We need even more connectives ⊕𝑚, ⊕𝑠, ⊗𝑚, ⊗𝑠 For arbitrarily large 𝑙 ∈ ℕ, 𝑙 + 𝜕 = 𝜕 and 𝜕 ⋅ 𝑙 ⋅ 𝜕 = 𝜕 ⋅ 𝜕 Refjne error conditions to deal with this

Conclusion

The structure (𝜕𝜕; +, <, ⋅, =) is semiautomatic.

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Semiautomatic Ring Structures

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Overview

Rings are obtained by adding to an Abelian group a notion of multiplication.

Theorem (Jain, Khoussainov, Stephan, Teng and Zou)

For any 𝑜 ∈ ℕ the ring (ℤ(√𝑜), ℤ, +, <, =; ⋅) is semiautomatic.

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Illustrating square roots

We illustrate with the simplest case, let 𝑣 = 1 + √ 5 2 denote the golden ratio.

Theorem (Jain, Khoussainov, Stephan, Teng and Zou)

(ℤ[𝑣], +, <, =; ⋅) is semiautomatic.

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The ingredients

3 = 𝑣−2 + 𝑣2 for any 𝑦 + 𝑧𝑣 ∈ ℤ[𝑣], so update coeffjcients until 𝑦 + 𝑧𝑣 = ∑

𝑗

𝑏𝑗𝑣𝑗 but now each |𝑏𝑗| ≤ 2. Tail bound – ∑𝑗≤2 𝑣𝑗 is a geometric series

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Sign test

To check if 𝑏 + 𝑐 = 𝑑 we perform sign test on 𝑏 + 𝑐 − 𝑑. Given a list of coeffjcients, we slide over them and use the equation 𝑣2 = 𝑣 + 1 to update our memory The fjrst time any coeffjcient overshoot 4 ⋅ ∑𝑗≤2 𝑣𝑗, by our earlier tail bound we will know the sign If all input is processed, do a fjnite case distinction Use the characterisation that automatic functions are also computable by linear-time one-tape Turing machines where input and output start at the same position (Case, Jain, Seah and Stephan) to look through coeffjcients in order.

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Generalising to cube roots

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The case of

3

√ 7

Let 𝑣−1 = 2 −

3

√ 7 and we use a similar strategy to show (ℤ[𝑣], +, <, =; ⋅) is semiautomatic. To achieve our coeffjcient bound we have 𝑣3 − 12𝑣2 + 6𝑣 − 1 = 0 where 12 dominates. The tail bound is also geometric, but since we update two coeffjcients at once in every step of the sign test, we need more refjned bounds.

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The case of

3

√ 7

Denote our current coeffjcients 𝑏𝑗, 𝑏𝑗+1, 𝑏𝑗+2 and we terminate if any condition here is broken

|𝑏𝑗+2| ≤ 16𝑙′, |𝑏𝑗+1| ≤ 4𝑙′, or |𝑏𝑗| ≤ 𝑙′, or

where 𝑙′ = 360. This gives us a semiautomatic ring containing

3

√ 7.

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Conclusion

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Contributions

Incorporate notion of semiautomaticity into small ordinals (below 𝜕𝜕). Found semiautomatic rings containing cube roots, generalising existing result.

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Future directions

On ordinals

Question

For how large 𝛽 can we fjnd semiautomatic representations of (𝛽; +, <, ⋅, =)?

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Future directions

On semiautomatic rings

Question

For every algebraic number 𝑑 is there a semiautomatic ring with order which contains 𝑑?

Question

Is there a semiautomatic ring with order that simultaneously contains √ 2, √ 3 and 1

2?

When these 3 numbers are simultaneously present, the ring has enough information to encode the angle 15 degrees.

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