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Semiautomatic Ordinal and Ring Structures Qi Ji NUS School of Computing 13th November 2019 Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 1 / 37

Slides https://m4th.b0ss.net/semi/semi.pdf Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 2 / 37

Background Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 3 / 37

Automata theory Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 4 / 37

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. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 5 / 37

Check multiple of 3 Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 6 / 37

Automatic relations Encode idea of process tuples in parallel Add a padding symbol # 𝑑𝑝𝑜𝑤(010, 01236) = ( 0 0 )( 1 1 )( 0 2 )( # A function is automatic ifg its graph (encoded this way) is automatic Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 7 / 37 3 )( # 6 )

Verifying addition # # 2 2 13th November 2019 Semiautomatic Ordinal and Ring Structures Qi Ji (NUS School of Computing) i i n n n n n n c n # 1 5 5 2 4 7 0 # 1 1 2 Primary school algorithm 3 3 3 3 2 3 5 8 Incorrect addition Correct addition i – wrong c – carry n – correct and no carry 8 / 37

Automatic structures A structure (𝐵, 𝑔 1 , … , 𝑔 𝑜 , 𝑆 1 , … , 𝑆 𝑛 ) is automatic ifg 𝐵 is a regular set, Example (ℕ, +) is automatic but (ℕ, +, ⋅) is not automatic. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 9 / 37 𝑔 1 , … , 𝑔 𝑜 are automatic functions, 𝑆 1 , … , 𝑆 𝑛 are automatic relations.

Semiautomatic structures (𝐵, 𝑔 1 , … , 𝑔 𝑜 , 𝑆 1 , … , 𝑆 𝑛 ; 𝑕 1 , … , 𝑕 𝑞 , 𝑇 1 , … , 𝑇 𝑟 ) is semiautomatic ifg 𝐵 is a regular set, relations, semiautomatic relations, resultant 𝐵 → 𝐵 function is automatic. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 10 / 37 𝑔 1 , … , 𝑔 𝑜 are automatic functions and 𝑆 1 , … , 𝑆 𝑛 are automatic 𝑕 1 , … , 𝑕 𝑜 are semiautomatic functions and 𝑇 1 , … , 𝑇 𝑛 are where 𝑔 ∶ 𝐵 𝑜 → 𝐵 is semiautomatic ifg fjxing 𝑜 − 1 inputs, the

Semiautomatic structures Example (ℕ, +, <, =; ⋅) is not automatic. For any constant 𝑜 , implement multiplication by 𝑜 as repeated addition. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 11 / 37

Set theory Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 12 / 37

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. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 13 / 37

Ordinals Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 14 / 37

Semiautomatic Ordinal Structures Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 15 / 37

Semiautomatic ordinals with automatic addition Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 16 / 37

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 We can defjne + on the 𝑜 -ary convolution of an automatic copy of (ℕ, +, <) . Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 17 / 37 𝜕 𝑜−1 ⋅ 𝑑 𝑜−1 + ⋯ + 𝜕 ⋅ 𝑑 1 + 𝑑 0 for coeffjcients 𝑑 0 , … , 𝑑 𝑜−1 ∈ ℕ .

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 Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 18 / 37 For any ordinal 𝛽 < 𝜕 𝜕 the structure (𝛽, +, <, =; ⋅) is semiautomatic.

Left multiplication Let expanding the giant expression, we get … Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 19 / 37 𝛾 = 𝜕 𝑙 ⋅ 𝑐 𝑙 + 𝜕 𝑙−1 ⋅ 𝑐 𝑙−1 + ⋯ + 𝜕 ⋅ 𝑐 1 + 𝑐 0 𝛿 = 𝜕 𝑚 ⋅ 𝑑 𝑚 + 𝜕 𝑚−1 ⋅ 𝑑 𝑚−1 + ⋯ + 𝜕 ⋅ 𝑑 1 + 𝑑 0 𝛾 ⋅ 𝛿 = 𝛾 ⋅ 𝜕 𝑚 ⋅ 𝑑 𝑚 + 𝛾 ⋅ 𝜕 𝑚−1 ⋅ 𝑑 𝑚−1 + ⋯ + 𝛾 ⋅ 𝜕 ⋅ 𝑑 1 + 𝛾 ⋅ 𝑑 0 = 𝜕 𝑙+𝑚 ⋅ 𝑑 𝑚 + 𝜕 𝑙+𝑚−1 ⋅ 𝑑 𝑚−1 + ⋯ + 𝜕 𝑙+1 ⋅ 𝑑 1 + (𝜕 𝑙 ⋅ (𝑐 𝑙 ⋅ 𝑑 0 ) + 𝜕 𝑙−1 ⋅ 𝑐 𝑙−1 + ⋯ + 𝜕 𝑐 1 + 𝑐 0 ) ⋅ 1 𝑑 0 ≠0 where 1 𝑑 0 ≠0 is 1 is 𝑑 0 ≠ 0 and 0 otherwise.

Right multiplication Ordinal multiplication distributes on the right, so we get a fjnite composition of right-multiplication by 𝜕 , right-multiplication by fjxed constants, ordinal addition. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 20 / 37

Right multiplication by 𝜕 𝜕 4 13th November 2019 Semiautomatic Ordinal and Ring Structures Qi Ji (NUS School of Computing) otherwise 0 𝜕 𝜕 2 𝜕 3 21 / 37 ⎩ { { { ⎨ { { { ⎧ = (𝜕 3 ⋅ 𝑐 3 + 𝜕 2 ⋅ 𝑐 2 + 𝜕 ⋅ 𝑐 1 + 𝑐 0 ) ⋅ 𝜕 if 𝑐 3 > 0 if 𝑐 3 = 0, 𝑐 2 > 0 if 𝑐 3 = 0, 𝑐 2 = 0, 𝑐 1 > 0 if 𝑐 3 = 0, 𝑐 2 = 0, 𝑐 1 = 0, 𝑐 0 > 0

Semiautomatic Ordinal and Ring Structures Qi Ji (NUS School of Computing) 13th November 2019 22 / 37 Semiautomatic ordinals at 𝜕 𝜕 and beyond

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. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 23 / 37

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 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”. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 24 / 37 1, 10, 2 ⊕ 0, 4 represents the expression (2𝑦 2 + 10𝑦 + 1) ⋅ (4𝑦)

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. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 25 / 37

Semiautomatic Ring Structures Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 26 / 37

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. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 27 / 37

Illustrating square roots √ 5 2 denote the golden ratio. Theorem (Jain, Khoussainov, Stephan, Teng and Zou) (ℤ[𝑣], +, <, =; ⋅) is semiautomatic. Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 28 / 37 We illustrate with the simplest case, let 𝑣 = 1 +

The ingredients 𝑦 + 𝑧𝑣 = ∑ 𝑗 𝑏 𝑗 𝑣 𝑗 but now each |𝑏 𝑗 | ≤ 2 . Qi Ji (NUS School of Computing) Semiautomatic Ordinal and Ring Structures 13th November 2019 29 / 37 3 = 𝑣 −2 + 𝑣 2 for any 𝑦 + 𝑧𝑣 ∈ ℤ[𝑣] , so update coeffjcients until Tail bound – ∑ 𝑗≤2 𝑣 𝑗 is a geometric series