Computable Euclidean Domains
Asher M. Kach
(Joint Work with Rod Downey and with Paul Ellis and Reed Solomon)
Southern Wisconsin Logic Colloquium 9 October 2012
Asher M. Kach Computable Euclidean Domains 9 October 2012 1 / 43 Outline
1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains
2 Transfinite Euclidean Domains and Rings
3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I
4 Reverse Mathematics
5 Open Questions
Asher M. Kach Computable Euclidean Domains 9 October 2012 2 / 43 The Division Algorithm...
Problem Divide 18 into 218 (over Z).
Answer. Perform long division 12 18 218 180 38 36 2 and so 218 = 12 · 18 + 2.
Asher M. Kach Computable Euclidean Domains 9 October 2012 3 / 43 The Division Algorithm...
Problem Divide x + 2 into x3 + 18x2 + 2x + 18 (over Q).
Answer. Perform long division x2 + 16x − 30 x + 2 x3 + 18x2 + 2x + 18 − x3 − 2x2 16x2 + 2x − 16x2 − 32x − 30x + 18 30x + 60 78 and so x3 + 18x2 + 2x + 18 = (x2 + 16x − 30)(x + 2) + 78.
Asher M. Kach Computable Euclidean Domains 9 October 2012 4 / 43 Euclid’s Algorithm...
Proposition The algorithm function gcd(a,b) if (a < b) swap(a,b) if (b == 0) return a return gcd(a-b,b)
computes the greatest common divisor of nonnegative integers a and b.
Problem Find the greatest common divisor of 18 and 10.
Answer. Note gcd(18, 10) = gcd(8, 10) = gcd(10, 8) = gcd(2, 8) = gcd(8, 2) = gcd(6, 2) = gcd(4, 2) = gcd(2, 2) = gcd(0, 2) = gcd(2, 0) = 2.
Asher M. Kach Computable Euclidean Domains 9 October 2012 5 / 43 Euclid’s Algorithm...
Proposition The algorithm function gcd(a,b) if (a < b) swap(a,b) if (b == 0) return a return gcd(a-b,b)
computes the greatest common divisor of nonnegative integers a and b.
Problem Find the greatest common divisor of 18 and 10.
Answer. Note gcd(18, 10) = gcd(8, 10) = gcd(10, 8) = gcd(2, 8) = gcd(8, 2) = gcd(6, 2) = gcd(4, 2) = gcd(2, 2) = gcd(0, 2) = gcd(2, 0) = 2.
Asher M. Kach Computable Euclidean Domains 9 October 2012 5 / 43 The Division Algorithm and Euclid’s Algorithm...
Remark In both Z and Q[X], the division algorithm (Euclid’s algorithm) terminates because the dividend (either a or b) decreases in size at every step. Within Z, the size of an integer is its magnitude. Within Q[X], the size of a polynomial is its degree. Generalizing this requirement of remainders decreasing in size yields the (traditional) definition of a Euclidean ring.
Asher M. Kach Computable Euclidean Domains 9 October 2012 6 / 43 Defining Euclidean Rings and Domains...
Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 → ω (where R0 := R \{0}) satisfying
(∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a − qd = 0 or φ(a − qd) < φ(d).
The function φ is termed a Euclidean function for R.
Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors).
Remark Note the hidden function (existential) quantifier in the definition.
Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains...
Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 → ω (where R0 := R \{0}) satisfying
(∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d).
The function φ is termed a Euclidean function for R.
Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors).
Remark Note the hidden function (existential) quantifier in the definition.
Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains...
Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 → ω (where R0 := R \{0}) satisfying
(∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d).
The function φ is termed a Euclidean function for R.
Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors).
Remark Note the hidden function (existential) quantifier in the definition.
Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Defining Euclidean Rings and Domains...
Definition A ring R (always commutative with multiplicative identity) is a Euclidean ring if there is a function φ : R0 → ω (where R0 := R \{0}) satisfying
(∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d).
The function φ is termed a Euclidean function for R.
Definition A Euclidean ring R is a Euclidean domain if it is also an integral domain (i.e., there are no zero divisors).
Remark Note the hidden function (existential) quantifier in the definition.
Asher M. Kach Computable Euclidean Domains 9 October 2012 7 / 43 Euclidean Functions for Z...
Example The integers Z are a Euclidean domain.
Proof. The functions
φ1(z) := |z|
φ2(z) := blog2 |z|c ( |z| if z 6= 5 φ3(z) := 13 otherwise
are Euclidean functions for Z.
Note that φ3 serves as an example where the implication
x divides y implies φ(x) ≤ φ(y)
fails (consider x = 5 and y = 10).
Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 Euclidean Functions for Z...
Example The integers Z are a Euclidean domain.
Proof. The functions
φ1(z) := |z|
φ2(z) := blog2 |z|c ( |z| if z 6= 5 φ3(z) := 13 otherwise
are Euclidean functions for Z.
Note that φ3 serves as an example where the implication
x divides y implies φ(x) ≤ φ(y)
fails (consider x = 5 and y = 10).
Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 Euclidean Functions for Z...
Example The integers Z are a Euclidean domain.
Proof. The functions
φ1(z) := |z|
φ2(z) := blog2 |z|c ( |z| if z 6= 5 φ3(z) := 13 otherwise
are Euclidean functions for Z.
Note that φ3 serves as an example where the implication
x divides y implies φ(x) ≤ φ(y)
fails (consider x = 5 and y = 10).
Asher M. Kach Computable Euclidean Domains 9 October 2012 8 / 43 An Alternate Definition...
Definition (∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d).
Definition
If R is a commutative ring, define a sequence of sets {Rn}n∈ω via recursion by
0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + qd = 0 or a + qd ∈ R S 0 where R Theorem (Motzkin 1949, Samuel 1971) A ring R (respectively integral domain) is a Euclidean ring 0 S (respectively Euclidean domain) if and only if R = n∈ω Rn. Asher M. Kach Computable Euclidean Domains 9 October 2012 9 / 43 An Alternate Definition... Definition (∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d). Definition If R is a commutative ring, define a sequence of sets {Rn}n∈ω via recursion by 0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + qd = 0 or a + qd ∈ R S 0 where R Theorem (Motzkin 1949, Samuel 1971) A ring R (respectively integral domain) is a Euclidean ring 0 S (respectively Euclidean domain) if and only if R = n∈ω Rn. Asher M. Kach Computable Euclidean Domains 9 October 2012 9 / 43 An Alternate Definition... Definition 0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R Remark 0 R0 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0]}. Thus R0 is the set of units of R as the existential witness q must be the a element − d . In other words, the set R0 consists of those elements d that exactly divide every other a ∈ R. Remark 0 R1 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R0]}. Thus R1 consists of those elements d that either exactly divide or divide modulo a unit remainder every other a ∈ R. Asher M. Kach Computable Euclidean Domains 9 October 2012 10 / 43 An Alternate Definition... Definition 0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R Remark 0 R0 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0]}. Thus R0 is the set of units of R as the existential witness q must be the a element − d . In other words, the set R0 consists of those elements d that exactly divide every other a ∈ R. Remark 0 R1 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R0]}. Thus R1 consists of those elements d that either exactly divide or divide modulo a unit remainder every other a ∈ R. Asher M. Kach Computable Euclidean Domains 9 October 2012 10 / 43 An Alternate Definition... Definition 0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R Remark 0 R0 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0]}. Thus R0 is the set of units of R as the existential witness q must be the a element − d . In other words, the set R0 consists of those elements d that exactly divide every other a ∈ R. Remark 0 R1 := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R0]}. Thus R1 consists of those elements d that either exactly divide or divide modulo a unit remainder every other a ∈ R. Asher M. Kach Computable Euclidean Domains 9 October 2012 10 / 43 Examples... Example For R = Z, it is not hard to show that R0 = {±1}: These are the units of the ring. It is also not hard to show that R1 = {±2, ±3}: Every integer is even (divisible by (plus or minus) two) or odd (leaves remainder one [a unit] when divided by (plus or minus) two). Example For R = Q[X], it is not hard to show that R0 = Q: These are the units of the ring. It is also not hard to show that R1 is the set of degree one polynomials: Every polynomial leaves remainder a rational number (possibly zero) when divided by a degree one polynomial. Asher M. Kach Computable Euclidean Domains 9 October 2012 11 / 43 Examples... Example For R = Z, it is not hard to show that R0 = {±1}: These are the units of the ring. It is also not hard to show that R1 = {±2, ±3}: Every integer is even (divisible by (plus or minus) two) or odd (leaves remainder one [a unit] when divided by (plus or minus) two). Example For R = Q[X], it is not hard to show that R0 = Q: These are the units of the ring. It is also not hard to show that R1 is the set of degree one polynomials: Every polynomial leaves remainder a rational number (possibly zero) when divided by a degree one polynomial. Asher M. Kach Computable Euclidean Domains 9 October 2012 11 / 43 The Least Euclidean Function φR... Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (Euclidean ring), define φR : R → ω by φR(d) := n where n is least so that d ∈ Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean ring, the function φR is a Euclidean function for R. Moreover, it is the pointwise least Euclidean function for R. Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Asher M. Kach Computable Euclidean Domains 9 October 2012 12 / 43 The Least Euclidean Function φR... Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (Euclidean ring), define φR : R → ω by φR(d) := n where n is least so that d ∈ Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean ring, the function φR is a Euclidean function for R. Moreover, it is the pointwise least Euclidean function for R. Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Asher M. Kach Computable Euclidean Domains 9 October 2012 12 / 43 The Least Euclidean Function φR... Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (Euclidean ring), define φR : R → ω by φR(d) := n where n is least so that d ∈ Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean ring, the function φR is a Euclidean function for R. Moreover, it is the pointwise least Euclidean function for R. Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Asher M. Kach Computable Euclidean Domains 9 October 2012 12 / 43 A Basic Inequality... Theorem (Folklore (Samuel 1971)) If R is an integral domain and B, T ∈ R are nonzero, then φR(T ) ≤ φR(BT ). Proof. Define φ(T ) := min06=B∈R φR(BT ). Then φ is a Euclidean function for R: Given nonzero D ∈ R, there is a nonzero B ∈ R with φ(D) = φR(BD). Thus for any A ∈ R, there is a Q ∈ R such that either: A + QBD = 0 or φR(A + QBD) < φR(BD). Considering BQ, we have either A + (BQ)D = 0 or φ(A + (BQ)D) ≤ φR(A + (BQ)D) < φR(BD) = φ(D). Asher M. Kach Computable Euclidean Domains 9 October 2012 13 / 43 A Basic Inequality... Theorem (Folklore (Samuel 1971)) If R is an integral domain and B, T ∈ R are nonzero, then φR(T ) ≤ φR(BT ). Proof. Define φ(T ) := min06=B∈R φR(BT ). Further φ = φR since φR(T ) ≤ φ(T ) ≤ φR(1T ) = φR(T ) as a consequence of the minimality of φR and taking 1 for B. It is clear that φ, and thus φR, has the desired property. Asher M. Kach Computable Euclidean Domains 9 October 2012 13 / 43 A Basic Inequality... Proposition (Folklore (Samuel 1971)) If R is an integral domain, B, T ∈ R are nonzero, and B is a nonunit, then φR(T ) < φR(BT ). Proof. Since B is a nonunit, it follows BT does not divide T . Thus min{φR(T + QBT )} < φR(BT ) Q∈R by virtue of the definition of Rα. As 1 + QB 6= 0 for all Q ∈ R, by the previous proposition, we have min{φR(T + QBT )} = min{φR(T (1 + QB))} ≥ φR(T ). Q∈R Q∈R Asher M. Kach Computable Euclidean Domains 9 October 2012 14 / 43 Moving Past N... Remark Note that, for all these definitions and algorithms, it is unnecessary that the range of the ranking function be ω. Any ordinal would do, indeed a well-founded partial order would suffice. Asher M. Kach Computable Euclidean Domains 9 October 2012 15 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 16 / 43 Transfinite Euclidean Domains... Definition A commutative ring R is a transfinite Euclidean ring if there is a function φ : R0 → ON (where ON is the class of ordinals) satisfying (∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d). The function φ is termed a transfinitely-valued Euclidean function for R. Theorem (Motzkin 1949, Samuel 1971) An integral domain R is a transfinite Euclidean domain if and only if 0 S R = α∈ON Rα. Indeed, it suffices for the union to be over those ordinals α with α < |R|+. Asher M. Kach Computable Euclidean Domains 9 October 2012 17 / 43 Transfinite Euclidean Domains... Definition A commutative ring R is a transfinite Euclidean ring if there is a function φ : R0 → ON (where ON is the class of ordinals) satisfying (∀d ∈ R0)(∀a ∈ R)(∃q ∈ R)a + qd = 0 or φ(a + qd) < φ(d). The function φ is termed a transfinitely-valued Euclidean function for R. Theorem (Motzkin 1949, Samuel 1971) An integral domain R is a transfinite Euclidean domain if and only if 0 S R = α∈ON Rα. Indeed, it suffices for the union to be over those ordinals α with α < |R|+. Asher M. Kach Computable Euclidean Domains 9 October 2012 17 / 43 Properly Transfinite Euclidean Domains... Question (Motzkin 1949, Samuel 1971, Schrieber 1985) Is there an integral domain that is a properly transfinite Euclidean domain? In other words, is there an integral domain having transfinitely-valued Euclidean functions but not finitely-valued Euclidean functions? Asher M. Kach Computable Euclidean Domains 9 October 2012 18 / 43 Transfinite Euclidean Functions... Theorem (Samuel 1971) The commutative ring Z ⊕ Z has a transfinitely-valued Euclidean function (indeed, one with range ω2 + ω2) but no finitely-valued Euclidean function. Proposition (Samuel 1971) A finite product of transfinite Euclidean rings is a transfinite Euclidean ring. Proposition (Samuel 1971) The function φ(z) = ω · i + j + 1 is a (nontrivial) Euclidean function for Z, where z = ±2i (2j + 1). Asher M. Kach Computable Euclidean Domains 9 October 2012 19 / 43 Transfinite Euclidean Functions... Theorem (Samuel 1971) The commutative ring Z ⊕ Z has a transfinitely-valued Euclidean function (indeed, one with range ω2 + ω2) but no finitely-valued Euclidean function. Proposition (Samuel 1971) A finite product of transfinite Euclidean rings is a transfinite Euclidean ring. Proposition (Samuel 1971) The function φ(z) = ω · i + j + 1 is a (nontrivial) Euclidean function for Z, where z = ±2i (2j + 1). Asher M. Kach Computable Euclidean Domains 9 October 2012 19 / 43 Transfinite Euclidean Functions... Theorem (Samuel 1971) The commutative ring Z ⊕ Z has a transfinitely-valued Euclidean function (indeed, one with range ω2 + ω2) but no finitely-valued Euclidean function. Proposition (Samuel 1971) A finite product of transfinite Euclidean rings is a transfinite Euclidean ring. Proposition (Samuel 1971) The function φ(z) = ω · i + j + 1 is a (nontrivial) Euclidean function for Z, where z = ±2i (2j + 1). Asher M. Kach Computable Euclidean Domains 9 October 2012 19 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 20 / 43 Effective Algebra Questions... Question Is there a difference between a computable domain having a computable Euclidean function and having a computable transfinite Euclidean function? Question If R is a computable Euclidean domain, how complicated is it to compute φR? To compute any ranking function φ? Question What is the proof-theoretic strength (reverse mathematical strength) of the existence of a pointwise least ranking function? Asher M. Kach Computable Euclidean Domains 9 October 2012 21 / 43 Effective Algebra Questions... Question Is there a difference between a computable domain having a computable Euclidean function and having a computable transfinite Euclidean function? Question If R is a computable Euclidean domain, how complicated is it to compute φR? To compute any ranking function φ? Question What is the proof-theoretic strength (reverse mathematical strength) of the existence of a pointwise least ranking function? Asher M. Kach Computable Euclidean Domains 9 October 2012 21 / 43 Effective Algebra Questions... Question Is there a difference between a computable domain having a computable Euclidean function and having a computable transfinite Euclidean function? Question If R is a computable Euclidean domain, how complicated is it to compute φR? To compute any ranking function φ? Question What is the proof-theoretic strength (reverse mathematical strength) of the existence of a pointwise least ranking function? Asher M. Kach Computable Euclidean Domains 9 October 2012 21 / 43 Counting Quantifiers... Definition 0 Rn := {d ∈ R :(∀a ∈ R)(∃q ∈ R)[a + dq = 0 or a + dq ∈ R Remark 0 0 Though R0 is seemingly Π2(R), it is Σ1(R): It is enough to know whether q exists for the choice of 1 for a. Remark 0 Consequently, for n > 0, the set Rn is Π2n(R). Note that the existential quantifier for R1 and the existential quantifier for R0 combine when analyzing the case n = 1. Asher M. Kach Computable Euclidean Domains 9 October 2012 22 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 23 / 43 Computing R0 = {z : φR(z) = 0}... Theorem (Schrieber 1985) 0 There is a computable Euclidean domain R ⊆ Q for which R0 ≡T ∅ . Proof. Identify the Halting Problem K with a subset of the prime numbers. Have R consist of those fractions a/b ∈ Q with b a multiple of elements of K . Then a prime p is a unit if and only if p ∈ K . a On the other hand, writing z = b ∈ R ⊆ Q in lowest terms, the function φ(z) := a is a Euclidean function for R. Asher M. Kach Computable Euclidean Domains 9 October 2012 24 / 43 Computing R0 = {z : φR(z) = 0}... Theorem (Schrieber 1985) 0 There is a computable Euclidean domain R ⊆ Q for which R0 ≡T ∅ . Proof. Identify the Halting Problem K with a subset of the prime numbers. Have R consist of those fractions a/b ∈ Q with b a multiple of elements of K . Then a prime p is a unit if and only if p ∈ K . a On the other hand, writing z = b ∈ R ⊆ Q in lowest terms, the function φ(z) := a is a Euclidean function for R. Asher M. Kach Computable Euclidean Domains 9 October 2012 24 / 43 Computing Some φ... Theorem (Folklore) Let D be a Euclidean domain. Let R ⊆ Q(D) be a subring of the field of fractions of D. Then R is a Euclidean domain via the function φ(z) = φ(a/b) := φD(a), with z expressed in lowest terms. Remark Thus, there are examples where there is a difference in complexity between computing φR and computing some φ. Asher M. Kach Computable Euclidean Domains 9 October 2012 25 / 43 Computing Some φ... Theorem (Folklore) Let D be a Euclidean domain. Let R ⊆ Q(D) be a subring of the field of fractions of D. Then R is a Euclidean domain via the function φ(z) = φ(a/b) := φD(a), with z expressed in lowest terms. Remark Thus, there are examples where there is a difference in complexity between computing φR and computing some φ. Asher M. Kach Computable Euclidean Domains 9 October 2012 25 / 43 Computing R1 = {z : φR(z) = 1}... Remark 0 00 Fix R. Since R1 is Π2(R), the set R1 is ∅ -computable in any computable presentation of R. Theorem (Ellis, Kach, and Solomon) There is a computable coinfinite set of primes P such that any 0 computable ring R with Z(P) ⊆ R ⊆ Q has φR ≤T ∅ . Proof. Choose P so that φR(z) = φR(a/b) is the number of nonunit prime factors in a. Note that φR(z) is always at least this big. Note that if P has primes in every modular congruence class, then φR(z) is always at least this small. Asher M. Kach Computable Euclidean Domains 9 October 2012 26 / 43 Computing φR... Theorem (Ellis, Kach, and Solomon) 0 If a (technical) number-theoretic conjecture is true, then φR ≤T ∅ for all computable R ⊆ Q. Asher M. Kach Computable Euclidean Domains 9 October 2012 27 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 28 / 43 The Schrieber Ring K hXi ii∈I Definition (Schrieber 1985) If K is a field and {Xi }i∈I is a set of variables, denote by K hXi ii∈I the commutative ring of reduced fractions p/q with p, q ∈ K [Xi ]i∈I and Xi 6 | q for all i. Remark Thus, every element z of the commutative ring K hXi ii∈ω is the product of a monomial m and a unit u. Theorem (Schrieber 1985) The function φ(z) = φ(mu) := deg(m), where m is a monomial and u is a unit, is the least Euclidean function for K hXi ii∈I. In particular, K hXi ii∈I is a Euclidean domain. Asher M. Kach Computable Euclidean Domains 9 October 2012 29 / 43 The Schrieber Ring K hXi ii∈I Definition (Schrieber 1985) If K is a field and {Xi }i∈I is a set of variables, denote by K hXi ii∈I the commutative ring of reduced fractions p/q with p, q ∈ K [Xi ]i∈I and Xi 6 | q for all i. Remark Thus, every element z of the commutative ring K hXi ii∈ω is the product of a monomial m and a unit u. Theorem (Schrieber 1985) The function φ(z) = φ(mu) := deg(m), where m is a monomial and u is a unit, is the least Euclidean function for K hXi ii∈I. In particular, K hXi ii∈I is a Euclidean domain. Asher M. Kach Computable Euclidean Domains 9 October 2012 29 / 43 No Computable Euclidean Function... Theorem (Schrieber 1985) There is a Euclidean computable domain R with no computable Euclidean function. Proof. Begin with Q hXi ii∈ω. In order to defeat ϕe, wait for ϕe(Xe) ↓. When (if) it does, introduce a (ϕe(Xe) + 1)th root of Xe. Note that any ranking function must satisfy φ(Xi ) ≥ ϕi (Xi ) + 1 (provided this computation converges) for all i. Corollary (Downey and Kach 2011) There is a Euclidean computable domain R having no computable finitely-valued Euclidean function but having a computable transfinitely-valued Euclidean function. Asher M. Kach Computable Euclidean Domains 9 October 2012 30 / 43 No Computable Euclidean Function... Theorem (Schrieber 1985) There is a Euclidean computable domain R with no computable Euclidean function. Proof. Begin with Q hXi ii∈ω. In order to defeat ϕe, wait for ϕe(Xe) ↓. When (if) it does, introduce a (ϕe(Xe) + 1)th root of Xe. Note that any ranking function must satisfy φ(Xi ) ≥ ϕi (Xi ) + 1 (provided this computation converges) for all i. Corollary (Downey and Kach 2011) There is a Euclidean computable domain R having no computable finitely-valued Euclidean function but having a computable transfinitely-valued Euclidean function. Asher M. Kach Computable Euclidean Domains 9 October 2012 30 / 43 Theorems... Theorem (Downey and Kach 2011) There is a Euclidean computable domain R having no transfinitely-valued computable Euclidean function φ. Theorem (Downey and Kach 2011) There is a Euclidean computable domain R for which the set R1 is 0 Π2-complete. Theorem (Downey and Kach 2011) There is a Euclidean computable domain R for which there is no finitely-valued ∅0-computable Euclidean function φ. Asher M. Kach Computable Euclidean Domains 9 October 2012 31 / 43 Proofs... Theorem (Downey and Kach 2011) There is a Euclidean computable domain R for which there is no finitely-valued ∅0-computable Euclidean function φ. Proof. Wish to prevent ϕe(x) = lims ϕe(x, s) from being a ranking function for all e. At stage s, we compute ϕe(Xe, s) (call this value ne). If this value has changed, we make Xe the product of ne + 1 many new variables Xe,s,0,..., Xe,s,ne . Make the previous Xe,t,1,..., Xe,t,ne all units. Asher M. Kach Computable Euclidean Domains 9 October 2012 32 / 43 Proofs... Theorem (Downey and Kach 2011) There is a Euclidean computable domain R for which the set R1 is 0 Π2-complete. Proof. 0 Fix a Π2-complete set S and an appropriate predicate P(i, s). The idea is to start with the rationals Q and expressions {Zi }i∈ω. As the construction proceeds, each expression Zi will be declared equal to a product of two variables Zi = Xi,j Yi,j (starting with j = 0). Every time i appears in S, we make Xi,j a unit and declare Zi also equal to the product Xi,j+1Yi,j+1. <∞ The point is that if ∃ s [P(i, s)], then Zi will have rank two (being a ∞ product of two variables); and if ∃ s [P(i, s)], then Zi will have rank one (being a product of only a variable and a unit). Asher M. Kach Computable Euclidean Domains 9 October 2012 33 / 43 Proofs... Theorem (Downey and Kach 2011) There is a Euclidean computable domain R having no transfinitely-valued computable Euclidean function φ. Proof. Because of issues with ordinal notations, it is easier to defeat computable suborderings of Q than ordinal valued ranking functions. Use variables Xi and Yi to defeat Li : s+1 If Xi Asher M. Kach Computable Euclidean Domains 9 October 2012 34 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 35 / 43 Minimal Ranking Functions (Where?)... Remark It would seem that, since we have a computable domain for which 0 φR ≥T 0 , the statement asserting that minimal ranking functions exist 0 would prove ACA0 (essentially that X exists if X exists). However, a priori, there is no reason for the minimal Euclidean function in the model to be the minimal Euclidean function φR. Asher M. Kach Computable Euclidean Domains 9 October 2012 36 / 43 Reverse Mathematics... Lemma (Downey and Kach 2011) (RCA0) Fix a Euclidean domain R and a non-minimal (finitely-valued) Euclidean function φ for R. Let α be the least ordinal for which there is a T ∈ R with α = φR(T ) < φ(T ). Then (fixing such a T ) ( φ(z) if z 6= T φˆ(z) = φR(T ) if z = T is a (finitely-valued) Euclidean function for R and satisfies φˆ < φ. Proof. Since φˆ(T ) = φR(T ) < φ(T ), it is immediate that φˆ < φ. As α was chosen minimal, for any A ∈ R, there exists a Q ∈ R with φ(A + QT ) < φ(T ) as φ(A + QT ) = φR(A + QT ) < φR(T ) = α. Asher M. Kach Computable Euclidean Domains 9 October 2012 37 / 43 Theorem (Downey and Kach 2011) (RCA0) The statement MEF: Every Euclidean domain has a minimal Euclidean function proves ACA0. Proof. Fix X in the model. Consider the X-computable ring whose units are 0 Σ1(X)-complete constructed by relativizing Schrieber’s construction. As φ(a/b) = a is a computable Euclidean function, the ring R is a Euclidean domain, i.e., it has a Euclidean function in the model. Consequently, by MEF, we may fix a minimal Euclidean function φ 0 0 for R (so that φ ≤ φ for all φ in the model). Then φ = φR as a consequence of the lemma. 0 As φR computes X , the model must be closed under the Turing Jump. Asher M. Kach Computable Euclidean Domains 9 October 2012 38 / 43 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Effective Algebra: Computing φ and φR Subrings of Q Subrings of K (Xi )i∈I 4 Reverse Mathematics 5 Open Questions Asher M. Kach Computable Euclidean Domains 9 October 2012 39 / 43 Questions... Question Is there a properly transfinite Euclidean domain? Question Is there a Euclidean computable domain in which R0 is computable 0 and R1 is Π2-complete? 0 Is there a Euclidean computable domain in which R2 is Π4-complete? 0 Is there a Euclidean computable domain in which Rn is Π2n-complete for all n? Question + Does MEF prove ACA0 over RCA0? 1 Does TMEF prove Π1 − CA over RCA0? Asher M. Kach Computable Euclidean Domains 9 October 2012 40 / 43 Questions... Question Is there a properly transfinite Euclidean domain? Question Is there a Euclidean computable domain in which R0 is computable 0 and R1 is Π2-complete? 0 Is there a Euclidean computable domain in which R2 is Π4-complete? 0 Is there a Euclidean computable domain in which Rn is Π2n-complete for all n? Question + Does MEF prove ACA0 over RCA0? 1 Does TMEF prove Π1 − CA over RCA0? Asher M. Kach Computable Euclidean Domains 9 October 2012 40 / 43 Questions... Question Is there a properly transfinite Euclidean domain? Question Is there a Euclidean computable domain in which R0 is computable 0 and R1 is Π2-complete? 0 Is there a Euclidean computable domain in which R2 is Π4-complete? 0 Is there a Euclidean computable domain in which Rn is Π2n-complete for all n? Question + Does MEF prove ACA0 over RCA0? 1 Does TMEF prove Π1 − CA over RCA0? Asher M. Kach Computable Euclidean Domains 9 October 2012 40 / 43 More Questions... Question What more can be said [classically] about (transfinite) Euclidean domains? What more can be said about Euclidean computable (transfinite) domains? Asher M. Kach Computable Euclidean Domains 9 October 2012 41 / 43 The End... Thank You! Asher M. Kach Computable Euclidean Domains 9 October 2012 42 / 43 References Rodney G. Downey and Asher M. Kach. Euclidean Functions of Computable Euclidean Domains. Notre Dame Journal of Formal Logic, 52(2):163–172, 2011. Th. Motzkin, The Euclidean algorithm. Bull. Amer. Math. Soc., 55:1142–1146, 1949. Pierre Samuel. About Euclidean rings. J. Algebra, 19:282–301, 1971. Leonard Schrieber. Recursive properties of Euclidean domains. Ann. Pure Appl. Logic, 29(1):59–77, 1985. V. Stoltenberg-Hansen and J. V. Tucker. Computable rings and fields. In Handbook of Computability Theory, volume 140 of Stud. Logic Found. Math., pages 363–447. North-Holland, Amsterdam, 1999. Asher M. Kach Computable Euclidean Domains 9 October 2012 43 / 43