Euclidean Domains and Euclidean Functions Rod Downey (Joint Work with Asher Kach) Chicago May 2010 Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 1 / 44 References Th. Motzkin, The Euclidean algorithm. Bull. Amer. Math. Soc., 55:1142–1146, 1949. Leonard Schrieber. Recursive properties of Euclidean domains. Ann. Pure Appl. Logic, 29(1):59–77, 1985. Pierre Samuel. About Euclidean rings. J. Algebra, 19:282–301, 1971. 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. Rod Downey and Asher Kach, Euclidean Functions of Computable Euclidean Domains, submitted. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 2 / 44 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Computing Any Euclidean Function φ for R and φR 4 Open Questions Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 3 / 44 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 4 / 44 The Division Algorithm... Problem Divide x + 2 into x3 + 18x2 + 2x + 18 (over Q). Answer. Perform long division and so x3 + 18x2 + 2x + 18 = (x2 + 16x − 30)(x + 2) + 78. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 5 / 44 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 non-negative 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 6 / 44 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 non-negative 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 6 / 44 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 domain. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 7 / 44 Defining Euclidean Domains... Definition A commutative ring R is a Euclidean ring if there is a function φ : R0 ! N (where R0 := R nf0g) satisfying (8a; d 2 R0)(9q 2 R)a + qd = 0 or φ(a + qd) < φ(d): The function φ is termed a Euclidean function for R. If the ring is also an integral domain (i.e., there are no zero divisors) then it becomes a Euclidean Domain. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 8 / 44 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) = jzj φ2(z) = dlog2 jzje ( jzj 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 9 / 44 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) = jzj φ2(z) = dlog2 jzje ( jzj 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 9 / 44 Euclidean Functions for Z... Example The integers Z are a Euclidean domain. Proof. The functions φ1(z) = jzj φ2(z) = dlog2 jzje ( jzj 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. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 9 / 44 Euclidean without Euclidean functions Definition If R is a commutative ring (with 1), define a sequence of sets fRngn2N via recursion by 0 0 Rn := fd 2 R :(8a 2 R )(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g S 0 where R<n = m<n Rm and R = R − f0g. Remark Thus R0 consists of the units, R1 consists of those elements which exactly divide every other a 2 R0 or leave remainder a unit, etc. (NB if you read Samuel, R1 = R2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R0 = S R . n2N n Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 10 / 44 Euclidean without Euclidean functions Definition If R is a commutative ring (with 1), define a sequence of sets fRngn2N via recursion by 0 0 Rn := fd 2 R :(8a 2 R )(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g S 0 where R<n = m<n Rm and R = R − f0g. Remark Thus R0 consists of the units, R1 consists of those elements which exactly divide every other a 2 R0 or leave remainder a unit, etc. (NB if you read Samuel, R1 = R2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R0 = S R . n2N n Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 10 / 44 Euclidean without Euclidean functions Definition If R is a commutative ring (with 1), define a sequence of sets fRngn2N via recursion by 0 0 Rn := fd 2 R :(8a 2 R )(9q 2 R)[a + dq = 0 or a + dq 2 R<n]g S 0 where R<n = m<n Rm and R = R − f0g. Remark Thus R0 consists of the units, R1 consists of those elements which exactly divide every other a 2 R0 or leave remainder a unit, etc. (NB if you read Samuel, R1 = R2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R0 = S R . n2N n Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 10 / 44 The Least Euclidean Function φR Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (ring), define φR : R ! N by φR(d) = n where n is least so that d 2 Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φR is a Euclidean function for R. Moreover, it is the least Euclidean function for R; i.e., if φ 0 is a Euclidean function for R, then φR(d) ≤ φ(d) for all d 2 R . Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 11 / 44 The Least Euclidean Function φR Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (ring), define φR : R ! N by φR(d) = n where n is least so that d 2 Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φR is a Euclidean function for R. Moreover, it is the least Euclidean function for R; i.e., if φ 0 is a Euclidean function for R, then φR(d) ≤ φ(d) for all d 2 R . Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 11 / 44 The Least Euclidean Function φR Definition (Motzkin 1949, Samuel 1971) 0 If R is a Euclidean domain (ring), define φR : R ! N by φR(d) = n where n is least so that d 2 Rn. Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φR is a Euclidean function for R. Moreover, it is the least Euclidean function for R; i.e., if φ 0 is a Euclidean function for R, then φR(d) ≤ φ(d) for all d 2 R . Consequently, the function φR satisfies φR(d) = inf φ(d) φ where φ ranges over all Euclidean functions for R. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 11 / 44 I am sure that, as logicians, you immediately notice that it is unnecessary in the definition of a Euclidean ring that the range of the ranking function is N: Any ordinal will do, and maybe even well founded partial orders. Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 12 / 44 Outline 1 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 2 Transfinite Euclidean Domains and Rings 3 Computing Any Euclidean Function φ for R and φR 4 Open Questions Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 13 / 44 Transfinite Euclidean Domains..