Euclidean Domains and Euclidean Functions Rod Downey (Joint Work - PowerPoint PPT Presentation

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 The Division Algorithm, Euclid’s Algorithm, and Euclidean Domains 1 Transfinite Euclidean Domains and Rings 2 Computing Any Euclidean Function φ for R and φ R 3 Open Questions 4 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 x 3 + 18 x 2 + 2 x + 18 (over Q ). Answer. Perform long division and so x 3 + 18 x 2 + 2 x + 18 = ( x 2 + 16 x − 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 : R 0 → N (where R 0 := R \{ 0 } ) satisfying ( ∀ a , d ∈ R 0 )( ∃ q ∈ 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 φ 1 ( z ) = | z | ⌈ log 2 | z |⌉ φ 2 ( z ) = � | z | if z � = 5 φ 3 ( z ) = 13 otherwise are Euclidean functions for Z . Note that φ 3 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 φ 1 ( z ) = | z | ⌈ log 2 | z |⌉ φ 2 ( z ) = � | z | if z � = 5 φ 3 ( z ) = 13 otherwise are Euclidean functions for Z . Note that φ 3 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 φ 1 ( z ) = | z | ⌈ log 2 | z |⌉ φ 2 ( z ) = � | z | if z � = 5 φ 3 ( z ) = 13 otherwise are Euclidean functions for Z . Note that φ 3 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 { R n } n ∈ N via recursion by R n := { d ∈ R 0 : ( ∀ a ∈ R 0 )( ∃ q ∈ R ) [ a + dq = 0 or a + dq ∈ R < n ] } m < n R m and R 0 = R − { 0 } . where R < n = � Remark Thus R 0 consists of the units, R 1 consists of those elements which exactly divide every other a ∈ R 0 or leave remainder a unit, etc. (NB if you read Samuel, R 1 = R 2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R 0 = � n ∈ N R 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 { R n } n ∈ N via recursion by R n := { d ∈ R 0 : ( ∀ a ∈ R 0 )( ∃ q ∈ R ) [ a + dq = 0 or a + dq ∈ R < n ] } m < n R m and R 0 = R − { 0 } . where R < n = � Remark Thus R 0 consists of the units, R 1 consists of those elements which exactly divide every other a ∈ R 0 or leave remainder a unit, etc. (NB if you read Samuel, R 1 = R 2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R 0 = � n ∈ N R 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 { R n } n ∈ N via recursion by R n := { d ∈ R 0 : ( ∀ a ∈ R 0 )( ∃ q ∈ R ) [ a + dq = 0 or a + dq ∈ R < n ] } m < n R m and R 0 = R − { 0 } . where R < n = � Remark Thus R 0 consists of the units, R 1 consists of those elements which exactly divide every other a ∈ R 0 or leave remainder a unit, etc. (NB if you read Samuel, R 1 = R 2 there) Theorem (Motzkin 1949, Samuel 1971) An integral domain R (resp. ring) is a Euclidean domain (resp. ring)if and only if R 0 = � n ∈ N R n . Rod Downey (VUW) Computable Euclidean Domains and Euclidean Functions May 2010 10 / 44

The Least Euclidean Function φ R Definition (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (ring), define φ R : R 0 → N by φ R ( d ) = n where n is least so that d ∈ R n . Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φ R is a Euclidean function for R . Moreover, it is the least Euclidean function for R ; i.e., if φ is a Euclidean function for R , then φ R ( d ) ≤ φ ( d ) for all d ∈ R 0 . Consequently, the function φ R satisfies φ R ( d ) = inf φ φ ( d ) 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 φ R Definition (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (ring), define φ R : R 0 → N by φ R ( d ) = n where n is least so that d ∈ R n . Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φ R is a Euclidean function for R . Moreover, it is the least Euclidean function for R ; i.e., if φ is a Euclidean function for R , then φ R ( d ) ≤ φ ( d ) for all d ∈ R 0 . Consequently, the function φ R satisfies φ R ( d ) = inf φ φ ( d ) 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 φ R Definition (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (ring), define φ R : R 0 → N by φ R ( d ) = n where n is least so that d ∈ R n . Theorem (Motzkin 1949, Samuel 1971) If R is a Euclidean domain (resp. ring), the function φ R is a Euclidean function for R . Moreover, it is the least Euclidean function for R ; i.e., if φ is a Euclidean function for R , then φ R ( d ) ≤ φ ( d ) for all d ∈ R 0 . Consequently, the function φ R satisfies φ R ( d ) = inf φ φ ( d ) 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