The Cardinality of an Oracle in Blum-Shub-Smale Computation Russell Miller Queens College & CUNY Graduate Center New York, NY. Seventh International CCA Conference Jiangsu University Zhenjiang, China, 23 June 2010 (Joint work with Wesley Calvert, Murray State University, and Ken Kramer, CUNY.) Slides available at qc.edu/˜rmiller/slides.html Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 1 / 13
BSS Computation on R Roughly, a BSS machine M on R operates like a Turing machine, but with a real number in each cell, rather than a bit. M can compute full-precision + . − . · , and ÷ on numbers in its cells. M can compare 0 to the number in any cell, using = or < , and fork according to the answer. M is allowed finitely many real numbers z 0 , . . . , z m as parameters in its program. The input and output (if M halts) are tuples y ∈ R ∞ = { finite tuples from R } . � A subset S ⊆ R ∞ is BSS- decidable iff its characteristic function χ S is computable by a BSS machine, and BSS- semidecidable iff S is the domain of some BSS-computable function. Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 2 / 13
Basic Facts about BSS Computation For a machine M with parameters � z , running on input � y , only elements of the field Q ( � z ,� y ) can ever appear in the cells of M . Cell: 0 · · · m m + 1 · · · m + n m + n + 1 · · · z 0 · · · z m y 1 · · · y n z 0 · · · z m y 1 · · · y n z m + y n . . . . . . . . . . . . . . . f 0 , s ( � · · · f m , s ( � f m + 1 , s ( � · · · f m + n , s ( � f m + n + 1 , s ( � · · · y ) y ) y ) y ) y ) . . . . . . . . . . . . . . . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 3 / 13
Basic Facts about BSS Computation For a machine M with parameters � z , running on input � y , only elements of the field Q ( � z ,� y ) can ever appear in the cells of M . Cell: 0 · · · m m + 1 · · · m + n m + n + 1 · · · z 0 · · · z m y 1 · · · y n z 0 · · · z m y 1 · · · y n z m + y n . . . . . . . . . . . . . . . f 0 , s ( � · · · f m , s ( � f m + 1 , s ( � · · · f m + n , s ( � f m + n + 1 , s ( � · · · y ) y ) y ) y ) y ) . . . . . . . . . . . . . . . For each input � y , every f i , s ( Y 1 , . . . , Y n ) is a rational function with coefficients from the field Q ( � z ) . If the input { y 1 , . . . , y n } is algebraically z ) , then each f i , s ( � independent over Q ( � Y ) is uniquely defined. Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 3 / 13
Restrictions on BSS Computation Given a machine M with parameters � z , choose any input � y algebraically independent over Q ( � z ) . If M ( � y ) halts after t steps, then only finitely many functions f i , s appear. So there is an ǫ > 0 such that for all inputs � x within ǫ of � y , M at stage s contains: f 0 , s ( � f m , s ( � f m + 1 , s ( � f m + n , s ( � f m + n + 1 , s ( � x ) · · · x ) x ) · · · x ) x ) · · · with the same functions f i , s as for � y . Therefore, on an ǫ -ball around � y in R n , M always halts after t steps, and computes the function � f 0 , t ( � x ) , . . . , f m + n + t , t ( � x ) � . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 4 / 13
Restrictions on BSS Computation Given a machine M with parameters � z , choose any input � y algebraically independent over Q ( � z ) . If M ( � y ) halts after t steps, then only finitely many functions f i , s appear. So there is an ǫ > 0 such that for all inputs � x within ǫ of � y , M at stage s contains: f 0 , s ( � f m , s ( � f m + 1 , s ( � f m + n , s ( � f m + n + 1 , s ( � x ) · · · x ) x ) · · · x ) x ) · · · with the same functions f i , s as for � y . Therefore, on an ǫ -ball around � y in R n , M always halts after t steps, and computes the function � f 0 , t ( � x ) , . . . , f m + n + t , t ( � x ) � . Corollary : No BSS-decidable set can be dense and codense within any nonempty open subset of R n . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 4 / 13
Oracle BSS-Machines To do the same for a machine M with parameters � z and an oracle C ⊆ R ∞ , we would have to ensure that | � x − � y | < ǫ and also, for all s , � � f i k , s ( � ⇒ � f i k , s ( � � ( ∀ i 0 , . . . , i m ) x ) : k ≤ m � ∈ C ⇐ y ) : k ≤ m � ∈ C . Then the computation will fork exactly the same for � x as for � y , and will output � f i , t ( � x ) � . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 5 / 13
Oracle BSS-Machines To do the same for a machine M with parameters � z and an oracle C ⊆ R ∞ , we would have to ensure that | � x − � y | < ǫ and also, for all s , � � f i k , s ( � ⇒ � f i k , s ( � � ( ∀ i 0 , . . . , i m ) x ) : k ≤ m � ∈ C ⇐ y ) : k ≤ m � ∈ C . Then the computation will fork exactly the same for � x as for � y , and will output � f i , t ( � x ) � . Theorem : Let H = {� � x � : Program � p ; � p halts on input � x } be the BSS Halting Problem. If χ H is computable by a BSS program with oracle C ⊆ R ∞ , then | C | = 2 ℵ 0 . This answers a question from Meer and Ziegler. Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 5 / 13
Proving the Theorem Assume that the oracle C ⊆ R ∞ has | C | < 2 ℵ 0 . For any oracle z and oracle C , we claim that M C does machine M with parameters � not compute χ H . Let p be the program which, on input � a , b � , halts iff b is algebraic over Q ( a ) . Fix any y 0 , y 1 ∈ R algebraically independent over the field E (of size < 2 ℵ 0 ) generated by � z and p and all tuples in C . Let R be the finite set of rational functions f ∈ E ( Y 0 , Y 1 ) such that f ( y 0 , y 1 ) appears in a cell during this computation. Fix n ∈ N such that each f ∈ R is a quotient of polynomials of degree < n . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 6 / 13
Proving the Theorem Assume that the oracle C ⊆ R ∞ has | C | < 2 ℵ 0 . For any oracle z and oracle C , we claim that M C does machine M with parameters � not compute χ H . Let p be the program which, on input � a , b � , halts iff b is algebraic over Q ( a ) . Fix any y 0 , y 1 ∈ R algebraically independent over the field E (of size < 2 ℵ 0 ) generated by � z and p and all tuples in C . Let R be the finite set of rational functions f ∈ E ( Y 0 , Y 1 ) such that f ( y 0 , y 1 ) appears in a cell during this computation. Fix n ∈ N such that each f ∈ R is a quotient of polynomials of degree < n . ∈ H , by algebraic independence, so M C ( p , y 0 , y 1 ) = 0. Now � p , y 0 , y 1 � / We want to choose � p , x 0 , x 1 � ∈ H close to � p , y 0 , y 1 � to fool M C into computing M C ( p , x 0 , x 1 ) = 0 as well. Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 6 / 13
Proving the Theorem Recall: y 0 , y 1 ∈ R independent over E ; finite set R ⊂ E ( Y 0 , Y 1 ) ; all f ∈ R have f = g h of degree < n . √ x 0 + q , with m > n Now choose x 0 transcendental over E , and x 1 = m prime and q ∈ Q so that x 0 , x 1 are sufficiently close to y 0 , y 1 . So x 1 has degree m over E ( x 0 ) . Now for f = g h ∈ R , f ( � ⇒ g ( � x ) − ch ( � x ) = c ∈ E = x ) = 0 = ⇒ ( g − ch ) = 0 in E [ Y 0 , Y 1 ] . So f = g h = c is constant. Thus f ( x 0 , x 1 ) ∈ E ⇐ ⇒ f is constant ⇐ ⇒ f ( y 0 , y 1 ) ∈ E . So the computation by M C on input � p , x 0 , x 1 � follows the same path as on � p , y 0 , y 1 � , and outputs the same answer: � p , x 0 , x 1 � / ∈ H . This is wrong! Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 7 / 13
Shall We Generalize? When can a countable set decide an uncountable (and co-uncountable) set? Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 8 / 13
Shall We Generalize? When can a countable set decide an uncountable (and co-uncountable) set? Easy answer: { x ∈ R : x > 0 } is BSS-decidable. (Is there a similar subset of C , for BSS-computation on C ?) Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 8 / 13
Shall We Generalize? When can a countable set decide an uncountable (and co-uncountable) set? Easy answer: { x ∈ R : x > 0 } is BSS-decidable. (Is there a similar subset of C , for BSS-computation on C ?) Indeed, { x ∈ R : x ∈ ( 0 , 1 ] & x begins with an even number of 0’s } is BSS-decidable. This is the set � 1 � � 1 � � 1 � 32 , 1 8 , 1 · · · ∪ ∪ 2 , 1 . 16 4 Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 8 / 13
Local Bicardinality Defn.: A set S ⊆ R is locally of bicardinality ≤ κ if there exist two open subsets U and V of R with | R − ( U ∪ V ) | ≤ κ and | U ∩ S | ≤ κ and | V ∩ S | ≤ κ . The local bicardinality of S is the least cardinal κ such that S is locally of bicardinality ≤ κ . So both S and S are open, up to a set of size κ . Notice that the open set ( U ∩ V ) is empty, since | U ∩ V | ≤ | U ∩ S | + | V ∩ S | ≤ κ. (Question: is there an equivalent but simpler definition?) Example: The Cantor middle-thirds set has local bicardinality 2 ℵ 0 . Russell Miller (CUNY) Cardinality of an Oracle CCA 2010 9 / 13
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