- Email: [email protected]
P≠NC over the p-adic numbers
Journal of Complexity 19 (2003) 125–131
http://www.elsevier.com/locate/jco
PaNC over the p-adic numbers$ Michael Mallera,* and Jennifer Whiteheadb b
a Department of Mathematics, Queens College, Flushing, New York 11367, USA Department of Computer Science, Queens College, Flushing, New York 11367, USA
Received 15 March 2002; revised 25 June 2002; accepted 26 June 2002
Abstract We show that in the Blum–Shub–Smale model of computation, over the p-adic numbers Qp ; the class NCQp is strictly contained in the class PQp : That is, there exist sets of p-adic numbers which can be recognized in sequential polynomial time, but which cannot be recognized in 2n polylogarithmic parallel time. We use the sets ðx1 ; x2 ; y; xn ÞAQnp CQN p such that x1 ¼ x2 : We also show that the inclusion PARQp CEXPQp is strict. These results extend previous work of Cucker, and of Blum, Cucker, Shub and Smale in the real case. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction Let R be a ring, ht : R-Rþ a height function, with htðxÞ representing the cost of computing with xAR; and let w:R-f21; 1g be a characteristic function controlling branching in computations. Blum, Shub and Smale defined machines and complexity classes over (R; ht; w) [2,3]. In this paper we will be concerned only with the case htðxÞ ¼ 1 for all xAR. A machine M over R runs in polynomial time on X CRN if M halts on all inputs xAX in time TM ðxÞ; and there are constants CAR; qAN such that, for all xAX ; x ¼ ðx1 ; x2 ; y; xn ÞARn CRN TM ðxÞpCnq ¼ CðsizeðxÞq Þ
$
Partially supported by PSC-CUNY Research award 63452 00 32. *Corresponding author. Tel.: +718-997-5851; fax: +718-997-5861. E-mail addresses: [email protected] (M. Maller), [email protected] (J. Whitehead).
0885-064X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0885-064X(02)00002-X
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
126
A decision problem SCRN is in the class PR if its characteristic function is computed by a polynomial time machine. A model of parallel computation over the real numbers is defined in [2, Chapter 18] in two equivalent ways, using PRAM’s and using algebraic circuits. We adapt their definitions to describe PRAM’s over Qp, with modifications to accommodate branching on squares and non-squares in Qp, which is an unordered field. As in the real case, an equivalent model could be defined using algebraic circuits over Qp. Let xAQp ; xa0; vp ðxÞ ¼ k; x ¼ pk ða0 þ a1 p þ Þ; ai AZ=p: Recall that provided pa2; x is a square in Qp if and only if k is even and a0 is a square in the cyclic group (Z/pZ)*, i.e. a0 is an even power of a generator s of (Z/pZ)*. Therefore, for pa2 and xa0; exactly one of the following four elements is a square in Qp: x; px; sx; spx: The Macintyre language for the p-adics [8] introduced the predicates Pk ðxÞ3ð(yAQp Þðyk ¼ xÞ
and
Rk ðxÞ3ðxa04Pk ðxÞÞ
In [10] we defined (sequential) machines over the p-adic numbers, pa2; which branch on P2 ðxÞ; LP2 ðxÞ and use ht ðxÞ ¼ 1 for all xAQp : (Some technical details differ in the case p ¼ 2; see remark at the end of this paper.) A processor M over Qp will be a p-adic machine as in [10] enhanced with communication nodes which permit processors in computations in a parallel machine to access registers of other activated processors. A parallel machine M is a sequence of identical processors fMi g; iAN; with an activation function p : N-N, one input space IM ¼ QN p ; one output space OM ¼ h N QN p ; and a state space SM ¼ ,h¼1 ðSM Þ : These parallel machines compute synchronously, all processors following a global clock. Given an input ðx0 ; y; xn1 ÞAQnp CQN p to M the input to the state space of processor Mi is
ðy; 0; i; n; xi d0; yÞ;
if
ion
and ðy; 0; i; n; 0d0; yÞ;
if
npiopðnÞ
where is a place marker. The class PLkQp ; parallel polylogarithmic time, is defined to be the class of sets SCQN p that are decided by a parallel Qp machine in time Oðlogk ðnÞÞ using a number of active processor pðnÞ polynomial in n: Parallel computations over Qp can also be defined as in [2, 18.4] in terms of k algebraic circuits, leading to the class NCQp ¼ ,k NCQ : As in the real case the two p approaches are equivalent, so we can define NCQp ¼
[ k
k NCQ ¼ p
[ k
PLkQp :
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
127
Lemma 1. Let M be a parallel machine over a ring R, and let L be the maximum degree of the polynomials computed at the computation nodes of the (identical) processors of M: Then the maximum degree of a polynomial computed by M in time T is LT. Proof If. hðx1 ; x2 ; y; xn Þ is computed by M in time T; unroll the computation into a tree. Any path from the root is of length at most T; and at each node the maximum degree computed is L: Therefore the Lemma follows by the law of exponents. & We will adapt the problem used in [2] in the real case (‘‘Cucker’s Problem’’), n recognizing roots of f ðx; yÞ ¼ x2 y: To formulate this as a uniform problem with input size n, consider for input x ¼ ðx1 ; x2 ; y; xn ÞARn CRN n
F ðx1 ; x2 ; y; xn Þ ¼ x21 x2 ¼ x21
sizeðxÞ
x2 :
The proofs in [2, Section 19.1, 5] use the Hilbert Nullstellensatz to show that a parallel machine which recognizes these roots would have to compute a polynomial of degree 2n and therefore could not run in polylogarithmic time. There certainly are p-adic versions of the Nullstellansatz, but they are less accessible for this problem (see for example [4]). Therefore our proof will take the argument back down into two dimensions where more elementary algebraic geometry will suffice. Cucker’s problem gives a satisfying proof that PaNC over R since it reflects the intuition that some problems, like iterated squaring, are inherently sequential. Observe that over Z the same proof shows that a parallel machine deciding Cucker’s problem would have to compute a polynomial of degree 2n, but in the bit model Cucker’s decision problem is not in P.
2. Proof of main theorem n
n 2 Let CðnÞDQnp CQN p be the decision problem CðnÞ ¼ fðx1 ; x2 ; y; xn ÞAQp jx1 ¼ x2 g: We prove that CðnÞAPQp but CðnÞeNCQp :
Theorem 1. The inclusion of complexity classes NCQp CPQp is strict. Lemma 2. CðnÞAPQp : Proof The. size of input ðx1 ; x2 ; y; xn Þ is n. Iterate squaring x1 ; n times to compute n n x21 in linear time. Test x21 x2 ¼ 0: Qp machines detect z ¼ 0 in two steps as P2 ðzÞ4P2 ðpzÞ: & Lemma 3. Let F be a field, x, y, indeterminants and dAZ+. Then gðx; yÞ ¼ xd 2yAF ½x; y is irreducible.
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Proof By. Eisenstein’s criterion [7, p. 164]. Since F is a field, F ½y is a UFD, and P F ½x; y is canonically isomorphic to F ½x ½y DF ½y ½x : Consider hðxÞ ¼ di¼0 ai xi ¼ xd yAF ½y ½x : The coefficients ad ¼ 1; a0 ¼ y; ai ¼ 0; 0oiod; are relatively prime in F ½y so hðxÞ is primitive in F ½y ½x : The element yAF ½y is irreducible, and :yjad ; :y2 ja0 and yjai ; 0piod: Therefore by Eisenstein’s criterion hðxÞ is irreducible in F ½y ½x ; whence gðx; yÞ ¼ xd 2y is irreducible in F ½x; y : & We will need the following elementary fact from algebraic geometry. Let k be a field and fi Ak½x1 ; x2 ; yxn ; i ¼ 1; y; m; be polynomials in n variables, V ðf Þ ¼ fxAkn jf ðxÞ ¼ 0g and V ðffi j1pipmgÞ ¼ fxAkn jfi ðxÞ ¼ 0; i ¼ 1ymg: Lemma 4. Let k be a field, f ðx; yÞ; gðx; yÞ polynomials T in k½x; y with no common factors. Then the set of common roots V ðf ; gÞ ¼ V ðf Þ V ðgÞ is finite. Proof See. [6, p. 18].
&
Lemma 5. CðnÞeNC Qp : Proof Suppose. CðnÞANCQp : Let M be a parallel Qp machine which decides CðnÞ in polylogarithmic time tðnÞ: So (D; k such that tðnÞpD logk ðnÞ: Let L be the maximum degree of the polynomials computed at the computation nodes of M; and choose N sufficiently large so LtðNÞ pLD log
k
ðNÞ
o2N :
Fix this N and consider the computation of M on inputs ðx1 ; x2 ; y; xN ÞAQN p : m j This computation takes place in a finite dimensional subspace Z Qp of the state space of M: As in [2], we unroll the computations of M on these inputs into a be the accepted set X¼ computation tree T: Let X CQnp S N N 2 fðx1 ; x2 ; y; xN ÞAQp jx1 x2 ¼ 0g; and write X ¼ gAG Xg where G is the set of computation paths in T from the root to an accepting leaf. For each gAG; the points in Xg are defined by a system of conditions rg
rg þug
i¼1
i¼rg þ1
4 P2 ðQi ðx1 ; x2 ; y; xN ÞÞ4 4 :P2 ðQi ðx1 ; x2 ; y; xN ÞÞ;
where the Qi ðx1 ; x2 ; yxN Þ are rational Qp functions of the inputs ðx1 ; x2 ; y; xN Þ computed by M along the path g: Here tautological relations of the form 2kN 2k2 1 P2 ðx2k 1 ; x2 ; y; xN Þ may be removed, without loss of generality. For zAQp ; P2 ðzÞ2R2 ðzÞ3z ¼ 0 and if s is a generator of ðZ=pZÞ then :P2 ðzÞ2P2 ðpzÞ3P2 ðszÞ3P2 ðspzÞ: Similarly, for z; wAQp ; R2 ðz=wÞ2
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ðR2 ðzÞ4R2 ðwÞÞ3ðR2 ðpzÞ4R2 ðpwÞÞ3ðR2 ðszÞ4R2 ðswÞÞ3ðR2 ðspzÞ4R2 ðspwÞÞ: Replacing the rational functions Qi ðx1; x2 ; y; xN Þ by pairs of relatively prime polynomials hðx1 ; x2 ; y; xN Þ and distributing by the equivalences above, we obtain a new disjunction of conjunctions characterizing the points of X : VZ
VZ þWZ
ZAY i¼1
i¼VZ þ1
3 ð4 R2 ðhi ðx1 ; x2 ; y; xN ÞÞ4 4 hi ðx1 ; x2 ; y; xN Þ ¼ 0Þ;
where the polynomials hi ðx1 ; x2 ; y; xN Þ are compositions of computation node polynomials computed by M along the paths in G: Now the polynomials hi ðx1 ; x2 ; y; xN Þ are all continuous, and the set of non-zero 2N squares fxjR2 ðxÞgDQp is open, but the set X ¼ fðx1 ; x2 ; y; xN ÞAQN p jx1 x2 g has no interior in QN p : Therefore each conjunction above must include at least one equation hi ðx1 ; x2 ; y; xN Þ ¼ 0; with hi not the zero polynomial, so WZ 40 all ZAY: N Now f ðx; yÞ ¼ x2 y has infinitely many roots (x; y) in Q2p ; and the index set Y above is finite, so for some Z0 AY there are infinitely many points of the form ðx1 ; x2 ; 1; y1ÞAQN p which satisfy the conjunction for Z0 and satisfy f ðx1 ; x2 Þ ¼ 0: Choose any of the polynomials hi ðx1 ; x2 ; y; xN Þ ¼ 0 in Z0 and let hðx; yÞ ¼ hi ðx; y; 1; y; 1Þ: Then f ðx; yÞ and hðx; yÞAQp ½x; y and have infinitely many common roots ðx; yÞ: By Lemma 4, f ðx; yÞ and hðx; yÞ must have a common factor. But N f ðx; yÞ ¼ x2 y is irreducible. Therefore f ðx; yÞ is a factor of hðx; yÞ in Qp ½x; y : In this case in fact it is easy to see directly that f ðx; yÞjhðx; yÞ: Applying the N division algorithm (since f ðx; yÞ ¼ x2 y) we obtain hðx; yÞ ¼ f ðx; yÞgðx; yÞ þ rðxÞ; N
where rðxÞ ¼ hðx; x2 Þ is a polynomial (see [1, p. 10]). Since hðx; yÞ and f ðx; yÞ have infinitely many roots, so does rðxÞ; so rðxÞ is the zero polynomial. N Since f ðx; yÞ ¼ x2 y is a factor of hðx; yÞ; degree hi ðx1 ; x2 ; y; xN Þ X degree N hðx; yÞX2 : But hðx1 ; x2 Þ ¼ hi ðx1 ; x2 ; 1; y; 1Þ is computed by M in polylogarithmic k time tðNÞ; so degree hðx; yÞpLtðnÞ pLD log ðnÞ o2N : This completes the proof of Theorem 1. & The same proof yields the following general result: Theorem 2. Let F be an infinite topological field, and consider computations over F. Assume htðxÞ ¼ 1 over F, and the branching map w : F -f21; 1g satisfies both w21 ð1Þ and w21 ð21Þ are open in F\\0}, and assume F does not have the discrete topology. Then P and NC are both defined over F and the inclusion NCF D PF is proper. Remark 1. From the proof of Theorem 1, we obtain the following lower bound. Let M be a parallel Qp machine which decides CðnÞ in time tðnÞ: Let L be, as above, the maximum degree of the polynomials computed at the computation nodes of M:
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Then n : tðnÞX log2 ðLÞ Remark 2. Observe that in the proof of Lemma 5, as in Cucker’s original proof over R, no use is made of the restriction in PLK Qp to polynomially many processors, only finiteness of the number of processors is used. This could suggest there is still some unused power in these arguments. Remark 3. In the case p ¼ 2; we defined Qp machines which have two sorts of branch nodes, branching on both P2 ðxÞ and P3 ðxÞ [9]. There are eight cosets of the squares in Q2 ; and three cosets of the cubes. Therefore in the proof of Lemma 5, more clauses are generated in obtaining a disjunction of conjunctions characterizing the points of the accepted set X ; but the rest of the proof is unchanged. As in the real case ([2, p. 364]) some further separations follow from the proof of Theorem 1. Let PARQp be the class of sets X CQN p that are decided by parallel machines in polynomial time, using exponentially many processors. Let EXPQp be the class of sets X CQN p that are decided by sequential Qp machines in exponential time. As in the real case: Theorem 3. The inclusion PARQp CEXPQp is strict. Proof Consider. the decision problem 2n
fðx1 ; x2 ; y; xn ÞAQnp jx21 x2 ¼ 0g and argue as above.
&
References [1] R. Bix, Conics and Cubics: A Concrete Introduction to Algebraic Curves, Undergraduate Texts in Mathematics, Springer, Berlin, Heidelberg, New York, 1998. [2] L. Blum, F. Cucker, M. Shub, S. Smale. Complexity and Real Computation, Springer, Berlin, Heidelberg, New York, 1998. [3] L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers: NPcompleteness, recursive functions and universal machines, Bull. Amer. Math. Soc. 21 (1) (1989) 1–46. [4] S. Bosch, U. Gu¨ntzer, R. Remmert, Non-Archimidean Analysis, Springer, Berlin, Heidelberg, New York, 1984. [5] F. Cucker, PRa NCR, J. Complexity 8 (1992) 230–238. [6] W. Fulton, Algebraic Curves, Benjamin, New York, 1969. [7] T.W. Hungerford, Algebra, Holt, Rinehart & Winston, New York, Chicago, San Fancisco, 1974.
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[8] A. Macintyre, On definable subsets of p-adic fields, J. Symbolic Logic 41 (3) (1976) 605–610. [9] M. Maller, J. Whitehead, Computational Complexity Over the 2-adic Numbers, Lectures in Applied Mathematics, Vol. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 513–521. [10] M. Maller, J. Whitehead, Computational Complexity Over the p-adic Numbers, J. Complexity 13 (2) (1997) 195–207.
http://www.elsevier.com/locate/jco
PaNC over the p-adic numbers$ Michael Mallera,* and Jennifer Whiteheadb b
a Department of Mathematics, Queens College, Flushing, New York 11367, USA Department of Computer Science, Queens College, Flushing, New York 11367, USA
Received 15 March 2002; revised 25 June 2002; accepted 26 June 2002
Abstract We show that in the Blum–Shub–Smale model of computation, over the p-adic numbers Qp ; the class NCQp is strictly contained in the class PQp : That is, there exist sets of p-adic numbers which can be recognized in sequential polynomial time, but which cannot be recognized in 2n polylogarithmic parallel time. We use the sets ðx1 ; x2 ; y; xn ÞAQnp CQN p such that x1 ¼ x2 : We also show that the inclusion PARQp CEXPQp is strict. These results extend previous work of Cucker, and of Blum, Cucker, Shub and Smale in the real case. r 2002 Elsevier Science (USA). All rights reserved.
1. Introduction Let R be a ring, ht : R-Rþ a height function, with htðxÞ representing the cost of computing with xAR; and let w:R-f21; 1g be a characteristic function controlling branching in computations. Blum, Shub and Smale defined machines and complexity classes over (R; ht; w) [2,3]. In this paper we will be concerned only with the case htðxÞ ¼ 1 for all xAR. A machine M over R runs in polynomial time on X CRN if M halts on all inputs xAX in time TM ðxÞ; and there are constants CAR; qAN such that, for all xAX ; x ¼ ðx1 ; x2 ; y; xn ÞARn CRN TM ðxÞpCnq ¼ CðsizeðxÞq Þ
$
Partially supported by PSC-CUNY Research award 63452 00 32. *Corresponding author. Tel.: +718-997-5851; fax: +718-997-5861. E-mail addresses: [email protected] (M. Maller), [email protected] (J. Whitehead).
0885-064X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. doi:10.1016/S0885-064X(02)00002-X
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
126
A decision problem SCRN is in the class PR if its characteristic function is computed by a polynomial time machine. A model of parallel computation over the real numbers is defined in [2, Chapter 18] in two equivalent ways, using PRAM’s and using algebraic circuits. We adapt their definitions to describe PRAM’s over Qp, with modifications to accommodate branching on squares and non-squares in Qp, which is an unordered field. As in the real case, an equivalent model could be defined using algebraic circuits over Qp. Let xAQp ; xa0; vp ðxÞ ¼ k; x ¼ pk ða0 þ a1 p þ Þ; ai AZ=p: Recall that provided pa2; x is a square in Qp if and only if k is even and a0 is a square in the cyclic group (Z/pZ)*, i.e. a0 is an even power of a generator s of (Z/pZ)*. Therefore, for pa2 and xa0; exactly one of the following four elements is a square in Qp: x; px; sx; spx: The Macintyre language for the p-adics [8] introduced the predicates Pk ðxÞ3ð(yAQp Þðyk ¼ xÞ
and
Rk ðxÞ3ðxa04Pk ðxÞÞ
In [10] we defined (sequential) machines over the p-adic numbers, pa2; which branch on P2 ðxÞ; LP2 ðxÞ and use ht ðxÞ ¼ 1 for all xAQp : (Some technical details differ in the case p ¼ 2; see remark at the end of this paper.) A processor M over Qp will be a p-adic machine as in [10] enhanced with communication nodes which permit processors in computations in a parallel machine to access registers of other activated processors. A parallel machine M is a sequence of identical processors fMi g; iAN; with an activation function p : N-N, one input space IM ¼ QN p ; one output space OM ¼ h N QN p ; and a state space SM ¼ ,h¼1 ðSM Þ : These parallel machines compute synchronously, all processors following a global clock. Given an input ðx0 ; y; xn1 ÞAQnp CQN p to M the input to the state space of processor Mi is
ðy; 0; i; n; xi d0; yÞ;
if
ion
and ðy; 0; i; n; 0d0; yÞ;
if
npiopðnÞ
where is a place marker. The class PLkQp ; parallel polylogarithmic time, is defined to be the class of sets SCQN p that are decided by a parallel Qp machine in time Oðlogk ðnÞÞ using a number of active processor pðnÞ polynomial in n: Parallel computations over Qp can also be defined as in [2, 18.4] in terms of k algebraic circuits, leading to the class NCQp ¼ ,k NCQ : As in the real case the two p approaches are equivalent, so we can define NCQp ¼
[ k
k NCQ ¼ p
[ k
PLkQp :
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
127
Lemma 1. Let M be a parallel machine over a ring R, and let L be the maximum degree of the polynomials computed at the computation nodes of the (identical) processors of M: Then the maximum degree of a polynomial computed by M in time T is LT. Proof If. hðx1 ; x2 ; y; xn Þ is computed by M in time T; unroll the computation into a tree. Any path from the root is of length at most T; and at each node the maximum degree computed is L: Therefore the Lemma follows by the law of exponents. & We will adapt the problem used in [2] in the real case (‘‘Cucker’s Problem’’), n recognizing roots of f ðx; yÞ ¼ x2 y: To formulate this as a uniform problem with input size n, consider for input x ¼ ðx1 ; x2 ; y; xn ÞARn CRN n
F ðx1 ; x2 ; y; xn Þ ¼ x21 x2 ¼ x21
sizeðxÞ
x2 :
The proofs in [2, Section 19.1, 5] use the Hilbert Nullstellensatz to show that a parallel machine which recognizes these roots would have to compute a polynomial of degree 2n and therefore could not run in polylogarithmic time. There certainly are p-adic versions of the Nullstellansatz, but they are less accessible for this problem (see for example [4]). Therefore our proof will take the argument back down into two dimensions where more elementary algebraic geometry will suffice. Cucker’s problem gives a satisfying proof that PaNC over R since it reflects the intuition that some problems, like iterated squaring, are inherently sequential. Observe that over Z the same proof shows that a parallel machine deciding Cucker’s problem would have to compute a polynomial of degree 2n, but in the bit model Cucker’s decision problem is not in P.
2. Proof of main theorem n
n 2 Let CðnÞDQnp CQN p be the decision problem CðnÞ ¼ fðx1 ; x2 ; y; xn ÞAQp jx1 ¼ x2 g: We prove that CðnÞAPQp but CðnÞeNCQp :
Theorem 1. The inclusion of complexity classes NCQp CPQp is strict. Lemma 2. CðnÞAPQp : Proof The. size of input ðx1 ; x2 ; y; xn Þ is n. Iterate squaring x1 ; n times to compute n n x21 in linear time. Test x21 x2 ¼ 0: Qp machines detect z ¼ 0 in two steps as P2 ðzÞ4P2 ðpzÞ: & Lemma 3. Let F be a field, x, y, indeterminants and dAZ+. Then gðx; yÞ ¼ xd 2yAF ½x; y is irreducible.
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
128
Proof By. Eisenstein’s criterion [7, p. 164]. Since F is a field, F ½y is a UFD, and P F ½x; y is canonically isomorphic to F ½x ½y DF ½y ½x : Consider hðxÞ ¼ di¼0 ai xi ¼ xd yAF ½y ½x : The coefficients ad ¼ 1; a0 ¼ y; ai ¼ 0; 0oiod; are relatively prime in F ½y so hðxÞ is primitive in F ½y ½x : The element yAF ½y is irreducible, and :yjad ; :y2 ja0 and yjai ; 0piod: Therefore by Eisenstein’s criterion hðxÞ is irreducible in F ½y ½x ; whence gðx; yÞ ¼ xd 2y is irreducible in F ½x; y : & We will need the following elementary fact from algebraic geometry. Let k be a field and fi Ak½x1 ; x2 ; yxn ; i ¼ 1; y; m; be polynomials in n variables, V ðf Þ ¼ fxAkn jf ðxÞ ¼ 0g and V ðffi j1pipmgÞ ¼ fxAkn jfi ðxÞ ¼ 0; i ¼ 1ymg: Lemma 4. Let k be a field, f ðx; yÞ; gðx; yÞ polynomials T in k½x; y with no common factors. Then the set of common roots V ðf ; gÞ ¼ V ðf Þ V ðgÞ is finite. Proof See. [6, p. 18].
&
Lemma 5. CðnÞeNC Qp : Proof Suppose. CðnÞANCQp : Let M be a parallel Qp machine which decides CðnÞ in polylogarithmic time tðnÞ: So (D; k such that tðnÞpD logk ðnÞ: Let L be the maximum degree of the polynomials computed at the computation nodes of M; and choose N sufficiently large so LtðNÞ pLD log
k
ðNÞ
o2N :
Fix this N and consider the computation of M on inputs ðx1 ; x2 ; y; xN ÞAQN p : m j This computation takes place in a finite dimensional subspace Z Qp of the state space of M: As in [2], we unroll the computations of M on these inputs into a be the accepted set X¼ computation tree T: Let X CQnp S N N 2 fðx1 ; x2 ; y; xN ÞAQp jx1 x2 ¼ 0g; and write X ¼ gAG Xg where G is the set of computation paths in T from the root to an accepting leaf. For each gAG; the points in Xg are defined by a system of conditions rg
rg þug
i¼1
i¼rg þ1
4 P2 ðQi ðx1 ; x2 ; y; xN ÞÞ4 4 :P2 ðQi ðx1 ; x2 ; y; xN ÞÞ;
where the Qi ðx1 ; x2 ; yxN Þ are rational Qp functions of the inputs ðx1 ; x2 ; y; xN Þ computed by M along the path g: Here tautological relations of the form 2kN 2k2 1 P2 ðx2k 1 ; x2 ; y; xN Þ may be removed, without loss of generality. For zAQp ; P2 ðzÞ2R2 ðzÞ3z ¼ 0 and if s is a generator of ðZ=pZÞ then :P2 ðzÞ2P2 ðpzÞ3P2 ðszÞ3P2 ðspzÞ: Similarly, for z; wAQp ; R2 ðz=wÞ2
M. Maller, J. Whitehead / Journal of Complexity 19 (2003) 125–131
129
ðR2 ðzÞ4R2 ðwÞÞ3ðR2 ðpzÞ4R2 ðpwÞÞ3ðR2 ðszÞ4R2 ðswÞÞ3ðR2 ðspzÞ4R2 ðspwÞÞ: Replacing the rational functions Qi ðx1; x2 ; y; xN Þ by pairs of relatively prime polynomials hðx1 ; x2 ; y; xN Þ and distributing by the equivalences above, we obtain a new disjunction of conjunctions characterizing the points of X : VZ
VZ þWZ
ZAY i¼1
i¼VZ þ1
3 ð4 R2 ðhi ðx1 ; x2 ; y; xN ÞÞ4 4 hi ðx1 ; x2 ; y; xN Þ ¼ 0Þ;
where the polynomials hi ðx1 ; x2 ; y; xN Þ are compositions of computation node polynomials computed by M along the paths in G: Now the polynomials hi ðx1 ; x2 ; y; xN Þ are all continuous, and the set of non-zero 2N squares fxjR2 ðxÞgDQp is open, but the set X ¼ fðx1 ; x2 ; y; xN ÞAQN p jx1 x2 g has no interior in QN p : Therefore each conjunction above must include at least one equation hi ðx1 ; x2 ; y; xN Þ ¼ 0; with hi not the zero polynomial, so WZ 40 all ZAY: N Now f ðx; yÞ ¼ x2 y has infinitely many roots (x; y) in Q2p ; and the index set Y above is finite, so for some Z0 AY there are infinitely many points of the form ðx1 ; x2 ; 1; y1ÞAQN p which satisfy the conjunction for Z0 and satisfy f ðx1 ; x2 Þ ¼ 0: Choose any of the polynomials hi ðx1 ; x2 ; y; xN Þ ¼ 0 in Z0 and let hðx; yÞ ¼ hi ðx; y; 1; y; 1Þ: Then f ðx; yÞ and hðx; yÞAQp ½x; y and have infinitely many common roots ðx; yÞ: By Lemma 4, f ðx; yÞ and hðx; yÞ must have a common factor. But N f ðx; yÞ ¼ x2 y is irreducible. Therefore f ðx; yÞ is a factor of hðx; yÞ in Qp ½x; y : In this case in fact it is easy to see directly that f ðx; yÞjhðx; yÞ: Applying the N division algorithm (since f ðx; yÞ ¼ x2 y) we obtain hðx; yÞ ¼ f ðx; yÞgðx; yÞ þ rðxÞ; N
where rðxÞ ¼ hðx; x2 Þ is a polynomial (see [1, p. 10]). Since hðx; yÞ and f ðx; yÞ have infinitely many roots, so does rðxÞ; so rðxÞ is the zero polynomial. N Since f ðx; yÞ ¼ x2 y is a factor of hðx; yÞ; degree hi ðx1 ; x2 ; y; xN Þ X degree N hðx; yÞX2 : But hðx1 ; x2 Þ ¼ hi ðx1 ; x2 ; 1; y; 1Þ is computed by M in polylogarithmic k time tðNÞ; so degree hðx; yÞpLtðnÞ pLD log ðnÞ o2N : This completes the proof of Theorem 1. & The same proof yields the following general result: Theorem 2. Let F be an infinite topological field, and consider computations over F. Assume htðxÞ ¼ 1 over F, and the branching map w : F -f21; 1g satisfies both w21 ð1Þ and w21 ð21Þ are open in F\\0}, and assume F does not have the discrete topology. Then P and NC are both defined over F and the inclusion NCF D PF is proper. Remark 1. From the proof of Theorem 1, we obtain the following lower bound. Let M be a parallel Qp machine which decides CðnÞ in time tðnÞ: Let L be, as above, the maximum degree of the polynomials computed at the computation nodes of M:
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Then n : tðnÞX log2 ðLÞ Remark 2. Observe that in the proof of Lemma 5, as in Cucker’s original proof over R, no use is made of the restriction in PLK Qp to polynomially many processors, only finiteness of the number of processors is used. This could suggest there is still some unused power in these arguments. Remark 3. In the case p ¼ 2; we defined Qp machines which have two sorts of branch nodes, branching on both P2 ðxÞ and P3 ðxÞ [9]. There are eight cosets of the squares in Q2 ; and three cosets of the cubes. Therefore in the proof of Lemma 5, more clauses are generated in obtaining a disjunction of conjunctions characterizing the points of the accepted set X ; but the rest of the proof is unchanged. As in the real case ([2, p. 364]) some further separations follow from the proof of Theorem 1. Let PARQp be the class of sets X CQN p that are decided by parallel machines in polynomial time, using exponentially many processors. Let EXPQp be the class of sets X CQN p that are decided by sequential Qp machines in exponential time. As in the real case: Theorem 3. The inclusion PARQp CEXPQp is strict. Proof Consider. the decision problem 2n
fðx1 ; x2 ; y; xn ÞAQnp jx21 x2 ¼ 0g and argue as above.
&
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[8] A. Macintyre, On definable subsets of p-adic fields, J. Symbolic Logic 41 (3) (1976) 605–610. [9] M. Maller, J. Whitehead, Computational Complexity Over the 2-adic Numbers, Lectures in Applied Mathematics, Vol. 32, Amer. Math. Soc., Providence, RI, 1996, pp. 513–521. [10] M. Maller, J. Whitehead, Computational Complexity Over the p-adic Numbers, J. Complexity 13 (2) (1997) 195–207.