Reducibility and Degree Outline General Remark Many-One Reduction

Home , Reduction (complexity)

Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Outline

∗ Reducibility and Degree 1 Reduction and Degree Many-One Reduction Degrees Xiaofeng Gao m-Complete r.e. Set

Department of Computer Science and Engineering Shanghai Jiao Tong University, P.R.China 2 Relative Computability CS363-Computability Theory 3 Turing Reducibility

∗ Special thanks is given to Prof. Yuxi Fu for sharing his teaching materials. CSC363-ComputabilityTheory@SJTU XiaofengGao ReducibilityandDegree 1/64 CSC363-ComputabilityTheory@SJTU XiaofengGao ReducibilityandDegree 2/64

Reduction and Degree Many-One Reduction Reduction and Degree Many-One Reduction Relative Computability Degrees Relative Computability Degrees Turing Reducibility m-Complete r.e. Set Turing Reducibility m-Complete r.e. Set General Remark Many-One Reduction

A problem is a set of numbers.

A reduction is a way of deﬁning a solution of a problem with the help The set A is many-one reducible (m-reducible) to the set B if there is a of the solutions of another problem. total computable function f such that x ∈ A iff f (x) ∈ B for all x.

There are several inequivalent ways of reducing a problem to another We shall write A ≤m B or more explicitly f : A ≤m B. problem. If f is injective, then we are talking about one-one reducibility . The differences between different reductions consists in the manner and extent to which information about B is allowed to settle questions about A.

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1. K is m-reducible to {x | φx = 0}, {x | c ∈ Wx} and {x | φx is total }. Let A, B, C be sets.

0 if x ∈ Wx y if x ∈ Wx 1. ≤m is reﬂexive: A ≤m A. f (x, y) = fN(x, y) = 0 ↑ if x ∈ W ↑ if x ∈ W x x f : A ≤m A is the identity function. B 2. Rice Theorem is proved by showing that K ≤m {x | φx ∈ }. 2. ≤m transitive: A ≤m B, B ≤m C ⇒ A ≤m C. B g(y) if x ∈ Wx x ∈ Wx ⇒ φk(x) = g ∈ Let f : A ≤m B, g : B ≤m C, then g ◦ f : A ≤m C. fg(x, y) = ↑ if x ∈ Wx x ∈ Wx ⇒ φk(x) = f∅ ∈ B 3. A ≤m B iff A ≤m B. 3. {x | φx is total } ≤ m {x | φx = 0}. If g : A ≤m B, then x ∈ A ⇔ f (x) ∈ B; so x ∈ A ⇔ g(x) ∈ B. φk(x) = 0 ◦ φx Hence g : A ≤m B.

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4. If A is recursive and B ≤m A, then B is recursive. 7. (i). A ≤m N iff A = N; (ii). A ≤m ∅ iff A = ∅. g : B ≤m A; then cB(x) = cA(g(x)) . So cB is computable. (i).“ ⇐": By reﬂexivity, N ≤m N. (i).“ ⇒": Let f : A ≤m N, then x ∈ A ⇔ f (x) ∈ N. Thus A = N. 5. If A is recursive and B = ∅, N, then A ≤m B. (ii). A ≤m ∅ ⇔ A ≤m N ⇔ A = N ⇔ A = ∅. b if x ∈ A; Let b ∈ B, c ∈ B, f (x) = ; then f is c if x ∈ A. 8. (i). N ≤m A iff A = ∅; (ii). ∅ ≤m A iff A = N. computable. (i). “ ⇒": Let f : N ≤ A, then A = Ran (f ), so A = ∅ (f is x ∈ A ⇔ f (x) ∈ B. m total). (i). “ ⇐": If A = ∅, choose c ∈ A. If g(x) = c, we have 6. If A is r.e. and B ≤m A, then B is r.e. g : N ≤m A. C Let g : B ≤m A, A = Dom (h), ( h ∈ 1); then B = Dom (h ◦ g) (ii). ∅ ≤m A ⇔ N ≤m A ⇔ A = ∅ ⇔ A = N. (B is r.e.)

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Fact . If A is r.e. and is not recursive, then A ≤ m A and A ≤ m A. Corollary . Neither {x | φx is total } nor {x | φx is not total } is m-reducible to K. Proof . “ A ≤ m A": By contradiction, if A ≤m A, then A is r.e., then A is recursive! Proof . By contradiction, if {x | φx is total } ≤ m K, and K is r.e., then {x | φx is total } is r.e. (same as {x | φx is not total }). “A ≤ m A": By contradiction, if A ≤m A, then A ≤m A, then A is recursive! However, by Rice-Shapiro Theorem, Neither {x | φx is total } nor {x | φ is not total } is r.e. x Notation: It contradicts to our intuition that A and A are equally difﬁcult.

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Theorem . A is r.e. iff A ≤m K. Deﬁnition . Two sets A, B are many-one equivalent , notation A ≡m B (abbreviated m-equivalent), if A ≤m B and B ≤m A. Proof . “ ⇐". Since A ≤m K, and K is r.e., then A is r.e. 1, if x ∈ A, Suppose A is r.e. Let f (x, y) be f (x, y) = ↑, if x ∈/ A. Theorem . ≡m is an equivalence relation.

By s-m-n Theorem ∃s(x) : N → N such that f (x, y) = φs(x)(y). Proof . (1). Reflexivity: A ≤m A ⇒ A ≡m A. It is clear that x ∈ A iff φs(x)(s(x)) is defined iff s(x) ∈ K. Hence A ≤m K. (2). Symmetry: A ≡m B ⇒ B ≤m A, A ≤m B ⇒ B ≡m A. (3). Transitivity: A ≡m B, B ≡m C ⇒ A ≤m C, C ≤m A ⇒ A ≡m C. Notation . K is the most difficult partially decidable problem.

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1. {x | c ∈ Wx} ≡ m K. 3. If A is r.e. but not recursive, then A ≡ m A. y if x ∈ Wx “⇐": fN(x, y) = ⇒ K ≤m {x | c ∈ Wx} A is r.e. but not recursive ⇒ A ≤ m A, A ≤ m A. ↑ if x ∈ Wx “⇒": {x | c ∈ Wx} is r.e., so {x | c ∈ Wx} ≤ m K. 4. {x | φx = 0} ≡ m {x | φx is total }. Thus {x | c ∈ Wx} ≡ m K. “⇐": φk(x) = 0 ◦ φx ⇒ { x | φx is total } ≤ m {x | φx = 0}. 0 if φ (y) = 0; 2. If A is recursive, A = ∅, N, then A ≡m A. x “⇒": Let φk(x)(y) = . ↑ if φx(y) = 0. A = ∅, N ⇒ A = ∅, N. Then φx = 0 ⇔ φk(x) is total ⇒ { x | φx = 0} ≤ m {x | φx is total }. A is recursive, by previous theorem A ≤m A. Similarly, A ≤m A.

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Deﬁnition . The set of m-degrees is ranged over by a, b, c,... .

Deﬁnition . Let dm(A) be {B | A ≡m B}. Deﬁnition (Partial Order on m-Degree) . Let a, b be m-degrees.

Deﬁnition . An m-degree is an equivalence class of sets under the (1). a ≤m b iff A ≤m B for some A ∈ a and B ∈ b. relation ≡m. It is any class of sets of the form dm(A) for some set A. (2). a

Notation . From the deﬁnition of ≡m, a ≤m b ⇔ ∀ A ∈ a, B ∈ b, A ≤m B.

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1. o and n are respectively the recursive m-degrees {∅} and {N}. Theorem . The relation

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4. An r.e. m-degree consists of only r.e. sets.

If A is r.e. and B ≤m A, then B is r.e.

5. If a ≤m b and b is an r.e. m-degree, then a is also an r.e. m-degree.

If A is r.e. and B ≤m A, then B is r.e.

′ 6. The maximum r.e. m-degree dm(K) is denoted by 0m. A set A is r.e. iff A ≤m K.

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1. Excluding o and n, there is a minimum r.e. m-degree 0m (in fact 0m is minimum among all m-degrees).

2. The r.e. m-degrees form an initial segment of the m-degrees; i.e., Theorem . The m-degrees form an upper semi-lattice . anything below an r.e. m-degree is also r.e.

′ 3. There is a maximum r.e. m-degree 0m.

4. While there are uncountably many m-degrees, only countably many of these are r.e.

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In mathematics, a group is an algebraic structure consisting of a set In mathematics, a lattice is a partially ordered set (poset) (L, ≤) in together with an operation (G, •) that combines any two of its which any two elements have a unique supremum (also called a least elements to form a third element. upper bound or join) and a unique inﬁmum (also called a greatest lower bound or meet). To qualify as a group, the set and the operation must satisfy four conditions ( group axioms ), namely closure , associativity , identity and To qualify as a lattice, the set and the operation must satisfy tow invertibility . conditions: join-semilattice , meet-semilattice .

closure : a, b ∈ G ⇒ a • b ∈ G. join-semilattice : ∀a, b ∈ L, the set {a, b} has a join a ∨ b. associativity : (a • b) • c = a • (b • c). (the least upper bound) identity : ∀a ∈ G, ∃ identity element e ∈ G, s.t. e • a = a • e = a. meet-semilattice : ∀a, b ∈ L, the set {a, b} has a meet a ∧ b. − (the greatest lower bound) invertibility : ∀a ∈ G, ∃inverse b ∈ G s.t. a • b = b • a = e (b = a 1).

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The name "lattice" is suggested by the form of the Hasse diagram Theorem . Any pair of m-degrees a, b have a least upper bound; i.e. depicting it. I.e., the right pic- there is an m-degree c such that ture is the lattice of partitions of (i). a ≤m c and b ≤m c (c is an upper bound);

a four-element set {1, 2, 3, 4}, or- (ii). c ≤m any other upper bound of a, b. dered by the relation "is a reﬁne- ment of".

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(ii). Let d is an m-degree such that a ≤m d, and b ≤m d. (i). Pick A ∈ a, B ∈ b, and let C = A ⊕ B, i.e., ∀D ∈ d, suppose f : A ≤m D and g : B ≤m D. Then

C = {2x | x ∈ A} ∪ { 2x + 1 | x ∈ B}. x x − 1 x ∈ C ⇔ (x is even & ∈ A) ∨ (x is odd & ∈ B) 2 2 Then x x − 1 x ∈ A ⇔ 2x ∈ C =⇒ A ≤ C; ⇔ (x is even &f ( ) ∈ D) ∨ (x is odd &g( ) ∈ D) m 2 2

x ∈ B ⇔ 2x + 1 ∈ C =⇒ B ≤m C; x f ( 2 ) if x is even ; Thus c is an upper bound of a, b. Thus we have h : C ≤m D if we deﬁne h = x−1 g( 2 ) if x is odd .

Hence c ≤m d.

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Deﬁnition . An r.e. set is m-complete if every r.e. set is m-reducible to Theorem . The following statements are valid. it. (i) K is m-complete.

′ (ii) A is m-complete iff A ≡m K iff A is r.e. and K ≤m A. Notation . 0m, the m-degree of K is maximum among all r.e. ′ m-degrees, and thus K is m-complete r.e. set (or just called (iii) 0m consists exactly of all the m-complete sets. m-complete set ).

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The following sets are m-complete. Theorem . Any m-complete set is creative . (i) {x | c ∈ Wx}.

(ii) Every non-trivial r.e. set of the form {x | φx ∈ B}. Proof . If A is m-complete, A is r.e. set.

(iii) {x | φx(x) = 0}. Also, K ≤m A, so K ≤m A. Thus A is productive.

(iv). {x | x ∈ Ex}.

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Corollary . If a is the m-degree of any simple set, then ′ Myhill’s Theorem . A set is m-complete iff it is creative. 0m

Proof . Simple sets are designed to be neither recursive nor creative.

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Comparison Relative Computability

m-reducibility has two unsatisfactory features: Suppose χ is a total unary function. (i) The exceptional behavior of ∅ and N. Informally a function f is computable relative to χ, or χ-computable, (ii) The invalidity of A ≡ m A in general. if f can be computed by an algorithm that is effective in the usual sense, except from time to time during computations f is allowed to The problem is due to the restricted use of oracles. consult the oracle function χ. Such an algorithm is called a χ-algorithm. E.g. x ∈ A iff x ∈ A

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A URM with oracle , URMO for short, can recognize a ﬁfth kind of instruction, O(n), for every n ≥ 1.

If χ is the oracle, then the effect of O(n) is to replace the content rn of Rn by χ(rn).

Pχ denote the program P when used with the function χ in the oracle.

Pχ(a) ↓ b means the computation Pχ(a) with initial conﬁguration a1, a2, , an, 0, 0, stops with the number b is register R1.

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility URMO-Computable Facts

(i) χ ∈ C χ. Let χ be a unary total function, and suppose the f is a partial function Use URMO program O(1). from Nn to N. (ii) C ⊆ C χ. (a) Let P be a URMO program, then P URMO-computes f relative Any URM program is a URMO program. Nn N to χ (or f is χ-computed by P) if, for every a ∈ and b ∈ , (iii) If χ is computable, then C = C χ. Pχ(a) ↓ b iff f (a) ≃ b. Since C ⊆ C χ, we need to prove C χ ⊆ C . χ is computable, then whenever a value of is requested simply compute it by the (b) The function f is URMO-computable relative to χ (or χ χ-computable) if there is a URMO program that algorithm for χ. By Church’s thesis, f is computable. URMO-computes it relative to χ. (iv) C χ is closed under substitution, recursion and minimalisation. Construct corresponding URMO programs. χ C is the set of all χ-computable functions. (v) If ψ is a total unary function that is χ-computable, then C ψ ⊆ C χ. By Church’s thesis.

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The class Rχ of χ-partial recursive functions is the smallest class of functions such that Let’s ﬁx an effective enumeration of all URMO programs (a) the basic functions are in Rχ. Q0, Q1, Q2,.... (b) χ ∈ Rχ. χ, n Let φm be the n-ary function χ-computed by Qm. Rχ (c) is closed under substitution, recursion, and minimalisation. χ χ, 1 Let φm be φm .

-recursive , -primitive recursive are deﬁned in the obvious way. χ χ χ χ χ χ Wm is Dom (φm) and Em is Ran (φm).

Theorem. For any χ, Rχ = C χ.

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Numbering URMO programs Universal Programs for Relative Computability

S-m-n Theorem . For each m, n ≥ 1 there is a total computable Universal Function Theorem . For each n, the universal function m (m + 1)-ary function sn (e, x) such that for any χ χ, n ψU for n-ary χ-computable functions given by χ, n χ, m+n χ, n φe (x, y) ≃ φ m (y). χ, n sn (e,x) ψU (e, x) ≃ φe (x)

m is χ-computable. Notice that sn (e, x) does not refer to χ.

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Let A be a set

Once we have the S-m-n Theorem and the Universal Function (a) A is χ-recursive if cA is χ-computable. Theorem, we can do the recursion theory relative to an oracle. (b) A is χ-r.e. if the partial characteristic function 1 if x ∈ A, f (x) = is χ-computable. ↑ if x ∈ A

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility χ-Recursive and χ-r.e. Sets Computability Relative to a Set

Computability relative to a set A means computability relative to its Theorem . The following statements are valid. characteristic function cA. (i) For any set A, A is χ-recursive iff A and A are χ-r.e. For example: A c (ii) For any set A, the following are equivalent. P for P A (if P is a URMO program), A c A is χ-r.e. C for C A , χ A cA A = Wm for some m. φm for φm . χ A cA A = Em for some m. Wm for Wm , A cA A = ∅ or A is the range of a total χ-computable function. Em for Em , A c For some χ-decidable predicate R(x, y), x ∈ A iff ∃y.R(x, y). K for K A , A-recursive for cA-recursive χ def χ (iii) K = {x | x ∈ Wx } is χ-r.e. but not χ-recursive. A-r.e. for cA-r.e.

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Suppose A ≤T B and P is the URMO program that computes cA relative to B. Then ∀x, PB(x) converges and The set A is Turing reducible to B, notation A ≤ B, if A has a T PB(x) ↓ 1 if x ∈ A B-computable characteristic function c . A PB(x) ↓ 0 if x ∈ A B The sets A, B are Turing equivalent , notation A ≡T B, if A ≤T B and When calculating P (x) there will be a ﬁnite number of requests to B ≤T A. the oracle for a value cB(n) of cB. These requests amount to a ﬁnite number of questions of the form ‘ n ∈ B?’.

So for any x, ‘ x ∈ A?’ is settled in a mechanical way by answering a ﬁnite number of questions about B.

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Facts. Facts. (2)

(i) ≤T is reﬂexive and transitive. A B A ≤T B iff C ⊆ C ; (v) If A is recursive, then A ≤T B for all B. χ (ii) ≡T is an equivalence relation. Since C ⊆ C . A B A ≡T B iff C = C ; (vi) If B is recursive and A ≤T B, then A is recursive. χ (iii) If A ≤m B then A ≤T B. If χ is computable, then C = C .

If f : A ≤m B and P is URM program to compute f , then the (vii) If A is r.e. then A ≤T K. URMO program P, O(1) is B-compute cA. If A ≤m B then A ≤T B; A set A is r.e. iff A ≤m K. (iv) A ≡T A for all A.

cA = sg ◦ cA, A is A-recursive =⇒ A ≤T A. (Similarly A ≤T A.)

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A set A is T-complete if A is r.e. and B A for every r.e. set B. ≤T The set of degrees is ranged over by a, b, c,... .

The equivalence class dT (A) = {B | A ≡T A} is called Turing degree a ≤ b iff A ≤T B for all A ∈ a and B ∈ b. of A, or T-degree of A. a < b iff a ≤ b and a = b. A T-degree containing a recursive set is called a recursive T-degree . The relation ≤ is a partial order. A T-degree containing an r.e. set is called an r.e. T-degree .

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Theorem Jump Operation

Theorem . The following statements are valid. (i) There is precisely one recursive degree 0, which consists of all the A def A (i) K = {x | x ∈ Wx } is A-r.e. recursive sets and is the unique minimal degree. Since Kχ is χ-r.e. If A is recursive, then A ≤T B for all B; If B is recursive and A (ii) If B is A-r.e., then B ≤T K . A ≤T B, then A is recursive. By relativised s-m-n theorem, if B is A-r.e., then B ≤ KA. ′ ′ ′ m (ii) Let 0 be the degree of K. Then 0 < 0 and 0 is a maximum A among all r.e. degrees. (iii) If A is recursive then K ≡T K. ′ ′ “⇐" K ≤ KA since K is A-r.e. for any A; From (i), 0 ≤ 0 ; 0 = 0 since K is not recursive. Since A is r.e. ⇒ T ′ “⇒" If A is recursive then A-computable partial characteristic A ≤T K, we have if a is any r.e. degree, a ≤ 0 . function of KA is actually computable (if χ is computable, then χ A A (iii) dm(A) ⊆ dT (A); and if dm(A) ≤m dm(B) then dT (A) ≤ dT (B). C = C ). Hence K is r.e., and K ≤T K. A If A ≤m B then A ≤T B. (iv) A

KA is a T-complete A-r.e. set. Also called the completion of A, or the jump of A, and denoted as A′. A B (v) If A ≤T B then K ≤T K . Definition. The jump of a, denoted a′, is the degree of KA for any If A ≤ B, then since KA is A-r.e. it is also B-r.e., so KA ≤ KB. T T A ∈ a. A B (vi) If A ≡T B then K ≡T K . Follows immediately from (v). Notation (1). By Relativization jump is a valid definition because the degree of KA is the same for every A ∈ a. Notation (2). The new definition of 0′ as the jump of 0 accords with our earlier definition of 0 ′ as the degree of K.

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Reduction and Degree Reduction and Degree Relative Computability Relative Computability Turing Reducibility Turing Reducibility Basic Properties Important Results

Theorem . Any degrees a, b have a unique least upper bound. Theorem . Any non-recursive r.e. degree contains a simple set. Theorem . For any degree a and b, the following statements are valid. Theorem . There are r.e. sets A, B s.t. A ≤ T B and B ≤ T A. Hence, if (i) a a′. < a, b are dT (A), dT (B) respectively, a ≤ b and b ≤ a, and thus ′ ′ (ii) If a < b then a′ < b′ 0 < a < 0 and 0 < b < 0 . (iii) If B ∈ b, A ∈ a and B is A-r.e. then b ≤ a′. Degrees a, b such that a ≤ b and b ≤ a are called incomparable degrees, denoted as a | b. Theorem . For any r.e. degree a > 0, there is an r.e. degree b such that b | a.

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Sack’s Density Theorem . For any r.e. degrees a < b there is an r.e. degree c with a < c < b. Sack’s Splitting Theorem . For any r.e. degrees a > 0 there are r.e. degrees b, c such that b < a c < a and a = b ∪ c (hence b | c). Lachlan, Yates Theorem . (a). ∃ r.e. degrees a, b > 0 such that 0 is the greatest lower bound of a and b. (b). ∃ r.e. degrees a, b having no greatest lower bound (either among all degrees or among r.e. degrees). Shoenﬁeld Theorem . There is a non-r.e. degree a < 0′. Spector Theorem . There is a minimal degree. ( A minimal degree is a degree m > 0 such that there is no degree a with 0 < a < m).

Theorem . For any r.e. m-degree a >m 0m, ∃ an r.e. m-degree b s.t. b | a. CSC363-ComputabilityTheory@SJTU XiaofengGao ReducibilityandDegree 64/64