Volume 15. Number 2 INFOR~WTIION PROCESSING LETTERS 6 wtember 1982

A NQT’E Qlfl A ‘I’HEQREM BY LADNER

J.E. BALCAZAR and J. DIAZ

Fact&at d)Informdtica, UPB, Jordi Gimna 31, Borcelorla 34, Spain

Received 23 April 1982

Keywor&: P, NP, NP-completc, NP-intermediate, T-reducibility, T-quivalence, T-incomparable

1. Mmduction

Irz [2] Ladner answered affirmatively to the question of the existence, under the hypothesis P+NP, of languages that are neither P nor NP- complete. In this note, we generaliz Ladner’s main theorem, using similar hinds of techniques to the ones he used. We prove that if P# NP, then there exists an infinite number of non-comparable languages in NP.

Let us start recalling some well-known d.efii- tions and facts. For a more extensive exposition of these, see [l] and [4]. As it i( known, P is ‘the class of languages accepted by a deterministic Turing machine (DTM) working in time polynomial, and NP is the class of languages accepted by non-de terministic Turing machine (NDTM) in poly- nomial time. Let q* (PO, P, ,... } and %t (M,, M,, . . . ) be, respectively, effective enumerations of DTM that run in polynomial time, and oracle Turing machines that also run in polynomial time. We denote by L = Pi the fact that the language L is ampted by some machine PiEq and in the same manner L = ML’ denotes the fact that the language L is accepted by an oracle Turing mat- hine Mi E Gilt with oracle L’. If z E L, Pi(Z) t means that z is not accepted by ma&he Pi, in the same manner M:*(z) t means that z is not accepted by Mi E % with oracle Li. Notice that L E P iff 3 Pi E 9 SUCh that L = Pi* c?h4% alphabets Zr and Z,, and two languages L, E 27 and L, E Zz, L, is T-reducible in polynomial time to L, (L, a Lz) iff 3 M, E % L, = Mkz. L, is m-reducible in poly-

nomial time to L, iff 3f: Zf --+ 2; which is compu- table by some PiE9 811c1 such that xELr+*f(x) E L,Vx Ext. Throughout this paper, we shall assume that C, = E, = ‘(1,O). A hquage LENP belongs to the class NP-complete (NPC) if Li E NP implies LiQc,L. TWO languages!, L, and L,, are T-incomparables if neither Lr a,.L, nor L2 aTLI, they are T-equivalent~(L, =rLz:) if LI =rL, and L, =rLr. The following lemma is well known [3].

Lemma 1.1, rfLI c&L2 + L, arLz. (1)

Given languages L,, L, E Z we define the join of L, and L, by

L,UL,=‘{OxifxEL,&lxifxEL,).

Clearly, L, aTLr 0 Lz and Lz a:TL, 0 L,.

%.Tkmaintheorem

In this section we state and prove our main theorem, that is an extension of the theorem by Ladner [2].

Theorem 2.1. Let (Li)?! 1, Li IL Z*, veri& the f02- lowing:

(i) Li ENP-P Vi E’(0, l,...,m), (ii) LiaT9,.,‘~iE’(P,2,...,m),

(iii) L,&TLi Vi E’(l,2,...,m). Then, there exists an L,, , C 27’ such that:

69 L*+* ENP-P,

84 0020~0 190/82/OOOO-0000/$02.75 @ 1982 North-Holland

Volume IS, Number 2 INFORMATION PROCESSING LE’ITERS 6 September 1982

09 Lm+ t a-&o, w Lo*&m-t 19

60 Lm+I and Li are T-incomparable Vi E ’

(1 2 9 ,...,m).

Proof. We construct a polynomial time transducer T, with entrance x E Z* and range c ( 1 )* using a language L,+l defined by

L m+l =‘(xEL,,:(T(x)] mod(2m+2)am+ 1)

whe;;::: IT(x)1 is the length of output when T has an input x, and mod denotes the remainder of the integer division. Then T is described by

(a) T(X) = A, (b) if 1x1 = n and x #On, then T(x) = T(O”), (c) on inputs of the form O”, n a 1, T works as

follows: 0)

(2)

(3)

For n moves, using self-simulation, it computes the sequence T(X), T(O), T(02),...,T(01). Divide IT( by 2m f 2, let k, i E Z + be such that IT( = k(2m + 2) + i. For n moves, T will try to find a z E Z* which satisfies one of the following con- ditions: 00

09

Cc)

(4

Ccrsei=O: z E L* @ Mkm+i(z) t . Case 1 GiGm: z E I+ - MkL”“(2) t * Ca8em-t-lGib;2m: z E Lo - M+(z) t .

Casei=2m+l: z E Lo - Pk(Z) t .

If such a a: is found, T(0’) = lk(2m+2)3-i+ ‘, otherwise T(0’) = lk(2m+2)+i.

As T works polynomially, the function

if IT(x)1 mod(2m + 2) 3 2m + 1,

otherwise,

(2) is computable in polynomial time, therefore L,, 1 a,Lo which implies

L m+l aTLo*

It should be noticed that

Vn IT( 4 IT(O”“)] 4 ]T(O”)l + 1 l

(3)

We also need to prove that

rangeT=‘(l}*. (43

This is done by induction. Clearly T(X) = X f { 1 }*. Suppose that 1’ E range T but, l’+* @range T.

Then for every x but a finite number, we have T(X) = 1’. Let k, i E WI such that t = k(2m + 2) + i. Then we have that Vi, i < m, L,, 1 E P (because L m+, is finite), and Vi, i 5 m t 1, L,+] smLo.

But if the length IT(x)1 is never increased after a certain step, it means that we can ever find a z satisfying the corresponding condition. Therefore such a z cannot exist. This implies the following:

If O<iSm,

L~=M~~+~=+L~~TL,+~J,L~EP.

If m+ 1 <i<2m,

(5)

L m+l= Mki-m*LO ~TL,+I cTLi__m* (6)

If i=2m+ 1,

L m+l=Pk*L()=TPkdLoEP. (7)

But (5), (6) and (7) are contradictions to the hy- pothesis and therefore lt+l E range T, proving statement (4).

In view of the above, we can assert that Vk it is possible to find 2m + 2 z’s which ensure

Lo # Mkm+t, i.e., LO &L,+ 1 9

Li # Mkm+l, i.e., Li#*L,+l Vi E’( 3 ,..., m},

L m+ 1 # Mki, i.e., L,, , @TLi Vi E ‘( i, . . .,m) ,

L m+l epk? i.e., L,,, B P,

which, together with (3), gives the starnent of the theorem. Now, we just have to prove the existence

of Lrn+ I. and T. But as the descriptions given of ‘I’

and Lxn+1 are effective constructions, let, say, f and g be such constructions. By the recursion theorem [4] 3So such that

Ws, = W(f(so))

and the theorem is proved. 0

3. Clonsequences of the mlaiu theorem

Corollm 3.1. If L, E P/Sip -- P, then there exists an L, E NP - P, such tha@L, aTLO but Lo +TL,.

5

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Volume 15, Number 2 INFORMATION PROCESSING LETTERS 6 September 1982

Proof. Just apply Theorem 2.1 with i = 0. •l

Corollary 3.1 is the theorem of Ladner as it was proved in 1975 12). In that paper Ladner also shows that between any two non-equivalent NP languages there exist two languages that are. in- comparable. The following corollary to the main theorem is an extension of that result.

Ccrrsll~ 3.2. For all L 3 E NP - P, there is an infinite family Of languages, 9= ‘(Li)iEN--(O) S&l

that

(i) LiENIP-PViEN’(O}, (ii) Lia,L, but L,#rLi Vi E N z {0),

(iii) Li and Lj are incomparable Vi, j E N A (0) and i #j.

Proof. Let % = @. Inductively, if %#,, = ’

(L ,,..., E,), then apply the main theorem, to get L m+ . Form $Q+r = (L,,...,L,, L,,,). Then let + b % % is the desired family of incom- nEN n’ parable languages.

We are also able to answer negatively to the question of finding an equivalence class in NIP - (P U NPC), that is comparable with all the other equivalences classes in NP.

Corolhuy 3.3. Let L, E NP be such that for all L, E NP either L, o+L, or L, arL,. Then we have that L, E P or L, E NPC.

Pro&‘. Let L, be such that L, E NP - (P U NPC), and let L, be any language NP-complete. Then we have that L, a,L, but L,&L,. By the main theorem (with m = 1) we get L, incomparable with L,. q

It fan be noticed that Theorem 2.1 produces a descending (towards P) sequence of elements. It is possible to state a prove of an ascending version of Theorem 2.1.

Theorem 3.4. Let {Li)~!,, Li e S,*, verify the fol- lowing:

(i) Li E VP - NPC Vi E ‘(0, l,...,m), (ri) L,a.,L, Vi E’tl,..., m),

(iii) Li &TL, Vi E ‘{l,...,m). Then, there exist L, + , c Z* such that

86

o-0 Ll+l E NP - NPC,

(b) L, arLIn+ 1, (c) L,+1 @*LcV (d) LFn+1 and Li are incomparables Vi E ’

(1 ,...,m).

Sketch of the proof. Let Lb be any NPC language. In a similar manner to the proof of Theorem 2.1, we define L’,,, = ‘(x E L’, : AT\ mod 2n < m) and construct the transducer T in such a way that, in each step k, it checks that the following is satisfied:

L, # M~‘,+I,

L, += M;ooLk+l,

L’,+r = Mki.

Finally it will do to define L,+, = L,o L’,, ,. 0

4. Ady& of th reslllffs with m-mdwibility

In view or” (1) we can substitute the T-reducibil- ity by the m-reducibility. in all the affirmative hypotheses and in all negative conclusions. Also note that, in (2), we have not only showed L,+, =rL,, but also that Lm+l a,L,,. With respect to proving results with the weaker hypothesis of the m-reducibility, it is easy to reduce the construction of the transducer in such a way that it finds z E 2* such that

zEL,c+f&) BL,.

Notice that in Corollaries 3.2 and 3.3, the aT could be directly substituted by am without prob- lem, and in Corollary 3.1, L, cc=L, can also be substituted immediately by L, amLo.

References

M.R. Garey and D.S. Johnson, Computer and Intractabil- ity. A Guide to the Theory of NP-completeness (Freeman, San Francisco, 1979). R. Ladner, 0n the structlie of polynomial time reducibil- ity, 3. ACM 22 (1975) 155-171. R. Ladner, N.A. Lynch and A.L. Serlman, A comparison of polynomial time reducibilities, T’heoret. Cornput. &ii. E (1976) 103-123. H. Rogers, Theory of recursive functions and effective computability (McGraw-Hill, New ‘York, 1967).