On the Existence of Optimal Propositional Proof Systems and Oracle-Relativized Propositional Logic
Shai Ben-David
Anna Gringauzey
Abstract
We investigate sucient conditions for the existence of optimal propositional proof systems. We concentrate on conditions of the form CoN F = N F . We introduce a purely combinatorial property of complexity classes - the notions of slim vs. fat classes. These notions partition the collection of all previously studied time-complexity classes into two complementary sets. We show that for every slim class an appropriate collapse entails the existence of an optimal propositional proof system. On the other hand, we introduce a notion of a propositional proof system relative to an oracle and show that for every fat class there exists an oracle relative to which such an entailment fails. As the classes P (polynomial functions), E (2O(n) functions) and EE (2O(2n ) functions) are slim, this result includes all the previously known suciency conditions for the existence of optimal propositional proof systems. On the nother hand, the classes N EX P , QP (the class of quasi-polynomial functions) and 2 EEE (2O(2 ) functions), as well as any other natural time-complexity class which is not covered by our suciency result, are fat classes. As the proofs of all the known suciency conditions for the existence of optimal propositional proof systems carry over to the corresponding oracle-relativized notions, our oracle result shows that no extension of our suciency condition to non-slim classes can be obtained by the type of reasoning used so far in proofs on these issues.
CS Dept., Technion, Haifa 32000, Israel, email:
[email protected] y IBM Haifa Research Center, Haifa, Israel, email:
[email protected]
1
1 Introduction
The questions concerning the relationships between the classes P ; NP and CoNP are of the most important open problems in computational complexity. It turns out that the proof theory of propositional calculus provides a framework in which some of these questions can be reformulated and analyzed from a somewhat di erent perspective. The rst step in this direction is due to Cook and Reckhow [CR79] who have posed the following proof-theoretic question: \Does there exist a propositional proof system in which every tautology has a short proof (of size polynomial in the size of the tautology)?" To formulate this question precisely Cook and Reckhow in [CR79] introduced a following abstract de nition of a propositional proof system:
De nition 1 A propositional proof system is a polynomial time computable function f : f0; 1g 7!
onto
Taut.
A propositional proof system that allows short proofs to all tautologies is called a super propositional proof system. More formally, Let the size of a formula or a proof be the total number of symbols in it.
De nition 2 A propositional proof system f is called super if there exists a polynomial p : N 7! N s.t. for every 2 Taut there exists w 2 f0; 1g of size p(j j) s.t. f (w) = . In the same paper, Cook and Reckhow show that the existence of a super propositional proof system is equivalent to the assertion NP = CoNP . A natural step in the analysis of such a problem is to introduce a relevant partial order, in our case, a partial order re ecting the relative strength of propositional proof systems.
De nition 3 Let f , g be propositional proof systems. f simulates g if there exists a polynomial p s.t for every w 2 f0; 1g, there exists w0 2 f0; 1g of length p(jwj) s.t. f (w0) = g (w). We say that f p-simulates g if such a w0 can be found eciently, that is, if there exists a poly-time computable function h : f0; 1g 7! f0; 1g s.t. for all w, g (w) = f (h(w)). A \strongest" propositional proof system is called optimal.
A propositional proof system is called optimal if it simulates every other propositional proof system.
A propositional proof system is called p-optimal if it p-simulates every other propositional proof system.
Here again, the existence of a propositional proof system having such a property is unknown. Moreover, no natural open problems in computational complexity theory are known to be equivalent to the existence of such a propositional proof system. The strongest result on this topic is due to Kobler and Mener [KM98]:
Theorem [Kobler and Mener [KM98]]:
2
If TallyCoNEE NEE then there exists a optimal propositional proof system. If TallyCoNEE EE then there exists a p-optimal propositional proof system. This improves an earlier result of Pudlak [Pu84] and of Mener and Toran [MT97] deriving the existence of an optimal propositional proof system from the assumption NE = CoNE , and SparseCoNEE NEE , respectively. Here NE and NEE are the complexity classes of all languages recognized by nondeterministic Turing machines running in 2 ( ) and 2 (2n) time respectively. (where n denotes the size of an input), and EE and CoNEE denote the corresponding deterministic time and co-nondeterministic time classes. Tally stands for the restriction of a class to its unary langages. Krajcek and Pudlak, [KP89] present several proof theoretic statements which are equivalent to the existence of an optimal propositional proof system. The only known necessary condition for existence of optimal propositional proof system (relating it to a natural computational complexity issue) was shown (implicitly) by Razborov [Ra94]: Theorem [Razborov [Ra94]]: If there exists a optimal propositional proof system, then there is a complete (under polynomial reduction) problem is DisNP . Where DisNP is de ned as follows: O n
O
De nition 4 For every pair of languages L1; L2, such that L1 \ L2 = ;, de ne a problem D given an input x, nd an i 2 f1; 2g s.t. x 62 L . DisNP = fD : L1 ; L2 2 NP ; L1 \ L2 = ;g.
L1 ;L2
:
i
def
L1 ;L2
It is an open question whether the existence of optimal propositional proof systems implies the existence of complete problems for either NP \ CoNP or DisNP (or for any other complexity class for which the existence of a complete language is an open question). In this paper we take one more step towards clarifying the relationship between the collapse of complexity classes and the existence of optimal propositional proof systems. We introduce a purely combinatorial property of complexity classes - the notions of slim vs. fat classes. These notions partition the collection of all previously studied time-complexity classes into two complementary sets. We show that for every class in one of these sets an appropriate collapse entails the existence of an optimal propositional proof system, while, for every class in the other set there exists an oracle relative to which such an entailment fails. More precisely; On one hand we generalize the Kobler - Mener suciency condition and show:
Theorem 1: For every slim class F , TallyCoN F N F implies the existence of optimal propositional proof systems, and TallyCoN F DF implies the existence of p-optimal propositional proof systems. (DF stands for the class of languages computable in deterministic time belonging to F , N F and CoN F stand for the corresponding non-deterministic and co-non-deterministic classes). As the classes P (polynomial functions), E (2 ( ) functions) and EE (2 (2n) functions) are slim, this result includes all the previously know suciency conditions for the existence of an optimal propositional proof system. On the other hand, we introduce a notion of a propositional proof system relative to an oracle, and show: O n
O
Theorem 2: For every fat class F , there exists an oracle relative to which CoN F = DF and yet there is no optimal propositional proof system.
3
2n
The classes NEXP , QP (the class of quasi-polynomial functions) and EEE (2 (2 ) functions),as well as any other natural time-complexity class which is not covered by Theorem 1, are fat classes. As the proofs of all the known suciency conditions for the existence of optimal pps's (including Theorem 1 here) carry over to the corresponding oracle-relativized notions, this result shows that the reason that Theorem 1 does not extend to non-slim classes is deeper than just a technicality in the existing proof. Of independent interest is the following result, showing that the common knowledge rule of thumb, stating that a collapse of low-level complexity classes can be shown, by padding tricks, to imply a collapse of any higher complexity class, is not always true. O
Corollary 1: For every fat class of functions, F , there exists an oracle relative to which N F = DF and yet relative to the same oracle, for every slim class F 0, TallyCoN F 0 6 NF 0. (In particular, there is an oracle relative to which NEEE = DEEE and yet there are Tally languages in CoNEE which are not in NEE ).
2 De nitions and Notation
We say that a function f is super-polynomial if 8k 8 n f (n) n (where 8 n means `for all but nitely many n's'). We say that f is non-polynomial if 8k :(8 n f (n) n ), that is, f exceeds every polynomial in nitely often. De nition 5 Let F be a class of functions (from the set of natural numbers to itself). F is slim if 9f 2 F 8g 2 F 9c 8n g(n + 1) (f (n)) k
k
c
For a function H : N 7! N , F is H -fat if 8f 2 F 9g 2 F 8n g(n + 1) > H (f (n)) We say that F is fat if there exists some super-polynomial H such that F is H -fat. Note that the notions of slim and fat classes of functions are almost complementary; Every non-slim class is H -fat for some non-polynomial function H , and every class of functions used to bound some complexity measure in the de nition of `natural' complexity classes is either slim or fat.
Example 1: The classes E = ff : 9c8nf (n) 2 g, EE = ff : 9c8nf (n) 2 n g and P = ff : 9c8nf (n) n g are slim classes. The classes EXP = ff : 9c s:t: 8n f (n) n 2 c g, QP = ff : 9c s:t: 8n f (n) 2 c g, as well as EEE = ff : 9c s:t: 8n f (n) 2 g are all fat classes. Notation: For a class of functions F , DF denotes the complexity class of all languages accepted by deterministic algorithms whose running time is a function in F . c2
cn
c
n
c2
2
4
log (n)
N F denotes the complexity class of all languages accepted by non-deterministic algorithms whose running time is a function in F . CoN F denotes the complexity class of all languages accepted by co-non-deterministic algorithms whose running time is a function in F . A language L f0; 1g is Tally if it is a subset of f0g. TallyN F is the class of all tally languages in N F . We de ne the classes TallyCoN F and TallyDF similarly.
3 A General Sucient Condition The following theorem is a straightforward generalization of the Kobler and Mener [KM98] suciency theorem. Theorem 1 Let F be a slim class of functions then 1. Tally CoNF N F implies the existence of an optimal pps.
2. Tally CoNF DF implies the existence of a p- optimal pps.
Proof: We concentrate on the rst claim of the theorem. The proof of the second claim is just a variation on this. We shall comment below on the changes needed to obtain that claim. Let (S : i 2 N ) be a recursive enumeration of all Turing machines mapping binary strings to binary strings and running in time n2 on input of size n. Assume further that, on input (i; x; 0 ) the rst k steps of the run of S on input x can be eciently simulated. Say that a machine S is an n-sound pps if for every 2 f0; 1g , S ( ) is a propositional tautology. Given a function f : N 7! N , de ne the language Sound by i
n
i
n
f
Sound = f0 : S ( where i(n) is maxfi : 2 divides ng. n
i n)
f
is f (n)-soundg
i
It is easy to see that given a class of functions F , a function f 2 F and an enumeration (S : i 2 N ) as above, Sound is in Tally CoNF . Assuming Tally CoNF N F , there exists a polytime-computable relation R and a function g 2 F so that Sound = f0 : 9y g(n) R(0 ; y)g Let f be a function that demonstrates the slimness of F , that is, for every g 2 F there is some constant c such that for all n, g (n + 1) f (n) . Let us de ne a proof system S by having i
f
f
n
n
c
8 >
s
:
if jwj < s and, for some odd k; s = f (k2 ); y g (k2 ) and R(0 2i ; y ) holds. A _ :A otherwise
S (w)
i
i
i
k
Let us show that S is indeed an optimal pps; Clearly S is a sound (that is, for every , S ( ) is a tautology). Note also that, as g (n) f (n) for some constant c, S is eciently computable. Now, given any proof system S , for every string w, let k be the minimal odd k s.t. jwj f (k2 ), and let y g (k 2 ) be an R-witness to the f (k 2 )-soundness of S . We readily get S (0 ( w 2i); w; y) = S (w). All that is needed to complete the proof is to show that, for some polynomial p , for every w, k satis es g (k 2 ) + f (k 2 ) + jwj p(jwj). By the minimallity of k , f ((k ? 1)2 ) < jwj. Let c be such that for all n, f (n + 1) + g(n + 1) f (n) . It follows that c
i
i
w
f k
w
i
i
i
i
w
w
i
w
w
w
i
w
i
i
c
5
2i
for all n and i, g (n + 2 ) + f (n + 2 ) f (n) . By substituting n = (k ? 1)2 we conclude that, i g(k 2 ) + f (k 2 ) jwj . To prove part 2 of the theorem, just note that under the assumption of that claim Sound 2 DF so one can repeat the above construction without the `witness' strings y and get an e ective construction of the S proof from the given S proof, w. i
w
i
w
i
i
c
i
w
2
c
f
i
4 Oracle-Relativized Propositional Calculus
We wish to have a class of languages that are parametrized by oracles fTautA g, so that for every oracle A, the language TautA is CoN P A -complete. Furthermore, we wish to de ne a notion of a relativized proof system, such that all the previous results concerning proof systems lift up to these relativized notions. It turns out that this can be achieved by extending propositional calculus to allow an extra connective, a connective whose semantics depends upon the oracle relative to which our complexity notions are de ned. De nition 6 The set of X -well-formed formulas (X ? WFF s) are de ned inductively: a) Every propositional variable is an X ? WFF . b) If and are X ? WFF s, then so are (: ); ( ^ ); ( _ ); ( ! ). c) For n 2 N , if 1; 2; : : :; are X ? WFF s, then so is X ( 1; 2; : : :; ). De nition 7 Let A be an oracle (that is, A f0; 1g) , v - a truth assignment to propositional variables. The truth value v of X ? WFF s is de ned inductively: a) If A is a propositional variable then v(A) = v (A). b) If ; are X ? WFF s, 2 f^; _; !g then v( ) = TT(v ( ); v( )); v(: ) = TT:(v ( )). (Where, TT stands for the usual truth-table of the connective ). c) Let us interpret truth values as binary bits (so True becomes 1 and False becomes 0). For n 2 N , if 1; 2; : : :; are X ?WFF s, then v(X ( 1; 2; : : :; )) = 1 i ((v( 1)); (v( 2)); : : : (v( ))) 2 A. De nition 8 Let A be an oracle. SATA is the set of all A-satis able X ? WFF s. TautA is the set of all X ? WFF s which are tautologies with respect to A. n
n
n
n
n
De nition 9 Let A be an oracle. A propositional proof system w.r.t. A is a function f : f0; 1g ! TautA such that f 2 FP A (the class of functions computable by polynomial Turing machines with oracle A). Lemma 1 For every oracle A, SATA is NP A-complete for polynomial time reductions. Proof: Slightly modi ed Cook's proof of \SAT is NP -complete". Corollary 2: [A Relativised Cook-Reckhow Theorem] For every oracle A, there exists an Arelativized super propositional proof system i N P A = CoN P A.
onto
It is also straight-forward to check the the suciency results for the existence of optimal propositional proof systems (in particular, Theorem 1 above) and Razborov's necessary condition carry over to the relativized notions. 6
5 An Oracle-Relativized Insuciency Condition
De nition 10 A class F is reasonable if For every f 2 F and every k 2 N , the function f (n) is also in F . For every pair of functions, f; g; if f 2 F and fn : f (n) 6= g(n)g is nite, then g 2 F . For every f; g 2 F the function max(f; g), de ned by max(f; g )(n) = maxff (i); g (i) : i ng, is also in F . Theorem 2 Let F be a reasonable class of functions. If F is fat then there exists an oracle A relative to which CoN F = DF and yet there is no optimal propositional proof system. Proof: We shall construct the oracle A inductively. At each stage i we handle some task by adding a nite number of strings of length i to A. Let A denote the set of strings de ned to be in A in stages i, A will be de ned as [ A . The tasks are of two types; Making CoN F A = DF A and making sure that no optimal A-propositional proof system exists. There will be a function Pl k
i
i
i
that will map tasks to numbers, determining which task will be taken care of at which stage of the construction.
Tasks of the rst type. Let fM : i 2 N g enumerate all nondeterministic oracle machines, and let L denote the CoN F language de ned by M . So, for some g 2 F and some poly-time computable relation R , L = fx : 8y g (jxj) R (x; y )g. Let r 2 N be such that n i i
i
i
i
i
i
i
i
r
i
dominates the running time of R . For every pair (k; n) 2 N N , the task T( ) is to guarantee that, relative to constructed oracle, some DF machine for the language L computes the correct outputs for input strings of length n. We carry out this task by de ning, at stage Pl(k; n), the strings of length Pl(k; n) in A to be exactly f0x0 ( )?( +1) : jxj = n and x 2 LAi? g. i
k;n k
P l k;n
1
n
k
Claim 1 If for every k, the function Pl(k; ) is a member of F , and for all but nitely many of the pairs f(k; n) : n 2 Ng, the membership of strings of length n in LA is determined by the membership of strings of length < Pl(k; n) in A, then, carrying out T for all but nitely many n's guarantees that LA 2 DF A . Proof: Immediate from the construction. Tasks of the second type. Let (S : i 2 N ) be a recursive enumeration of all Turing machines mapping binary strings to binary strings and running in time n on input of size n (that is, k
(k;n)
k
i
2
an enumeration of all candidate propositional proof systems). Furthermore, we require that for every i, fj : S = S g is an in nite set computable in polynomial time. For each number t the task R is to guarantee that, relative to constructed oracle, S A is not an A-optimal propositional proof system. For each n 2 N , let denote :X (True; x1; : : :; x ). is an X ? WFF and, for every oracle A, it is an A-tautology i A contains no strings of length n + 1 that have 1 as their rst bit. We shall de ne below a super-polynomial function h, satisfying 8n(h(n))2 < 2 , and a function l. At stage Pl(t) = i we check whether, relative to A , there exists some string w of length h(l(t)) such that S Ai? (w) = ( ). That is, whether relative to the oracle at this stage, S has a `short' proof for ( ). If no such w exist we do nothing, that is, de ne i
j
t
t
n
n
n
n
i
1
l t
t
t
l t
7
A = A ? . However, if such a w exists, we add to A some string of the form 1y of length l(t) that was not queried by S Ai? on its run on input w. As S runs in time n , for w h(l(t)) its running time is (h(l(t)) which, by the choice of h, is less than 2 , so necessarily such i
i
1
1
t
t
2
a string y exists.
2
l(t)
Claim 2 Let h be a super-polynomial function s.t. h(n) < H (n) for all n, and let A be constructed according to the above recipe. If for every t no strings of length (h(l(t)) are added to A at stages later than stage Pl(t), then there are no optimal A-propositional proof 2
2
system.
Proof: As mentioned above, the output of each of the machines S on inputA strings of length h(l(t)) is determined by A . It follows that for w of length h(t), S P l t (w) = t
( )
(h(l(t))2
S A(w). Thus, for every propositional proof system S , 1. If there exists some t s.t. S = S and there exists a short (that is, of length h(l(t))) proof of ( ) in S A , then, by the construction, there exists a word y of length l(t) s.t. 1y 2 A. (Recall that ( ) is a tautology i there is no word 1y in A, where jy j = l(s).) Thus, ( ) is not a tautology. Therefore, S A is not sound. 2. Otherwise, for every k s.t. S = S there exists no S A -proof of ( ) of size h(l(k)). It follows that in S A there are no polynomial proofs of the set of A-tautologies t
t
t
l t
t
l s
l t
k
l k
R = f ( ) : S = S g S
l k
k
R 2 P , so there exists the propositional proof system U which has polynomial proofs for every ( ) 2 R . Thus S A does not simulate U . Therefore, S A is not optimal. S
l k
S
Lemma 2 If
1. Pl is a one-to-one function mapping (N N ) [ N to N , and it is increasing (that is, on the sub-domain N N it is an increasing function whenever one coordinate is xed, and it is also increasing on the sub-domain N ). 2. For all i; n, Pl(i; n) > g (n) i . r
i
3. h is super-polynomial and h(n)2 H (n) for all n. 4. For all s, l(s + 1) > (h(l(s)))2. 5. For every i, fn : 9s [g (n) i ; Pl(i; n)] l(s); h(l(s))2 g is nite. i
r
Then the oracle de ned according to the above recipe witnesses the main theorem.
By Claims 1, 2 above, it suces to show that for each of our tasks, once it is taken care of, no later addition to the oracle will interfere with it. We establish that by the following four simple claims. Claim 3 If Pl(i; n) < Pl(r; s) then the machine M , while computing membership in L of strings of size n, cannot read the oracle values of strings of length Pl(r; s). i
8
i
Proof:
On input of size n, M computes a poly-time relation R on strings of length g (n). The computation time of R on input of size m is bounded by m i . So the requirement that Pl(i; n) > g (n) i takes care of the claim. i
i
i
i
i
r
r
Claim 4 If Pl(t) < Pl(s) then the machine S , while running on input of size h(l(t)) (which t
bounds the length of the proofs of the tautology ( ) that we killed at stage Pl(t)) cannot read the oracle values for strings of length l(s) (which is where the task handled at stage Pl(s) may write). Proof: As Pl is a monotone increasing function, t s ? 1. Recalling that, for all n, l(n + 1) > h(l(n))2 and that (h(l(t))2 is an upper bound on the range that S can access on h(l(t))-size proofs establishes the claim. l t
t
Claim 5 If Pl(t) < Pl(i; n) then the machine S , while running on input of size h(l(t)), cannot read the oracle values for strings of length Pl(i; n). Proof: On proofs of length h(l(t)) the proof system S cannot read beyond lengthPl(t). t
t
Claim 6 Under the assumption of Lemma 2, for every i, for all suciently large n and for all t, if
Pl(i; n) < Pl(t) then machine M , while running on input of size n, cannot read the oracle values of strings of length l(t) (which is where the task handles at stage Pl(t) may write). Proof: Let n be large enough to make 8s [g (n) i ; Pl(k; n)] 6 [l(s); h(l(s))2]. For such n's, the assumption Pl(i; n) < Pl(t) implies that g (n) i < l(t). and g (n) i is an upper bound on the length of strings that M can access on input of size n. Now the proof of our theorem boils down to the following purely combinatorial problem of nding functions Pl, h and l so that the assumptions of Lemma 2 are met. The existence of these functions follow easily from Lemma 3 and Lemma 5 of the next section. i
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6 The Combinatorics
Lemma 3 Let F be an H -fat and reasonable class of functions. Let fg : i 2 Ng F , and fr : i 2 Ng a sequence of natural numbers. Then there exists a sequence of functions ff : k 2 N g F i
i
k
s.t .: 1. Every function in the sequence is monotone increasing. 2. For every n 2 N , the sequence (f (n) : k 2 N ) is increasing. 3. The ranges of the functions f are mutually disjoint and are all contained in the set of odd numbers. 4. For every k, for all n, f (n) g (n) k . 5. f +1 (n + 1) > H (f (n)), for every k; n 2 N . k
k
k
k
k
r
k
Proof: De ne the sequence ff : k 2 Ng by induction on k. At stage k +1, apply the H -fatness of F to nd a function f 0 so that f 0 (n +1) > H (f (n)) (for all n), and then apply the `reasonability' k
property to nd an increasing f
above f , f 0 and g k
k +1
k
k +1
(n) k . r +1
We shall also need the following lemma, that may be of independent interest. 9
Lemma 4 For every suer-polynomial functions H , there exists a super-polynomial function h such that, for every k 2 N , for all suciently large n H (n) > h (n). (where h is the k'th iterate of (k )
(k )
h, that is, h(1)(n) = h(n) and h( +1) (n) = h(h( ) (n))). Proof: For every n, let r def = maxfk : 8m n; m( k ) H (m)g Note that, for all n, r +1 r , and, as H eventually exceeds every polynomial, r goes to 1 as n i
i
k
n
n
n
n
does. De ne a function g by (strong) induction on n, as follows: g(n) = minfg(blog(n)c) + 1; r g It is easy to see that, 1. lim !1 g (n) = 1. (g is the minimum of two functions, both growing to in nity). 2. For every polynomial p, 8 n g (n ( ( ))) g (n) + 1. (This follows from the requirement g(n) g(blog(n)c) + 1). We now de ne h by h(n) = n ( ) As g increases to in nity, h is super-polynomial. n
n
p g n
g n
Let us now show that,
8k 8n 2k r =) h (n) H (n) (k )
n
As r grows unboundedly with n, this will establish the lemma. Claim 7 For all suciently large n, h( )(n) n( ( ) k ) Proof: [of the claim] By induction on k, h( ) (n) = h(h( ?1) (n)) By the induction hypothesis, for all suciently large n, n
k
g n
k
2
k
h(h( ?1) (n)) h(n( (
g n)
k
?1) )
) = (n( ( ) k? )) ( By property 2 of g , for all suciently large n, this is
h(n( (
g n)
(n
2(k
?1) )
2(k
?
g n
(g (n)2(k 1) )
Which, for suciently large n's is n( (
g n)
)(
g n)+1
2k )
2(
1)
= n( (
g n)
g n
2(k
)
(g (n)2(k
?1) ) )
?1) )(g(n)+1)
.
Having proved the claim, the proof of the lemma is now immediate; As g (n) r , if 2k r we get, for suciently large n's, r h( ) (n) n nn Which, by the de nition of r is H (n). Finally, the following De nition and Lemma will supply all the remaining ingredients in the construction of our oracle. n
k
r
n
10
n
De nition 11 A sequence of large intervals is a sequence of pairs of numbers f(a ; b ) : i 2 Ng i
such that, for some super-polynomial function F , for all i F (a ) < b < a +1 . i
i
i
i
Lemma 5 Let Q = fq ; i; j 2 Ng be an in nite matrix of numbers. If 1. For every i, the sequence (q : j 2 N ) is increasing. 2. For every j , the sequence (q : i 2 N ) is increasing. i;j
i;j
i;j
3. There exists a super-polynomial function H such that q +1
> H (q ), for all i; j . Then there exists a sequence of large intervals, f(a ; b ) : i 2 Ng such that for every k, ;j +1
i
i
fn : 9i [q ; q k;n
k +1;n
i;j
i
] [a ; b ]g is nite: i
i
Proof: Applying Lemma 4, let F be a super-polynomial function such that for every k 2 N , for
all suciently large n, H (n) > F ( ) (n). We shall actually prove a slightly stronger result; For every k there exists some n s.t. for all m n the interval [q (m); q +1(m + 1)] contains an interval of the form [n; F (n)] that contains no interval of the form [q (r); q +1(r)] for any i < k. Fix k and m. Let I = [q (r ); q +1 (r +1)] be a containment- minimal interval in the ( nite) set f(i; r) : [q (r); q +1(r +1)] [q (m); q +1(m +1)]g. Note that for every i at most one interval of the form [q (n); q +1(n)] is contained in I . Otherwise, if both [q (n); q +1(n)] and [q (l); q +1(l + 1)] are contained in I (for some l > n) then, as q +1 (n +1) q +1 (l +1), we'll have [q (n); q +1(n +1)] I , contradicting the of I . Consider the sequence of intervals k
k
k
k
k
i
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i
i
i
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k
k
i
i
i
i
i
i
i
S = [q (r); F (q (r))]; : : :: : : [F ( ) (q (r )); F ( +1)(q (r))] i
k
i
k
i
i
i
i
There are k many disjoint (except for their endpoints) intervals in S . Recall that
q +1 (r + 1) > H (q (r)) > F ( +1) (q (r )) k
i
i
i
(assuming r is suciently large). It follows that all of these k intervals are contained in I . As there are only k ? 1 many intervals in the set f[q (n); q +1(n)] I : i < kg, one of the intervals in S contains no interval of the form [q (r); q +1(r)] for any i < k. Completing the proof of Theorem 2 { The de nitions of the functions Pl, h and l: Apply Lemma 3 (for the given class F and function H ) with the g 's being the functions bounding the witness-length in the CoN F languages L and the r 's being the exponents in the time bound for the computation of the relations R in the de nitions of these languages (as detailed in the second paragraph of the Proof of Thm. 2). Let ff : k 2 Ng be the sequence of functions whose existence is guaranteed by the lemma. Apply Lemma 5 to the matrix de ned by q = f (n), and let F be the `largeness' function of the sequence of large intervals whose existence is concluded. i
i
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k
k;n
k
We shall de ne Pl separately for the domain N N (for tasks of the rst type, indexed by a pair (k; n)) and for the domain N (for tasks of the second type, indexed by t). { For the sub-domain N N , de ne Pl : N N 7! N , de ne Pl(k:n) = f (n). { Let us de ne the missing part of Pl by Pl(s) = F (l(s)), k +1
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Let l be any increasing function whose range is contained in fa : i 2 Ng where the a 's are i
the left-most points in the intervals (a ; b ) whose existence is concluded by Lemma 5. p p Finally, de ne h by h(n) = minf F (n); H (n)g. i
i
i
It is straightforward to verify that these functions, Pl, h and l satisfy the requirements of Lemma 2. This completes the proof of Theorem 2.
Acknowledgements We wish to thanks Jochen Mener for suggesting the use of techniques from [KM98] to obtain an improvement in our Theorem 1.
References [CR79]
Cook, S., A., and Reckhow, R., A., \The relative eciency of Propositional Proof Systems" Journal of Symbolic Logic 44: 36-50, 1979. [KM98] Kobler J., and Mener J., \Complete Problems for Promise Classes by Optimal Proof Systems for Test Sets" To appear in Proceedings of the Computational Complexity Conference, 1998. [KP89] Krajcek, J., and Pudlak, P., \Propositional Proof Systems, the Consistency of First Order Theories and the Complexity of Computation" Journal of Symbolic Logic, 54(3): 1063-1079, 1989. [MT97] Mener, J., and Toran, J., \Optimal Proof Systems for Propositional Logic and Complete Sets" ECCC report series Technical Report, June 1997. [Pu84] Pudlak, P., \On the Length of Proofs of Finitistic Consistency Statements in First Order Logic" Logic Colloquium '84, pp 165-195. [Ra94] Razborov, A., A., \On Provably Disjoint NP-pairs. Technical Report, BRICS november, 1994.
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