No Feasible Monotone Interpolation for Cut-free. Gentzen Type Propositional Calculus with. Permutation Inference. Noriko H. Arai 3

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1 No Feasible Monotone Interpolation for Cut-free Gentzen Type Propositional Calculus with Permutation Inference Norio H. Arai 3 Department of Computer Science, Hiroshima City University 151 Ozua, Asaminami-u, Hiroshima Japan Abstract The feasible monotone interpolation method has been one of the main tools to prove the exponential lowerbounds for relatively wea propositional systems. In [2], we introduced a simple combinatorial reasoning system, GCNF+permutation, as a candidate for an automatizable, though powerful, propositional calculus. We show that the monotone interpolation method is not applicable to prove the superpolynomial lower bounds for GCNF+permutation. At the same time, we show that Cutting Planes, Hilbert's Nullstellensatz and the polynomial calculus do not p-simulate GCNF+permutation. eywords: automated theorem proving, proof complexity, Craig's interpolation theorem, proof theory 1 Introduction In Arai (1996) [2], we introduced a new system for propositional calculus, which gives a natural framewor for combinatorial reasoning using \without loss of generality" argument and brute force induction. This system, called GCNF+permutation, is a fusion of the cut-free sequent calculus and the substitution Frege, and it inherits two virtues from its origin. It is a well-nown result by Gentzen that cut-free sequent calculus proofs enjoy the subformula property, which Frege and LK with cuts fail to have. GCNF+permutation inherits this nice property from cut-free sequent calculus, hence it is liely for machines to nd the shortest proof by simply breaing down a given formula in the \bottom-up" manner. 3 arai@cs.hiroshima-cu.ac.jp 1

2 The substitution Frege system is nown to be a quite powerful system: no tautology is suggested as a candidate to show this system is not polynomially bounded. GCNF+permutation inherits the eciency from the substitution Frege. Amazingly, GCNF+permutation is strong enough to prove pigeonhole principle, mod principles, Bondy's theorem and many other combinatorial theorems in polynomial time [3]; these are notorious tautologies which have been used to show the exponential lower bounds for analytic tableaux, resolution, Hilbert's Nullstellensatz, and polynomial calculus. There is no sequence of tautologies which requires superpolynomial size proofs in GCNF+permutation found so far. It is a well-nown result in classical logic that when A(~p; ~q) B(~q; ~r) is a tautologies with the occurrences of variables fully indicated, there exists a formula C called interpolant such that the variables in C are from q's, and both A C and C B are tautologies. The question whether or not the interpolant is obtainable in polynomial time algorithm is answered by Mundici somewhat negatively [10]: interpolation functions are not always computable in polynomial-time unless P = NP \ co 0 N P. Nevertheless, it is possible to nd such a procedure or to bound the (circuit) size of the interpolants polynomially for particular propositional systems. In some cases, one can pic monotone circuits as interpolants: the resolution and the cutting plane system are among of those which enjoy such property. This fact is used to show that these propositional systems do not have polynomial-size proofs for a sequence of tautologies, called -T est(n), which expressing the positive and the negative test for -clique problem [6], [11], [12]. In this paper, we use -T est(n) to show the opposite result. GCNF+permutation is powerful enough to polynomially prove -T est(n), hence it does not enjoy feasible monotone interpolation. At the same time, -T est(n) witnesses the fact that the cutting plane system does not polynomially simulate GCNF+permutation. The paper is organized as follows: Section 2 contains denitions of the system of GCNF+permutation. Section 3 reviews the results we acheived so far. Section 3 contains polynomial-size proofs of GCNF+permutation for -T est(n). Section 4 contains some conjectures and open problems. 2 The system of GCNF+pewrmutation The system GCNF is a subsystem of cut-free Gentzen type propositional calculus, called cut-free LK. GCNF is designed exclusively for the conjunctive normal form formulas. One can understand that GCNF is a generalization of analytic tableaux written as directed acyclic graphs. A literal is a propositional variable p or a conjugate p. A clause is a nite set of literals, where the meaning of the clause is the disjunction of the literals in the clause. A nite set of clauses is called a cedent. For simplicity, we express 2

3 as if cedents as a sequence of clauses. In the rest of our argument, literals are denoted by l's, clauses by C's and cedents by capital Gree letters. GCNF refutation is a sequence of cedents in which every sequent is an initial sequent of the form, fpg; fpg or derived from previous cedents by one of following inference rules. structural inference 0 0 [ 1 logical inference 0; C 1 ; : : : ; C 5; flg 0 [ 5; C 1 [ flg; : : : ; C [ flg (l) l is an arbitrary literal, which is called the auxiliary literal of this inference. The clauses C 1 [ flg; : : : ; C [ flg are called the principal formulas. Now we introduce a new inference rule, permutation, to GCNF. permutation 0(p 1 ; : : : ; p m ) 0((p 1 )=p 1 ; : : : ; (p m )=p m ) is a permutation on fp 1 ; : : : ; p m g and 0((p 1 )=p 1 ; : : : ; (p m )=p m ) is the result of replacing every occurrence of p i (1 i m) in 0(p 1 ; : : : ; p m ) by (p i ). When P is a sequence of symbols (such as formulas and proofs). The size of P is the number of all the symbols used in P, that is denoted by size(p). Next we dene a scale to measure the eciency of a proof system. We say that a propositional system S 1 polynomially simulates (p-simulates) another propositional system S 2 if there is a polynomial-time algorithm which, given an S 2 -proof of a formula A, produces an S 1 -proof of A. Proposition 1 GCNF is a sound and complete propositional system. When GCNF is written in tree form, it polynomially equivalent to the tree resolution system. Proposition 2 1. Let P be a tree GCNF refutation of C 1 ; : : : ; C n. Then, there exists a tree resolution refutation R of C 1 ; : : : ; C n with size(r) size(p ): 2. Let R be a tree resolution refutation of C 1 ; : : : ; C n. Then, there exists a GCNF refutation of C 1 ; : : : ; C n with size(p ) (size(r)) 2 : When a clause C consists of a single literal l, we express C by l instead of flg for the sae of simplicity. 3

4 3 GCNF+permutation is powerful In the previous section, we dened the systems of GCNF and GCNF+permutation, and show that GCNF is polynomially equivalent to the system of resolution when they are written in tree form. In this section, we review the results we achieved so far on the system of GCNF+permutation written in DAG, and show that how the permutation inference rule maes GCNF powerful. Denition 1 (Pigeonhole principle) The pigeonhole principle states that for each n, if f : f0; : : : ; ng! f0; : : : ; n01g then f is not one-to-one. For each i and j with 0 i n and 0 j n 0 1 we will have the variable p i;j which `means' f(i) = j. W P HP n : 0in 0jn01 p i;j ; 0i<mn 0jn01 fp i;j ; p m;j g V 0in C i is an ab- 0in p i is an abbreviation for the clause fp 0 ; : : : ; p n g. breviation for the cedent C 0 ; : : : ; C n. The number of all the literals contained in P HP n is n 3 + 2n 2 + n. Denition 2 (-equipartition) The -equipartition states that if an integer n is not evenly divisible by, then there is no partition of f1; : : : ; ng into disjoint sets of size. Let Jn = f(j 1 ; : : : ; j ) : 1 j 1 < : : : j ng For ~j 2 Jn, we write i 2 ~j to mean that there exists 1 l such that i = j l. Suppose that n 6 0(mod ). We introduce new variables x i;(j1;:::;j ) for 1 i; j 1 ; : : : ; j n to mean that (j 1 ; : : : ; j ) is a partition of f1; : : : ; ng and i 2 fj 1 ; : : : ; j g. -Eq(n) is dened as the following cedent; 0Eq(n) : 1in ~ j 2 J n i 2 ~ j x i; ~j ; ~j2j n ; i 1;i 2 2~j; i 1 6=i 2 fx i1 ;~j ; x i 2 ;~j g; ~ j 1 ; ~j 2 2 J n i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6= ~j 2 The number of all literals contained in -Eq(n) is n n n + n n n n 0 1 V The rst of clauses expresses V that \each i is contained in some partition whose size is." The second of clauses expresses that \if (i 1 ; : : : ; i ) is a partition V containing i 1, then it is also a partition containing i 2,... and i." The last of clauses means that \if i s = j t for some 1 s and 1 t and if (i 1 ; : : : ; i ) 6= (j 1 ; : : : ; j ), then either (i 1 ; : : : ; i ) or (j 1 ; : : : ; j ) is not a partition." We show that GCNF+permutation has polynomial size refutations for P HP n and -Eq(n). fx i; ~j 1 ; x i; ~j 2 g 4

5 Theorem 1 There exists a GCNF+permutation refutation of P HP n whose length is O(n) and the size is O(n 4 ). (Proof.) We prove P HP n bacwards and reduce it to P HP n01. Then, we show that the length of the proof of P HP n is bounded by O(n). First, we decompose the clause fp n;0 ; : : : ; p n;n01 g by applying logical inferences bacwards. As a result, we obtain the cedents 0 j (0 j n 0 1) p n;j ; 0in01 0jn01 p i;j ; 0i<mn 0jn01 fp i;j ; p m;j g: Each 0 j for 0 j n 0 2 is obtainable by exchaning p i;j by p i;n01 for each 0 i n. Hence, we only need to consider 0 n01. By applying a structural inference and a logical inference bacwards, of which auxiliary literal is p n;n01 to 0 n01, we obtain an initial cedent, and a cedent of the form p 0;n01 ; : : : ; p n01;n01 ; p n;n01 ; p n;n01 ; 0in01 0jn01 p i;j ; 0i<mn01 0jn02 fp i;j ; p m;j g: By applying ligical inferences bacwards, of which auxiliary literals are p 0;n01 ; : : : ; p n01;n01, we obtain a cedent of the form 0in01 0jn02 p i;j ; 0i<mn01 1jn02 fp i;j ; p m;j g; which is P HP n01. By examing closely, we can conclude the length of the proof obtained above is bounded by a linear function of n. Since the size of the each line is bounded by O(n 3 ), the size of the proof is bounded by O(n 4 ). 2 Theorem 2 There exists a polynomial function p, independent from n, and a GCNF+permutation refutation of -Eq(n) whose size is bounded by p(n). (Proof.) We prove -Eq(n) bacwards and reduce it to -Eq(n 0 ). Then, we show that the length of the proof of W -Eq(n) is bounded by O(n ). First, we decompose the clause ~j2j ;n2 ~j x n;~j by applying logical inferences bacwards. Then, we obtain cedents, 0 ~ j n+, of the form x n; ~j ; 1in01 ~j2j n ;i2 ~j x i; ~j ; ~j 2 J n i 1 ; i 2 2 ~ j; i 1 6= i 2 fx i1;~j ; x i 2;~j g; ~j 1 ; ~j 2 2 J n i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6= ~j 2 fx i; ~j 1 ; x i; ~j 2 g; 5

6 for ~ j 2 Jn and n 2 ~ n 0 1 j. Note that the number of such cedents are 0 1 All of them are obtainable by applying permutation to 0 j ~ 0, x n; ~j 0 ; 1in01 ~j2j n ;i2 ~j x i; ~j ; ~j2j n ; i1;i22 ~j; i 16=i 2 fx i1;~j ; x i 2;~j g; ~ j 1 ; ~j 2 2 J n i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6= ~j 2 where ~ j 0 is (n 0 + 1; : : : ; n). Hence, we only need to consider 0 ~ j 0. Now we apply a logical inference bacwards of which auxiliary literal is x n; ~j 0. Then we obtain an initial cedent, x n; ~j 0 ; x n; ~j 0 ;. fx i; ~j 1 ; x i; ~j 2 g and a cedent of the form x i; ~ j0 ; x i; ~ j ; fx i1 ; ~ j ; x i2 ;~ j g; x n; ~ j ; fx i; ~ j1 ; x i; ~ j2 g: n0+1in01 1in01 ~ j2j n ;i2 ~ j ~ j 2 J n 0 f ~j 0 g i 1 ; i 2 2 ~ j; i 1 6= i 2 ~ j2j n 0f ~ j0g ~ j 1 ; ~j 2 2 J n i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6=~j 2 By applying a structural inference and logical inferences bacwards of which auxiliary literals are x n0+1; ~j 0 ; : : : ; x n01; ~j 0, we obtain a cedent of the form 1 i n 0 ~ j 2 J n 0 J n0 x i; ~j ; 1in0 ~j2j n ;i2 ~j x i; ~j ; ~ j 2 J n0 i 1 ; i 2 2 ~ j; i 1 6= i 2 fx i1;~j ; x i 2;~j g; ~ j 1 ; ~j 2 2 J n0 i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6= ~j 2 fx i; ~j 1 ; x i; ~j 2 g: By applying logical inferences bacwards of which auxiliary literals are x i; ~j for every ~ j 2 Jn 0 Jn0, we obtain 1in0 ~j2j ;i2 ~j n0 x i; ~j ; ~ j 2 J n0 i 1 ; i 2 2 ~ j; i 1 6= i 2 fx i1;~j ; x i 2;~j g; ~ j 1 ; ~j 2 2 J n0 i 2 ~j 1 ; i 2 ~j 2 ; ~j 1 6= ~j 2 fx i; ~j 1 ; x i; ~j 2 g; which is -Eq(n 0 ). By examing closely, we can conclude the length of the proof obtained above is bounded by O(n ). The size of every line in the proof is bounded by O(n 2 ). Hence the size of the whole proof is bounded by a polynomial of n. 2 Theorem 1 and theorem 2 witness the fact that none of resolution, bounded depth Frege, polynomial calculus, Hilbert's Nullstellensatz do not p-simulate the system of GCNF+permutation. Proposition 3 [9] There exists a constant c, c > 1 such that, for suciently large n, every resolution refutation of P HP n contains at least c n dierent cedents. 6

7 Proposition 4 [1] There exists a constant c, c > 1 so that, for suciently large n, every constant-depth Frege proof of :(-Eq(n)) contains at least c n dierent cedents. Note that any refutation of -Eq(n) in resolution with limited extension can be converted to a constant-depth Frege proof of :(-Eq(n)) within a linear factor. Hence, it also gives an exponential lower bound for the system of resolution with limited extension. Liewise, it gives an exponential lower bound for the system of cut-free LK or cut-free LK with analytic cut (cuts of which cut-formulae must be subformulas of the end-sequent) written in DAG form. Proposition 5 [14] Polynomial calculus refutations of P HP n must have degree at least dn=2e + 1 over any eld. Proposition 6 [5] There is no Hilbert's Nullstellensatz refutation of -Eq(n) of constant degrees. Corollary 1 The following systems of propositional calculus do not p-simulates GCNF+permutation. Resolution, Resolution with limited extension, Bounded depth Frege, Cut-free LK, Cut-free LK with analytic cut, Polynomial calculus of constant depth, and Hilbert's Nullstellensatz of constant depth. 4 No feasible interpolation for GCNF+permutation We say that a propositional system S admits feasible interpolation when there exists a polynomial function f satisfying the following property. When a formula A(~p; ~q) B(~q; ~r), with the variables fully indicated, has an S-proof P, there exists a formula C(~q), with variables fully indicated, such that 1. both A C and C B are valid, and 2. the DAG (circuit) size of C is bounded by f(size(p )). 7

8 Some proof systems, such as resolution and Cutting Planes, admit even a stronger version of feasible interpolation. When ~q occurs only positively either in A or in B, we can pic a monotone circuit C as an interpolant. This property is called feasible monotone interpolation. We show that GCNF+permutation does not admit feasible monotone interpolation by using Razborov's theorem on the lowerbounds for monotone circuits size. Dene a cedent -Clique(n) by the set of the following clauses. 1. fq i;1 ; : : : ; q i;n g for 1 i, 2. fq i;m ; q j;m g for 1 m n and 1 i < j, and 3. fq i;m ; q j;l ; p m;l g for 1 m < l n and 1 i; j. The above clauses encode a graph which has n vertices and contains -clique as follows. We enumerate all the vertices of the graph f1; : : : ; ng. The q's encode a function f from f1; : : : ; g to f1; : : : ; ng. The literal q i;l means that f(i) = l. (The intuitive meaning of f(i) = l is that the vertex named i in the graph is actually the vertex named l in the -clique.) The p m;l encode that there exists an edge between m and l. Hence, the rst clause means that the function f is dened for all i (i = 1; : : : ; ). The second clause means that f is one-to-one. The third clause means that if there exists i; j such that f(i) = m and f(j) = l, then there exists an edge between m and l. Note that -Clique(n) corresponds to the positive test graph in [13] and [7]. Now we dene a cedent 0 -Color(n) by the set of the following clauses. 1. fr m;1 ; : : : ; r m; 0g for 1 m n, 2. fr m;i ; r m;j g for 1 m n and 1 i < j 0, and 3. fr m;i ; r l;i ; p m;l g for 1 m < l n and 1 i 0. The above clauses encode a graph which is a 0 -partite graph as follows. The r's encode a coloring function g from f1; : : : ; ng to f1; : : : ; 0 g. The literal r m;i means that the vertex named m is colored by i. Hence, the rst clause means that every vertex is colored. The second clause means that none of the vertices has more than one color. The third clause means that the coloring is proper: when the vertices m and l has the same color, then there is no edge between m and l. Note that 0 -Color(n) corresponds to the negative test graph. We dene -T est(n) by the cedent consists of all the clauses in -Clique(n) and (01)-Color(n) together. The size of -T est(n) is O(n 4 ). It is obvious that -T est(n) is unsatisable as follows. When all the clauses in -Clique(n) is true, that means the graph contains a -clique. Any -clique does not have a proper (01) coloring. Than means at least one of the clauses in (01)-Color(n) must be false. 8

9 Theorem 3 -T est(n) has a proof of length O(n 5 ) and size O(n 9 ) in GCNF + permutation. (Proof.) We prove -T est(n) bacwards and reduce it to propositional pigeonhole principle. Then, we show that the length of the proof of -T est(n) is bounded by O(n 5 ). The cedent -T est(n) consists of the clauses listed below. 1. fq i;1 ; : : : ; q i;n g for 1 i, 2. fq i;m ; q j;m g for 1 m n and 1 i < j, and 3. fq i;m ; q j;l ; p m;l g for 1 m < l n and 1 i; j. 4. fr m;1 ; : : : ; r m;01 g for 1 m n, 5. fr m;i ; r m;j g for 1 m n and 1 i < j 0 1, and 6. fr m;i ; r l;i ; p m;l g for 1 m < l n and 1 i 0 1. First, we decompose the clause fq 1;1 ; : : : ; q 1;n g in -T est(n) by applying logical inferences bacwards. Then, we obtain n-many cedents of the form q 1;m ; 0 1 (1 m n), where 0 1 denote the cedent obtained from -T est(n) by deleting the clause fq 1;1 ; : : : ; q 1;n g. Note that q 1;m ; 0 1 is obtainable from q 1;1 ; 0 1 by exchanging q 1;m by q 1;1. Hence, we only need to consider q 1;1 ; 0 1. Secondly, we decompose the clause fq 2;1 ; : : : ; q 2;n g in 0 1 by applying logical inferences bacwards. Then, we obtain n-many cedents of the form q 2;m ; q 1;1 ; 0 2 (1 m n), where 0 2 denote the cedent obtained from 0 1 by deleting the clause fq 2;1 ; : : : ; q 2;n g. For m = 1, the cedent q 2;m ; q 1;1 ; 0 2 is reducible to the cedent q 2;1 ; q 1;1 ; fq 2;1 ; q 1;1 g, which has a simple proof. For m > 1, q 2;m ; q 1;1 ; 0 2 is obtainable from q 2;2 ; q 1;1 ; 0 2 by applying a permutation. Hence, we only need to consider q 2;2 ; q 1;1 ; 0 2. We eep go on until we obtain the cedent q ; ; : : : ; q 1;1 ; 0 where 0 consists of the following clauses. 1. fq i;1 ; : : : ; q i;n g for + 1 i, 2. fq i;m ; q j;m g for 1 m n and 1 i < j, and 3. fq i;m ; q j;l ; p m;l g for 1 m < l n and 1 i; j. 4. fr m;1 ; : : : ; r m;01 g for 1 m n, 5. fr m;i ; r m;j g for 1 m n and 1 i < j 0 1, and 6. fr m;i ; r l;i ; p m;l g for 1 m < l n and 1 i

10 The cedent q ; ; : : : ; q 1;1 ; 0 intuitively means that the vertices f1; : : : ; g forms a clique, and it has a proper (01)-coloring. The length of the proof up to here is bounded by O(n 2 ). By applying logical inferences of which auxiliary literals are q i;i (1 i ) and a structural inference bacwards, we obtain the cedent 1 which consists of the following clauses. 1. fq i;1 ; : : : ; q i;n g for + 1 i, 2. fp m;l g for 1 m < l, 3. fr m;1 ; : : : ; r m;01 g for 1 m, 4. fr m;i ; r m;j g for 1 m and 1 i < j 0 1, and 5. fr m;i ; r l;i ; p m;l g for 1 m < l and 1 i 0 1. By applying logical inferences of which auxiliary literals are p m;l (1 m < l ), we obtain a propositional pigeonhole principle P HP. In the previous section, we already showed that P HP has a GCNF+permutation proof of length O() and size O( 4 ). The length of the proof combined together is bounded by O(n 2 ). The size of every line is O(n 4 ). Consequently the size of the proof given above is bounded by O(n 6 ). 2 Razborov showed that any small size monotone circuit either almost always outputs 0 for the positive test graph, or almost always outputs 1 for the negative test graph. That means there is no small-size monotone circuit C(~p m;l ) such that C(~p m;l ) is false! -Clique(n) is false, and C(~p m;l ) is true! (01)-Color(n) is false. Consequently, we have the following corollary. Corollary 2 GCNF+permutation does not admit feasible monotone interpolation. The cutting plane system CP is an extension of resolution, which has polynomialsize proofs for the pigeonhole principle, the s0t connectivity, and the non-unique endnode principle [8]. No exponential lower bound was nown for CP until recently: Pudla proved an exponential lower bound by showing that CP admits feasible monotone interpolation property [11], [12]. It requires exponential size proofs in CP for -T est(n) (under an adequate translation). This technique has been applied for other propositional calculi, including polynomial calculus, Hilbert's Nullstellensatz, and generalized cutting plane system to show that they do not have polynomial-size proofs for -T est(n). Corollary 3 Any propositional calculus which admits feasible monotone interpolation does not p-simulate GCNF+permutation. More specically, CP and generalized CP do not p-simulate GCNF+permutation. 10

11 5 Open problems Corollary 1 and corollary 2 suggest that GCNF+permutation may be more ecient than resolution, cut-free LK and many algebraic systems, which are popular systems for automated theorem proving: there is no doubt that we can obtain a much faster theorem prover by implementing GCNF+permutation. We implemented GCNF in tree form [4]. Amazingly, even tree form GCNF (without permutation) already has the ability to produce a proof of P HP n of size O(n!) automatically [4], and it is as short as the best nown DAG resolution refutation of P HP n. Our next goal is to design an algorithm (deterministic or probablistic) to nd suitable permutations without the exhaustive search. There are many open problems on the relative eciency of this system. Does GCNF+permutation p-simulates resolution? Does GCNF+permutation p-simulates CP? Note that is a well-nown open problem whether or not GCNF (or cut-free LK in DAG) p-simulates resolution. We conjecture that GCNF+permutation does not p-simulate Frege system. In [3], we showed that Frege system p-simulates GCNF+renaming if and only if Frege system p-simulates extended Frege system. However, it may be possible that Frege p-simulates GCNF+permutation, since permutation rule is a very restricted form of renaming rule. We conjecture that Frege+permutation does not p-simulate extended Frege. It is also an open problem whether or not GCNF+permutation admits feasible nonmonotone interpolation. References [1] M.Ajtai, \The complexity of the pigeonhole principle", 29th Annual Symposium on the Foundations of Computer Science (1988), [2] N.H. Arai, \Tractability of cut-free Gentzen type propositional calculus with permutation inference", Theoretical Computer Science, Vol.170 (1996), pp [3] N.H. Arai, \Tractability of cut-free Gentzen type propositional calculus with permutation inference II", submitted for publication. [4] N.H. Arai and R. Masuawa, \An implementation of cut-free Gentzen type propositional calculus", manuscript. 11

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