On the Deduction Rule and the Number of Proof Lines (Extended Abstract)
Maria Luisa Bonet∗ Samuel R. Buss∗
Department of Mathematics Department of Mathematics U.C. Berkeley U.C. San Diego Berkeley, California 94720 La Jolla, California 92093
Abstract inference are also sound. A notable example of this is We introduce new proof systems for propositional the deduction rule which states that if a formula B has logic, simple deduction Frege systems, general deduc- a proof from an additional, extra-logical hypothesis A ` ⊃ tion Frege systems and nested deduction Frege systems, (in symbols, A B ) then there is a proof of A B . which augment Frege systems with variants of the This paper considers various strengthenings of this deduction rule. We give upper bounds on the lengths deduction rule and establishes upper bounds on the of proofs in these systems compared to lengths in proof-speedups obtained with these deduction rules. Frege proof systems. As an application we give a near-linear simulation of the propositional Gentzen By a “speedup” of a proof, we mean the amount sequent calculus by Frege proofs. The length of a that proofs can be shortened with additional inference proof is the number of steps or lines in the proof. rules. In this paper, the length of a proof is the number of lines in the proof; where a line consists A general deduction Frege proof system provides at either a formula or a sequent (depending on the proof most quadratic speedup over Frege proof systems. A system). We write B (and A ,...,As B) nested deduction Frege proof system provides at most k 1 k a nearly linear speedup over Frege system where by to indicate that the formula B has Frege proof of ≤ k lines (from the hypotheses A1,...,As). More “nearly linear” is meant the ratio of proof lengths is T O(α(n)) where α is the inverse Ackermann function. generally, we write “ k ” to mean “provable in proof A nested deduction Frege system can linearly simulate system T with ≤ k lines”. If S and T are proof the propositional sequent calculus and hence a Frege systems we say that S can linearly (respectively, proof system can simulate the propositional sequent quadraticly) simulate T if, for any T -proof of k calculus with proof lengths bounded by O(n · α(n)). lines, there is an S -proof of the same (or sometimes As a technical tool, we introduce the serial transitive an equivalent) formula of O(k) lines (respectively, of closure problem: given a directed graph and a list of O(k2) lines). We say that T provides at most linear closure edges in the transitive closure of the graph, (respectively, quadratic speedup) over S if S can the problem is to derive all the closure edges. We give linearly (respectively, quadraticly) simulate T . a nearly linear bound on the number of steps in such a derivation when the graph is tree-like. An alternative, commonly used measure of the length of a propositional proof is the number of symbols in 1 Introduction the proof. This is the approach used, for instance, by Cook-Reckhow [3] and Statman [12]. It should be A Frege proof system is an inference system for noticed that the minimum number of lines in a Frege propositional logic in which the only rule of inference proof of a formula is polynomially related to the min- is modus ponens. Although it suffices to have modus imum number of symbols in an extended Frege proof. ponens as the single inference rule to obtain a complete On the other hand, prior work on proof lengths in proof system, it is well-known that other modes of first-order logic has frequently measured the number ∗Supported in part by NSF Grant DMS-8902480 of lines in proofs; this includes [2, 4, 6, 7, 8, 10, 11] and others. We next define extensions to the Frege proof system We begin by defining the main propositional proof that incorporate the deduction theorem as a rule of systems used in this paper. The logical connectives inference. For this purpose, the systems defined below of all our systems are presumed to be ¬, ∨, ∧ and have proofs in which the lines are sequents of the ² ⊃; however, our results hold for any complete set of form Γ A; intuitively, the sequent means that the connectives. formulas in Γ tautologically imply A: operationally, a sequent Γ ² A means that A has been proved using Definition A Frege proof system (denoted F )is the formulas in Γ as assumptions. characterized by: A general deduction Frege system (denoted dF ) incor- porates a strong version of the deduction rule. Each (1) A finite set of axiom schemata. For example, a line in a general deduction Frege proof is a sequent of ⊃ ⊃ possible axiom schema is (A (B A)); A and the form Γ ² A where A is a formula and Γ is a set B represent arbitrary formulas. of formulas. When Γ is empty, we write just ² A. (2) The only rule of inference is Modus Ponens (MP): The four valid axioms and inference rules in a general deduction Frege proof are: AA⊃B B ² A − A an axiom
(3) A proof in this system is a sequence of formulas {A} ² A − Hypothesis A1,...,An (also called ‘lines’) where each Ai is Γ ² A ⊃ B Γ ² A either a substitution instance of an axiom schema 1 2 − ∪ ² Modus Ponens or is inferred by Modus Ponens (MP) from some Γ1 Γ2 B Aj and Ak with j, k < i. Γ ² B −Deduction Rule Γ \{A}²A⊃B (4) The proof system must be consistent and com- dF plete. We write A1,...,An k B to indicate that {A1,...,An}²B has a general deduction Frege proof The length of an F -proof is the number of lines in of ≤ k lines. the proof; we write A to indicate that A has k Deduction Frege systems are quite general since they a Frege proof of length ≤ k . We further write allow hypotheses to be “opened” and “closed” (i.e., A ,...,As B to mean that B is provable from 1 k “assumed” and “discharged”) in arbitrary order. A the hypotheses Ai with a Frege proof of ≤ k lines; in more restrictive system is the nested deduction Frege other words, that there is a sequence of ≤ k formulas proof system which requires the hypotheses to be each of which is one of the Ai ’s, is an axiom, or is used in a ‘nested’ fashion. The nested deduction inferred by modus ponens from earlier formulas such Frege systems are quite natural since the correspond that B is the final formula of the proof. Although we to the way mathematicians actually reason while have not specified the axiom schemata to be used in carrying out proofs. A second reason the nested a Frege proof system, it is easy to see that different deduction Frege system seems quite natural is that we choices of axiom schemata will change the lengths of shall prove below that nested deduction Frege proof proofs only linearly. systems can simulate with linear size proofs both the The simplest form of the deduction theorem states propositional Gentzen sequent calculus and tree-like that if A ` B then ` A ⊃ B . This can be informally general deduction Frege proofs. phrased as a rule in the form The primary feature of the nested deduction Frege A ` B proof system is that hypotheses must be closed in A ⊃ B reverse order of their opening. And after a hypothesis which is called the 1-simple deduction rule; more is closed, any formula proved inside the scope of generally, the simple deduction rule is the hypothesis is now longer available. For example, a nested deduction Frege proof may look like the ` A1,...An B following: A1 ⊃(···(An ⊃B)···) hA1i²A1 Hypothesis A1 opened by an example; the fragmentary ndF -proof given . . above would be pictorially represented as: hA1,A2i²A2 Hypothesis A2 opened A1 . . . . hA1,A2i²A3 A2 hA1i²A2 ⊃A3 Deduction Rule; A2 closed . . . . h i² A3 A1,A4 A4 Hypothesis A4 opened ⊃ A2 A3 . . A4 hA ,A i²A . 1 4 5 . hA1i²A4 ⊃A5 Deduction Rule; A4 closed A5 . . A4 ⊃ A5 h i² . A1 A6 . hi²A1 ⊃ A6 Deduction Rule; A1 closed A6 A1 ⊃ A6 A sequent in a nested deduction Frege (ndF ) proof Nested deduction Frege proofs are conceptually simple is of the form Γ ² A where now Γ is a sequence of and natural and, in practice, seem to simplify the pro- formulas. An ndF proof is a sequence of sequents cess of discovering proofs. Thus, it is surprising that Γi ² Ai (i =1,2,...,n) such that Γ is taken to 0 a Frege proof system can simulate nested deduction be the empty sequence and, for each i, one of the Frege proofs with near linear size proofs: this fact is following holds: the content of our main theorems below.
(a) Γi =Γi−1 and Ai is an axiom. 2 Summary of Results
(b) Γi =Γi−1∗Ai. This opens an assumption, ∗ In this section we outline our primary results giving denotes concatenating a formula to the end of fairly sharp bounds on how much the deduction rule the sequence. can shorten proofs. Most of our results are stated in the form “If proof system X can prove a formula in ∗ ⊃ (c) Γi−1 is Γi B and Ai is B Ai−1 . This is the n lines, then the formula has a Frege proof of f(n) deduction rule. lines”. Obviously f depends on the system X .
(d) Γi =Γi−1 and Ai is inferred from Aj and Ak First recall the usual proof of the deduction theorem by Modus Ponens where each of Γj ² Aj and (see e.g. Kleene [5]) which establishes the following: Γk ² Ak are available to sequent i.Wesay sequent j is available to sequent i if j Proof Given an n line proof P of B from assump- We now consider the simulation of proof systems with 0 tions A1,...,Am, we construct a Frege proof P of B more powerful versions of the deduction rule. from the single assumption A1 ∧ A2 ∧···∧Am (where the conjunction is to be associated from left-to-right). Theorem 4 dF If n B then O(n2) B . For any Frege proof system, there is a constant k such ∧ ∧ 0 that C D k C and C D k D. Thus, P can be Theorem 4 states that a general deduction Frege proof constructed to (1) first derive each of A1,...,Am in system can provide no more than a quadratic speedup 2k(m − 1) lines and (2) then derive B in ≤ n lines 0 over a Frege proof system; whether this quadratic (via P ). Clearly P has O(m + n) lines and by one bound is optimal is an open question. application of Theorem 1, there is a Frege proof of m A1 ∧···∧Am ⊃ B with O(m + n) lines. Finally it V Proof For the proof, we let Ai denote any can be shown by induction on m that i=1 conjunction of the formulas Ai ordered and associated ∧···∧ m ⊃ ⊃ O(m) [A1 A B] arbitrarily (each Ai should occur exactly once as a conjunct). To prove Theorem 4, we prove the [(A1 ⊃ (A2 ⊃···⊃(Am ⊃B)···))]. more general result that if {A1,...,Am} ² B has a Vm By combining these last two proofs with a Modus dF -proof P of n lines then ( Ai) ⊃ B has a Frege Ponens inference, Theorem 2 is proved. 2 i=1 proof P 0 of O(n2) lines. To form the proof P 0 replace each sequent {A1,...,Am} ²B of P by the formula We use the name simple deduction Frege proof system Vm for the system in which all hypotheses must be opened ( Ai) ⊃ B ; it will suffice to “fill in the gaps” to i=1 at the beginning a proof and closed at the end of a make P 0 a valid proof. First, an axiom in P becomes proof. Theorem 2 shows that a Frege proof system w.l.o.g. an axiom of the Frege system. Second, a can simulate simple deduction Frege proofs with linear hypothesis {A} ² A in P becomes the tautology size proofs; or equivalently, that the simple deduction A ⊃ A which has a constant length Frege proof. Frege proof system provides only a linear speedup (i.e., Third, the sequents in a Modus Ponens inference in P a constant factor speedup) over Frege proof systems. ² ⊃ ² An interesting corollary to Theorem 1 is that conjunc- Γ1 A B Γ2 A ∪ ² tions may be arbirarily reordered and reassociated Γ1 Γ2 B V V with linear size Frege proofs: ⊃ ⊃ ⊃ becomeV the formulas Γ1 (A B) and Γ2 A and (Γ ∪ Γ ) ⊃ B . It will suffice to show that the Corollary 3 1 2 Let B be any conjunction of A1,...,Am third formula can be proved from the first formulas in that order but associated arbitrarily. Let i1,...,in with a Frege proof of O(nV) lines. By Corollary 3 be any sequence from {1,...m} and let C be any there are Frege proofs of (Γ1 ∪ Γ2) ⊃ Γi for i = conjunction of Ai1 ,...,Ain again in the indicated 1, 2 containing O(m) lines where Γ1 ∪ Γ2 contains order and associated arbitrarily. Then Vm formulas. From theseV latter two formulas and from ⊃ ⊃ ⊃ ⊃ ΓV1 (A B) and Γ2 B there is a Frege proof O(m+n) B C. of (Γ1 ∪ Γ2) ⊃ B with a constant number of lines. It is easily shown that the number of formulas in the such that A(i, i) >n. Equivalently, α(n) is equal to ∗i− lefthand side of sequent in a dF -proof is bounded by the least i such that log( 1) n The proof of (Γ \{A})∗¬(A⊃C)|=p∧¬p is: 0 Case 2: The last line of P is {A}|=A. Then P is \{ } Γ A an ndF -proof of hA, ¬Ai|=p∧¬p which has a ¬(A⊃C) constant number of lines, say c lines. 2 . . A ¬C ) Case 3: The last line of P is . . from P ∧¬ 1 Γ1 |= A ⊃ B Γ2 |= A p p Γ ∪ Γ |= B 0 0 1 2 This proof has size ≤ c(n − 1) + d where d is a constant. So taking c ≥ d0 , the proof has size Assume the proof of Γ1 |= A ⊃ B has n1 lines ≤ c · n. and the proof of Γ2 |= A has n2 lines, so that n = n1 + n2 + 1 since P is treelike. By the Lemma 9 follows from cases 1-4, by taking ≥ 0 2 induction hypothesis, there are ndF -proofs P1 c c1,c2,d,d and Theorem 7 is proved. and P2 of the sequents Γ1 ∗(¬(A ⊃ B)) |= p ∧¬p and Γ2 ∗ (¬A) |= p ∧¬p of lengths ≤ c · n1 and The next theorem gives a linear simulation of the 0 ≤ c · n2 lines, respectively. The proof P of propositional Gentzen sequent calculus by the nested V ⊃ deduction Frege system. For this theorem, it is cru- equivalent formula ( Γ) B where conjunctionV cial that Gentzen sequent calculus proofs are always is associated from left to right; thus Γ is the ··· ∧ ∧···∧ ∧ treelike. For the definition of the Gentzen sequent formula ((V (A1 A2) Ak−1) Ak). When Γ calculus, see Takeuti [13]; we are concerned with only is empty, Γ is a fixed tautology. This translation of the propositional fragment of the sequent calculus. sequents into equivalent formulas gives us a sequence The following theorems also hold for many variations of formulas P 0 ; unfortunately, P 0 is not a valid Frege of the sequent calculus, for instance with the mix proof and so it remains to show how P 0 can be made rule, or with a rule that allows arbitrary reordering of into a valid Frege proof with only a relatively small cedents, or with either the multiplicative or additive increase in the number of lines. versions of rules. (But the treelike property is crucial To make P 0 into a valid Frege proof we shall add for our proofs.) additional lines. There are four rules of inference for ndF : Axiom, Hypothesis, Deduction Rule and Modus Theorem 10 → Suppose the sequent Γ ∆ has a Ponens. For each rule of inference, we explain what Gentzen sequent calulus proof of length n lines. Then 0 ndF V W lines need to be added to P ; we save Modus Ponens Γ ⊃ ∆. O(n) for last since it is by far the most difficult case. First consider an axiom inference in P which is Corollary 11 If →A has a Gentzen sequent calcu- of the form Γ ² B where B is an axiom, so P 0 lus proof of length n lines, then A. V O(n·α(n)) contains Γ ⊃ B as the corresponding formula. Since a Frege proof system is axiomatized with axiom Theorem 10 is proved by showing by induc- schemata, there is a proof of B ⊃ (X ⊃ B) with a tion on n that, if A ,...,Ak→B ,...,B` has a 1 1 constant number of lines (independent of the formulas Gentzen sequent calculus proof of n lines, then ndF B and X ). Thus thereV is a constant length Frege hA ,...,Ak,¬B ,...¬B`i²p∨¬p. Because of O(n) 1 1 proof of the formula Γ ⊃ B ; namely, take the space constraints, the proof of Theorem 10 is omitted V axiom B , derive B ⊃ ( Γ ⊃ B) and then use modus from this extended abstract. ponens. This constant length Frege proof is inserted Corollary 11 improves a theorem of Orevkov [10] which into the sequence P 0 . → states that if A has a proof in a sequent calculus Second, consider a hypothesis inference where KGI of n lines and height h, then A. 0 O(n log h) P contains a sequent Γ ∗ B ² B and P contains The proof system KGI is a reformulation of the ((∧Γ) ∧ B) ⊃ B . It is easy to derive (X ∧ B) ⊃ B usual sequent calculus [9]; although KGI proofs need in a Frege proof in a constant number of lines where not be tree-like, Gentzen proofs must be tree-like in the constant is independent of the formulas B and order to be linearly translated in KGI. Orevkov, like X , so this sequent in P 0 can be derived in a constant us, does not need to count structural inferences. number of lines. 3 Proof of the Main Theorems Third, consider a deduction rule inference in P ; here P contains a sequent Γ ∗ A ² B followed immediately In this section we reduce the main theorems to a serial by Γ ² A ⊃ B and P 0 contains the corresponding transitive closure problem which will be considered in formulas ((∧Γ) ∧ A) ⊃ B and (∧Γ) ⊃ (A ⊃ B). the next section. Again there is a constant length Frege proof of the Suppose that we have a natural deduction Frege latter formula in P 0 from the former one; this constant proof P of a sequent ² B such that P contains n lines length proof is to be inserted into P 0 . and uses the hypothesis rule m times. To prove Main Fourth and hardest, we consider a line in P 0 that Theorem 5 for a fixed value of i, we will translate P ∗i corresponds to a sequent of P obtained by Modus into a Frege proof of B containing O(n + m log( ) m) Ponens. Suppose that in P there are lines Γ1 ² A lines. Likewise, for Main Theorem 6, P is translated and Γ ² A ⊃ B from which Γ ² B is inferred by · 2 into a Frege proof of B of O(n α(n)) lines. Modus Ponens. Since P is a nested deduction Frege ² 0 Each line in the proof P is of the form Γ B proof, Γ1 and ΓV2 are initial subsequencesV of Γ. In P where Γ is a sequence of formulas hA1,...,Aki. the formulas ( Γ1) ⊃ A and ( Γ2) ⊃ (A ⊃ B) ¿From the sequent Γ ² B we form the logically appear and from them we wish to derive the formula V ( Γ) ⊃ B in a small number of lines. Note that there tured in section 1, the directed graph of tautologies is a constant size Frege proof of X ⊃ B from the is: ⊃ ⊃ ⊃ ⊃ hypotheses X1 A and X2 (A B) and X X1 V and X ⊃ X where the constant is independent of the hA1,A2i 2 V V formulas X , X1 , X2 , A and BV. Thus weV will modify hi hA i 0 1 P by adding the formulas ( Γ) ⊃ ( Γi) at the V V hA ,A i beginning and inserting a Frege proof of ( Γ) ⊃ B 1 4 from these new formulas and from the other two 4 Serial Transitive Closure Problem formulas. The serial transitive closure problem is the problem It remainsV now toV give short Frege proofs of the formulas ( Γ) ⊃ ( Γ0) which have been added to of deriving a set of “closure edges” in the transitive beginning of P 0 . By examining the fourth case above closure of a directed graph. If X and Y are nodes → we see that there are < 2n such formulas and they in a directed graph we write X Y to indicate the always have Γ0 an initial subsequence of Γ and thus presence of an edge from X to Y . Of course, any edge they are tautologies of the form in the transitive closure of a graph can be obtained by a series of closure steps which are inferences of the ((···(A ∧A )∧···)∧Ak)⊃((···(A ∧A )∧···)∧A`) form 1 2 1 2 A → BB→C where without loss of generality ` ? ? ? ? ? s - s - s - s - s - s - s - s - s X0 X1 X2 X3 X4 X5 X6 X7 X8 6 6 6 6 6 6 & %& %& %& % Figure 1 Round `: For round number ` we add auxiliary General Case: The proof of Theorem 15 for G a tree edges incident on the nodes Xbk·m/2`c for odd val- uses a construction similar to the proof of the linear ues of k . Specifically, for each odd value k<2` case. For the general case, we shall use Lemma 14 to the following auxiliary edges¥ are derived:¦ (1) the split G into multiple subtrees of size less than half the → − ` ≤ ¥edges Xj¦ Xbk·m/2`c for (k 1)m/2 j< size of G (one of these is scarred); then we similarily · ` → k m/¥ 2 and¦ (2) the¥ edges Xbk·m/¦ 2`c Xj split these subtrees into subtrees of size less than one for k · m/2` 2 · m + c · (N · α(m)+m·α(m)) + 2 · N. The rounds are iterated until all the subtrees have ∗i size ≤ 1; namely, in no more than log( ) m rounds. Instead of iterating the reduction process, we instead At the end every closure edge has been derived. The just directly derive the rest of the closure edges. If 0 total number of closure steps used is bounded by there are remaining N closure edges then a total of N 0 · α(m) closure steps suffice; this is because each (∗i) logX m remaining closure edge connects two nodes inside a (1 + 2i)(n` + m) subtree of size <α(m). `=1 0 P Since n = N+N , the serial transitive closure problem and since n` ≤ n, the total number of closure edges has a solution of size bounded by is bounded by (c + 2)(n + m) · α(m). ∗i (1 + 2i)(n + m log( ) m). Q.E.D. Theorems 13 and 6. That completes the proofs of Theorems 16, 12 and 5. We do not know if the theorems above provide 2 the best possible asymptotic bounds for the serial transitive closure problem or on the size of Frege proof It remains to prove Theorem 13. To do this we first simulations of nested deduction Frege proof systems. note the following simple corollary of Theorem 16: References Lemma 17 If the directed graph G is a tree, then the [1] R. P. 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