REDUCTION OF HILBERT-TYPE PROOF SYSTEMS TO THE IF-THEN-ELSE EQUATIONAL LOGIC. 1. Introduction

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1 J. Appl. Math. & Computing Vol. 14(2004), No. 1-2, pp REDUCTION OF HILBERT-TYPE PROOF SYSTEMS TO THE IF-THEN-ELSE EQUATIONAL LOGIC JOOHEE JEONG Abstract. We present a construction of the linear reduction of Hilbert type proof systems for propositional logic to if-then-else equational logic. This construction is an improvement over the same result found in [4] in the sense that the technique used in the construction can be extended to the linear reduction of first-order logic to if-then-else equational logic. AMS Mathematics Subject Classification : 03B22,03F99 Key words and phrases : Proof systems, reduction, if-then-else equational logic 1. Introduction We consider a proof system efficient if every statement in the system has a short proof. In [6], Cook and Reckhow investigated the relative efficiency of various proof systems for classical propositional proof systems. (By classical, we mean the logic is determined by a single valuation system with only two distinct truth values {, }, as opposed to intuitionistic, modal etc.) A handful of researchers followed this line of research, e.g., Krajíček-Pudlák [8], Urkuhart [12], Arai [1, 2] and Messner-Torán [11] to name a few. The key concepts in this research field are given in definition 1, 2 and 3 they are adopted from [6] with a slight modification. Then we generalize the notion of p-simulation in definition [4]. Definition 1. Let Σ be an alphabet and L Σ be a recursively enumerable language. A proof system for L is a (deterministic) polynomial time surjective function f : P L Received September 24, Revised November 12, This work was supported by grant No. R from the Basic Research Program of the Korea Science & Engineering Foundation. c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 69

2 70 Joohee Jeong for some recursive language P Σ 1 in some alphabet Σ 1.Wesay x is a proof for y just in case y = f(x). The membership property for P is required to be determined in polynomial time. It is reasonable to assume that the objects that we want to prove in a proof system are strings over an alphabet Σ, and they form a language L Σ. Normally this language L consists of formulas or sequents. The formal proofs or derivations are usually finite sequences or trees or dags with nodes labelled by members in L, satisfying certain conditions (called the rules of inference), and hence we can view them as strings over some alphabet Σ 1, which is normally a superset of Σ, under suitable encoding if necessary. Henceforth, we will use the word derivation exclusively in place of proof to prevent possible confusions between (object level) formal proofs and (metalevel) informal proofs. (Maybe the term proof system should really be called derivation system.) Definition 2. A proof system f : P L is polynomially bounded if there exists a polynomial p(n) such that ( ) ( y L)( x P ) y = f(x) and x p( y ), where denotes the length of string. def def Definition 3. Let P 1 = f 1 : P 1 L and P 2 = f 2 : P 2 L be proof systems for L. We say P 2 p-simulates P 1 just in case there exists a polynomial time computable function g : P 1 P 2 such that f 2 (g(x)) = f 1 (x), x P 1 Definition 2 is talking about the efficiency of proof systems. We consider a derivation x of y short if x p( y ), and we may say that a polynomially bounded proof system is efficient. Definition 3 captures the notion of the translation of derivations and the relative efficiency: i.e., the derivation x in a proof system P 1 is uniformly translated via the map g to yield a derivation g(x), for the same formula f 1 (x), in another proof system P 2. When P 2 p-simulates P 1, we may say that P 2 is as efficient as P 1 modulo a polynomial. This research field has been mainly interested in the classical propositional proof systems: i.e., L is almost always taken to be the set TAUT of (classical) tautologies. It is not known whether there exists an efficient (i.e., polynomially bounded) proof system for TAUT. In fact this problem is equivalent to the question of whether TAUT is in NP as mentioned in [6]. In this paper, we generalize the notion of p-simulation to polynomial reduction, and this generalization enables us to deal with the cases when L TAUT. 2. Generalizing the notion of p-simulation

3 Reduction of Hilbert-type proof systems 71 def def Definition 4. Let P 1 = f 1 : P 1 L 1 and P 2 = f 2 : P 2 L 2 be proof systems. We call an ordered pair g, h of maps g : P 1 P 2 and h : L 1 L 2 such that ( x P 1 ) ( f 2 (g(x)) = h(f 1 (x)) ) and (1) ( y L 1 ) ( ( x 2 P 2 )(f 2 (x 2 )=h(y)) ( x 1 P 1 )(f 1 (x 1 )=y) ) (2) a reduction of P 1 to P 2. g, h is called a weak reduction if it satisfies (1) only. A reduction is said to be polynomial (resp. linear) just in case both g and h are computed in polynomial (resp. linear) time. The condition (1) in above definition forces the following diagram commute: f 1 P 1 L 1 g f 2 h P 2 L 2 But condition (2) apparently does not appear in definition 3. This is because (1) implies (2) in case L 1 = L 2 = TAUT, which will be explained shortly. One of the reasons why we would want a reduction g, h of P 1 to P 2 is to reduce the problem of determining the provability relation α A in a proof system P 1 to another proof system P 2. Here, A is a formula in P 1, α is a finite set {A 1,...,A n } of formulas in P 1, and L 1 consists of all such expressions α A that are provable in P 1. In order to define h, it is usual that each formula A is assigned a formula Ā in P 2, and then we let h(α A) def = ᾱ Ā (3) where {Ā1,...,Ān} is denoted by ᾱ. Let us write α P1 A just in case there exists a derivation ψ P 1 such that f 1 (ψ) =α A. Similarly, we may write β P2 B just in case there exists a derivation verifying β B in P 2. If we write g(ψ) := ψ, then condition (1) says ψ is a derivation of α P1 A ψ is a derivation of ᾱ P2 Ā. In many occasions we are only interested in whether derivation of α P1 A exists derivation of ᾱ P2 Ā exists which may be written more succinctly as α P1 A ᾱ P2 Ā. (4)

4 72 Joohee Jeong Condition (1) implies of (4), while condition (2) implies. In case L 1 = L 2 = TAUT, h is the identity map and hence Ā = A, and moreover α =. Thus ᾱ P2 Ā P2 A A TAUT P1 A α P2 A follows easily from (1) alone: i.e., in this case the notion of weak reduction coincides with the notion of reduction. In this paper we are particularly interested in the proof system of if-then-else equational logic, denoted by ITE, for the target proof system P 2. Then, we find that the propositional calculus can be linearly reduced to ITE. In [4], it is shown that the predicate calculus is reduced to ITE in the sense of (4). But the proof is model theoretic and there is no mention on translation of proofs let alone efficiency of proof systems. The reduction consists of g : L 1 L 2 alone (i.e., h : P 1 P 2 is the identity map) in [4]. In [5], the propositional calculus is linearly reduced to ITE, which is the same as the main theorem of this paper. However, the construction in this paper is more flexible and powerful since the same technique can be utilized to linearly reduce more complex proof systems, such as first-order predicate calculus to ITE. In fact the linear reduction of first-order predicate calculus to ITE has been done recently by the author of this paper in [7]. We use the symbol to denote syntactic equality: i.e., if x and y are syntactic objects such as symbols, terms, formulas, derivations etc., then we write x y just in case they are the same. Sometimes we may get deliberately sloppy in notation. For instance, if u 1,...,u n are strings, or any mathematical object (such as integers) that can be represented by strings, then we consider the finite sequence u 1,...,u n as a string. As another such example, if A 1,A 2,... are formulas and α 1,α 2,... are finite sequences of formulas, then we write A 1,A 2,...,α 1,α 2... to denote the finite sequence of formulas obtained in the obvious way. 3. The proof systems FL and ITE In this section we give formal definitions of two proof systems, together with some relevant facts. The proof systems we deal with here are the propositional calculus denoted by FL after Frege- Lukasiewicz, and the if-then-else equational logic denoted by ITE. Propositional calculus, i.e., the Hilbert type proof system for classical propositional logic is usually presented in the following way: Axiom schemes (FL-axioms) A (B A) (FL1 )

5 Reduction of Hilbert-type proof systems 73 (A (B C)) ((A B) (A C)) (FL2 ) ( A B) (B A) (FL3 ) Rules of inference (modus ponens) A B, A (MP) B We pick a fixed set of propositional letters, which are normally assumed to be countable. We assume that the set of propositional formulas is built from the propositional letters and the two connectives and in the usual way. Normally, we consider above as an adequate formal definition of propositional calculus. But we can further formalize the proof system as in the following definition. Definition 5. Define the proof system FL def = f 1 : P 1 L 1 by L 1 is the set of all finite sequences A 1,...,A n,b, n 0 of propositional formulas satisfying (A 1 A n B) TAUT. Obviously, our intended interpretation of f 1 (x) = A 1,...,A n,b is x is a derivation of {A 1,...,A n } B. Let P1 0 be the set of all finite sequences n, C 1,...,C m where n 0, m 1 and C i s are propositional formulas. Define a relation f 1 P1 0 L 1 by n, C 1,...,C m f 1 A 1,...,A n,b holds iff the following conditions are satisfied (1) n = n <mand C i A i for each i =1,...,n, (2) C m B, (3) for each i = n +1,...,m, C i is an instance of FL-axioms or C i {A 1,...,A n } or for some j, k {n +1,...,i 1}, we have C j C k C i. def Let P 1 = {x P1 0 (x, y) f 1 for some y L 1 }. (Then we see that f 1 is a function from P 1 onto L 1.)

6 74 Joohee Jeong It is easy to see that this definition fits in 1. Of course other presentations for propositional calculus are possible, but the construction we use here is insensitive to such variations. We could proceed our work in a more general setting. For instance, instead of working with this particular proof system FL, we could adopt the definition of Frege systems, given in [6], (which is a synonym for Hilbert system) for propositional logic. But we believe that such a generalization only makes our work lengthy (and messy) without providing any useful insight. Now we will describe the proof system ITE F for if-then-else equational logic, where F is any set of function symbols. (We follow the convention that constants are 0-ary function symbols.) By an equation we mean an ordered pair t, s where t and s are first-order terms defined in the usual way. An equation t, s will be written as t s. In equational logic, formulas are synonym for equations. In if-then-else equational logic, we must have a distinguished 4-ary function symbol [ ] F, called the switching function, where the intended interpretation of the term t : [x,y,z,w] is { z if x = y t = (5) w otherwise. The defining equation (5) is not an equation in the strict sense. If S is an F-algebra, then 5 holds in S if and only if the following holds: S (t z x y) (t w (x y)). (6) Thus a quantifier-free formula (or a first-order universal formula, you might say) shown in (6) seems to be necessary in order to define a switching function. But in 1975, McKenzie showed that switching function (in an equational class of algebras) can be defined by a set of equations [9]. This set of equations, modified slightly by Burris, is as follows. Definition 6. Define the McKenzie-Burris axioms MB F to be the set of equations [x, x, y, x] y (MB1 ) [x, y, x, x] x (MB2 ) [x,y,y,x] x (MB3 ) [x, [x,y,z,x],y,x] y (MB4 ) [x, y, u, v] [[x, y, u, x], [x,y,v,x],v,[x, y, u, x]] (MB5 ) [u, v, f(x 1,...,x n ),u] [u, v, f([u, v, x 1,u],...,[u, v, x n,u]),u] (MB6 ) where f is any n-ary function symbol in F other than [ ]. The variables x, y, z, u, v, x 1,... are all assumed to be distinct. In [5], it is shown that an algebra in the signature F satisfies all the equations in MB F if and only if it satisfies the defining equation (5) for switching function.

7 Reduction of Hilbert-type proof systems 75 Similar results can be found in [3, 10], but the number of axioms (schemes) used there are greater 12 and 9 respectively. Definition 7. For a signature F of if-then-else equational logic and a recursive set Φ of F-equations, we define the proof system ITE F (Φ) def = f 2 : P 2 L 2 to be the well-known equational proof system for signature F augmented by the extralogical axioms Φ and the McKenzie-Burris axioms MB F. Above definition is a little informal compared to 5. But it is easy to see that this definition can be made enough formal to fit in 1. Curious readers are referred to [7]. Also it is clear, by the Birkhoff s theorem combined with the result of McKenzie-Burris, that the soundness theorem and the completeness theorem hold for the if-then-else equational logic. We state this fact as a theorem without proof. Theorem 1. Let F be a signature of if-then-else equational logic: i.e., the 4-ary function symbol []belongs to F. LetA be an F-equation and α be a finite set of F-equations. Let Φ be any set of F-equations. Then α ITEF (Φ) A if and only if for every F-algebra S in which []is interpreted as a switching function and S Φ α, we have S A. 4. Reduction of propositional calculus to ITE Now we describe a linear reduction g, h, as defined in definition 4, from FL def = f 1 : P 1 L 1 to ITE F (Φ) def = f 2 : P 2 L 2, where F def = {0, 1, [], f} {,,c 1,c 2,...}. 0 and 1 are distinct constants, [ ] is a 4-ary function symbol, f is a distinguished unary function symbol, is a unary function symbol, is a binary function symbol, and c 1,c 2,... are denumerably many distinct constants. Then Φ is defined to be a set of some suitably chosen equations in the signature F. For the purpose of defining h : L 1 L 2 and g : P 1 P 2, assume without loss of generality that we have denumerably many distinct propositional letters {X 1,X 2,...}. First, for each propositional sentence A, we obtain an F-term A t by replacing each occurrence of X i by c i. Then define a ground equation Ā by Ā def = f(a t ) 1.

8 76 Joohee Jeong Definition 8. Define h : L 1 L 2 by h(a 1,...,A n,b) def = Ā1,...,Ān, B. It is routine to check that h is linear time computable. Definition 9. Fix two distinct variables x, y, and let A 01 : [0, 1,x,y] y, A mp : [f(x), 1, [f(x y), 1, f(y), 1], 1] 1, FL def = {Ā A is an FL-axiom}, and define Φ def = {A 01,A mp } FL. The construction of g : P 1 P 2 takes a little longer. Definition 10. Given n, A 1,...,A n,c 1,...,C k P 1, we will first define Ĉi for each i, which could be either an equation or a finite sequence of equations, and then let g(n, A 1,...,A n,c 1,...,C k ) def = n, Ā 1,...,Ā n, Ĉ 1,...,Ĉ k. We define Ĉi for each i =1,...,k as follows. If C i belongs to {A 1,...,A n } or is one of the FL-axioms, then just let Ĉ i : C i. Otherwise, C i must be the result of applying modus ponens, to two earlier formulas C j and C k : C j C i. In this case we let Ĉi be the sequence of the following 9 equations: (The rightmost comments are used in lemma 2.) [f(cj), t 1, [f(cj t Ci t ), 1, f(ci t ), 1], 1] 1 subst.,a mp (7) [f(cj t ), 1, [f(ct j Ct i ), 1, f(ct i ), 1], 1] [1, 1, [1, 1, f(ct i ), 1], 1] (8) repl., f(cj t ) 1, f(ct j Ct i ) 1 [1, 1, [1, 1, f(ci t ), 1], 1] [f(cj), t 1, [f(cj t Ci t ), 1, f(ci t ), 1], 1] (9) symm., (8) [1, 1, [1, 1, f(ci t ), 1], 1] 1 trans., (9), (7) (10) [1, 1, [1, 1, f(ci t ), 1], 1] [1, 1, f(ci t ), 1] subst., MB1 (11) [1, 1, f(ci t ), 1] f(ct i ) subst., MB1 (12) [1, 1, [1, 1, f(ci t ), 1], 1] f(ct i ) trans., (11), (12) (13) f(ci t ) [1, 1, [1, 1, f(ci t ), 1], 1] symm., (13) (14) f(ci t ) 1 trans., (14), (10) (15)

9 Reduction of Hilbert-type proof systems 77 We have completed the construction of F, Φand g, h First we need to check that the translation g : P 1 P 2 really takes a derivation in FL to a derivation in ITE F (Φ). Lemma 2. Let n, A 1,...,A n,c 1,...,C k be a derivation in FL, and let α := {A 1,...,A n }. Then n, ᾱ, Ĉ1,...,Ĉk is a derivation in ITE F (Φ) with conclusion ᾱ C k. Proof. The proof proceeds by induction on i =1,...,k. We know, from the hypothesis of this proposition, that for each i, n, α, C 1,...,C i is an FL-derivation with conclusion α C i. We want to show that, for each i, n, ᾱ, Ĉ1,...,Ĉi is a derivation in ITE F (Φ) with conclusion ᾱ C i For i = 1, there are two cases to consider. First, if C 1 α, then Ĉ1 : C 1 ᾱ and hence n, ᾱ, Ĉ1 is a derivation in ITE F (Φ) with conclusion ᾱ C 1. Second, if C 1 is an FL-axiom, then Ĉ1 : C 1 FL Φ and hence we are done again. For i>1, there are three cases to consider. The first two cases are the same as above. If C i is obtained by modus ponens, then Ĉi is the sequence of the nine equations (7) (15). The rightmost comments of these equations show that n, ᾱ, Ĉ1,...,Ĉi is a derivation. In (8), f(cj t) 1 and f(ct j Ct i ) 1 hold from the induction hypothesis. Now, by noting that the last equation in Ĉi is nothing but C i, we are done. Corolloary 3. FL def = f 1 : L 1 P 1 linearly weak-reduces to ITE F (Φ) def = f 2 : L 2 P 2. Proof. In fact, essentially everything we need to prove this corollary has been shown already in lemma 2. To clarify what we have achieved, let x P 1 be given. Then we may choose n 0, finite sets α := {A 1,...,A n } and ψ := {C 1,...,C k } of FL-formulas so that x = n, α, ψ. We need to show h(f 1 (x)) = f 2 (g(x)). But h(f 1 (x)) is nothing but h(α, C k ) which is ᾱ, C k. Lemma 2 says that n, ᾱ, ψ, which is g(x) by the definition of g, is a derivation of this ᾱ, C k in ITE F (Φ): i.e., f 2 (g(x)) = ᾱ, C k = h(f 1 (x)) as was to be shown. The straightforward verification of the fact that g and h are computable in linear time will be omitted. Theorem 4. FL linearly reduces to ITE F (Φ).

10 78 Joohee Jeong Proof. Let FL = f 1 : L 1 P 1 and let ITE F (Φ) = f 2 : L 2 P 2 as before. Since we already have corollary 3, we only need to check the condition (2) ( y L 1 ) ( ( x 2 P 2 )(f 2 (x 2 )=h(y)) ( x 1 P 1 )(f 1 (x 1 )=y) ) of definition 4. So let y def = α B L 1 be given where α def = {A 1,...,A n } is a finite set of propositional formulas and B is a proposional formula. Suppose that h(α B) def = ᾱ B is proved in ITE F (Φ), with a witnessing derivation ψ: i.e., f 2 (ψ) = ᾱ B. Then we want to show that some derivation ϕ P 1 witnesses α B. The problem is that there may not exist ϕ P 1 for which ψ = ϕ: i.e., the map ϕ ϕ from P 1 into P 2 may not be onto. So it is hopeless to construct such a ϕ from ψ. Therefore we must resort to some model theoretic methods. Basically, we have to show ᾱ B ITEF (Φ) α FL B. (We know that the converse of above holds by corollary 3.) We will construct a model S of ᾱ MB F Φ so that for all propositional-formula B, S B α FL B. (16) Suppose that such an F-algebra S is found. If ᾱ ITEF(Φ) B, then since S is a model of ᾱ MB F Φ, we get S B by applying theorem 1. Therefore α FL B by (16). Let S 0 be the absolutely free term algebra in the signature F {[], f}. Then let S def = S,... be the F-expansion of S 0 obtained by interpreting the function symbols [ ] as the switching function, and f by { f S 1 if t = A t for some formula A such that α FL A, (t) = 0 otherwise. We want to verify that S is a model of ᾱ MB F Φ, and (16) holds. First of all, an element of ᾱ is of the form Ā with A α. So fs (A t )=1, which is exactly the F-equation Ā. Thus S is a model of ᾱ. S MB F is trivial since [ ] is a switching function on S def = the domain of discourse of S. If A Φ, then A A 01 or A A mp or A B for some FL-axiom B. S A 01 holds trivially simply because 0 and 1 are distinct symbols. (Hence they are interpreted as distinct individuals in S.) Next we will show that S A mp is impossible. Suppose that [f(x), 1, [f(x y), 1, f(y), 1], 1] 1 does not hold in S. Then f S (b 1 ) = 1, f S (b 1 b 2 ) = 1 and f S (b 2 ) 1 for some b 1,b 2 S. Thus, by construction of S, we must have b 1 B1 t and

11 Reduction of Hilbert-type proof systems 79 b 2 b 2 (B 1 B 2 ) t for some formulas B 1, B 2 such that α FL B 1 and α FL B 1 B 2. (We are using the fact that the map A A t is one-to-one.) But then α FL B 2 which contradicts f S (b 2 ) 1. If A B for some FL-axiom B, then since α FL B, we should have f S (B t )= 1inS, which means exactly S B. (Recall that B t is a (F {[], f})-term and at the same time a member of S.) We have shown that S is a model of ᾱ MB F Φ as desired, and hence it remains to show (16). But S B S f(b t ) 1 B t B1 t for some B 1 with α FL B 1. Since the map A A t is one-to-one, we must have B B 1. Thus we have shown (16) and consequently we are done. 5. Conclusion P-simulation of proof systems, first appeared in [6], is an interesting notion to study, because it gives us a tool to measure the relative efficiency of various proof systems. In this paper we generalize this notion of p-simulation to polynomial reduction and present an appropriate definition in [4]. Then we show that the if-then-else equational proof system ITE can prove all theorems of propositional calculus without losing any substantial efficiency. We could state this efficiency issue as follows: there exist constants k 1,k 2 > 0 such that for any proof ψ in FL with size l, there exists a corresponding proof ψ in ITE with size k 1 l+k 2. Moreover the translation ψ ψ is algorithmic where the number of steps required in the translation is asymptotically bounded by a linear function of size of ψ. Although we presented the reduction of only one particular proof system FL to ITE, it should be clear that our method is applicable to any Hilbert type proof system for propositional logic. In fact the construction developed in this paper is strong enough to linearly reduce the first-order logic to if-then-else equational logic, which is presented in [7]. References 1. N. Arai, A proper hierarchy of propositional sequent calculi, Theoretical Computer Science 159 (1996), N. Arai, Relative efficiency of propositional proof systems: resolution vs. cut-free LK, Annals of Pure and Applied Logic 104 (2000), 3 16

12 80 Joohee Jeong 3. S. Bloom and R. Tindell, Varieties of if-then-else, Siam J. Computing 12 (1983), S. Burris, Discriminator varieties and symbolic computation, J. Symbolic Computation 13 (1992), S. Burris, manuscript, S. Cook and R. Rechkow, The relative efficiency of propositional proof systems, J. of Symbolic Logic 44 (1979), no. 1, J. Jeong, Linear reduction of first-order logic to if-then-else equation logic, in preparation 8. J. Krajicek and P. Pudlak, Propositional proof systems, the consistency of first order theories and the complexity of computations, J. of Symbolic Logic 54 (1989), no. 3, R. McKenzie, On the spectra, and negative solution of the decision problem for identities having finite nontrivial model, J. Symbolic Logic 40 (1975), A. Mekler and E. Nelson, Equational Bases for if-then-else logic Siam J. of Computing, 18 (1989), J. Messner and J. Torán, Optimal proof systems for propositional logic and complete sets, Lecture Notes in Computer Sci. 1373, Springer, Berlin, 1998, STACS 98 (Paris), A. Urquhart, The complexity of Gentzen systems for propositional logic, Theoretical Computer Science 66 (1989), Joohee Jeong received his BS from Seoul National University and Ph.D. at the University of California at Berkeley under the direction of Prof. R. McKenzie. His research interests focus on logic, universal algebra and their application to programming language semantics. Department of Mathematics Education, Kyungpook National University, Puk-gu Sankyuk-dong 1370, Daegu , Korea. jhjeong@mathed.knu.ac.kr