Proof-terms for classical and intuitionistic resolution - Semantic Scholar

Proof-terms for classical and intuitionistic resolution Eike Ritter David Pym Lincoln Wallen School of Computer Science Queen Mary & West eld College Computing Laboratory University of Birmingham University of London Oxford University Abstract

We extend Parigot's -calculus to form a system of realizers for classical logic which re ects the structure of Gentzen's cut-free, multiple-conclusioned, sequent calculus LK when used as a system for proof-search. Speci cally, we add (i) a second binding operator,  , which realizes classical, multipleconclusioned disjunction, and (ii) explicit substitutions, , which provide sucient term-structure to interpret the left rules of LK. A necessary and sucient condition is formulated on realizers to characterize when a given (classical) realizer for a sequent witnesses the intuitionistic provability of that sequent. A translation between the classical sequent calculus and classical resolution due to Mints is used to lift the conditions to classical resolution, thereby giving a characterization of the intuitionistic force of classical resolution. One application of these results is to allow standard resolution methods of uniform proof-search to be used directly for intuitionistic logic but, more signi cantly, they support a type-theoretic analysis of search spaces in both classical and intuitionistic logic.

1 Introduction 1.1 Local methods for intuitionistic logic It is standard practice to draw a sharp distinction between local methods of automated deduction for classical logic, inspired by techniques such as Robinson's resolution [28] and Maslov's inverse method [13], and global methods, those inspired by Gentzen's sequent calculus [10] and Smullyan's tableaux systems [29]. Local methods have the advantage of deriving small independent objects like clauses bearing a simple logical relationship to their parents and allowing wholesale dynamic simpli cation of the search space through operations such as subsumption. Global methods, on the other hand, are typically analytic (e.g., [9]), bene t from complex implementation methods (e.g., [1, 3]) and produce search spaces which are deep, but well-focussed on the shortest proofs available; proofs which are nevertheless (usually) exponentially longer than the shortest proofs available in their local, non-analytic, counterparts (cf. [5]). The characterization outlined above is broadly coherent for a truth-functional logic such as classical propositional logic. For a non-classical logic, such as intuitionistic propositional logic, global methods are more easily developed (see e.g., [8, 23, 17]) and, as Mints points out in his [15], many attempts to formulate local methods fail to preserve the essential properties of local methods for classical systems. He goes on to formulate a list of criteria by which a system can qualify as \resolution", and to present a local method which satis es them. We have pointed out before in [31, 33, 23] that local methods are extended to the classical predicate calculus by means of Herbrand's Theorem and strong quanti er normal forms; the logical properties of the derived objects arising from the strong de nability properties of classical logic (Skolem-Godel Theorem). A careful treatment of this generalization, as contained in Bibel's paper [4] for example, reveals that the  This research was supported in part by UK EPSRC grants GR/J46616 and GR/K41687 under the common title, \Search Modules I: Representation and Combination of Proof Procedures".

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essential step in achieving a local treatment of quanti ers is the localization to connections (atomic clashes) of a global condition on instantiating terms.1 We continued in this vein to show how to transform global conditions for modal connectives, easily stated in terms of global methods such as tableaux, into local conditions on connections [32, 33]. The step to acceptable resolution calculi satisfying Mints' criteria is then straightforward, q.v. [16]. It is the propositional structure of the resolution method that gives it its combinatorial strength. The viewpoint outlined above suggests that in obtaining a local method for a non-classical logic we try to preserve the propositional structure of the standard method as far as is possible, modifying only the condition under which a particular clash or connection is sound. The complexity of the local soundness check should be small compared with the complexity of the propositional search space. For intuitionistic propositional logic this approach is of particular signi cance. Gentzen [10] formulates the intuitionistic sequent calculus LJ as a restriction of the sequent calculus for classical logic LK. The restriction concerns the use of weakening on the right. Since LJ is a restriction of LK, the latter is complete for intuitionistic logic, but not sound. By studying the structure of LK derivations under permutation of inference rules, it is possible to assess their intuitionistic force and hence use classical search with its simple combinatorics to determine intuitionistic provability. In [25, 26], we developed a system of terms based on Parigot's -calculus to serve as a system of realizers for sequent derivations in a disjunction-free fragment of classical and intuitionistic logic. Combinatorial operations on the realizers were used to enlarge the set of classical derivations that could be considered to have non-trivial intuitionistic force leading to a simpli cation of the search space. In [27] we used Mints' Maslov-inspired translations between resolution systems and the sequent calculus, to extend the analysis to resolution yielding a characterization of the search space for that fragment of intuitionistic logic in terms of the search space generated by classical resolution. In this paper we extend the results of [25, 26] and [27] to include disjunction thereby simplifying and completing the picture of the intuitionistic search space in terms of classical resolution. This involves two steps. The rst is the formulation of a suitable system of realizers for full classical logic and the characterization of the intuitionistic force of such realizers; the second is the lifting of that a characterization to resolution via Mints' translations. An overview of the technical results follows.

1.2 Overview of technical results In x 2 we develop the key properties of the -calculus: a system of realizers for full classical logic

re ecting the properties of Gentzen's sequent calculus LK. This extends the results presented in [25, 26]. After a motivational discussion, we establish local con uence, strong normalization and con uence properties for reductions in the fragment of the calculus without explicit substitution (Theorem 6). The incorporation of explicit subsitution is then considered and proved to be of only syntactic signi cance (Proposition 7). In x 3, we summarize and extend the results of [25] to account for disjunction, as presented in [26]. A translation of sequent derivations into  terms is given for classical logic. The terms are seen as realizers for the derivations. A realizer is said to be intuitionistic if it satis es a certain structural condition related to weakening and rule permutation. A sequent is intuitionistically provable if there is a classical sequent proof of it whose realizer is intuitionistic (Theorem 12). By de ning a ( nite) restricted operation of permutation on realizers we obtain a completeness result (Theorem 15): if a sequent is intuitionistically provable, then there is a classical derivation for which some permutation of its realizer is intuitionistic. True to the spirit of the outline above, the intuitionistic search space is rendered as a restriction of the classical search space together with a computable test for intuitionistic soundness. In x 4, we show that (inessential variants of) Mints' translations establish tight connections between uniform proofs and resolution derivations (Theorem 23). We also show that for classical logic permutations in the resolution search space correspond to permutations in the sequent search space (Proposition 26). The results of x 3 then give realizers for (classical) resolution derivations. In x 5, we use the results summarized in x 3 to assess the intuitionistic force of (classical) resolution derivations. An intuitionistic soundness result for resolution is proved as Theorem 29. By a careful analysis, 1

Popularly known as the \occurs-check" in the uni cation algorithm.

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and modi cation of Mints' translation, the class of resolution derivations that can be deemed to have nontrivial intuitionistic force can be extended. With respect to this extended class of derivations a completeness result is established as Theorem 30.

2 Classical disjunction and the -calculus In this section, we describe our variation on Parigot's -calculus, which we call the -calculus (cf. [25, 26]). Our pattern of development is a standard one, similar to that found in, for example, [11]. The -calculus provides a term calculus for the (; :)-fragment of classical propositional natural deduction: i.e., realizers for a calculus in which multiple-conclusioned sequents can be derived without impure constraints [6]. Consequently, the form of the typing judgement in the -calculus is ? ` t : A ;
, where ? is a context familiar from the typed -calculus and
is a context containing types indexed by names, ; ; : : :, distinct from variables in ?. These types are written as A etc.. Names can be bound and replaced by substitution according to the various inference and reduction rules which we describe. The idea is that each -sequent has exactly one principal formula, A, on the right-hand side, i.e., the leftmost one, which is the formula upon which all introduction and elimination rules operate. This formula is the type of the term t. The basic grammar of  is as follows:

t ::= x j x: A : t j tt j []t j  : t The corresponding inference rules are given below in Figure 1. ?; x: A ` x: A;
Ax ?; x: A ` t: B;
? ` t: A  B;
? ` s: A;
 E ? ` x: A:t: A  B;
 I ? ` ts: B;
? ` t: A ;
 ? ` t: A;
[ ]  ? ` []t: A ;
? ` :t: A;
? ` t:
? ` t: A; A ;
[ ] ? ` []t: A ;
? ` :t: A;
 ? ` t: ?;
? ? ` t:
? ` [ ]t:
? ` :t: ?;
? The second instances of the rules [ ] and  model contraction and weakening respectively. In the the last rule for ?, the name  does not occur in
. Figure 1: Rules for well-typed -terms The term []t realizes the introduction of a name. The term :[ ]t realizes the exchange operation: ? ` t: B; A ;
? ` :[ ]t: A; B ;
exc i.e., if A was part of \side-context" of the succedent before the exchange, then A is the principal formula of the succedent after the exchange. Taken together, these terms also provide a notation for the realizers of contractions and weakenings on the right of a multiple-conclusioned calculus. It is also easy to detect whether a formula B in the right-hand side is, in fact, super uous, i.e., there is a derivation of ? ` t: A;
0 where
0 does not contain B ; it is super uous if is not a free name in t. The negation of a formula A is modelled in the -calculus as A ?, and the two rules for ? express the fact that ? can be added and removed to the succedent at any time. 3

The variation presented below has two aspects. Firstly, in addition to implicational types, we include both conjunctive (product) and disjunctive types. The addition of the conjunctive types follows the standard method for adding products to the simply-typed -calculus. The addition of disjunctive types requires a more subtle approach, as we will see below. For the conjunctive types we add the following constructs to the grammar for terms:

t ::= ht; ti j (t) j 0 (t) The rules of inference associated with these terms are given below in Figure 2. ? ` t: A;
? ` s: B;
^ I ? ` ht; si: A ^ B;


? ` t: A ^ B;
^ E ? ` (t): A;


? ` t: A ^ B;
^ E ? ` 0 (t): B;


Figure 2: Conjunctive rules for well-typed -terms The key point in the addition of disjunctive types is naturally explained in the setting of the multipleconclusioned sequent calculus. One possible formulation, with a single minor formula in the premiss, follows the form of the right rule for disjunction in LJ,2 ? ?! Ai ;
(1) ? ?! A1 + A2 ;
i = 1; 2; yielding the usual addition of sums (coproducts) to the realizing -terms:

t ::=

inl(t)

j

inr(t)

j

case t of inl(x)

) t j inr(y) ) t

An alternative formulation exploits the presence of multiple conclusions thus: ? ?! A1 ; A2 ;
: (2) ? ?! A1 _ A2 ;
This formulation is the more desirable as a basis for proof-search as it maintains a local representation of the global choice between A1 and A2 . In particular, this is the form of disjunction that is exploited by classical (and hence intuitionistic) resolution, which is the main application of the realizers in this paper. But from the point of view of the -calculus, this latter formulation presents a new diculty. Suppose the -sequent ? ` t: A; B ;
is to be the premiss of the _I rule. In forming the disjunctive active formula A _ B , we move the named formula B from the context to the active position. Consequently, _I is a binding operation. Therefore, we introduce the following additional constructs, to form the grammar of  -terms:

t ::= h it j  : t The term  : t introduces a disjunction and the term h it eliminates one. The associated inference rules are given below in Figure 3. ? ` t: A _ B;
t: A; B ;
_I ? ?` ` :t : A _ B;
_E ? ` h it: A; B ;
2

Figure 3: Disjunctive rules for well-typed  -terms Note that we use ` in typing judgements and ?! in the sequent calculus. 4

The de nition of the reduction rules requires not only the standard substitution t[s=x], but also a substitution for names t[s=[]u], which intuitively indicates the term t with all occurrences of a subterm of the form []u replaced by s. To de ne this notion, we need the notion of a term with holes. Such a term C with holes of type A is a  -term which may have also the additional term constructor with the rule ? ` : A;
. The term C (u) denotes the term C with the holes textually (with possible variable capture) replaced by u. Then we de ne t[C (u)=[]u], where  is a name and u is a term, by x[C (u)=[]u] = x ([]t)[C (u)=[]u] = C (t[C (u)=[]u]) (hit)[[C (u)=[]u]] =  :C (:[ ]hit[C (u)=[]u]) and de ned on all other expressions by pushing the replacement inside. The reduction rules appear in Figure 5. Remark To avoid loops during reduction, the rules  ,  ^ and   do not apply if the term t in which the name  is changed is equal to hit0 , and  does not occur in t0 . Although the LK-like _ can derived from the LJ-like + (and vice versa ) via weakening, they are not \isomorphic" in the sense that there exists a bijection between ft j ? ` t : A1 + A2 ;
is provableg and ft j ? ` t : A1 _ A2 ;
is provableg: which respects the congruence on terms induced by normalization. Indeed, the imposition of such a bijection forces that, for every ?,
, t and A, ft j ? ` t : A;
is provableg have at most one element. The proper formulation of this result requires more sophisticated semantic techniques and will be presented by some of us (Pym and Ritter) elsewhere. The second aspect concerns the addition of explicit substitutions, ft=xg. These are crucial for an analysis of proof-search in that they provide representative realizers for possibly incomplete sequent derivations. The raw terms of the -calculus are given by extending the grammar with the following -construct: t ::= t fu=yg The corresponding inference rule is given below in Figure 4.3 ?; w: A ` t: B;
? ` s: A;
 ? ` t fs=wg : B;
Figure 4: Explicit substitution for well-typed -terms The reduction rules for the -calculus, which are those of  together with those necessary to avoid interference between  -reductions and explicit substitution, are given in Figure 5. Note that in the noninterference reductions, there is no case of the form tfu=xgfu0=x0 g ; : : : : we do not compose explicit substitutions.4 3 In [25, 27], we took a rule for  based on L, ?; w : B ` t : C;
? ` s : A;
L: ?; x : A  B ` tfxs=wg Such a rule is not adequate for the interpretation _L (see De nition 8) which requires -explicit substitutions that are not of the form fxs=wg. 4 In the corresponding de nitions of [25, 27], we also excluded the case xft=xg. In fact, such an exclusion is inessential and unnecessarily complicates the metatheory.

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We use explicit substitution in two places. The rst is in the translation of the  L-rule, in which we have to delay the substitution of the term corresponding to the left branch in the derivation ? ?! A;
?; B ?!
?; A  B ?!
The delay ensures that in the -calculus there is a subterm corresponding to the left branch even if the formula B in the sequent ? ?!
was introduced by weakening. The second place is when we translate the _L-rule. For the same reason again, we have to delay the substitution for the terms corresponding to the two sequent derivations which are the hypothesis of the _L-rule.

   ^ ^  

?

; ; ; ; ; ; ; ; ; h i :t ; ? :[ ? ]t ;

(x: A:t)s (AB :t)s :[]s [ ](:s) (A^B :s) 0 (A^B :s) (ht; si) 0 (ht; si) h i(:s)

xfu=xg ; ; ; ; ; ; ;

(x: A:t)fu=z g (ts)fu=z g ([]t)fu=z g (:t)fu=z g (h it)fu=z g ( :t)fu=z g

t[s=x]  B :t[[ ]us=[]u] s if  not free in s s[ =]  A :s[[ ](u)=[]u]  B :s[[ ]0 (u)=[]u] t s s[ =] :t[[]his=[ ]s] t u x: A:tfu=z g tfu=z gsfu=z g []tfu=z g :tfu=z g hitfu=z g  :tfu=z g

We assume standard variable-capture conditions. We also assume that any name of type ? is dierent from any other name in the term. The names in the right hand side of the  -rules are fresh names. Figure 5: Reduction rules of the -calculus The presentation of the -,  - and -calculi taken here is not the only possible one. An alternative would be to reformulate the E-, ^E- and _E-rules as left rules. It would then be necessary to take not only the usual cut rule, and then establish cut-elimination, 5 but also the -rule given above. The advantage of such an approach would be that we would obtain specializations of the -rule which would then amount to explicit versions of the L-rule and _L-rule. Parigot gives only reduction rules for -reduction. To model uniform proofs, we also need extensionality, i.e., we must have the -rules. We will work with long -normal forms in the sequel. We introduce them here as expansions; that is, each term of functional type is transformed into a -abstraction, each term of product type into a product and each term of sum type into a term  :t0 . These rules are  t ; x: A:tx ^ t ; h(t); 0 (t)i

_ t ; :hit In these rules, we assume that t is neither a -abstraction, nor a product, nor a term :t0 , nor that t occur as the rst argument of an application, or as the argument of a projection  or 0 or of some term h i . In the  -, ^ - and _ -rules, we also assume that t is of function type, product type and sum type respectively. 5

Such a programme was executed for the (dependently-typed) -calculus in [21].

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These -rules generate critical pairs which give rise to additional reduction rules. As an example, consider the term t = []:s, where  is a name of type A  B . This term can reduce via an -expansion to []x: A:(:t)x, and via a  -rule to t. The reduction from []x: A:(:t)x to t can be seen as a generalized renaming operation. This operation is denoted by t f g and is de ned as follows:

De nition 1 De ne the generalized renaming of a  -term t by a name , written t f g, by induction over the type of the name as follows: Atomic type: (:t) f g = t[ =]; A  B : (x: A:t) f g = t f 0 g [[ ]x: A:u=[ 0 ]u] for some fresh name 0 if x occurs in t f 0 g only within the scope of [ 0 ]u, otherwise (x: A:t) f g is unde ned; A ^ B : If t = ht1 ; t2 i and for some names 1 and 2 of type A and B respectively, t2 f 2 g arises from t1 f 1 g by replacing each subterm [ 1 ]s1 recursively by some subterm [ 2 ]s2 , then t f g = t1 f 1 g [[ ]hs1 ; s2 i=[ 1 ]s1 ]; A _ B : (:t) f g = t f 0 g [ ]:u=[ 0 ]u] for some fresh name 0 if  occurs in t f 0 g only within the scope of [ 0 ]u, otherwise (:t) f g is unde ned. The additional reduction rule, which is called   , can now be stated as:



[]t ; t fg

(3)

Note that this reduction rule specializes to the rule  if  is a name of atomic type. Because the outermost bindings : of names of atomic type disappear by an application of the   -rule, this rule cannot give rise to reduction sequences t ; t. Logically, the   -rule amounts to taking a right-rule and moving it above a structural rule (i.e., weakening, contraction) applied to its principal formula. Our rst lemma gives the local con uence of  , extending Parigot's result for  [19].

Lemma 2 The notion of reduction in the  -calculus is locally con uent. Proof We show that all critical pairs can be completed. For critical pairs arising from the rules ,   and

 this is part of the con uence of Parigot's -calculus. We show only a few characteristic cases for the rule   (3). The rst case is an overlap with the  -rule. The term

can reduce via   to

u = (:


[](x: A:


0 :


[0 ]t


)


)s

0 :


[0 ](x: A:


0 :


[0 ]t


)s


which in turn reduces via and  to 0 :


[0 ]t[s=x]


: The other reduction sequence via the additional rule is

 u ; (:


[]x: A:t


)s  ; 0 :


[0 ](x: A:t)s


; :


:[0 ]t[s=x]


: The second case we consider is the overlap of the  -rule with the -expansion. This is the case which gives rise to the additional reduction rule   . For this, consider the term w = []:t, which can be reduced via the  -rule to t. The reduction sequence via the rule   is as follows:  w ; [](x: A:(:t)x) ; []x: A:0 :t[[0 ]ux=[]u]  ; t[[]x: A:ux=[]u];

which is t modulo some -expansions and/or -reductions. 7

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The choice of a distinguished formula on the right-hand side of the sequent is required to ensure strong normalization and con uence. We follow the original proof of Tait [30]. The key point is to de ne a subset of strongly normalising terms, the so-called reducible terms with additional closure properties which make it possible to show that a term is reducible. Parigot's proof for the -calculus [20] uses Girard's reducibility candidates because he presents an impredicative second-order calculus. as we consider a rst-order calculus only, we can de ne the set of reducible terms by induction over the type structure. This proof has been presented in [26], but it is reproduced here for completeness.

De nition 3 Assume ? ` t: A;
. By induction over the structure of types in A and
we de ne sets of reducible  -terms of type A;
, written Red(A;
), and for each term ? ` t: A;
closure terms of type A;
, written clA;
(t) or cl(t) for short, as follows:  If A and
are all atomic types or ?, then Red(A;
) = ft j ? ` t: A;
and t is SNg and clA;
(t) = ;;  If one of the types in A or
is not a atomic type or ?, de ne Red(A;
) to be the set of all terms ? ` t: A;
such that all terms in cl(t) are reducible;  The set of closure terms clA;
(t) is de ned as the union of the sets

fts j s 2 Red(B;
)g f(t); 0 (t)g f:[ ]hit; hitg f(:[ ]t)s j s 2 Red(B;
)g f(:[ ]t)); 0 (:[ ]t)g f :[]h i:[ ]t; h i:[ ]tg

if A = B  C if A = B ^ C if A = B _ C if BC 2
if B^C 2
if B_C 2
:

n (t) to be the set of Next we de ne the closure properties of the set of reducible terms. We de ne clA;
all terms tn such that there exists a sequence t0 ; t1 ; : : : ; tn with ti 2 cl(ti?1 ) and t = t0 , for all 1  i  n.

Lemma 4 Every set of reducible  -terms has the following properties: S1 If t is reducible, then t is strongly normalizing; S2 For all variables x, each element in cln(x) is reducible; S3  If t[s=x] is reducible, so is each element of cln(x: A:t);  If t and s are reducible, so is each element of cln (ht; si);  If t[ =] is reducible, so is each element of cln(:t). S4 If t is reducible, so is cln(:[ ]t). Proof We split each of conditions S2, S3 and S4 into two conditions, which we prove by induction, and

which together imply the original conditions. We use simultaneous induction over the types of A and
to show the following properties: S1 If t with ? ` t: A;
for some ? is reducible, then t is strongly normalizing; S2' If for any element t of cln(x) with ? ` t: A;
for some ? all elements of clm(t) are SN for any m  0, then t is reducible; S2" If ? ` x: A;
for some ?, then all elements of clm(x) are SN for any m  0; S3'  If all elements of clm(x: A:t) are SN for all m  0, then each element of cln(x: A:t) is reducible if ? ` cln (x: B:t): A;
for some ?; 8

 If all elements of clm (t) and clm(s) are SN for all m  0, then each element of cln(ht; si) is reducible if ? ` cln (ht; si): A;
for some ?;  If all elements of clm(:t) are SN, then each element of cln (:t) is reducible if ? ` cln(:t): A;


for some ?. S3"  If ? ` x: B:t: A;
for some ? and t[s=x] is reducible for all reducible ? ` s: B;
, then each element of clm (x: B:t) is SN;  If ? ` ht; si: A;
for some ? and t and s are reducible, then each element of clm (ht; si) is SN;  If ? ` :t: A;
for some ? and t[ =] is reducible for each name , then each element of clm (:t) is SN; S4' If t is reducible and clm(:[ ]t) is SN for all m  0, then cln(:[ ]t) is reducible if ? ` cln(:[ ]t): A;
for some ?. S4" If ? ` t: A;
for some ? and t is reducible, then clm(:[ ]t) is SN for all m  0. The induction proceeds now as follows: S1 If A and
are all atomic or ?, then t is SN by de nition. If not, one does a case analysis of A and
. We consider here only the cases of A = B _ C and A = B  C . In the rst case, if t ; t0 , then either hit ; hit0 , or t0 = :hit00 , and t ; t00 via all reduction rules except top-level -expansions. Hence any in nite reduction sequence starting with t can be extended to an in nite reduction sequence of hit. This is a contradiction because by induction hypothesis, hit is SN. Now assume A = B  C . Choose a variable x of type B which does not occur freely in t. By S2" and S2', x is reducible. Hence by de nition, tx is reducible, and SN by S1 by induction hypothesis. Hence all reduction sequences of t which do not involve outermost -expansions terminate. The case of an outermost -expansion is treated in the same way as in the case of A = B _ C ; S2' If A and
are all atomic or ?, the claim is trivial. If not, we have to show that all elements of cln+1(x) are reducible. This follows directly from the induction hypothesis; S2" Here we do an induction over m and use the fact that by induction hypothesis for all reducible terms s which occur in clm(x), cl(k (s) is SN for all k. In particular, the restriction of the  -rules mentioned in the remark on page 5 prevents an in nite loop in the term (:[ ]hix)s; S3' Same argument as for S2'; S3" Here we again use induction over m. We consider only one case; all other cases are similar. Consider a reduction sequence (:[ ](x: B:t)s)u ; 0 [ ](x: B:t[[0 ]wu=[]w])s[[0 ]wu=[]w] ; 0 :[ ]t[[0 ]wu=[]w][s[[0 ]wu=[]w]=x] = 0 :[ ]t[s=x][[0 ]wu=[]w] for an element of cl2 (x: A:t). By induction hypothesis (S1) (:[ ]s)u, (:[ ](x: A:t)x)u and 0 :[ ]t[s=x][[0 ]wu=[]w] are SN. Hence (:[ ](x: B:t)s)u is SN; S4' Same argument as for [S2']; S4" One shows that any in nite reduction sequence for s 2 clm(:[ ]t) yields an in nite reduction sequence for an element of clm (t), which is SN by the induction hypothesis (S1).

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The key theorem states that every term is reducible. For this we need a generalized induction hypothesis which includes all possible substitutions of reducible terms for free variables and all mixed substitutions for free names. Mixed substitutions arise as contracta of the  -rules in the same way as the ordinary substitution arises as a contractum of the -rule. 9

Theorem 5 For each  -term t such that ? ` t: A;
and reducible terms si and ui, all terms t[si =xi ;

[0j ]wuj =[j ]w ; [0k ](u)=[k ]u ; [0m ]0 (u)=[m ]u ; [0n ]h n iu=[n]u ; r =r ];

are reducible, where the names j , k , m and n range over all subsets of names in
of implication type, conjunction type, conjunction type and disjunction type respectively and each of the m is dierent from each of the k . The names r form some subset of the names in
.

Proof We write f for the substitution si =xi ;

[0j ]wuj =[j ]w ; [0k ](u)=[k ]u ; [0m ]0 (u)=[m ]u ; 0n ]h n iu=[n]u ; r =r

and write t[f ] for the application of the substitution f to t. The proof proceeds by induction over the derivation of t. xi : Obvious, as xi [f ] = si , which is reducible by assumption. x: A:t: By induction hypothesis t[f; s=x] is reducible for every reducible term s, hence (x: A:t)[f ] is reducible by S3. ts: By induction hypothesis, t[f ] and s[f ] are reducible, hence by de nition of reducibility t[f ]s[f ] = (ts)[f ] is reducible. A :t: By induction hypothesis, t[f ] is reducible. Hence by S4, :t[f ] is reducible as well. [A ]t: If  occurs in f only as part of a substitution [ =] or not at all, then ([]t)[f ] = [](t[f ]) or [ ](t[f ]), depending whether  occurs in f or not. By induction hypothesis all elements of cl(t[f ]) are reducible. Because cl([](t[f ]))  cl(t[f ]) and cl([ ](t[f ]))  cl(t[f ]) respectively, []t[f ] is reducible. So now assume that  does occur in f in a dierent position. In this case  :(([]t)[f ]) is an element of cl(t[f ]), hence it is reducible by induction hypothesis. ht; si: By induction hypothesis, t[f ] and s[f ] are reducible, hence by S3, ht[f ]; s[f ]i is reducible. (t), 0 (t): By induction hypothesis, t[f ] is reducible, hence (t[f ]) and 0 (t[f ]) are reducible by de nition. :t: By induction hypothesis, t[f; =] is reducible. Hence property S3 now implies the claim. h it: By induction hypothesis, t[f ] is reducible, and hence by de nition h it[f ] is reducible, too.

2 Now we are in a position to deduce con uence from local con uence and termination. Theorem 6 The  -calculus is con uent.

Proof The proof is a straightforward application of Newman's Lemma [12], which states that a locally con uent and terminating notion of reduction is con uent. 2 Now we turn to the addition of explicit substitution, to get the -calculus. Normally, substitution is a meta-operation and not part of the calculus itself. As explained earlier, for the modelling of the  L- and the _L-rule we need substitution to be part of the calculus. Hence we add a term t fs=xg for explicit substitution (corresponding to the implicit t[s=x]) and turn the inductive de nition of substitution as a meta-operation into reduction rules. We call the rules for explicit substitution -reduction rules. The following lemma shows that the normal forms of the extended calculus are the normal forms of the  -calculus, and that con uence is preserved. Proposition 7 The -calculus is con uent and normalizing. 10

Proof For each -term t we de ne a  -term [ t] as the term obtained by transforming all explicit substitutions into implicit ones, i.e., replacing each subterm s fu=xg by s[u=x]. Now show by an induction over the structure of t that the two following properties hold:   [ t] , and if t ; s, then also [ t] ; [ s] ; (i) For each -term t, we have t ; (ii) Each reduction t ; s in the  -calculus is also a reduction in the -calculus.

For con uence, assume that t ; t1 and t ; t2 . By (i), we have [ t] ; [ t1 ] and [ t] ; [ t2 ] . The con uence of the  -calculus implies the existence of a term s such that [ t1 ] ; s and [ t2 ] ; s. By (ii), this sequence is also a reduction sequence in the -calculus, hence we obtain t1 ; [ t1 ] ; s and t2 ; [ t2 ] ; s. Normalization follows by a similar argument: To compute the normal form of t, rst reduce t to [ t] and then apply the  -reduction to [ t] to obtain the normal form of t. 2 Normalization and con uence suce to show that t and s are equal if and only if [ t] and [ s] are equal. This means that the explicit substitution is only a syntactic addition and does not change the semantics of the  -calculus.

3 The intuitionistic force of classical search In [25, 26], we presented a characterization of when search by means of the classical sequent calculus yields sucient evidence for provability in a fragment of intuitionistic logic, namely propositional intuitionistic logic with implication, negation, conjunction and, in [26], disjunction. The characterization takes account of the rule permutability properties of both logics. We de ned a translation, [ ?] , from classical sequent derivations into terms of the -calculus [18, 25, 26] and gave a feasible criterion for the corresponding -term to determine intuitionistic validity of the endsequent. Based on this characterization, we de ned a proof procedure which extends the notion of uniform proof as de ned by Miller et al. [14]. The procedure was shown to be sound and complete for the fragment considered. Here we con ne our attention to a summary of the main points. In the analysis of the relationship between classical and intuitionistic logics, the r^ole of disjunction is informative: there are sequents ? ?! B , provable in LJ, in whose cut-free LK proof the last inference is necessarily an _L. A simple example is A _ B ?! (C  B ) _ B : the only cut-free sequent proof is

A; C ?! A R B ?! B A ?! C  A _R _R A ?! (C  A) _ B B ?! (C  A) _ B _L: A _ B ?! (C  A) _ B

The same is true if we adopt a multiple-conclusioned intuitionistic calculus. In this case, the sequent

A _ B ?! C  A; B is provable only by reducing A _ B rst. The search procedures presented in [25] rely heavily on the possibility of reducing the RHS rst, hence they cannot immediately be generalized to the full fragment. Miller et al. also encountered this issue and de ned a smaller class of formul, called hereditary Harrop formul, whose cut-free sequent proofs have the property that no disjunction occurs in any antecedent of any sequent. Hence these sequent proofs contain no instance of the _L-rule, and each sequent has a proof which starts with the appropriate right introduction rule if the RHS is not an atom. This is also true for the multiple-conclusioned intuitionistic sequent calculus: each sequent derivation ? ?!
, where ? and
are hereditary Harrop, has as the last rule an appropriate right (introduction) rule for some formula in
if not all formul in
are atoms. Hence the results of [25] carry over to the intuitionistic sequent calculus with disjunction if the formul are restricted to the hereditary Harrop fragment. These results are reported in [26]. 11

As we shall see below, translations of resolution derivations do not only involve hereditary Harrop formul; consequently Miller's results are not directly applicable for our purposes. In order to capture the case of resolution derivations, we must generalize the notion of uniform proof to include all proofs that have all possible occurrences of the _L-rule as close to the root as possible. We call such proofs weakly uniform. Corresponding to weakly uniform proofs, we have a class of weakly hereditary Harrop formul for which weakly uniform proofs are complete.6 We show in this section that each sequent has a weakly uniform proof which arises as the translation in sequent calculus of a normal -term of the appropriate type. In general, even the existence of such a weakly uniform proof is not enough to derive a search procedure, as formulated in [25]: we also need the ability to swap right introduction rules in the multiple-conclusioned intuitionistic sequent calculus. This is true for weakly hereditary Harrop formul, but not for all disjunctive formul. But for resolution derivations the class of weakly uniform derivations is small enough; two weakly uniform derivations dier only in the order of the  L-rules, and hence the search procedure from [25] still applies. A -term is identi ed as intuitionistic if the free names of the term model weakening. For example, consider the sequent B ?! A  B; D  E . An intuitionistic search for a proof of this sequent based on LJ will be successful only if we reduce the formula A  B rst; i.e., closer to the root of the derivation. This is not so classically. If a search according to the multiple-conclusioned classical rules of LK results in a (classical) proof in which the formula D  E is reduced at the root of the proof, we would need to detect from the corresponding -term whether this reduction can be considered super uous. Therefore, if we can judge this property, we can use the classical proof to determine that the sequent is intuitionistically provable. In fact, an intuitionistic proof of the sequent can be constructed from the data to hand. In the -term x: A: :[ ]y: D::[ ]b, corresponding to the obvious classical proof outlined above, this amounts to determining that certain subterms, here the abstraction y: D, model weakening. Full details are provided in [25, 26]. Here we summarize the main de nitions and theorems. We start with the translation of LK-sequent derivations into -terms. We shall use the following notation: if  is a derivation whose last rule is R applied to the derivations 1 ; : : : ; n , we write (1 ; : : : ; n ); R for .

De nition 8 Let : ? ?! A;
be a classical sequent derivation and suppose that each moccurrence of a formula in ? and
has a label, i.e., we have ? = x1 : A1 ; : : : ; xn : An and
= B1 1 ; : : : ; Bm . (These labels turn into variables and names in the -calculus, hence we also use them for the derivations.) We de ne a -term  satisfying ? ` [ ] : A;
by induction over the structure of  as follows (note the clause for the exchange rule):

Axiom: Suppose  : ?; x: A ?! A;
is an axiom, then [ ] def = x; Exchange: Suppose : ? ?! A; B ;
, and

0 = ; exc: ? ?! B; A ;
: We de ne [ 0 ] to be the normal form of the term  :[][[] with respect to the rules  and  ; ^L: Suppose we have the derivation

: ?; x: A; y: B ?! A;
^L; ; ^L: ?; z : A ^ B ?! A;
then the corresponding -term is

[ ; ^L] def = [ ] [(z )=x; 0 (z )=y];

^R: Suppose we have the derivation 6

This class is described in x 4.

12

: ? ?! A;
: ? ?! B;
^R; (; ); ^R: ? ?! A ^ B;
then we de ne

[ (; ); ^R] def = h[ ] ; [ ] i;

 L: Suppose we have the derivation : ? ?! A; C ;
: ?; w: B ?! C;
L (; );  L: ?; x: A  B ?! C;
then we de ne [ (; );  L] to be the normal form of  :[ ][[ ] fx[ ] =wg with respect to the reduction rules  and  ;  R: Suppose we have the derivation

: ?; x: A ?! B;
R; ;  R: ? ?! A  B;
then we de ne [ ;  R] to be x: A:[ ] ; :L: Suppose we have the derivation

: ? ?! A; C ;
L ; :L: ?; x: :A ?! C;
then we de ne [ ; :L] to be  :[?]w fx[ ] =wg; :R: Suppose we have the derivation

: ?; x: A ?! B;
R; ; :R: ? ?! :A; B ;
then we de ne [ ; :R] to be the normal form of x: A:?:[ ][[] via the reduction rules  and ? ; _L: Suppose we have the derivation : ?; x: A ?!
: ?; y: B ?!
_L (; ); _L: ?; z : A _ B ?!
we de ne [ (; ); _L] to be the normal form of  :[ ][[ ] f:[ ][[] fhiz=yg =xg with respect to the reduction rules  and  ; _R: Assume we have a derivation : ? ?! A; B ;
; _R: ? ?! A _ B;
then we de ne [ ; _R] =  :[ ] .

13

A derivation constructed by a search can contain reductions which make no essential contribution to the existence of such a derivation. We want to be able to detect such reductions. For example, we want to be able to detect that the use of the  R rule to reduce the formula D  E in the derivation of B ?! A  B; D  E (the example above) is super uous and to do so by using the -term corresponding to this proof. We can then conclude that there is an intuitionistic proof of this sequent. For the -term representing this derivation x: A: :[ ]y: D::[ ]b ; this amounts to determining when a subterm (here the -abstraction over D) models weakening on the right. The technical details follow below.

De nition 9 We de ne weakening terms and weakening occurrences of names by induction over the structure of terms as follows: (i) :t is a weakening term if all occurrences of  in t are weakening occurrences; (ii) A term t of type ? is always a weakening term; (iii) ht; si is a weakening term if t and s are weakening terms; (iv) x: A:t is a weakening term if t is a weakening term and if x has only weakening occurrences in t; (v) The outermost occurrence of  in []t and hit is a weakening occurrence if t is a weakening term; (vi) :t is a weakening term if t is a weakening term and all occurrences of  are weakening occurrences; (vii) All occurrences of ? in t are weakening occurrences; (viii) The occurrence of the variable x in tx is a weakening occurrence if t is a weakening term and x is not free in t. In this case, the term tx is a weakening term as well; (ix) t fu=xg is a weakening term if t is a weakening term.

Let us reconsider the previous example. There are two -terms corresponding to the two derivations of B ?! A  B; D  E . The rst one, which corresponds to reducing A  B rst, is the term

x: A: :[ ]y: D::[ ]b ; and the second one, which corresponds to reducing D  E rst, is the term y: D::[]x: A:b : In both cases we have an intuitionistic -term because the -abstraction over D is a weakening term. This ex-

ample shows the parallelism obtained by using a classical sequent calculus: both intuitionistic subderivations of either of the classical proofs are considered simultaneously without any need for backtracking. Next we show the correctness of the criterion. The crucial point is that a weakening term corresponds to a super uous subderivation. The following lemma makes this precise. Lemma 10 Let  be a derivation : ?; A1; : : : ; An ?! A; B1; : : : ; Bm;
such that ?; a1 : A1 ; : : : ; an : An ` [ ] : A; B1 1 ; : : : ; Bm m ;


holds. If the variables ai do not occur in [ ] and if the j have only weakening occurrences, then there is a procedure to construct a sequent derivation of ? ?! A;
. Moreover, if [ ] is a weakening term, then there is a procedure to construct a derivation of ? ?!
. These procedures transform sequent derivations which have an intuitionistic subderivation into those with the same property.

14

Proof The proof proceeds by induction over the structure of sequent derivations; it has been presented in [25, 26] and we quote it here for completeness. We give the case of a  L-rule to illustrate the argument.

Suppose we are given a proof ending with ? ?! C; A ;
?; D ?! A;
L ?; x: C  D ?! A;
and suppose that its -term is :[]t fxs=wg. The only interesting case arises if this term is a weakening term. In this case, the name  has only weakening occurrences in t and in s, and t is a weakening term. By the induction hypothesis, we obtain derivations of ? ?! C;
and ?; D ?!
and hence also a derivation of ?; C  D ?!
. 2 Finally, we are in a position to show the correctness of the criterion, provided we rst give a formal de nition of intutionistic terms, presented in [25, 26] and described informally above.

De nition 11 Call a -term intuitionistic if in any subterm x: A : t which is not a weakening term, all occurrences of free names are weakening occurrences.

Theorem 12 Let : ? ?! A;
be a classical sequent derivation. If [ ] is an intuitionistic -term, then there exists an intuitionistic derivation of ? ?! A;
. Proof The proof proceeds by induction over the structure of sequent derivations; it has been presented in [25, 26] and we quote it here for completeness. Suppose the last rule is the rule  R to obtain a sequent ? ?! A  B;
. By the induction hypothesis, we have an intuitionistic sequent derivation of ?; A ?! B;
. Let [ ] = a: A:t. Either [ ] is a weakening term, in which case Lemma 10 implies that there is also an intuitionistic derivation of ? ?!
, and hence also of ? ?! A  B;
. If [ ] is not a weakening term, then there are no free names in [ ] that have a non-weakening occurrence. Hence by Lemma 10 again, there is an intuitionistic derivation ?; A ?! B . Now the intuitionistic  R-rule yields the result. 2 We continue in [25, 26] by giving a proof procedure for intuitionistic logic by extending a de nition of Miller et al. [14]: we de ned a uniform proof to be a sequent derivation where right rules are closer to the root than left rules. We call a proof fully uniform if right rules are preferred even over putative axioms, thereby ensuring that the succedents of all axioms consist of atomic formul only. To state our results, we need to recall the de nition of propositional hereditary Harrop formulae, cf. [14, 22, 25, 26], as follows: De nition 13 De ne goal formulae G and de nite formulae D by

G ::= A j G ^ G j D  G j G _ G D ::= A j G  A j D ^ D; where A is atomic. Call a sequent ? ?!
hereditary Harrop if ? consists of just D-formulae and
consists of just G-formulae. In [25], we stated our results for the disjunction-free fragment.

Lemma 14 Let the sequent ? ?!
be intuitionistically provable and hereditary Harrop. Then there exists a formula A 2
such that ? ?! A is intuitionistically provable too. Moreover, there exists a sequent derivation : ? ?! A;
such that [ ] has no free names and that in all applications of the  L-rule where the principal formula is G  B the right branch is the axiom ?; B ?! B . Proof By assumption, there exists a normal -term t with ? ` t: A1 _


_ An if
= A1 ; : : : ; An . Because the formulae in ? and
are hereditary Harrop formulae, the term t is derivable by the grammar

t ::= inl(t) j

inr(t)

j x: A:t j 1 (


m (x)


)t1 : : : tn j ht; ti 15

where 1 ; : : : ; m is any combination of  and 0 . Note that in case m; n = 0, the fourth clause reduces to a variable. disjunctive Now construct by induction over the structure of t a sequent derivation : ? ?! A;
such that [ ] has no free names and such that in all applications of the  L-rule the right branch is the axiom ?; A ?! A where G  A is the principal formula of the  L-rule. We just consider the case of a term ? ` inl(t): A _ B . By induction hypothesis we have a sequent derivation  ending in ? ?! A with all the desired properties. Now consider the derivation 0 which is  with B added to the right-hand side of all sequents. We have [ ] = [ 0 ] , and 0 has all desired properties as well. It is now easy to see that the derivation 0 ; _R with [ 0 ; _R] =  :[ ] has all desired properties. 2 We then establish the following completeness result for the fragment: Theorem 15 If the hereditary Harrop sequent ? ?! A;
is intuitionistically provable, then there exists a fully uniform proof of this sequent such that [ ] is intuitionistic.

Proof The sequent ? ?! A;
is intuitionistically provable; so, by Lemma 14, there exists a formula B in A;
such that  is a fully uniform LJ-proof of ? ?! B in which each leaf of  is atomic. Moreover, [ ]

has no free names. Now show by a double induction over the derivation  and the structure of formul on the right-hand side of  that for any such derivation  and any antecedent ?0 and succedent
0 , any order of right rules applied to B;
0 , there is a fully uniform proof : ?; ?0 ?! B;
0 , with the order of the right rules such that the following three conditions are met: (i) [ ] is intuitionistic; (ii) has only weakening occurrences of free names except possibly a name for the formula B , and all subterms corresponding to right rules reducing formulas in
0 are weakening terms; (iii) The variables occurring in ?0 do not occur in [ ] . We consider here only the case of the last rule in  being a  R-rule, and the case of the formula B  C 2
0 . In the rst case assume  = 0 ;  R and A  B being the principal formula of the  R-rule. Then by induction hypothesis there exists a derivation 0 : ?; x: A; ?0 ; ?! B;
;
0 where in [ 0 ] all free names and all free variables in ?0 have only weakening occurrences. Hence the derivation 0 ;  R: ?; ?0 ?! A  B;
satis es the desired properties. Now we turn to the case of B  C 2
0 . Here, by induction hypothesis there exists a derivation 0 : ?; ?0; x: B ?! A; C ;
;
0 such that x and have only weakening occurrences in [ 0 ] . So the derivation 0 ;  R with [ 0 ;  R] = x: B: :[][[ 0 ] has the desired properties. The proof is concluded by setting
0 =
00 , where
00 is obtained from
by possible exchange of A and B .

2

Hence to check provability of a sequent it is enough to construct a uniform proof and then to check, for all possible axiom instances and for all possible exchanges of  L and :L-rules, whether any of the corresponding -terms are intuitionistic. We go on in [25] to show a permutability theorem. Theorem 16 Let  be a classical sequent derivation such that [ ] is an intuitionistic -term. (i) If is the derivation resulting from interchanging two  R rules in , then [ ] is an intuitionistic term. (ii) If  is the derivation ?; A ?! B; C;
?; A ?! B; D;
^R ?; A ?! B; C ^ D;
 R; ? ?! A  B; C ^ D;
then the derivation obtained by permuting the  R rule over the ^R rule, towards the leaves, has an intuitionistic -term [ ] . Conversely, if we start with a such that [ ] is an intuitionistic -term, and permute the rules other way around, then at least one of the -terms that results from a dierent choice of axioms in the permuted derivation is intuitionistic.

16

Proof For (i), we use induction over the structure of derivations. The additional statement in (ii) arises from the fact that if the term x: A: :[ ]t is not a weakening term, then in [ ]t the name has only weakening occurrences. Now we use Lemma 10 to show that in this case ?; A ?! B has a intuitionistic 2

sequent proof. The derivation is now obvious.

The last two theorems are not true for intuitionistic logic with arbitrary disjunction. In fact, we obtain meaningful results only if we weaken the de nition of uniform proof and ask for _L-rules to occur as close to the root as possible. De nition 17 We de ne a weakly uniform proof to be a proof in which all possible _L-rules are as close to the root as possible. In addition, all axioms are only of ground types, and the rightmost branch of an  L-rule with principal formula A  B is always an axiom if B is an atom. Moreover, if the principal formula is (A  C )  B or (A ^ C )  B , then the rule directly preceding the  L-rule on the left branch is a  R-rule or ^R-rule respectively. We obtain only the existence of a weakly uniform proof, but not the additional statement that there exists a weakly uniform proof for every order of right-rules. Theorem 18 If ? ?!
is intuitionistically provable, then there exists a weakly uniform classical proof  such that [ ] is intuitionistic.

Proof ? ?!
is intuitionistically provable, hence there exists a term t in the simply-typed -calculus, with product and sum types,7 in long -normal form with ? ` t: B , where B = A1 _


_ An and A1 ; : : : ; An are

all formul of
. By induction over the structure of this normal form construct a weakly uniform classical sequent proof  such that [ ] is intuitionistic. This is a special case of the translation of natural deduction into sequent calculus, so we just list the case of an application xs1


sn with x of type A1 


 An  B with B atomic. By induction hypothesis we have weakly uniform derivations i : ? ?! Ai;1 ; : : : ; Ai;ki ;
, and ? ` si : Ai;1 _


_ Ai;ki . Then we construct the following derivation ?  An;1 ; : : : ; An;kn ;
_R ?  An ;
?; B ?! B;
L ?; An  B ?!



?  A1;1 ; : : : ; A1;k1 ;
_R L ? ?! A1 ;
?; A2 


 An  B ?! B;
L ?; A1 


 An  B ?! B;
The additional constraints on the  L-rule follow from the fact that if Ai is a function type, then si is a -abstraction and if Ai is a product type, then si is a product. Hence the translation of si into sequent derivations ends with a  R-rule and a ^R-rule respectively. The fact that we consider a -term in long -normal form ensures that each subterm ?; z : A _ B ` s: C is actually a term ? ` case z of inl(x) ) t(x)jinr(y) ) s(y) : C . Hence the translation of this subterm ends with an _L-rule. 2

4 Resolution in classical logic In this section, we show that, under inessential modi cations, Mints' translations between resolution systems and the sequent calculus establish tight connections between weakly uniform proofs and resolution derivations in both classical and intuitionistic logic with disjunction (Theorem 23). 7

and

The typing rules for sum types are given by ? ` t: A _I ? ` inl (t): A _ B

? ` t: B _ I ? ` inr (t): A _ B

B ?; x: A ` s: C ?; y: B ` u: C : _E ? ` t?: A` _case t of inl(x) ) ujinr(y) ) u: C

17

The key point relies on the following de nition of weakly hereditary Harrop formul, in which A ranges over atoms: D ::= A j D ^ D j :(A ^ A) j A _ A j G  H G ::= A j :A j G ^ G j G _ G j D  G H ::= A j A _ A Weakly uniform proofs are complete for weakly hereditary Harrop consequences D1 ; : : : ; Dm ?! G1 ; : : : ; Gn .8 The extensions to the usual class of hereditary Harrop formul [14] are the inclusion of binary disjunctions of atomic formul in implicational goal-formul, negated atoms in goal-formul and negated binary conjunctions of atoms in de nite formul. These extensions are required if we are to interpret resolution proofs for clauses that include disjunctions as weakly uniform proofs because the translation of De nition 22 (at least its modi cation in x 5.2) make essential use of them. In the absence of disjunction, the simpler notions of uniform proof and hereditary Harrop formula [25, 27] will suce. We also show that, for classical logic, permutations in the resolution search space correspond to permutations in the search space generated by the sequent calculus (Proposition 26). The results of [25] then give realizers for (classical) resolution derivations. We begin by recalling from [15] the construction of a set of clauses of bounded complexity from an arbitrary propositional formula.

De nition 19 A formula A is a classical clause if it is either ? or a disjunction A1 _


_ An, with n > 0 and each Ai , 1  i  n, a literal. Clauses which dier only in the numbering or order of literals are identi ed. Lemma 20 For any propositional formula A, a set XA of clauses of length  3 can be constructed in linear space and time (of the length of A) such that A is valid if and only if XA is inconsistent. Proof It is enough to show this for formul constructed using only of negation and disjunction. Firstly, we construct, by induction over the structure of A, a set of clauses for the formul :X _ A and :A _ X , where X is a propositional variable. If A is an atom, we simply take these two formul. For the case of a disjunction A _ B , we introduce new propositional variables A0 or B 0 for non-atomic formul A or B , otherwise let A0 or B 0 be A or B respectively. We add to the clauses obtained by the induction hypothesis applied to :A0 _ A, :A _ A0 , :B 0 _ B and :B _ B 0 the clauses :X _ A _ B , :A _ X , :B _ X . For a negation :A, let A0 be a new propositional variable if A is not atomic; otherwise let A0 be A. Add to the clauses obtained by the induction hypothesis applied to :A0 _ A and :A _ A0 the clauses :X _ :A and A _ X . It is easy to see that the set XA , de ned as the element :X together with the clauses for the formul :X _ A and :A _ X as constructed above, satis es the claims. 2 Resolution is de ned as a calculus for deriving a judgement ? ` C , where ? is a set of clauses and C is a clause. The precise de nition follows below.

De nition 21 Let ? be a set of clauses, let C be a clause and let A and B be atoms. A resolution derivation of a judgement ? ` C is given by: ?; C; ?0 ` C

Ax

? ` A _ :A

EM

? ` :A1 _ C1


? ` :An _ Cn ?; A1 _


_ An ` C1 _


_ Cn

Res:

where C1 ; : : : ; Cn are sets of clauses. In the last case, we call the formula A1 _


_ An the input formula of the resolution rule. We identify the clause C _ ? with C in the above rules. 8

Indeed, weakly uniform proofs are complete for a slightly larger class than this.

18

Note that weakening is admissible in this system: whenever ? ` C and also ?  ?0 , then also ?0 ` C . We call a clause A a weakening clause in a derivation if it is not introduced by one of the rules. Mints [15] proves the following:

Theorem (Mints) A formula A is classically provable if and only if there is a resolution derivation XA ` ?. This is proved by transforming a resolution derivation into a sequent derivation where formul consist only of disjunction and negation and vice versa. We start our proof of this theorem, which we will later generalize to intuitionistic logic, with a translation of a resolution proof into a derivation in the classical sequent calculus LK without cut. This translation is essentially the one given in [15]. De nition 22 De ne the concatenation of the n sequents ?1 ?!
1 ; : : : ; ?n ?!
n to be the sequent ?1 ; : : : ; ?n ?!
1 ; : : : ;
n . (i) By induction over the structure of clauses we de ne a sequent derivation of ? ?!
, for each clause C with a polarity f+; ?g (de ned as in tableaux calculi). A clause has positive (negative) polarity if is part of
(?). If C is the clause C1+ _ C2+ , then we de ne [ C1+ _ C2+ ] to be the concatenation of the two sequents [ C1+ ] = ?1 ?!
1 and [ C2+ ] = ?2 ?!
2 . For the remainder of the clauses the de nition is as follows: [ (:A _ :B _ :C )? ] [ (:A _ :B _ C )? ] [ (:A _ B _ C )? ] [ (A _ B _ C )? ] [ (:A _ :B )? ] [ (:A _ B )? ] [ (A _ B )? ] [ (:A)? ] [ (A)? ] [ (:A)+ ] [ (A)+ ]

= = = = = = = = = = =

:(A ^ B ^ C ) ?! (A ^ B )  C ?! A  (B _ C ) ?! (A _ B _ C ) ?! :(A ^ B ) ?! A  B ?! (A _ B ) ?! ?! A A ?! A ?! ?! A

(ii) If X is a set of clauses C1 ; : : : ; Cn and C is a clause, we denote the sequence resulting from concatenation of [ C1? ] ; : : : ; [ Cn? ] and [ C + ] by [ X ?] ?! [ C + ] . By induction over the derivation of X ` C , we de ne a classical sequent derivation of [ X ?] ?! [ C + ] as follows:  With each axiom ?; C; ?0 ` C , associate the appropriate sequent derivation from the axioms;  With each axiom X ` p _:p, associate the sequent derivation consisting of the axiom [ X ?] ; p ?! p;  If the input formula is :A_:B and if we have resolution derivations of X ` A_C1 and X ` B _C2 , then we construct the following sequent derivation: [ X ? ] ?! A; [ C1+ ] [ X ? ] ?! B; [ C2+ ] ^R [ X ? ] ?! A ^ B; [ C1+ ] ; [ C2+ ] :L; [ X ? ] ; :(A ^ B ) ?! [ C1+ ] ; [ C2+ ]  If the input formula is :A1 _ :A2 _ :A3 , then the construction is similar;  If the input formula is :A_:B _C and if we have resolution derivations of X ` A_C1 , X ` B _C2 and X ` :C _ C3 , then we construct the following sequent derivation: [ X ? ] ?! A; [ C1+ ] [ X ? ] ?! B; [ C2+ ] ^R [ X ? ] ?! A ^ B; [ C1+ ] ; [ C2+ ] [ X ? ] ; C ?! [ C3+ ]  L; [ X ? ] ; (A ^ B )  C ?! [ C1+ ] ; [ C2+ ] ; [ C3+ ] 19

 If the input formula is :A _ (B _ C ) and if we have resolution derivations of X ` A _ C1 , X; B ` C2 and X ` :C _ C3 , then we construct the following sequent derivation: [ X ? ] ; B ?! [ C2+ ] [ X ? ] ; C ?! [ C3+ ] _L [ X ?] ?! A; [ C1+ ] [ X ? ] ; B _ C ?! [ C2+ ] ; [ C3+ ]  L; [ X ?] ; A  (B _ C ) ?! [ C1+ ] ; [ C2+ ] ; [ C3+ ]  If the input formula is :A _ B and we have resolution derivations X ` A _ C1 and X ` :B _ C2 , then we construct the following sequent derivation:

[ X ? ] ?! A; [ C1+ ] [ X ? ] ; B ?! [ C2+ ]  L; [ X ? ] ; A  B ?! [ C1+ ] ; [ C2+ ]

 If the input formula is A _ B and if we have resolution derivations X ` :A _ C1 and X ` :B _ C2 , we obtain the following sequent derivation:

[ X ? ] ; A ?! [ C1+ ] [ X ? ] ; B ?! [ C2+ ] _L; [ X ? ] ; A _ B ?! [ C1+ ] ; [ C2+ ]

 If the input formula is A1 _ A2 _ A3 , then the construction is similar;  If the input formula is A and if we have a resolution derivation X ` :A _ C , then we have a sequent derivation of [ X ? ] ; A ?! [ C + ] , by assumption, which we simply take;  If the input formula is :A and if we have a resolution derivation X ` A _ C , then we have a sequent derivation of [ X ? ] ?! A; [ C + ] , by assumption, which we simply take. By applying the translation of sequent derivations into -terms, as given in [25], we obtain a -term for each resolution derivation. Moreover, this sequent derivation is weakly uniform. Theorem 23 The sequent derivation associated with a resolution derivation is weakly uniform.

Proof Note that the right-hand side of all root sequents of such a sequent derivation is atomic. Furthermore any intermediate non-atomic formula on the right is reduced as soon as it occurs. Hence the sequent derivation is weakly uniform. 2

As an example, we will give the resolution derivation and the corresponding -term for the formula A  A. According to Lemma 20 the set XAA is the set f:X _ :A _ A; A _ X; :A _ X; :X g. A resolution derivation of the empty clause from a subset of these clauses can be obtained as follows:

:X ` :A _ A :X ` :X :X; A _ X ` A :X ` :X : :X; A _ X; :A _ X ` ? The corresponding sequent derivation is

A ?! A; X X ?! X _L A _ X ?! A; X XX  L: A _ X; A  X ?! X

The corresponding -term, which is obtained by extending the translation of sequent calculus into natural deduction to the case of multiple formul on the right-hand side, is  :[ ]yfh ix=yg. Observe that the sequent derivations obtained by translating resolution derivations do not use weakening. Moreover, these derivations can be rewritten in such a way that the axioms have the form A ?! A, but 20

at the expense of introducing weakening at the root of the derivation. These properties are a consequence of the absence of a weakening rule in the resolution calculus. A translation of classical sequent derivations into resolution derivations can be given only for sequents without weakening in the middle of the derivation. Mints [15] gives such a translation. Because every sequent derivation where all formul are either clauses or elements of [ X ?] can be transformed into one in which weakening occurs at the root of the derivation only, for each derivable sequent ? ?!
with this property there is a subsequent ?0 ?!
0 which has a resolution proof. This translation is part of the following theorem.

Theorem 24 Consider a weakly uniform sequent derivation of [ X ?] ;
1 ?!
2 such that (a)
1 and
2 consist of atoms; (b) all weakenings occur only at the root of the derivation; and (c) all axioms have the form A ?! A. Then there is a resolution derivation of X ` :
01 _
02 , in which
01 and
02 are subsets of
1 and
2 respectively. Furthermore, all of the formul in
1 and
2 that are not obtained by weakening are in
01 and
02 respectively.

Proof Let be the subderivation above the last weakening rule; proceed by induction over . It is necessary to strenghten the induction hypothesis and construct a resolution derivation Y ` :
01 _
02 , where Y is a subset of X ,
01 and
02 are as above and Y contains no weakening formula. Moreover, we show that all formula in [ X ? ] which were not obtained by weakening are in Y . Because weakening is an admissible rule in the resolution calculus, this statement implies the claim. Suppose the last rule in is an axiom A ?! A. If both atoms are part of
1 and
2 respectively, then the axiom X ` :A _ A yields the claim. If both atoms are part of [ X ? ] , then the derivation A`A A; :A ` ? yields the claim. If only one atom, say the one on the LHS, is part of [ X ? ] , then take the resolution axiom A ` A. If the one atom is the atom on the RHS, take the resolution axiom :A ` :A. Suppose the last rule in is a :L-rule, with the principal formula :(A ^ B ) or :(A1 ^ A2 ^ A3 ). We consider the case of a formula :(A ^ B ); the other one is similar. The weakly uniform derivation looks like

?1 ?! A; ?01 ?2 ?! B; ?02 ^R ?1 ; ?2 ?! A ^ B; ?01 ; ?02 :L ?1 ; ?2 ; :(A ^ B ) ?! ?01 ; ?02 where, by hypothesis, neither A nor B is obtained by weakening. Note that ?1 ; ?2 ; :(A ^ B ) ?! ?01 ; ?02 is the sequent [ X 0? ; (:A _:B )? ] ;
1 ?!
2 , where X 0 [f:A _ :B g is a subset of X . Furthermore, neither A nor B is an element of [ X 0? ] . The induction hypothesis applied to the sequents ?1 ?! A; ?01 and ?2 ?! B; ?02 yields resolution derivations Y 0 ` A _ C1 and Y 0 ` B _ C2 respectively, where Y 0 is a subset of X 0. Hence we can construct the following derivation:

Y 0 ` A _ C 1 Y 0 ` B _ C2 : Y 0 [ f:A _ :B g ` C1 _ C2 Next we consider the case that the last rule in is an  L-rule. We start with the case where the principal formula is A  B . The weakly uniform derivation looks like ? ?! A;
B ?! B L ?; A  B ?! B;
where, by hypothesis, A was not obtained by weakening. The induction hypothesis applied to the sequent ? ?! A;
yields a resolution derivation X 0 ` A _ C , where X 0 [ f:A _ B g is a subset of X . The following resolution derivation now yields the claim:

X 0 _ ` A _ C X 0 ` :B _ B : X 0 [ f:A _ B g ` C _ B 21

Now consider the case where the principla formula is A  (B _ C ). The weakly uniform derivation looks like

?2 ; B ?! ?02 ?3 ; C ?! ?03 _L ? ?! A; ?01 ?2 ; ?3 ; B _ C ?! ?02 ; ?03 ?1 ; ?2 ; ; ?3 ; A  (B _ C ) ?! ?01 ; ?02 ; ?03 where, by hypothesis, neither A nor B _ C are obtained by weakening. The induction hypothesis yields resolution derivations Y ` A _ D1 , Y; B ` D2 and Y; C ` D3 . Hence we obtain the following resolution derivation

Y ` A _ D1

: Y ` :B _ D2

Y ` :C _ D3 Y [ f:A _ B _ C g ` D1 _ D2 _ D3

The remaining case is that we have an _L-rule as the last rule. The case of a principal formula A _ B is similar to the case of a principal formula A  B , and the case of a principal formula A _ B _ C is similar to the case of a principal formula A  (B _ C ). 2 Mints' Theorem can now be obtained as a corollary. Corollary 25 A formula A is classically provable if and only if there is a resolution derivation XA ` ?.

Proof Suppose? A is classically provable. By Lemma 20, the set XA is inconsistent, hence there is a derivation of [ XA ] ?! . Theorem 24 implies the existence of a resolution derivation XA ` ?. Conversely, given a resolution derivation of XA ` ?. The second part of De nition 22 yields a derivation [ XA? ] ?! ; hence XA is inconsistent. So A is provable. 2 A central idea of [25] is to investigate when permutations transform a weakly uniform sequent derivation which is non-intuitionistic into an intuitionistic derivation. Here we show how permutations in the sequent calculus are related to the choice of input formul in the resolution calculus. Later on we will transfer this connection to intuitionistic logic. Because the formul occurring in sequent derivations arising from resolution derivations have a rather simple structure, it suces to consider exchanges of  L- rules, :L-rules and _L-rules. These are the only rules whose exchange leads from a weakly uniform derivation to another weakly uniform derivation. The details are contained in the following proposition:

Proposition 26

(i) Let

X ` :A1 _ C1 X ` :An _ Cn X; A1 _


_ An ` C1 _


Cn X ` :B1 _ D1


X ` :Bm _ Dm X; A1 _


_ An ; :C1 _ B1


_ Bm ` C2 _


Cn _ D1 _


_ Dm be a resolution derivation and let

X ` :A1 _ C1 X ` :B1 _ D1


X ` :Bm _ Dm X; :C1 _ B1


_ Bm ` :A1 _ D1 _


_ Dm X ` :A2 _ C2


X ` :An _ Cn X; A1 _


_ An ; :C1 _ B1


_ Bm ` C2 _


Cn _ D1 _


_ Dm be the derivation in which the application of the two instances of the resolution rule are exchanged. The translation of the second resolution derivation into a sequent derivation is obtained by exchanging the two left-rules to which the two applications of the resolution rule are translated. (ii) Conversely, given a weakly uniform sequent derivation of a sequent ? ?!
, where ? consists only of clauses and
only of atoms, the exchange of :L and  L-rules corresponds to the exchange of two resolution rules.

22

Proof For rst part, check each resolution formula in turn. For the second part, calculate the resolution 2

derivations for all possible exchanges.

Intuitively, this proposition indicates that the search for a weakly uniform derivation of a sequent with formul in clausal form is as complicated as the search for a resolution derivation of the corresponding clauses. In other words, this proposition shows that the essential aspect of the resolution method is the transformation of formul into clausal form; the complexity of nding the right input formula in a resoution derivation is the same as nding the right permutation in the sequent derivation. This analysis carries over to the intuitionistic case (see next section), including the case of a resolution formula (A  B )  C . This is important because, in contrast to the classical case, in intuitionistic logic permutations of inferences do matter. Classically, but not intuitionistically, any permutation of a sequent derivation transforms a proof only into a proof and a non-proof only into a non-proof. Moreover, Egly [7] shows that the transformation of sequents into clausal form decreases the complexity of proof-search in intuitionistic logic signi cantly.

5 Resolution in intuitionistic logic In this section, we develop a resolution calculus for intuitionistic logic based on the ideas above. The idea is to retain the resolution calculus for classical logic, because this calculus has no constraints on the order in which input formul are taken. The translation of such resolution derivations into -terms is used to decide when the derivation provides sucient evidence that the formula is intuitionistically provable.

5.1 Mints' intuitionistic resolution Mints [15] also de nes a resolution calculus for intuitionistic logic. It is easily seen that his calculus corresponds to constructing weakly uniform proofs in LJ, with weakening pushed as close to the root as much as possible. It is important to note that Mints' calculus is not a restriction of classical resolution, but has special rules for each connective of the logic. For completeness, we sketch the main elements of his calculus. Clauses are no longer formul, but sequents of the form A  B   C , A  B _ C and A1 ; : : : ; An  B  with n  3, where all formul are propositional variables and B  means either a propositional variable or the symbol ? (falsehood). The resolution calculus has the following rules:

X;  ` 

X `? A

X `AA

X ` (?; Ao  B  ) ? X; ((A  B  )  C ) ` ?  C ( ) X ` ?1  A X ` (?2 ; B  D ) X ` (?3 ; C  D ) (_? ) X; (A  B _ C ) ` (?1 ; ?2 ; ?3  D ) X ` ?1  A X ` ? 2  B X; (A; B  C  ) ` (?1 ; ?2  C  ) (c)

X ` ? ? X ` ?  C (?) X`?A X; (A  B ) ` ?  B

In these rules,  is a clause, A, B , C are atoms, ?1 , ?2 and ?3 are sets of atoms, the superscript 0 means possible absence of the corresponding formula and in the rule (? ) it is required that either B  = B  or (B  =?6= B  and B 0 = B ); similarly in (_? ), D = D if at least one of D , D is D, and D =? if D = D =?. The rule (?) is allowed only as the last rule in the derivation. Mints constructs, for every formula A, a set of clauses XA , the translation of these clauses into one formula YA and an atom F such that A is intuitionistically provable if and only if YA ?! F is provable in LJ. Mints then gives translations between resolution derivations and LJ derivations with weakening pushed down to the root as much as possible, and obtains as a corollary that a formula A is intuitionistically provable if and only if XA ` F is derivable in the resolution calculus. 23

The rules for implication and negation cannot be obtained as special cases of the rules for classical resolution, hence it is not immediately possible to transfer the implementations of classical resolution to the intuitionistic case. The reason is that derivations may contain weakening at places other than at their roots. As an example, consider the LK-derivation ? ?! B;
WL ?; A ?! B;
R ? ?! A  B;
C ?! C  L; ?; (A  B )  C ?! C;
where the weakening of the formula A cannot be pushed to the root of the derivation. Because the construction of Theorem 24 works only for derivations where weakening is applied only as the last rule of the derivation, there can be no resolution derivation corresponding to this sequent derivation. Indeed, the method of the previous section, which uses the (classical) equivalence (A  B )  C  (A _ C ) ^ (:B _ C ), yields only the following resolution derivation: ? ` B _
? ` :C _ C ; ?; :B _ C `
where
is interpreted as the disjunction of its members, and the input formula A _ C is added by weakening at the end and not obtained by a resolution step.

5.2 The intuitionistic force of classical resolution In this section, we exploit the results given above and in [25] to assess the intuitionistic force of classical resolution. We take the association of -terms with resolution derivations, as developed in the previous section, and identify when they provide evidence for intuitionistic provability. The translation of formul into clauses, referred to in Lemma 20, produces clauses given by the BNF

C ::= A1 _ A2 j :A1 _ A2 j :A1 _ :A2 _ A3 j :A1 _ :A2 j :A1 _ A2 _ A3 where A1 , A2 , A3 are all atomic. In the sequel we restrict attention to such clauses. Note that the transformations leading from formul to the clauses arising in the sequent derivations are no longer equivalences: the formula (A _ C ) ^ (B  C ) implies (A  B )  C , but not the other way round. In all other cases, the transformations that lead from formul to clauses are intuitionistic equivalences. The correspondence between the -calculus and intuitionistic logic is based on a sequent calculus with multiple conclusions for intuitionistic logic, as presented in [6, 33]. This calculus is the same as the calculus LK [10] for classical logic except for the  R and :R-rules: ?; A; ?! B ? ?! A  B;


; A ?!  R ? ??! :A;
:R

The translation of resolution derivations into -terms leads directly to a criterion when a resolution derivation gives rise to an intuitionistic proof.

De nition 27 A resolution derivation is said to be intuitionistic if it translates into an intuitionistic -

term.

We want to transfer the soundness theorem for classical resolution to the intuitionistic case. As stated this does not work because the implications (A _ X ) ^ (B  X )  ((A  B )  X ) and (A _ X )  :A  X have the wrong order: for the classical proof to go through we need that the formula with implication implies the clausal form and not vice versa. We address the rst case by restricting the class of resolution derivations under consideration and modify the translation of resolution derivations into sequent derivations. 24

We permit for clauses A _ C; :B _ C arising from the translation of the formula (A  B )  C into clauses only derivations of the form

X ` :A _ B _ D X ` :C _ C X; A _ C ` B _ D _ C X ` :C _ C X; A _ C; :B _ C ` D _ C

Such a resolution derivation is translated into [ X ? ] ; A ?! B; [ D+ ] R [ X ? ] ?! A  B; [ D+ ] C ?! C  L: ? + [ X ] ; (A  B )  C ?! [ D ] ; C For the second case, i.e., the clause A _ B , we change its translation into [ (A _ B )? ] = :A  B ?!. We allow for this clause only resolution derivations of the form X ` :A _
X ` :B _ B X; A _ B ` B _
which we translate into the sequent derivation [ X ?] ; A ?!
:R [ X ? ] ?! :A;
B ?! B L ? [ X ] ; :A  B ?! B;
Next we want to show soundness for the translation. The key point is contained in the following Lemma, which is a modi cation of Lemma 20. Lemma 28 A formula A is intuitionistically provable if there is an intuitionistic sequent derivation of [ XA? ] ?! . The soundness theorem for the translation is as follows: Theorem 29 A formula A is intuitionistically provable if there is a resolution derivation of XA ` ? such that the -term corresponding to the modi ed translation into the sequent calculus is intuitionistic. Proof The translation of the resolution derivation produces a derivation [ XA?] ?!. By assumption the -term corresponding to this derivation is intuitionistic, hence there is an intuitionistic derivation of this sequent [25]. Now Lemma 28 yields the claim. 2 Looking at the example of the resolution derivation for the formula A  A again, we see that the modi ed translation yields a derivation

A ?! A R ?! A  A X ?! X  L; (A  A)  X ?! X with the -term w(a: A:a), which is in fact a -term and hence an intuitionistic -term.

We need one extra step for the completeness proof. In our previous paper [25] we show that a sequent ? ?!
is intuitionistically provable if there is a weakly uniform classical sequent derivation such that the corresponding -term is intuitionistic. We now have: Theorem 30 Suppose we have a weakly uniform classical sequent derivation of a sequent [ X ?] ;
1 ?!
2 such that the corresponding -term is intuitionistic, all formul in X are clauses, all formul in
1 and
2 are atoms, weakening is pushed as far as possible to the root of the derivation, and all axioms have the form A ?! A. Then there is an intuitionistic resolution derivation X ` :
01 _
02 , where
01 and
02 are subsets of
1 and
2 respectively. Furthermore, all of the formul in
1 and
2 that are not obtained by weakening are in
01 and
02 respectively. 25

Proof We use the proof of Theorem 24 to construct a resolution derivation except for the case of the principal formul :A _ C and :B _ C , if they arise from the translation of (A  B ) _ C , and for A _ B . So assume we are given a derivation

?; A ?! B;
R ? ?! A  B;
C ?! C L ?; (A  B )  C ?! C;
The weakening assumption implies that at most one of A and B can be obtained by weakening. Note also that neither A nor B can be contained in [ X ?] . If neither A nor B is obtained by weakening, we have the following resolution derivation:

X n f:A _ C; :B _ C g ` A _ B _ D X n f:A _ C; :B _ C g ` :C _ C X n f:B _ C g ` :C _ C X n f:B _ C g ` B _ D _ C X `C_D If A is obtained by weakening, then the resolution derivation is X n f:B _ C g ` B _ D X n f:B _ C g ` :C _ C X `C _D and if B is obtained by weakening, the resolution derivation is X n f:A _ C g ` A _ D X n fA _ C g ` :C _ C X `C _D

The modi ed translation ensures that the translation of the constructed resolution derivation is also an intuitionistic sequent derivation. Lastly, we have to consider the case of the modi cation for the clause A _ B . So assume we are given a derivation ?; A ?!
:R ? ?! :A;
B ?! B : ?; :A  B ?! B;
The weakening assumption implies that A is not obtained by weakening. Then we construct the following resolution derivation:

X n A _ B ` :A _


:

X n A _ B ` :B _ B X ` B _


The modi cation ensures that the translation of the constructed resolution derivation is also an intuitionistic sequent derivation. 2 Soundness and completeness now follow in exactly the same way as shown for classical logic. Corollary 31 A formula A is intuitionistically provable if and only if there is an intuitionistic resolution derivation of XA ` ?.

Proof One direction has already been shown; see Theorem 29. For the other, the argument as in Corollary 25 2

works for the modi ed translation.

Now we turn to the connection between the choice of input formul in the resolution calculus and permutations in the sequent calculus. Consider the translation of a resolution derivation and examine all the permutations of  L-rules and :L-rules. If one permutation yields an intuitionistic -term, then permutation of the order of introducing the input formul yields the image of an intuitionistic resolution derivation under the translation. Hence, the soundness and completeness properties (Corollary 31) imply that 26

the search for an intuitionistic resolution derivation amounts essentially to the search for a permutation of the  L and :L-rules which yields an intuitionistic -term. As an example of this phenomenon, consider the formula (A  B ^ (A  B )  B )  B . This example is the same one we gave in our previous paper [25] to demonstrate how a permutation can transform a classical sequent derivation with no intuitionistic force into one with such force. The crucial point is that in order to obtain a weakly uniform intuitionistic proof, the  L-rule with principal formula (A  B )  B has to be the rule closest to the root of the derivation. This is also true for the resolution derivation of the formula (A  B ^ (A  B )  B )  B in that the resolution step that uses the input formula corresponding to (A  B )  B must occur as late as possible; this gives rise to a -term which is intuitionistic.

6 Summary We have given an account of the search space of propositional intuitionistic logic in terms of classical resolution. The following are the key steps in our analysis:  The extension of Parigot's -calculus to include (classical) disjunction and explicit substitution;  The use of this extended calculus, , as a system of realizers for multiple-conclusioned sequent calculus;  The formulation of a necessary and sucient condition on realizers to characterize when a given (classical) realizer for a sequent witnesses the intuitionistic provability of that sequent;  The provision of a translation between the classical sequent calculus and classical resolution, due to Mints, is used to lift the conditions to classical resolution, thereby giving a characterization of the intuitionistic force of classical resolution;  An application of these results which allows standard resolution methods of uniform proof-search to be used directly for intuitionistic logic. Most signi cantly, our results support a type-theoretic analysis of search spaces in both classical and intuitionistic logic. Certain issues remain, perhaps the most obvious being the possibility of extending our analyses to the rst-order quanti ers, 8 and 9.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

P.B. Andrews. Theorem-proving with general matings. JACM, 28:193{214, 1981. A. Avron. Gentzen-type systems, resolution and tableaux. J. Automat. Reas., 10(2):265{282, 1993. W. Bibel. On matrices with connections. JACM, 28:633{645, 1981. W. Bibel. Computationally improved versions of Herbrand's theorem. In: Stern (ed.), Proc. of the Herbrand Colloquium, North-Holland, 1982. G. Boolos. Don't Eliminate Cut. In Logic, Logic, and Logic, Richard Jerey, editor, Harvard University Press, 1998, 365-369. M. Dummett. Elements of Intuitionism. Oxford University Press, 1980. U. Egly. Some pitfalls of LK-to-LJ translations and how to avoid them. In: W. McCune (ed.), Proc CADE-14, LNCS 1249, 116-130, 1997. M.C. Fitting. Resolution for intuitionistic logic. In: Z. W. Ras and M. Zemankova (eds.), Methodologies for intelligent systems, 400{407, Elsevier, 1987. M.C. Fitting. First-order modal tableaux. J. Automat. Reas., 4:191{214, 1988. G. Gentzen. Untersuchungen uber das logische Schlieen. Math. Zeit., 176{210, 405{431, 1934. J.-Y. Girard, P. Taylor, and Y. Lafont. Proofs and Types. Cambridge University Press, 1989. J.W. Klop. Term Rewriting Systems. In: S. Abramsky, Dov M. Gabbay and T. S. E. Maibaum (editors), Handbook of Logic in Computer Science , Volume 2, 1{116, Oxford Science Publications, 1992. S.Yu Maslov. Theory of Deductive Systems and its Applications. MIT Press, 1987. D. Miller, G. Nadathur, A. Scedrov and F. Pfenning. Uniform proofs as a foundation for logic programming. Ann. Pure Appl. Logic, 51:125{157, 1991. G. Mints. Gentzen-type systems and resolution rules part I: Propositional logic. In: P. Martin-Lof and G. Mints (eds.), Proc. COLOG-88, LNCS 417, 198{231, 1990. H.J. Ohlbach. A resolution calculus for modal logics. In: E. Lusk and R. Overbeek (eds.), Proc. 9th CADE, LNCS 310, 500{516, 1988. J. Otten and C. Kreitz. A Connection Based Proof Method for Intuitionistic Logic. LNAI 918, 122-137, 1995.

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[18] M. Parigot. -calculus: an algorithmic interpretation of classical natural deduction. In: Proc. LPAR '92, LNCS 624, 190{201, 1992. [19] M. Parigot. Church-Rosser property in classical free deduction. In: Logical Environments, G. Huet and G. Plotkin (editors), Cambridge University Press, 1993. [20] M. Parigot. Strong normalization for second order classical natural deduction. In: Proc. LICS 93 , 39{47, IEEE Computer Soc. Press, 1993. [21] D.J. Pym. A note on the proof theory of the -calculus. Studia Logica , 54:199{230, 1995. [22] D.J. Pym and J.A. Harland. A uniform proof-theoretic investigation of linear logic programming. Journal of Logic and Computation , 4(2):175{207, 1994. [23] D. Pym and L. Wallen. Proof-search in the -calculus. In: G. Huet and G. Plotkin (eds.), Logical Frameworks, 309{340, Cambridge Univ. Press, 1991. [24] D. Pym and L. Wallen. Logic programming via proof-valued computations. In: K. Broda (ed.), ALPUK '92, 253{262, WICS, Springer-Verlag, 1992. [25] E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search (extended abstract). In: P. Miglioli, U. Moscato, D. Mundici and M. Ornaghi (eds.), Proc. Tableaux '96, LNCS 1071, 295{311, 1996. [26] E. Ritter, D. Pym and L. Wallen. On the intuitionistic force of classical search. To appear, Theoretical Computer Science, 1999. [27] E. Ritter, D. Pym and L. Wallen. Proof-terms for classical and intuitionistic resolution (extended abstract). Proceedings of CADE-13 (eds. M.McRobbie and J.Slaney), LNCS 1104, 17{31, 1996. [28] J. Robinson. A machine-oriented logic based on the resolution principle. JACM, 12:23{41, 1965. [29] R. Smullyan. First-order logic. Ergebnisse der Math. 43, Springer-Verlag, 1968. [30] W. W. Tait. Intensional interpretation of functionals of nite type I. Journal of Symbolic Logic, 32:198{212, 1967. [31] L. Wallen. Generating connection calculi from tableau- and sequent-based proof systems. In: A.G. Cohn and J.R. Thomas (eds.), Arti cial Intelligence and its Applications, 35{50, John Wiley & Sons, 1986, (Proc. AISB'85, 1985). [32] L. Wallen. Matrix proof methods for modal logics. In: J. McDermott (ed.), Proc. 10th IJCAI, 917{923, Morgan Kaufmann Inc., 1987. [33] L. Wallen. Automated Deduction in Non-Classical Logics. MIT Press, 1990.

LNCS denotes Lecture Notes in Computer Science, Springer-Verlag.

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