Sequents, Frames, and Completeness Thierry Coquand1 and Guo-Qiang Zhang2?? 1
2
Department of Computer Science, University of Goteborg S 412 96, Goteborg, Sweden
[email protected]
Department of Computer Science, University of Georgia Athens, GA 30602, U. S. A.
[email protected]
Abstract. Entailment relations, originated from Scott, have been used
for describing mathematical concepts constructively and for representing categories of domains. This paper gives an analysis of the freely generated frames from entailment relations. This way, we obtain completeness results under the unifying principle of the spatiality of coherence logic. In particular, the domain of disjunctive states, derived from the hyperresolution rule as used in disjunctive logic programs, can be seen as the frame freely generated from the opposite of a sequent structure. At the categorical level, we present equivalences among the categories of sequent structures, distributive lattices, and spectral locales using appropriate morphisms.
Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [15{17]. It can be seen as a generalisation of the earlier consequence calculus of Hertz [9] to a multi-conclusion consequence relation. The notion of consequence relation, with only one conclusion, has been analysed by Tarski [20]. This consequence calculus has been used by Scott in order to give a concrete representation of domains, as in information systems [18]. It is thus natural to wonder if the more general notion of entailment relation, with multiple conclusions, can be used to represent larger categories of domains, such as those related to non-determinism. This is indeed the case, and it has been developed in [21, 22] and [5], in an independent way from Scott's work on entailment relations (in [21], a set together with an entailment relation is called a sequent structure). Another related reference, also independent from Scott's work, is [8]. In this paper we analyse various completeness theorems for sequent structures by embedding them into frames. A goal of this study is to provide a uni ed way to present completeness results in logic, such as those for resolution and hyperresolution. ??
Corresponding author.
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A number of recent developments serve as the motivation for the current paper. In [3, 4], it is shown that entailment relations are naturally connected to several mathematical structures. They can be used to give elegant constructive version of some basic mathematical concepts (and theorems), such as continuous linear forms, space of valuations, etc. One key point here is that it is often possible to get direct explicit descriptions of entailment relations generated by some rules, avoiding syntactical induction and case analysis on derivations. In order to understand appropriate domains for the semantics of disjunctive logic programs, [23] introduces clausal logic based on the so-called hyperresolution rule [12]. Completeness of hyperresolution provides the basis for this domain-theoretic semantics: it establishes the equivalence of the model-theoretic semantics and the proof-theoretic semantics. Here, a set of clauses closed under hyperresolution is called a disjunctive state; the collection of disjunctive states under inclusion forms a complete lattice, which, in the case of information systems, is isomorphic to the Smyth powerdomain [13, 23]. A natural question is whether the cpo of disjunctive states can be seen as a universal construction for sequent structures. Related to this question is the canonical embedding of a sequent structure into a frame. For this purpose we use Johnstone's coverage method [7] to study frames generated from a sequent structure as well as from its opposite. Interestingly, the frame generated from the opposite is precisely the complete lattice of disjunctive states. Moreover, in each case the universal map gives a way to capture a point of the frame as an ideal element of the underlying sequent structure. The completeness theorem of coherent logic states that any coherent (or spectral) frame is spatial [7]. It ensures that enough models exist to uniquely determine the partial order, where models correspond to completely prime lters. This means that when sequent structures are embedded into spectral frames, we have enough models to uniquely determine the entailment relation, and thus obtain certain completeness result \for free", such as the completeness of hyperresolution. In return, existing results [13, 23] related to hyperresolution suggest several explicit constructions for the sequent-structure-generated frames: a semantical one, a proof-theoretic one, and a third one based on the notion of \choice inference". A couple of results in this paper may be seen as \folklore"; their roots may be traced back eventually to Stone's representation theorem [19]. We feel however that our contribution lies in tying in the more discrete notion of sequent structures with the more complete notion of locales through the so-called coverage relation [7] in a concrete logical setting. This allows the importation of existing results in locales to sequent structures, shedding new light on the topic. It is, for instance, quite interesting that the hyperresolution rule appears naturally in solving the problem of embedding an entailment relation in a frame, and it may not be obvious a priori that the disjunctive states form a frame. We hope that this paper is a rst step in exploring completeness of various logical systems by means of canonical embedding to locales.
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1 Coverage and spatiality of spectral frames A frame is a poset with nite meets and arbitrary joins which satis es the in nite distributive law _ _ x ^ Y = fx ^ y j y 2 Y g: For frames F and G, a frame morphism is a function f : F ! G that preserves nite meet and arbitrary joins. Frames are also called locales. Johnstone ([7], page 57) provides a way to construct a frame from a meetsemi-lattice based on the notion of coverage relation. De nition 1. Let (S; ^; ) be a meet-semi-lattice. A coverage on S is a relation 2S S satisfying 1. if Y a then for any y 2 Y , y a. 2. if Y a then for any b a, fy ^ b j y 2 Y g b: A -ideal determined by coverage is a subset I of S which is 1. lower-closed: a 2 I & b a ) b 2 I , 2. covered: U a & U I ) a 2 I . A meet-semi-lattice S equipped with a coverage relation is called a site. A frame H with i : S ! H is said to be generated from a site (S; ) if { i preserves nite meets, { i transforms covers to joins: Y a ) i(a) = W i(Y ), and { H; i is universal, i.e., the following diagram commutes:
S i
?? ? H
-F ?? ?9!g f
Remark. For here and for the rest of the paper, all maps are assumed to preserve the respective structures they are acting on. This remark will be implicitly in force for all commutative diagrams and will not be repeated. We also remark in general that such a universal property guarantees that the generated structures are always unique up to isomorphism. Here is Johnstone's basic result for the coverage relation. Theorem 1 (Coverage Theorem [7], page 58). The collection of -ideals under inclusion is the frame generated from a site (S; ). Recall that a frame can be seen as a \point-free" description of the open sets of a topological space. In this view, points are not basic, but are de ned as collection ofWopens: a point of a frame is a completely prime lter, i.e. a lter such that if X 2 then there exists x 2 X such that x 2 : If H is generated
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from (S; ) then a point is determined by its restriction to S , which is a lter of S such that a 2 & Y a ) (9b 2 Y ) b 2 : A frame H is called spatial (or has enough points) if for any a; b 2 H , a b i 8 ; a 2 implies b 2 ; where ranges over points of H . Intuitively, if we regard a; b as sets of points, then a b exactly when a b. There is a standard way to generate a frame from a distributive lattice D. One de nes the coverage by letting U a if and only if U #a and there exists a nite subset X of U such that a = _X . By distributivity, this is a coverage relation. A -ideal is then exactly an ideal of D: a downward-closed subset of D closed under nite joins. The generated frame is precisely the so-called ideal completion of D, which is written as Idl(D). We say that a frame (locale) is coherent or spectral if it is isomorphic to the ideal completion of a distributive lattice1 . The following fact will be used in the rest of the paper. Theorem 2 (Page 65, [7]). Spectral frames are spatial.
2 Sequent structures, distributive lattices, and frames We are interested in the question of frames generated by sequent structures. There are two ways to construct the frame generated by a sequent structure. The rst construction, discussed in this section, is an implicit one built in two steps: obtaining the generated distributive lattice [3] rst, and then taking its ideal completion as mentioned above. The second, explicit construction, is obtained by de ning an appropriate coverage relation, which will be discussed in Section 5. Let's recall the notion of entailment relation introduced by Scott in [15]. De nition 2. An entailment relation (or a sequent structure) is a set A with a binary relation ` between nite subsets Fin(A) of A such that (I )
a`a SX
X`Y Y T S`T X ` Y; a a; X ` Y (C ) X `Y We use the notations X; Y; : : : for nite subsets of A, and X; Y for X [ Y while X; a for X [ fag: (W )
Several properties of entailment relations are self-evident. First, entailment relations are completely symmetric: (A; `) is an entailment relation i (A; a) is. Second, entailment relations are closed under arbitrary intersections. Third, since the largest relation on Fin(A) is an entailment relation, given a family 1
The term coherent is used in such a way in [7]. But it is used with another meaning in domain theory or even in [8]. The term spectral, used because such frames are exactly the ones that are spectrum of a commutative ring, is less ambiguous.
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(Xi ; Yi )i2I of pairs of nite subsets of A, the entailment relation generated by the rules Xi ` Yi can be seen to be the intersection of all entailment relations on A satisfying Xi ` Yi for all i 2 I . (Of course one can close (Xi ; Yi )i2I up by (I), (W), and (C) directly.) Last, information systems [18] can be seen as a special kind of entailment relation generated by rules of the form Xi ` Yi with Yi being either a singleton or empty (intuitionistically they are more complex, however). Distributive lattices freely generated from sequent structures make it possible to use lattice-theoretic constructions in sequent structures. The concept of freely generated lattices is introduced in [3].
De nition 3. For a distributive lattice D and a sequent structure (A; `), a map i : A ! D is said to preserve ` if X ` Y implies ^i(X ) _i(Y ). We say that the distributive lattice L(A) is generated by (A; `) if there is a `-preserving map i : A ! L(A) which is universal among all such maps: f
A i
??
-L ? ?
??9!g
L(A)
Theorem 3 (Cederquist and Coquand [3]). Any entailment relation (A; `) generates a distributive lattice (L(A); ) with a map i : A ! L(A) such that X ` Y , ^i(X ) _i(Y ) for all nite subsets X; Y of A, where i(X ) is the image of X under i.
We can study the similar topic of interpreting a sequent structure in a frame.
De nition 4. Let H be a frame. An interpretation of a sequent structure (A; `) in H is a map m : A ! H such that for every nite X; Y , X ` Y ) ^m(X ) _m(Y ): A frame Frm(A) is generated by (A; `) if there is a universal interpretation m0 : A ! Frm(A):
-H ? m0 ?9?!f ?? ? Frm(A) A
m
Given a sequent structure (A; `), one can rst generate the distributive lattice L(A) using Theorem 3 and then obtain the generated frame Frm(A) := Idl(L(A))
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by ideal completion (see the ending part of Section 1). Combining the two steps, we get the following commutative diagram:
- L(A) idl - Idl(L(A)) ? ? i?? 9!f ?9!g ? ? ? ? ? m ? -H A
A
i
This is exactly a proof of the following result:
Theorem 4. Every sequent structure (A; `) generates a frame Idl(L(A)) with interpretation m0 = idl i. One can prove additionally that the map m0 has the property that for any nite X; Y , X ` Y if and only if ^m0 (X ) _m0 (Y ): By Theorem 3 it is enough to show that if idl(u) idl(v) in Idl(L(A)) then u v in L(A), where idl(u) stands for the principal ideal generated by u 2 L(A). But this follows from the special construction of Idl(L(A)) as the ideal completion of L(A).
3 Ideal elements, prime and completely prime lters What makes the results given in the previous section useful is that we have a canonical correspondence between ideal elements of the sequent structure, prime lters of the distributive lattice, and completely prime lters of the generated frame. We establish the correspondence in this section. We must rst recall what is an ideal element. Ideal elements have been used for representing domains. Given a sequent structure, the set of all of its ideal elements forms a dcpo under inclusion. One can obtain di erent categories of domains by considering di erent (sub)classes of sequent structures [22].
De nition 5. A subset x A is called an ideal element with respect to a sequent structure A = (A; `) if it is closed under entailment (where n stands for \ nite
subset of"):
(X n x & X ` Y ) ) x \ Y inhabited: The set of all ideal elements of A is denoted as jAj. A co-element of a sequent structure (A; `) is an ideal element of (A; a): By logical transposition, one easily checks classically that y is a co-element of (A; `) i y is the complement of an ideal element x of (A; `); but our de nition of co-element is formulated in a purely positive way. As noted earlier, for any sequent structure (A; `), (jAj ; ) is a dcpo (not necessarily with bottom).
7 i L(A), A ?! m Frm(A) be the Let (A; `) be a sequent structure and A ?! universal maps for the generated distributive lattice L(A) and generated frame Frm(A), respectively. For x A, de ne Ix L(A) as
Ix := fu 2 L(A) j (9X x) ^ i(X ) ug and de ne Jx Frm(A) in exactly the same way: Jx := fu 2 Frm(A) j (9X x) ^ m(X ) ug: We have the following result, which shows that ideal elements, prime lters, and completely prime lters uniquely determine each other under their respective universal maps. A direct proof of the second item is given later in Proposition 1. Theorem 5. Let (A; `) be a sequent structure. 1. If I is a prime lter of L(A) then the restriction of I to A, that is the set i?1 (I ), is an ideal element. Conversely if x is an ideal element of (A; `) then Ix L(A) is a prime lter such that x = i?1(Ix ): 2. If J is a completely prime lter of Frm(A) then the restriction of J to A, that is the set m?1 (J ), is an ideal element. Conversely if x is an ideal element of (A; `) then Jx Frm(A) is a completely prime lter such that x = m?1 (Jx ): Proof. The rst item is stated in [3] and the second item follows from item 1, Theorem 4, and an exercise in ([7], page 66) which states that there is a bijection between prime lters of L(A) and completely prime lters of Frm(A). Note that ideal elements need not exist for an arbitrary sequent structure. In particular, if we allow ; ` ;, then there is no way to obtain an ideal element. However, we have this basic result: Theorem 6 (Completeness). Every sequent structure (A; `) has enough ideal elements: X ` Y i for all ideal elements x, the set x \ Y is inhabited whenever
X x:
This theorem is an immediate consequence of Theorem 2 and Theorem 5 above. A quite standard direct proof also exists by using classical logic and a weak form of the axiom of choice: one shows that if X 6` Y , then there is an ideal element x such that X x but x \ Y = ;. This is done by showing that the maximal lter F containing ^X and disjoint from #_Y in the generated lattice L(A) is prime. The ideas used in such a proof seem to come from Birkho [2]. It is worth noting a number of consequences of Theorem 6. First, if we start from a set of pairs f(Xi ; Yi ) j i 2 I g, then the least entailment relation generated by it can be described as X ` Y if and only if for any x, if X x, then x \ Y is inhabited, where x is an ideal element determined by f(Xi ; Yi ) j i 2 I g. Secondly, as a special case of Theorem 6, we have ; ` ; if and only if the sequent structure does not have any ideal element. This is precisely when the generated distributive lattice L(A) is degenerated, i.e., 0 = 1. (However, a direct proof of this and the next remark is possible.)
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Thirdly, from the proof of Theorem 6 we see that for any nite set X A, there is an ideal element containing X if and only if X 6` ;. Finally, notice that rule (C ) is a form of the resolution rule. Thus, we get as a consequence completeness of resolution: a clause X ` Y is a semantical consequence of a set of rules Xi ` Yi , that is is valid in any model satisfying these rules, i it can be deduced from these rules using (I ); (W ) and (C ):
4 Clausal logic and hyperresolution The notion of clause is a basic concept in logic programming. A natural framework for reasoning about clauses, called clausal logic, is demonstrated in [13, 23] to play a fundamental role in disjunctive logic programming semantics. With respect to a sequent structure (A; `), a clause is a nite subset of A, and a clause set is a collection of clauses. An ideal element x is a model of a clause u if x \ u 6= ;. x is a model of a clause set W if it is an model of every clause in W . There are three distinct notions of inference in clausal logic: j=, `hr , and the \choice inference" 99 K. For a clause set W and a clause u, we write 1. W j= u if every model of W is a model of u. This is a model-theoretic concept, capturing the semantics. 2. W `hr u if either ; 2 W , or u can be deduced from W using the so-called hyperresolution rule
a1 ; X 1
:::
an ; Xn a1 ; : : : ; an ` Y X1 ; : : : ; Xn ; Y
This is clearly a proof-theoretic, or operational, concept. 3. fX1 ; : : : ; Xn g 99 K u if fai j 1 i ng ` u for any choice a1 2 X1 ; a2 2 X2 ; : : : ; an 2 Xn . This is an intermediate notion: it uses the notion of arbitrary choice. A result of [23] is that the three distinct notions of inference are equivalent to each other. Theorem 7 (Rounds and Zhang). Let (A; `) be a sequent structure. Let W be a nite clause set, and u a clause. The following three items are equivalent: 1. W j= u, 2. W `hr u, 3. W 99 K u: For any clause set C , we write *C for the least clause set containing C and closed under hyperresolution. A disjunctive state is a clause set C such that C = *C . The concept of disjunctive state is well-behaved on sequent structures [23]: Theorem 8. For a sequent structure A, the set of all its disjunctive states under inclusion is a complete lattice. This theorem will be re ned later, by giving a universal property of the lattice of disjunctive states w.r.t. the sequent structure (A; `):
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5 Explicit construction of generated frame using coverage It is possible to give an explicit construction of the generated frame from a sequent structure (A; `) through an appropriate coverage relation de ned by a dual form of hyperresolution. For a sequent structure (A; `), consider the meet-semi-lattice (Fin(A); [; ) and the relation de ned by fa1; X; a2 ; X; : : : ; an ; X g X i X ` a1 ; : : : ; an . Note that no subscripts are used for the X s here. Note also that if X ` ;, then we have f g X (one can take this to be the n = 0 case). This is clearly a coverage relation, according to De nition 1. A -ideal is, by de nition, precisely a subset U Fin(A) such that { if X 2 U and Y X , then Y 2 U ; { if fa1; X; a2 ; X; : : : ; an; X g U and X ` a1; : : : ; an, then X 2 U . We call such -ideals conjunctive states and write H0 for the set of all conjunctive states. For a set U Fin(A), we write cU for the conjunctive state generated by U . Note that there is a conceptually simpler way to generate such a conjunctive state: rst close U under nite super sets, and then add in all the X s that are covered by some nite subset of the resulting set. We can do this because the only way to obtain a covered set is by removing at most one element from an existing set. There is also a useful proof-theoretic reading of the generated conjunctive state. For any set U Fin(A), its generated state cU consists of all X s that can be derive from assumptions from U by using supersets of sets in U and the unique rule of inference:
a1 ; X
::: X
an ; X
provided X ` a1 ; : : : ; an
By Theorem 1, we immediately obtain that the set of conjunctive states under inclusion is the frame generated from the meet-semi-lattice (Fin(A); [; ) with coverage , which depends on `. We show that this frame has the required universal property for an interpretation. Lemma 1. Let H be any frame. There is a bijection between ( nite) meetpreserving maps i : Fin(A) ! H that transforms covers to joins, and interpretations m : A ! H . Proof. Suppose i : Fin(A) ! H preserves nite meets and transforms covers to joins. De ne a map mi : A ! H by letting mi (a) := i(fag) for each a 2 A. We show that mi is an interpretation. Since i preserves nite meets and meet for Fin(A) is set union, we have, for any nite X A,
i(X ) = i(
[
a2X
fag) = ^a2X i(fag) = ^a2X mi (a) = ^mi (X ):
Suppose X ` Y , with Y = fa1 ; : : : ; an g. By the de nition of ,
fa1 ; X; a2 ; X2 ; : : : ; an ; X g X:
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Since i transforms covers to joins, we have i(X ) = i(X [ fa1g) _ _ i(X [ fang) i(fa1 g) _ _ i(fang) = _mi (Y ): Therefore, ^mi (X ) _mi (Y ), as needed. (Note that when X ` ;, the empty collection f g covers W X , by de nition. Transforming covers to joins in this case means i(X ) = ; = 0, which can be restated as ^mi (X ) _mi (;).) Suppose, on the other hand, that m : A ! H is an interpretation. We de ne a map im : Fin(A) ! H by letting im(X ) := ^m(X ) for each X 2 Fin(A). By this de nition, im automatically preserves nite meets. We show that it also transforms covers to joins. If fa1 ; X; a2 ; X; : : : ; an ; X g X then by de nition X ` a1 ; : : : ; an . Therefore, ^m(X ) m(fa1g) _ m(fa2g) _ _ m(fan g): By distributivity, we have ^m(X ) = (^m(X [ fa1 g)) _ _ (^m(X [ fan g)): This means im(X ) = im (X [ fa1g) _ _ im (X [ fang), which is exactly the required property of \transforming covers to joins". It is clear that the given transformations i 7?! mi and m 7?! im amount to a bijection. By the previous lemma and the Coverage Theorem, we arrive at the next conclusion, which says that H0 is the generated frame from (A; `). Theorem 9. For any sequent structure (A; `), the set of its conjunctive states H0 is a frame under inclusion. Moreover, the interpretation m0 : A ! H0 mapping a to cfag is universal. Furthermore we have X ` Y if and only if ^m0 (X ) _m0 (Y ) for all nite subsets X; Y of A. Lemma 2. Let X; Y 2 Fin(A). Then cfX g^ cfY g = cfX g\ cfY g = cfX [ Y g: Proof. We show the non-trivial part that cfX g \ cfY g cfX [ Y g: Suppose Z 2 cfX g \ cfY g. Then one has a derivation tree for Z with supersets of X as leaves/premises as well as a derivation tree for Z with supersets of Y as leaves. One can use structural induction on derivations to show that the two derivation trees can always be put together to obtain a derivation tree for Z with supersets of X [ Y as leaves. Therefore, Z 2 cfX [ Y g. The concrete notion of coverage allows a direct proof of the correspondence between ideal elements of (A; `) and completely prime lters of H0 , repeated as follows. Proposition 1. Let m0 : A ! H0 be the universal interpretation given in the previous theorem. If J is a completely prime lter of H0 then the restriction of J to A, that is the set m?0 1 (J ), is an ideal element of (A; `). Conversely, if x is an ideal element then Jx H is a completely prime lter such that x = m?0 1 (Jx ), where Jx := fu 2 H0 j (9X x) ^ m0 (X ) ug:
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Proof. Suppose J is a complete prime lter of H0 . We show that m?0 1 (J ) is an ideal element. Suppose X ` Y and X m?0 1 (J ). Then m0 (X ) J and so ^m0 (X ) 2 J since J is a lter. Now Theorem 9 implies ^m0 (X ) _m0 (Y ), and so _m0 (Y ) 2 J . As J is prime, we have m0 (b) 2 J for some b 2 Y . So Y \ m?0 1 (J ) is inhabited. On the other hand, suppose x is a ideal element. We show that Jx is a completely prime lter. It is easy to see that it is a lter. To show it is completely prime, we use the concrete representation of elements in H0 as conjunctive states. BySLemma 2, ^m0 (X ) = cfX g. It suces to show that if X x and cfX g c( i2I ui ), where ui are conjunctive states, then there exists some Y x and i 2 I such that Y 2 ui . For this it is enough to notice that whenever we can apply the rule
a1 ; X
::: X
an ; X
::: X
an ; X
provided X ` a1 ; : : : ; an
and X x then there exists i such that X; ai x: Indeed there exists i such that ai 2 x because x is an ideal element. As a result of Theorem 9, we can talk about joins and meets of nite subsets of A, with the understanding that such operations are always carried out in the generated frame, H0 (or in the generated lattice L(A)). This is indeed the notational convention we adopt for the rest of the paper: ^X stands for ^m0 (X ). Call ^X a semantical consequence of ^X1 ; : : : ; ^Xn if for any ideal element x, X x implies Xi x for some i. We have the following completeness result, which is dual to Theorem 7. Proposition 2. Let H0 be the frame generated by (A; `). The following are equivalent in H0 : 1. ^X is a semantical consequence of ^X1 ; : : : ; ^Xn . 2. ^X ^X1 _ _ ^Xn . 3. X ` a1 ; : : : ; an for any choice a1 2 X1 ; : : : ; an 2 Xn . If we apply the construction H0 to the opposite of the relation `, which is also an entailment relation, but still use the same underlying meet-semi-lattice, we get the following result. Theorem 10. The complete lattice of all disjunctive states of (A; `) is the frame generated by (A; a): Proof. The elements of the frame generated by a are sets U of nite sets of A such that X 2 U whenever we have X1 ; : : : ; Xn 2 U with _X1 ^ ^_Xn _X: This is the same as the complete lattice of disjunctive states (see Section 4). In particular, there is a canonical correspondence between points of the frame of all disjunctive states and co-elements of the sequent structure (A; `): It is clear that the hyperresolution rule (Section 4) is equivalent to the rule
a1 ; X
provided a1 ; : : : ; an ` X
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together with the rule
X Y
provided X Y :
A simple combinatorial argument on permutation of rules show that we can even suppose the use of this last rule limited to the leaves of the derivation tree. By duality, it follows from our results that X is derived by hyperresolution from X1 ; : : : ; Xn i
_X1 ^ ^ _Xn _X
holds in D or equivalently, in H0 : Using Theorem 2 for the spectral frame H0 , this is true if and only if any point of H0 containing _X1 ; : : : ; _Xn contains also _X , which means exactly that the clause X is a semantical consequence (Section 4) of the clauses X1 ; : : : ; Xn : We get in this way yet another derivation of the completeness of the hyperresolution rule, Theorem 7 (see [12, 23] as well). By soundness of the cut rule (C ), which is nothing else than a form of the resolution rule, this gives a constructive proof that transforms any resolution proof into a hyperresolution proof. In particular this shows the equivalence between `hr and the \choice inference" 99 K, as stated in Theorem 7. There is, however, a direct proof of this equivalence.
Proposition 3. We have X1; : : : ; Xn 99 K X if and only if X follows from X1 ; : : : ; Xn by the hyperresolution rule.
Proof. For the \if" part we refer to [23]. We prove the \only if" part by induction on the size jXi j: Let a1 2 X1 ; : : : ; an 2 Xn . We claim that we can deduce all the clauses X; ai (1 i n) from X1 ; : : : ; Xn using the hyperresolution rule. The result follows then from
a1 ; X
: : : an ; X provided a1 ; : : : ; an ` X X Let us prove X; a1 from X1 ; : : : ; Xn ; the other cases are similar. Notice that we have b1 ; : : : ; bn ` X; a1 for any choice b1 2 X ? fa1 g; b2 2 X2 ; : : : ; bn 2 Xn . By induction hypothesis, we can deduce X; a1 from X1 ? fa1g; X2 ; : : : ; Xn and hence from X1 ; : : : ; Xn:
6 Example: Spectrum of a ring Let us give an example in algebra, that illustrates some of the notions introduced here. Let A be a commutative ring, and consider the entailment relation generated by the axioms { ` 0,
{ x ` xy { x; y ` x + y
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{ xy ` x; y { 1`
We have the following direct description of `.
Theorem 11. X ` Y if and only if the product of elements in Y belong to the radical of the ideal generated by X .
Proof. We prove rst that the relation \the product of elements in Y belong to the radical of the ideal generated by X " is an entailment relation, which satis es all the rules above. We analyse only the rule (C ), the other rules being directly checked: assume that we have both X ` Y; a and a; X ` Y . Let y be the product of the elements in Y and I the ideal generated by X: We reason in A=I : by assumption ya is nilpotent (in A=I ) and y belongs to the radical of the ideal generated by a: So we have m; n and x such that yn = ax and (ya)m = 0: This implies ym (ax)m = ymn+m = 0 and hence y is nilpotent in A=I: Hence X ` Y as required. It is direct that this entailment relation satis es all the rules above. Conversely, if the product of elements in Y belong to the radical of the ideal generated by X , we can derive X ` Y using only the given axioms. Indeed, the rst third rules show that X ` y whenever y belongs to the ideal generated by X , while the two last rules show y1 : : : ym ` y1 ; : : : ; ym :
In the particular case where A is a ring of polynomials, notice that we recover \for free" the proof of the formal Nullstellensatz theorem presented in [10]: the following items { x1 ; : : : ; xn ` y is a consequence of the above axioms, { y belongs to the radical of the ideal generated by x1 ; : : : ; xn , { fyg can be derived from fx1g; : : : ; fxng by hyperresolution are equivalent. An ideal element of this entailment relation is then exactly a proper prime ideal of A. Furthermore, if I is a radical ideal of A; then the set of nite subsets whose product is in I is a disjunctive state UI . Conversely, if U is a disjunctive state, and I is the set of elements x such that fxg 2 U then I is a radical ideal such that U = UI :
7 Categorical equivalences We extend our terminology rst in order to adequately express categorical concepts related to sequent structures. We have a natural category Seq of sequent structures, where a map f : A ! B is simply a map which preserves entailment: X1 ` X2 in A implies f (X1 ) ` f (X2 ) in B . Furthermore, any distributive lattice D (and hence any frame) de nes a sequent structure G(D) by taking X ` Y to mean ^X _Y . This de nes a
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functor G : Spec ! Seq from the category of spectral frames and an interpretation m : A ! H is nothing else than a map A ! G(D) in the category
Seq:
If A; B are sequent structures, we de ne an approximable relation from A to B to be an interpretation m : A ! Frm(B ) of A in the frame generated by B .
Notice that, in view of Theorem 9, this can be seen as a relation ` between nite subsets of B and elements of A satisfying the following conditions: { for any x 2 A the set of all Y B such that Y ` x is a conjunctive state, { if we have x1 ; : : : ; xn `A u1; : : : ; um and Y ` xi ; 1 i n then there exists Y1 ; : : : ; Ym such that Yj ` uj ; 1 j m and Y `B y1 ; : : : ; ym for any choice yj 2 Yj , j = 1; : : : ; m. By standard categorical construction (see for instance [11], Chapter VI, 5) we get that sequent structures with approximable maps form a category RelSeq: Similarly, we can introduce the category RelLat of distributive lattices, and maps m : D ! Idl(E ), where Idl(E ) is the frame generated by E . Theorem 12. The categories RelSeq; RelLat; Spec are equivalent. Proof. The equivalence between RelLat; Spec is standard (see [1], page 120), while the equivalence between RelSeq and Spec follows from the universal properties of the free frame construction (see for instance [11], Chapter VI, 5, Exercise 2).
8 Concluding remarks Sequent structures are the skeletons of propositional theories. A propositional theory can be reduced to a sequent structure by translating an entailment instance '1 _ '2 ` 1 ^ 2 to simpler ones 'i ` j (i; j 2 f1; 2g) repeatedly until only ^ appear on the left, and only _ appear on the right (distributivity is used in this process). The remaining ^'s and _'s can then be removed by virtue of sequents. Of course this process can be reversed; but we believe that working at the sequent level can in many cases avoid tedious syntactic details. It is possible to provide a similar treatment to in nitary sequent structures. These structures consist rules of the form X ` Y , with X nite and Y arbitrary. Any such structure can still be canonically embedded into a frame. However, completeness and compactness fail in this case. Except for the purpose of representing L-domains [22] and of providing a connection to sober spaces, the signi cance of such a concept remains to be seen. We omit the treatment of them due to space limitations. We end by repeating the hope given in the introduction that this paper be a rst step in exploring completeness of various logical systems by means of canonical embedding to locales. It should be interesting to develop richer tools for this purposes, in order to handle additional logical operators. The well-known Henkin construction for instance, has been investigated in this setting [14] for linear logic.
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Acknowledgment. We would like to thank the anonymous referees for insightful comments which lead to the improved presentation.
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