2 logic, such as those for resolution and hyperresolution. A number of recent developments serve as the motivation for the current paper. In [2, 3], i

Transcription

1 Sequents, Frames, and Completeness Thierry Coquand 1 and Guo-Qiang Zhang 2?? 1 Department of Computer Science, University of Goteborg S , Goteborg, Sweden coquand@cs.chalmers.se 2 Department of Computer Science, University of Georgia Athens, GA 30602, U. S. A. gqz@cs.uga.edu 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. Key words: sequent structures, lattices, frames, domain theory, resolution, category. Introduction Entailment relations were introduced by Scott as an abstract description of Gentzen's sequent calculus [13{15]. It can be seen as a generalisation of the earlier consequence calculus of Hertz [8] to a multi-conclusion consequence relation. The notion of consequence relation, with only one conclusion, has been analysed by Tarski [17]. This consequence calculus has been used by Scott in order to give a concrete representation of domains, as in information systems [16]. 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 [18, 19] and [4], in an independent way from Scott's work on entailment relations (in [18], a set together with an entailment relation is called a sequent structure). Another related reference, also independent from Scott's work, is [7]. In this paper we analyse various completeness theorems for sequent structures with a goal of providing a unied way to present completeness results in?? Corresponding author. Phone: , Fax:

2 2 logic, such as those for resolution and hyperresolution. A number of recent developments serve as the motivation for the current paper. In [2, 3], 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, [20] introduces the notion of disjunctive state based on the socalled hyperresolution rule. Completeness of hyperresolution provides the basis for this domain-theoretic semantics: it establishes the equivalence of the modeltheoretic 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 (with top; see [11, 20]). A natural question is whether disjunctive states can be seen as a universal construction with respect to a sequent structure. Related to this question is the canonical embedding of a sequent structure into a frame. For this purpose we use Johnstone's coverage method [6] to study the frame generated from a sequent structure (A; `), as well as the frame generated from the opposite (A; a). Interestingly, the frame generated from (A; a) 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 element of the underlying sequent structure. The completeness theorem of coherent logic, which can be stated as the fact that any coherent (or spectral) frame is spatial [6] 1 ensures that there are enough points to uniquely determine the entailment relation, where \points" correspond to \models". We get in this way another proof of completeness of hyperresolution. In return, existing results related to hyperresolution suggest dierent ways to construct the generated frames: a semantical one, a proof-theoretic one, and a third one based on the notion of \choice inference". A number of results in this paper may be seen as \folklore". We feel that our contribution lies in tying in the more discrete notion of sequent structures with the more complete notion of locales. 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. Organisation. Section 1 recalls basic concepts related to frames. Following standard convention, we use the terms \frame" and \locale" interchangeably 1 We recall the meaning of this statement in Section 1.

3 except when categories are concerned. Section 2 discusses ideal elements of sequent structures, their relationship to prime lters of the embedded lattices, and completeness of sequent structures. Section 3 presents the notion of disjunctive states based on hyperresolution. By canonical embedding of sequent structures in frames, Section 4 studies the operational aspects of these frames in light of the completeness of coherent logic. In particular, completeness of hyperresolution becomes a by-product in this context, and the complete lattice of disjunctive states is the frame generated by the opposite of a sequent structure. Section 5 discusses the example of the spectrum of a ring to illustrate the concepts, although the example is interesting in its own right. Section 6 gives the categorical equivalences among the categories of sequent structures, distributive lattices, and spectral locales by introducing appropriate morphisms. Concluding remarks are given in the last section. 3 1 Preliminaries We recall some basic denitions related to frames and sequent structures. For details about frames we refer to [6], and for sequent structures [2, 18]. 1.1 Frames and coverage A frame is a poset with nite meets and arbitrary joins which satises the innite distributive law x ^_ Y = _ fx ^ y j y 2 Y g: For frames F and G, a frame homomorphism is a function f : F! G that preserves nite meet and innite joins. Johnstone [6] provides a way to construct a frame from a meet-semi-lattice based on the notion of coverage relation. Denition 1. Let (S; ^; ) be a meet-semi-lattice. A coverage C on S is a relation a 2 S S satisfying the property that 1. Y a a & y 2 Y ) y a, 2. Y a a & b a ) fy ^ b j y 2 Y g a b: A C-ideal determined by a coverage C is a subset I of S which is 1. lower-closed: a 2 I & b a ) b 2 I, 2. covered: U a a & U I ) a 2 I. A meet-semi-lattice S equipped with a coverage relation C is called a site. A frame H with i : S! H is said to be generated from a site (S; C) if { i preserves nite meets, { i transforms covers to joins: Y a a ) i(a) = W fi(y) j y 2 Y g, and { H; i is universal: for any other map f : S! F with the same property, there is a unique frame homomorphism f 0 such that f 0 i = f.

4 4 Recall that a frame can be seen as a \point-free" description of the open sets of a topological space. In this approach, points are not basic, but can be dened as collection of opens; more precisely, a point of a frame is a completely prime lter, i.e. a lter such that whenever W X 2 there exists x 2 X such that x 2 : If H is generated from (S; C) then a point is determined by its restriction to S, which is a lter of S such that, whenever a 2 and Y a a there exists b 2 Y such that b 2 : We say that a frame H is spatial or has enough points if u v holds in H exactly when any point containing u also contains v. Intuitively, it means that if we see u; v as sets of points, then u v holds in H i u is a subset of v. Here is Johnstone's basic result for the coverage relation. Theorem 1 (Coverage Theorem [6]). The collection of C-ideals under inclusion is the frame generated from a site (S; C). Let us mention an important example. If D is a distributive lattice, we dene a coverage C by letting U a a i U #a and there exists a nite subset X of U such that a = _X. By distributivity, this is a coverage relation. A C-ideal is then exactly an ideal of D: a downward closed subset of D containing 0 and closed by nite joins. We say that a frame (locale) is coherent or spectral if it is isomorphic to such an ideal completion of a distributive lattice 2. Similarly, one can generate a frame from an entailment relations [2], a result which we recall next. 1.2 Entailment relations and distributive lattices The notion of entailment relation was introduced by Scott in [13]. Denition 2. An entailment relation over a set A is a binary relation X ` Y between nite subsets Fin(A) of A such that (I) (W ) (C) X \ Y inhabited ; X ` Y X 0 X X ` Y Y Y 0 ; X 0 ` Y 0 X ` Y; a a; X ` Y : X ` Y We use the notations X; Y; : : : for nite subsets of A, and X; Y denotes X [Y while X; a denotes X [ fag: Entailment relations will also be called sequent structures in this paper. They are completely symmetric: (A; `) is an entailment relation i (A; a) is. The trivial entailment relation is the one such that X ` Y always hold. Given a family (X i ; Y i ) i2i of pairs of nite subsets of A, we can consider the entailment 2 The term coherent is used in such a way in [6]. But it is used with another meaning in domain theory or even in [7]. The term spectral, used because such frames are exactly the ones that are spectrum of a commutative ring, is less ambiguous.

5 relation generated by the rules X i ` Y i which is the intersection of all entailment relations on A satisfying X i ` Y i for all i 2 I. 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 [2]. If D is a distributive lattice and ` a binary relation on Fin(A) we say that i : A! D preserves ` i ^x2x i(x) _ y2y i(y) whenever X ` Y: We say that D; i : A! D is generated by the sequent structure (A; `) i i preserves ` and for any other map f : A! L preserving ` there is a unique lattice map f 0 : D! L such that f 0 i = f: We shall write ^X for ^x2x i(x) and _Y for _ y2y i(y). Every entailment relation (A; `) generates a distributive lattice. The following theorem is proved in [2]. Theorem 2. If D; i : A! D is the distributive lattice generated by an entailment relation ` then we have X ` Y i ^X _Y in D. We describe a construction of D here, since similar ideas are used elsewhere in this paper. The elements of D are nite sets fx 1 ; : : : ; X n g of nite subsets of A; representing intuitively ^X 1 ^X n : We then dene fx 1 ; : : : ; X n g fy 1 ; : : : ; Y m g by saying that for any i < n and any choice b 1 2 Y 1 ; : : : ; b m 2 Y m we have X i ` b 1 ; : : : ; b m : It is checked in [2] that this denes a reexive and transitive relation, and the quotient by the corresponding equivalence relation is a distributive lattice generated by ` if we dene i(a) = ffagg: Yet another proof will be to dene ^; _ formulae where atomic formulae are elements of A, and to prove a cut-elimination theorem for this logic. Finally, we shall give another denition of fx 1 ; : : : ; X n g fy 1 ; : : : ; Y m g using a deduction rule similar to hyperresolution [10]. It follows from this construction that given X; Y in Fin(A) we have ^X _Y in D i X ` Y. This shows also that for X; X 1 ; : : : ; X n in Fin(A) we have in D i ^X ^X 1 ^X n X ` a 1 ; : : : ; a n for any choice a 1 2 X 1 ; : : : ; a n 2 X n : Finally, we note that any element of D can be put on the form ^X 1 ^X n or dually on the form _X 1 ^ ^ _X n. 5 2 Domain of elements When representing domains, we use the set of ideal elements associated with an entailment relation, under inclusion, to represent a dcpo.

6 6 Denition 3. 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 denition of co-element is formulated in a purely positive way. Proposition 1. If ` is generated by the rules (X i ` Y i ) i2i, then x A is an ideal element of ` i for all i 2 I, x \ Y i is inhabited whenever X i x: Proof. For the \if" part, let x be an arbitrary subset of A: We consider the relation X `x Y dened by X n x ) x \ Y inhabited: This is an entailment relation. We have X i `x Y i for all i 2 I, by assumption. Hence ``x since ` is the least entailment satisfying X i ` Y i for all i 2 I. This means that x \ Y is inhabited whenever X n x and X ` Y; and hence x is an ideal element of `. For x A we let I x D be the set of elements u 2 D such that there exists X x satisfying ^X u. The following fact, stated in [2], connects ideal elements of (A; `) and prime lters of D. Proposition 2. If I is a prime lter of D 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 I x D is a prime lter such that x = i?1 (I x ): 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 the two basic results: Proposition 3. Let (A; `) be a sequent structure. Then (jaj ; ) is a dcpo (not necessarily with bottom). The proof is straightforward, making essential use of the fact that in a sequent structure, the left hand side of ` is a nite set. The next result is a form of the completeness of coherent logic [6]. Theorem 3 (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: The proof is quite standard, using classical logic and a weak form of the axiom of choice [6] (see also [13]).

7 Proof. The non-trivial part is to show that X ` Y, if for all ideal elements x, X x implies x \ Y 6= ;. In other words, we need to show that if X 6` Y, then there is an ideal element x such that X x and x \ Y = ;. In light of Proposition 2, it suces to show the existence of a prime lter in D which contains ^X and is disjoint from #_Y, given that X 6` Y. Consider the set of lters F containing ^X and disjoint from # _Y. It is straightforward to check that any intersection of such lters F is again a lter disjoint from #_Y. By Zorn's lemma, there is a minimal lter F 0 among such F s. We show that F 0 is prime. Clearly 0 D not in F 0. Suppose a _ b 2 F 0. Since F 0 is a lter, (a_b)^(^x) is a member of F 0. By distributivity, this means that ^(fag [ X) _ ^(fbg [ X) 2 F 0 : Therefore, we cannot have both ^(fag[ X) _Y and ^(fbg[ X) _Y; because F 0 is disjoint from #_Y. So we may assume that "^(fag [ X) \ #_Y is empty. This implies F 0 = "^(fag [ X), by minimality of F 0. Hence a 2 F 0. It is worth noting a number of consequences of this result. First, if we start from a set of pairs f(x i ; Y i ) j i 2 Ig, then the least entailment relation generated by it can be described as X ` Y i for any x, if X x, then x \ Y is inhabited, where x is an ideal element determined by f(x i ; Y i ) j i 2 Ig. Secondly, as a special case of Theorem 3, we have ; ` ; i the sequent structure does not have any ideal element. This is precisely when the generated distributive lattice D is degenerated, i.e., 0 D = 1 D. Thirdly, from the proof of Theorem 3 we see that for any nite set X A, there is an ideal element containing X i 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 X i ` Y i, that is is valid in any model satisfying these rules, i it can be deduced from these rules using (I); (W ) and (C): 7 3 Hyperresolution and disjunctive states The notion of clause is a basic concept in logic programming. A natural framework for reasoning about clauses, called clausal logic, is demonstrated in [11, 20] 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 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.

8 8 2. W ` hr u if either ; 2 W, or u can be deduced from W using the so-called hyperresolution rule a 1 ; X 1 : : : a n ; X n a 1 ; : : : ; a n ` Y X 1 ; : : : ; X n ; Y This is clearly a proof-theoretic, or operational, concept. 3. W 99 K u if for any choice a 1 2 X 1 ; a 2 2 X 2 ; : : : ; a n 2 X n, we have fa i j 1 i ng ` u, where W = fx 1 ; : : : ; X n g. This is an intermediate notion: it uses the notion of arbitrary choice, which is not syntactical. A result of [20] is that the three distinct notions are equivalent to each other. Theorem 4 (Completeness of hyperresolution). 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 Ku: 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 [20]: Theorem 5. For a sequent structure A, the set of all its disjunctive states under inclusion is a complete lattice. This theorem will be rened later, by giving an universal property of the lattice of disjunctive states w.r.t. the sequent structure (A; `): 4 Entailment Relations and Completeness results We can analyse the similar problem of interpreting a sequent structure in a frame, which is a poset with nite meets and arbitrary join such that meet are distributive over joins [6]. Let H be a frame. We dene an interpretation of a sequent structure to be a map m : A! H such that ^x2x m(x) _ y2y m(y) whenever X ` Y 3. We build now the frame generated by a sequent structure (A; `): It is a frame H 0 with an interpretation m 0 : A! H which is universal among all interpretations: if m : A! H is any interpretation there exists a unique frame morphism, i.e. map preserving nite meets and arbitrary joins, f : H 0! H such that m = f m 0 : Let U; V; : : : denote arbitrary subset of Fin(A): If X 2 Fin(A) we dene when U covers X: it is the case if we can build a derivation tree having X as a root using the unique rule of inference a 1 ; X : : : a n ; X X provided X ` a 1 ; : : : ; a n 3 Notice that this would make sense even if Y was an arbitrary subset, not necessarily nite.

9 where all leaves are superset of elements in U. Since this rule is nitary, a derivation tree of X from U is a nite tree and if X is covered by U it is covered by a nite subset of U. Notice also that these rules can be thought of as a dual form of hyperresolution [10]. An important remark is that this notion of covering has the localisation property: if U covers X then it covers any set containing X. Finally all leaves of a derivation tree of X contains X as a subset. If we start from an entailment relation ` generated by a set of rules X i ` Y i we can replace the rule for covering relation by the following rule inference a 1 ; X : : : a n ; X X provided X i X and Y i = a 1 ; : : : ; a n Let us say that a subset U Fin(A) is a conjunctive state i X 2 U whenever U covers X. This implies that X 0 2 U whenever X X 0 and X 2 U, because we have then that U covers X 0 : Intuitively U will represent W X2U ^x2xx: Let H 0 be the set of all conjunctive states. We dene the map m 0 : A! H 0 by taking for m 0 (a) the set of all sets X covered by fag: We take U ^ V to be fx; Y j X 2 U & Y 2 V g and _ i2i U i to be the set of all X covered by [ i2i U i. Notice that, if U; V are in H 0, the meet U ^ V is actually the same as the intersection of the sets U and V. Proposition 4. H 0 is a frame, and the interpretation H 0 ; m 0 : A! H 0 is universal. Furthermore we have X ` Y i ^x2x m 0 (x) _ y2y m 0 (y) in H 0. Proof. The rst assertion is a particular case of the denition of frames by coverage relation (see Section 1.1 and [6]). We check here the non-trivial point, the distributivity law, by proving U ^_ i U i _ i (U ^ U i ): One has to prove that if U and [ i U i both cover X then X is covered by [ i (U\U i ): This follows directly from the localisation property: in the derivation tree of X from U all leaves are also covered by [ i U i, because they contain X as a subset. Hence each leave is covered by [ i (U \ U i ) and X itself is covered by [ i (U \ U i ): To check the second assertion we prove that if X is covered by U = ffyg j y 2 Y g then we have X ` Y. This is direct by induction on the derivation tree showing that X is covered by U. We can build another solution of this universal problem, by considering the frame of all ideals of the distributive lattice generated by the sequent structure (A; `): By uniqueness of the solution, it follows that H 0 is isomorphic to the this frame. Proposition 5. A point (i.e., a completely prime lter) of the frame H 0 is completely determined by its restriction to A, and such a restriction is exactly an ideal element of the sequent structure (A; `): if X and X ` Y then Y \ is inhabited. 9

10 10 Proof. The fact that is determined by its restriction to A follows from the fact that any element of H 0 is a join of nite meets of elements of A: If X and X ` Y we have ^X 2 and ^X _Y. Hence _Y 2 and Y \ is inhabited. Conversely, let x A be an ideal element of (A; `) and dene x H 0 by: u 2 x i there exists X such that X x and ^X u: Then x is a point of H 0 : For this it is enough to notice that whenever we can apply the rule a 1 ; X : : : a n ; X X provided X ` a 1 ; : : : ; a n and X x then there exists i such that X; a i x: Indeed there exists i such that a i 2 x because x is an ideal element. This shows that if U covers X and X x then there is Y 2 U such that Y x: Finally, we have m?1 ( 0 x) = x because ^X m 0 (a) holds i X ` a and X ` a, X x imply a 2 x because x is an ideal element. Thus there is a canonical correspondence between ideal elements of A, prime lter of D and completely prime lter of H 0 : Let us say that ^X is a semantical consequence of ^X 1 ; : : : ; ^X n i any ideal element containing X i for some i contains also X: From this discussion and the completeness theorem of coherent logic, that any coherent, or spectral, frame is spatial [6], follows the completeness result Proposition 6. Let H 0 equivalent in H 0 : be the frame generated by (A; `). The following are 1. ^X is a semantical consequence of ^X 1 ; : : : ; ^X n. 2. ^X ^X 1 ^X n. 3. X ` a 1 ; : : : ; a n for any choice a 1 2 X 1 ; : : : ; a n 2 X n. 4. X is covered by X 1 ; : : : ; X n. Proof. That H 0 is spatial means that u v in H 0 i any point of H 0 containing u contains v: We illustrate the equivalence of (1) and (2) here. Given the equivalence between ideal elements and points (Proposition 2), this implies that ^X ^X 1 ^X n holds i for any ideal element x; we have X i x for some i whenever X x: As noticed above, this theorem uses classical logic and a weak form of the axiom of choice [6]. If we apply this construction to the opposite of the relation `, which is also an entailment relation, we get the following results Theorem 6. The complete lattice of all disjunctive states is the frame generated by a : Proof. The elements of the frame generated by a are sets U of nite sets of A such that X 2 U whenever X is covered by U, where X is covered by U i we have X 1 ; : : : ; X n 2 U such that _X 1 ^ ^ _X n _X: This is the same as the complete lattice of disjunctive states [20].

11 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 3) is equivalent to the rule together with the rule a 1 ; X : : : a n ; X X X 0 provided 11 a 1 ; : : : ; a n ` X provided X 0 X: X 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 X 1 ; : : : ; X n i _X 1 ^ ^ _X n _X holds in D or equivalently, in H 0 : Using the completeness theorem of coherent logic [6] for the spectral frame H 0, this is the case i any point of H 0 containing _X 1 ; : : : ; _X n contains also _X, that is i the clause X is a semantical consequence of the clauses X 1 ; : : : ; X n : We get in this way yet another derivation of the completeness of the hyperresolution rule, Theorem 4 (see [10, 20] 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 an hyperresolution proof. In particular this shows the equivalence between ` hr and the \choice inference" 99 K: There is, however, a direct proof of this equivalence. Proposition 7. We have X 1 ; : : : ; X n 99 KX i X follows from X 1 ; : : : ; X n by the hyperresolution rule. Proof. We prove the \only if" part by induction on jx i j: Choose a 1 2 X 1 ; : : : ; a n 2 X n. We claim that we can deduce all the clauses X; a i (1 i n) from X 1 ; : : : ; X n using the hyperresolution rule. The result follows then from a 1 ; X : : : a n ; X X provided a 1 ; : : : ; a n ` X Let us prove X; a 1 from X 1 ; : : : ; X n ; the other cases are similar. Notice that we have b 1 ; : : : ; b n ` X; a 1 for any choice b 1 2 X? fa 1 g; b 2 2 X 2 ; : : : ; b n 2 X n. By induction hypothesis, we can deduce X; a 1 from X 1? fa 1 g; X 2 ; : : : ; X n and hence from X 1 ; : : : ; X n : It should be interesting to note a connection with powerdomain constructions in the case of (at least) deterministic sequent structures 4. We have established in [20] that for a deterministic sequent structure, the frame generated from its opposite is isomorphic to the Smyth powerdomain of (jaj ; ). The frame generated by (A; `) is related to the Hoare powerdomain. 4 A sequent structure (A; `) is called deterministic if for each instance X ` Y, either Y = ; or there exists b 2 Y such that X ` b.

12 12 Proposition 8. The frame generated by a deterministic sequent structure (A; `) is isomorphic to the ideal completion of the Hoare preorder on nite sets of cocompact co-elements of (A; `). This result can be obtained from the well-known fact that the Hoare powerdomain can be obtained as the ideal completion of the lower order on the set of nite sets of compact elements of the underlying Scott domain. However, we need to use co-compact co-elements here. 5 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 { 1 ` { x ` xy { x; y ` x + y { xy ` x; y We have the following direct description of `. Theorem 7. X ` Y i the product of elements in Y belong to the radical of the ideal generated by X. Proof. We prove that this is an entailment relation. We check only the rule (C), the other rules being direct: 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 y n = ax and y m a m = 0: This implies y m (ax) m = y mn+m = 0 and hence y is nilpotent in A=I: Hence X ` Y as required. It is direct that this entailment relation satises 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 4 rules show that X ` y whenever y belongs to the ideal generated by X, while the last rule shows that y 1 : : : y m ` y 1 ; : : : ; y m : 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 [9]: the following items { x 1 ; : : : ; x n ` y is a consequence of the above axioms, { y belongs to the radical of the ideal generated by x 1 ; : : : ; x n, { fyg can be derived from fx 1 g; : : : ; fx n g by hyperresolution

13 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 U I. 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 = U I : Finally, the frame generated by this entailment relation is exactly the spectrum of A: 13 6 Categorical equivalences We extend our terminology rst in order to adequately express categorical concepts related to sequent structures. Let (A; `) be a sequent structure. In light of Proposition 4, the frame generated by (A; `) is precisely the set of conjunctive states ordered by inclusion. Denition 4. Let (A; `A), (B; `B) be sequent structures. A sequent map from A to B is an intepretation m : A! F rm(b) of A in the frame generated by B. Proposition 9. Sequent structures with sequent maps form a category. The identity map on (A; `) is the universal interpretation m A : A! F rm(a): The composition of two sequent maps m 1 : A! F rm(b) and m 2 : B! F rm(c) is the composition f m 1 where f : F rm(b)! F rm(c) is the unique map such that f m B = m 2 : Proof. This follows from standard categorical construction (see for instance [?], chapter VI, 5). We write Seq for the category of sequent structures and sequent maps. We introduce also the category RelLat of distributive lattices, and map m : D F rm(e), where F rm(e) is the free frames over E (frame of ideals of E). Finally let Spec be the category is the category of spectral locales with continuous maps. Theorem 8. The categories Seq; RelLat; Spec are equivalent. Proof. The equivalence between RelLat; Spec is standard (see [?]), while the equivalence between Seq and Spec follows from the universal properties of the free frame construction (see for instance [?], chapter VI, 5, exercice 2). 7 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.

14 14 It is possible to provide a similar treatment to innitary 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 [19] and of providing a connection to sober spaces, the signicance of such a concept remains to be seen. We omit the treatment of them due to space limitations. The equivalences among the categories of sequent structures, distributive lattices, and spectral locales seem to be such a basic result that it should have appeared earlier. We have only a slight hint that this seems not to have occurred ([1], page 228). 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 [12]. References 1. R. Amadio and P.-L. Curien. Domains and Lambda-Calculi. Cambirdge University Press, J. Cederquist and Th. Coquand. Entailment relations and distributive lattices. To appear in the Proceedings of Logic Colloquium Th. Coquand and H. Persson. Valuations and Dedekind Prague Theorem. To appear in the Journal of Pure and Applied Logic. 4. M. Droste and R. Gobel. Non-deterministic information systems and their domains. Theoretical Computer Science 75, , M. Fourman, R. Grayson. Formal Spaces. in L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), 107{122, North-Holland, Amsterdam-New York, P. Johnstone. Stone Spaces. Cambridge University Press, A. Jung, M.A.M. Moshier and M. Kegelmann. Multi lingual sequent calculus and coherent spaces. Fundamenta Informaticae, vol 37, 1999, pages G. Gentzen Collected Works. Edited by Szabo, Not-Holland, V. Lifschitz. Semantical completeness theorems in logic and algebra. Proc. Am. Math. Soc., vol. 79, 1980, p J.A. Robinson. The generalised gesolution principle. Machine Intelligence, vol. 3, p W. Rounds and G.-Q. Zhang. Clausal logic and logic programming in algebraic domains. Submitted. Copy at: G. Sambin. Pretopologies and completeness proofs. J. Symbolic Logic 60 (1995), no. 3, 861{ D. Scott. Completeness and axiomatizability. Proceedings of the Tarski Symposium, 1974, p D. Scott. Background to formalisation. in Truth, Syntax and Modality, H. Leblanc, ed., p , D. Scott. On engendring an illusion of understanding. Journal of Philosophy, p , 1971.

15 D. Scott. Domains for denotational semantics. in: Lecture Notes in Computer Science 140, 577{613, A. Tarski. Logic, semantics, metamathematics. Oxford, G.-Q. Zhang. Logic of Domains. Birkhauser Boston, Inc., Boston, MA, G.-Q. Zhang. Disjunctive systems and L-domains. 19th International Colloquium on Automata, Languages, and Programming(ICALP'92), Lecture Notes in Computer Science 623, 1992, pp. 284{ G.-Q. Zhang and W. C. Rounds. An information-system representation of the Smyth powerdomain. International Symposium on Domain Theory. Shanghai, China, October Copy at: