A Strong Completeness Theorem for the Gentzen systems associated with finite algebras

In this paper we study consequence relations on the set of many sided sequents. We deal with the consequence relations axiomatized by the sequent calculi deñned in [2] and associated with arbitrary fi¬ nite algebras. This consequence relations are examples of what we cali Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extensión of the Completeness Theorem for provable sequents stated in [2]. For the special case of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [11], as a consequence of McNaughton’s The¬ orem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems. Mathematics Subject Classiñcation: 03B50, 03F03, 03B22.

Introduction and outline of the paper A class of sequent calculi associated with finite algebras is defined in [2]. In this paper we continué the study of the Gentzen system determined by these sequent calculi. This study started in [11], where, generalizing a result of [14], the m-dimensional algebraizable Gentzen systems were characterized. The Strong Completeness Theorem for the Gentzen systems determined by the sequent calculi associated with the finite linear MV-algebras was proved in [11], by using algébrale methods.
The aim of this paper is to prove the Strong Completeness Theorem (The¬ orem 4.1) for the Gentzen systems associated with arbitrary finite algebras.
By using the notion of satisfaction of sequents defined in [16] and [2], we define, for any finite algebra, a semántica! consequence relation on the set of ro-sequents, where m is the cardinality of the algebra. These semantical consequence relations are finitary (Theorem 2.19) and satisfy the same deduction detachment theorem satisfied by the Gentzen systems mentioned above (Theorem 3.6). Then, by also using the already known Completeness Theorem for provable sequents ([2, Theorems 3.1 and 3.2]), we prove the Strong Completeness Theorem. It is worth noticing (see Theorem 3.7) that for every finite algebra only one Gentzen system is associated with it by means of the sequent calculi defined in [2]. The Computer system MULTLOG ( [3] and [4]) provides a way to obtain automatically an axiomatization of these consequence relations such that the rules satisfy certain optimality conditions. 1

Preliminary definitions and results
This section contains the basic definitions for this paper and some already known results about deductive systems and Gentzen systems.

Consequence relations and deductive systems
Let £ be a propositional language (i.e. a set of propositional connectives).
By an £-algebra we mean a structure A = (A, {üA : □ € £}), where A is a non-empty set, called the universe of A, and üA is an operation on A of arity k for each connective □ of rank k. A consequence relation on A is 2 a relation h between subsets of A and elements of A such that the following conditions hold for all X U Y U {a} C A: (i) a e X implies Aha; (ii) X\~a and X C Y implies Fba; (iii) X\~a and Yh-b for every b € X implies Fha; A consequence relation is finitary if (iv) X\-a implies X'\~a for some finite X' C X.
We denote by Fm¿ the absolutely free algebra of type £ freely generated by a countable infinite set of variables. Its elements are ealled £-formulas. If A is an £-algebra, the set of homomorphisms from Fm¿ to A will be denoted by Hom(Fmc, A). (i) A deductive system is a pair S = (£, [1][2][3][4][5], where hs is a fini¬ tary consequence relation on the set of £-formulas, Fmc, which is structural in the following sense: TN/? implies h(F)\-h(tp) for every h e Hom(Fmc,Fmc), where h(T) stands for {^(7) : 7 € T}. Deduc¬ tive systems have been studied, among other places, in [6] (where the concept of an algebraizable deductive system is defined), [5] and [9].
(ii) Let 1 < k. A k-dimensional deductive system S over £ is a pair (£, [1][2][3][4][5] where hs is a finitary consequence relation over Fmkc, ( The set Fmkc is ealled the set of fc-formulas. -dimensional deductive systems have been studied in [7]. 3 (iii) The theory of (2-dimensional) Gentzen systems was developed in [14], where sequents are defined to be pairs of finite sequences of formulas, possibly with some limitations on the length of the sequences given by the type of the sequents. Let us recall some of the definitions given in [14]: The notion of a matrix allows the introduction of a very general concept of a semantics for a deductive system. Let us recall the definitions of a matrix, a matrix model of a deductive system and a matrix semantics of a deductive system (cf. [17], [6] and [7]). An £-matrix is a pair (A, F), where A is an £-algebra and F is a subset of A. m-sequents and m-sequent calculi An m-sequent, also called m-dimensional sequent or m-sided sequent, is a sequence (r0, Tx,..., rm_x) where each I\ is a finite sequence of £-formulas, which is called the i-th component (or place) of the sequent. Those sequents have been taken into account in [16], [3], [2], [18] and [11]. As in these works we will write To [Ti | ... | Tm_i for (ro, Ti,..., rm_i). We denote by m-Seqc the set of m-sequents. 5 Thus in the 2-dimensional case we will write T | A instead of the more common notations TI-A or T -*• A. The use of the Symbol | as a separator of the components prevents us from thinking of entailment relations between the components of a sequent. Note that in our notation the Symbol h is only used, possibly with a subindex, to denote consequence relations on the sets considered (formulas, fc-formulas, sequents or m-sequents).
If we have two or more sequents, we will sepárate them by the symbol . In this way there will be no confusión between, for instance, the 3-sequent T,a; | A,y | II and the two 2-sequents r,a: | A; y | II. The comma will be reserved for the juxtaposition operation on sequences: that is, expressions such as T, 6 will stand for (70,..., ik-u <5), where T = (70,..., 7fe_i). If T is a sequence of formulas and <p occurs in T, we will write <p € I\ Also, we write T C {ip0,..., <pn} to denote that all the formulas that occur in T are in {tpo, • • •, To increase the readability of some of the results of this paper in which we use simultaneously formulas, sequences of formulas, sequents and sets of sequents, we will use the following notation: lowercase letters from the end of the alphabet, possibly with subindex and superindex (p,Q,p¡r • •) to de¬ note propositional variables; Greek letters (<¿?, ...,) to denote formulas; uppercase Greek letters (r, A,...) to denote sequences and sets of formu¬ las; boldface uppercase Greek letters (T, A, Ai}...) to denote sequents, and boldmath uppercase letters of the end of the alphabet (T, S,...,) to denote sets of sequents.
If r is an m-sequent and i < m, then F(i) denotes the z-th component of r. If A is a sequence of formulas and I = {¿1,..., z'"} Q {0,..., m -1}, we denote by [I : A] the m-sequent whose z-th component is A if z € / and is empty otherwise, that is: if\V¡.  An m-sequent calculus is a set of m-rules of inference.
• Weakening rule (w : i) for the place i <m:  Table 2 we show a 3-dimensional sequent calculus obtained from the three-element MV-algebra following [2].
i If LX is a sequent calculus that contains some of the cut rules, we say that the cut elimination theorem holds for LX if every sequent LX-provable can be proved without using any of the cut rules.
The VL-sequent calculi Each finite £-algebra of cardinal m induces a semantical interpretation on the set of m-sequents, in such a way that several m-sequent calculi are known to be complete with respect to this semantical interpretation. Since in this paper we will extend the definition of the semantical interpretation to a semantical consequence relation on the set of m-sequents we will now recall some of the basic definitions involved. (iv) T C m-Seqc is simultaneously L-satisfiable if / 0.
The above definition of validity is the restriction to the propositional case of [2, Def 3.2] and of the definitions given in [3] and [18].
The elements of L are called truth valúes and if T is an L-valid sequent we will write, following [18], [=l T.
A dual semantical interpretation of sequents, which corresponds to analytic tableaux is studied, for instance, in [18].
It is always possible to find sequent calculi complete with respect to this definition of L-validity (see [2] and [18] for histórica! remarks). The calculi we will deal with were defined by M. Baaz et al. in [2] and they play the same role with respect to the algebra L as the sequent ealculus LK does with respect to the two-element Boolean algebra 2, in which case, as is well known, a sequent T | A is LA'-provable iff for every interpretation of the variables of the sequent in 2, some formula in T is false or some formula in A is true; thus a sequent is LA-provable iff it is 2-valid (see [2, p. 336] and [18, p. 31 and 33]). 9 We will now recall the definition of the introduction rules of these ealculi, which are called in this paper VL-rules, that is, preceding the ñame of the algebra with the letter V. Definition 1.9 (cf. [2, Definition 3.3]) and [3]). A VL-introduction rule (□ : i) for a connective □ at place i is a schema of the form: .., for every l < m and j € I, □ is a propositional connective of rank n, I is a finite set, and, for each h € Hom(Fmc,L), the following properties are equivalent: (VL1) h L-satisfies the sequent A¿ | ... | &3m-i for every j 6 /.
The existence of such rules for an arbitrary algebra is proved in [16, Lemma 1]. As pointed out in [2], it should be stressed that for any connective This definition corresponds to the propositional fragment of the sequent calculi VL defined in [3] and [18], and in [2] with the ñame LM. Among the properties of the sequent calculi just defined we are interested in the restriction to the propositional case of the following result: Theorem 1.11 (Completeness and Cut Elimination) Let L be a finite algebra, then the following properties hold: (i) Ifan m-sequent is provable in a VL-sequent calculas, then it is L-valid.
(ii) If an m-sequent is L-valid, then it is provable in any VL-sequent cal¬ culas without cuts. i Since the VL introduction rules for a given connective are not unique, for any finite £-algebra L there may be several calculi that satisfy the definition of a VL-sequent calculus. However, it follows from Theorem 1.11 that each VL-sequent calculus has the same provable sequents. More generally, we will prove in Theorem 3.7 that each VL-sequent calculus determines the same consequence relation over the set of m-sequents, that is, the same Gentzen System. G. Rousseau defines in [16] another class of sequent calculi for each Calgebra L, which we refer to as RL-calculus, and which have a different axiom and no structural rules. Although for any finite £-algebra L the VL and RL-sequent calculi have the same provable sequents, the absence of the structural rules makes the later less appropriate for our purposes. In [11] some differences between the consequence relations associated with these classes of calculi are considered. Example 1.12 Let L = ({/,p, t}, A) be an algebra of three elements where the truth table of the connective is given by Let us now find a VL-rule (A : 1). First note that /i(<p A^O = p 4=>-(h(ip) = p or hty) -p) and (h((p) = p or h(tp) = t) and (h(ip) = t or h(ip) = p).
This expression corresponds to a conjunctive normal form. Now, each of the conjuncts can be expressed by the fact that the homomorphism satisfies a certain sequent; for instance

Gentzen Systems
In order to study the consequence relations determined by the VL-sequent cal culi, we first recall the abstract definition of an m-dimensional Gentzen system. These systems were introduced in [11] (with some limitations on the length of the sequent, not considered in this paper) and can be seen as a generalization of the 2-dimensional Gentzen systems introduced in [14] (allowing an arbitrary but fixed number of components in the sequents), and also as a generalization of the m-dimensional deductive systems (considering m-tuples of sequences of formulas, instead of m-tuples of formulas). Let Q be a Gentzen system. If Th^A and Ab^r we will say that T and A are ¿7-equivalent. If $\-gT we will say that F is ¿/-derivable. We will sometimes write T; Tbg A for T U {r}bg A. Every m-sequent calculus, LX, determines a Gentzen system QLX = (£, blx) by using the rules of the calculus to derive sequents from sets of sequents, not just from the axiom alone, as stated in the following definition (cf. [14, p.14] and [1, p. 267]): Definition 2.1 GivenT\J{T} C m-Seqc , we say thatT follows fromT in Glx, in symbols T\~lxF iff there is a finite sequence of sequents To,..., r"_i, (n > 1), called a proof ofV from T, such that F"_i = T and for each i <n one of the following conditions holds: (i) r< is an instance of an axiom; (ü) Ti e T; (iii) Ti is obtained from {r_¿ : j < i} by using a rule (r) of LX, i.e., (S, I\) 6 (r) for some S C {r,-: j < i).
13 n is called the length of the proof.
Example 2.2 The Gentzen system determined by the sequent calculus LK is studied in [14], where it is denoted by Qcpc■ This Gentzen system is equivalent to the Classical Propositional Calculus and is algebraizable, the variety of Boolean algebras being its equivalent quasivariety semantics (see [14] for the definitions of equivalence and algebraizability of 2-dimensional Gentzen systems). Other Gentzen systems, obtained modifying some of the rules of LK, are studied in [14] and [13].
i Deñnition 2.3 Let L be a finite C-algebra. A VL-Gentzen system is a Gentzen system determined by a VL-sequent calculus. Table 2 is studied in [11] and [10]. This Gentzen system is equivalent to the 3-valued Lukasiewicz propositional logic and is algebraiz¬ able, the variety generated by the three-element MV-algebra being its equiv¬ alent quasivariety semantics (see [11] for the definitions of equivalence and algebraizability of ra-dimensional Gentzen systems). The Gentzen systems determined by a VS(m)-sequent calculus (defined in [2]), where S(m) is the linear MV-algebra of m elements are also studied in [11].

Example 2.4 The Gentzen system determined by the VS(3)~sequent cal¬ culus given in
i Definition 2.5 An m-dimensional Gentzen system satisfies an m-rule (r) if T\-gT for every (T, T) e (r).

The structural rules
The structural rules play an important role in the proof of some basic theorems of this paper. We introduce now some technical lemmas that will help us to shorten some proofs where different structural rules are involved. [r0,ri, a0, Ai] The next lemma allows us to cut a formula not only in a pair of different components, but in a pair of disjoint sets of components. We will say that a Gentzen system is accumulative if it is i-accumulative for every i < m. Accumulative (2-)dimensional Gentzen systems have been defined and studied in [14]. (2-)dimensional systems satisfying a similar property have been studied by A. Avron in [1] with the ñame of "puré". Now we will give sufficient conditions for a Gentzen system to be accumulative. Proposition 2.11 Let Q = (£,\~c) be an m-dimensional Gentzen system and i <m. IfQ satisfies the following two properties: (i) Q satisfies the axiom and the weakening rule for the place i, (ii) Q can be defined by a set of rules such that an arbitrary sequence of formulas A appears at the end of the i-th component of all the sequents that appear in the rule. That is, Q can be defined by a set of rules of the following form: Thus, by using (4) and (5)  Proof: Since every VL-sequent calculus contains the axiom and the weak¬ ening and exchange rules, the Gentzen system determined satisfies these rules. The weakening, the contraction rules and all the VL-introduction rules can be written as in the hypothesis of Corollary 2.12, by using the ex¬ change rules. The exchange rules are already in the desired form and finally, by using the contraction and the exchange rules, the cut rule can be replaced by Our definition of the notion of a matrix model of a Gentzen system is similar to the corresponding definition for a deductive system.
Let A be an £-algebra. An m-relation on A is a set R C U{An°x ... x Anm~l : rii E üj, i < m}, that is, a set of m-tuples formed by finite sequences of elements of A; 7Zm(A) will be the set of all m-relations on A. If there is no risk of confusión we write 7Zm instead of T^^A).
An m-matrix, or just a matrix, is a pair (A, R} where R is an mr elation on A. Notice that instead of considering a set F C A, we consider an m-relation. Let h E Hom(Fmc, A). If T is the sequent Let Q be an m-dimensional Gentzen system and let (r) be an m-rule of inference. An element R E 7Zm is closed under the rule (r) if for every pair (T,r)e(r),and every h € Hom{Fmc, A), h(T) C R implies h(T) € R. A (y-filter is a set R € 7Zm such that for every set of sequents T U {T} and for every h E Hom(Fmc, A), T\~gF and h(T) C R imply h(T) E R. When Q is defined by means of some axioms and inference rules, R is a £-filter iff R contains all the interpretations of these axioms and is closed under each of these rules. A matrix (A, R) is called a matrix model of Q (or ¿/-matrix) if R is a ¿/-filter. If (A, R) is an m-matrix, let (=<a,í?> be the structural consequence relation on the set m-Seqc defined by the following condition: T |={A,fí) r iff for every h E Hom(Fm.c, A), h(T) C R implies h(F) E R. Now we are going to define a semantical consequence relation over the set of m-sequents based on the definition of L-satisfaction. This consequence relation is defined from an m-matrix on the algebra L. So we start by defining the following m-relation, which contains the interpretation of the valid sequents: Definition 2.14 Let L be a finite algebra with universe L = {uo,... ,um_i} of cardinal m, then (AT(0),... ,X(m -1)) E LA0 x ... x Lrhn~1 : ni E u for i < m, and exists i <m such that Vi E X(i) 20 The connection between the m-matrix (L, Di) and the definition of Lvalidity and L-satisfaction is shown in the following (ii) 0 r <*=>■ r is an L-valid sequent.
(iü) t hL,DL> r <=► fis(n)^s(r)- In order to prove that the consequence relation associated to any VLsequent calculus and the semantical consequence relation (=(L,dl> are equal, and since the first one is, by definition, finitary, we will show first that the second is also finitary. We will follow the topological proof given by Los and Suszko (and sketched by Wojcicki in [17, p. 262]) of the fact that the consequence relation determined by a finite class of finite matrices is finitary. The topological basis of the proof can be found in [16].
Let us assign to L the discrete topology and to LVar the product topology, where Var is the set of propositional variables. Since L is a compact Hausdorff space, LVar is also compact Hausdorff, by Tychonoff's Theorem. Now, by identifying each homomorphism h : Fm£ i->■ L with its restriction a : Var i-► L, we can identify the sets Hom(Fmc, L) and LVar, so Hom(Fm.c, L) is also a compact Hausdorff space. The definition we give of the deduction detachment theorem (DDT for short) is a generalization of the one given in [14] for (2-)dimensional Gentzen Sys¬ tems, which is, in turn, a generalization of the deduction detachment theorem for deductive systems. This DDT is formulated by means of a deduction de¬ tachment set (DD-set for short) which acts as a kind of implication between sequents. If a Gentzen system has the DDT, then the DD-set malees it possible to carry one sequent from left to right of the consequence relation by using a finite number of sequents. These sequents will be in as many variables as there are formulas in the two sequents involved. Thus we need different sets of sequents, according to the lengths of the components of the sequents.
After giving the definition, we obtain sufficient conditions for a Gentzen system to have the deduction detachment theorem. Let Q -(£, hg) be an m-dimensional Gentzen system. If k = (ko,km-1) e um and l = (lo,lm-1)^um, p¡¡ and will denote the result of replacing the variable p¡ by 7/ (i < m,j < k{), and the variable q{ by 7r¿ (i < m,j < li) in every sequent of E^.
In the case kt = U = 0 for allí < m, we will write E($rnt ®m) -■®(0,...,0),(0,...,0)(Po)' The set E is calíed deduction-detachment set (DD-set, for short) for Q if for all T U {r; n} c m-Seqc , T,T\-gn iff T\-gE{r,n). Definition 3.1 A Gentzen system Q has the deduction-detachment theorem (DDT, for short) if it possesses some deduction-detachment set. If a Gentzen system Q has the deduction detachment theorem, the consequence relation 1-g is determined by the (/-derivable sequents and the DD-set, as shown in the following Theorem 3.2 Let Q\ = {C,\-gf) and G2 = (£,l-£2) be two m-dimensional Gentzen systems. If Gi and G2 have the deduction detachment theorem uñth respect to the same deduction-detachment set and they have the same derivable sequents, then Gi and G2 are equal.
Proof: We prove that for any finite set of sequents T and any sequent T rh^r <*=>■ T\-g2r by induction on the cardinal of the set T.
If 01-^T, then T is C?i-derivable and, by hypothesis, 0b^2r. Let £ be a DD-set for both Gentzen systems Q\ and Q2. If T'; Ilb^r, then, by the deduction theorem, T,hg1¿2(n, T). By inductive hypothesis T>\-g2E(TÍ, T), and since has the same DD-set, T',U\-g2T, and this finishes the proof.
i Now we will give sufficient conditions for a Gentzen system to have the deduction detachment theorem. First we show that, in a Gentzen system that satisfies the axiom, weakening and exchange rules, we can associate to each sequent T a set of derivable sequents: those obtained adding a formula that occurs in any component of T to all the other components, that is:  Example 3.5 Let Q be a 3-dimensional Gentzen system that satisfies the hypotheses of Theorem 3.4. Then, by applying the deduction detachment theorem T\ y \ ij> | V Vi 11>i I Vi t^~9 <Pi I V>i, V I Vu V T\-g (pur¡> I I r/i.V' TVg ipUT) I iffUr¡ \r]i.
Since the VL-Gentzen systems satisfy all the structural rules, and are accumulative, they satisfy the deduction-detachment theorem. 26 Corollary 3.6 Let L be a finite C-algebra. LetQ = (C, \-g) be a VL-Gentzen system. Then Q satisfi.es the deduction detachment theorem given in Theorem 3.4. 1 Although all the VL-Gentzen systems satisfy this DDT, by using special properties of the algebras involved, it may be possible to prove other DDT.
For instance, in [11,Theorem 45 and 48] we give two different DDT for the VS(m)-Gentzen systems, where S(m) is the linear MV-algebra of melements, in which the sets E¡¿¡-consists of a single sequent, for all k, l 6 um. Theorem 3.7 Let L be a finite C-algebra. If Gi and Gz are VL-Gentzen systems, then Gi -Gz, that is, each VL-sequent calculas determines the same Gentzen system. Proof: All the VL-Gentzen systems satisfy the deduction detachment the¬ orem with respect to the DD-set given in Theorems 3.4 and Corollary 3.6, and they all have the same set of derivable sequents (see Theorem 1.11). Thus, by Theorem 3.2, all these Gentzen systems are equal. 1 This theorem proves that the consequence relation determined by any VL-sequent calculus is independent of the set of introduction rules we choose for each connective and place. The consequence relation associated with any VL-sequent calculus will be denoted by bvL> and the Gentzen system determined by any VL-sequent calculus will be denoted by the expression Gvl = (£, Fvl)-This Gentzen system will be called the Gentzen system associated with the algebra L.

The Strong cut elimination Theorem
In [1, pag. 270] there is a proof of the Strong Cut Elimination Theorem for the "Gentzen-type system for the Classical Propositional Logic". By using