Dual Systems of Sequents and Tableaux for Many-Valued Logics Matthias Baaz Christian G. Fermüller Richard Zach November 25, 1992 Technical Report TUW–E185.2–BFZ.2–92 1993 Workshop on Theorem Proving with Analytic Tableaux Technische Universität Wien Institut für Computersprachen E185.2 Abteilung für Anwendungen der formalen Logik Resselgasse 3/1, A–1040 Wien, Austria/Europe Phone: (+43 1) 588 01 x 4088 Fax: (+43 1) 504 1589 Dual Systems of Sequents and Tableaux for Many-Valued Logics Matthias Baaz Christian G. Fermüller Richard Zach The aim of this paper is to emphasize the fact that for all finitely-many-valued logics there is a completely systematic relation between sequent calculi and tableau systems. More importantly, we show that for both of these systems there are always two dual proof sytems (not just only two ways to interpret the calculi). This phenomenon may easily escape one's attention since in the classical (two-valued) case the two systems coincide. (In two-valued logic the assignment of a truth value and the exclusion of the opposite truth value describe the same situation.) We employ the usual definitons of first order languages, many-valued interpretations (M) and induced valuation functions (valM) (see e.g. Carnielli [1987]). In the following V = {v1, . . . , vm} always denotes the set of truth values of a logic. To stress the dualty of the two types of calculi we shall define them simultanously: 1. Definition An (m-valued) sequent is an m-tuple of finite sets Γi of formulas, denoted as Γ1 | Γ2 | . . . | Γm. (As usual we abbreviate Γ ∪∆ by Γ,∆ and Γ ∪ {A} by Γ,A.) 2. Definition An interpretation M is said to p(n)-satisfy a sequent Γ1 | . . . | Γm, if there is an i (1 ≤ i ≤ m) and a formula F ∈ Γi, s.t. valM(F ) = ( 6=)vi. A sequent is called p(n)-valid, if it is p(n)-satisfied by every interpretation. The concept of p-satisfiability was used by Rousseau [1967] (compare also Schröter [1955]) in his formulation of many-valued sequents, whereas nsatisfiablity essentially already appears in Carnielli [1991]. 3. Definition An introduction rule for a connective 2 at place i in the logic L is a schema of the form: 〈 Γ j1 ,∆ j 1 | . . . | Γ jm,∆jm 〉 j∈I Γ1 | . . . | Γi,2(A1, . . . , An) | . . . | Γm 2:i where the arity of 2 is n, I is a finite set, Γl = ⋃ j∈I Γ j l , ∆ j l ⊆ {A1, . . . , An} It is called p(n)-admissible, if for every interpretation M the following are equivalent: (1) 2(A1, . . . , An) takes (does not take) the truth value vi. (2) For all j ∈ I, M p(n)-satisfies the sequents ∆j1 | . . . | ∆jm. 1 4. Example We state rules for the implication of the three-valued Gödel logic G3 with V = {f, u, t}. Let the expression Av (A6=v) denote the statement "A takes (does not take) the truth value v". Since (A ⊃ B)t iff (Af ∨ Au ∨ Bt) ∧ (Af ∨ Bu ∨ Bt) we get the following p-admissible introduction rule for position t: Γ,A | ∆,A | Π,B Γ ′, A | ∆′, B | Π ′, B Γ, Γ ′ | ∆,∆′ | Π,Π ′, A ⊃ B ⊃:t Because of (A ⊃ B)t iff Af ∨ (Au ∧ Bu) ∨ Bt we get by negating both sides of the equivalence the following n-admissible introduction rule for the implication at position t: Γ,A | ∆ | Π Γ ′ | ∆′, A,B | Π ′ Γ ′′ | ∆′′ | Π ′′, B Γ, Γ ′, Γ ′′ | ∆,∆′,∆′′ | Π,Π ′,Π ′′A ⊃ B ⊃:t It should be stressed that admissible introduction rules for a connective at a given place are far from being unique: Every p(n)-admissible introduction rule for 2(A1, . . . , An) at place i corresponds to a conjunction of disjunctions of some Avl (A6=vl) which is true iff 2(A1, . . . , An) takes (does not take) the truth value vi. Any such conjunctive normal form for 2(A1, . . . , An)vi will do. In particular, the truth table 2 for a connective 2 immediately yields a complete conjunctive normal form. For p-sequents the corresponding rule is as in Definition 3, with: I ⊆ V n is the set of all n-tuples j = (w1, . . . , wn) of truth values such that 2(w1, . . . , wn) 6= vi; and ∆jl = {Ak | 1 ≤ k ≤ n, vl 6= wk}. For n-sequents we get: I ⊆ V n consists of all n-tuples j = (w1, . . . , wn) of truth values such that 2(w1, . . . , wn) = vi; and ∆jl = {Ak | 1 ≤ k ≤ n, vl = wk}. 5. Definition An introduction rule for a quantifier Q at place i in the logic L is a schema of the form: 〈 Γ j1 ,∆ j 1 | . . . | Γ jm,∆jm 〉 j∈I Γ1 | . . . | Γi, (Qx)A(x) | . . . | Γm Q:i where I is a finite set, Γl = ⋃ j∈I Γ j l , ∆ j l ⊆ {A(a1), . . . , A(ap)}∪{A(t1), . . . , A(tq)}. The al are metavariables for free variables (the eigenvariables of the rule) satisfying the condition that they do not occur in the lower sequent; the tk are metavariables for arbitrary terms. Q:i is called p(n)-admissible, if for every interpretation M the following are equivalent: (1) (Qx)A(x) takes (does not take) the truth value vi under M. (2) For all d1, . . . , dp ∈ D, there are e1, . . . , eq ∈ D s.t. for all j ∈ I, M p(n)satisfies ∆′j1 | . . . | ∆′ j m where ∆ ′j l is obtained from ∆ j l by instantiating the eigenvariable ak (term variable tk) with dk (ek). The truth function Q for a (distribution) quantifier Q immediately yields admissible introduction rules for place i in a way similar to the method described above for connectives: For p-sequents let I = {j ⊆ V | Q(j) 6= vi}. Then the rule is given as in Definition 5, with ∆jl = {A(ajw) | w ∈ j, w 6= vl} ∪ {A(tj) | vl ∈ V \ j}. In contrast, for n-sequents we take I = {〈j, i〉 | j ⊆ V ∧ i ∈ j ∧ Q(j) = vi} and ∆ 〈j,i〉 l = {A(a j l ) | l ∈ j} ∪ {A(tj) | i = l}. Again, it should be stressed that in general these are not the only possible rules. 2 6. Definition A p-sequent calculus for a logic L is given by: (1) Axioms of the form: A | A | . . . | A, where A is any formula, (2) For every connective 2 and every truth value vi a p-admissible introduction rule 2:i, (3) For every quantifier Q and every truth value vi a p-admissible introduction rule Q:i, (4) Weakening rules for every place i: Γ1 | . . . | Γi | . . . | Γm Γ1 | . . . | Γi, A | . . . | Γm w:i (5) Cut rules for every pair of truth values (vi, vj) s.t. vi 6= vj : Γ1 | . . . | Γi, A | . . . | Γm ∆1 | . . . | ∆j , A | . . . | ∆m Γ1,∆1 | . . . | Γm,∆m cut:ij A n-sequent calculus for a logic L is given by: (1) Axioms of the form: ∆1 | . . . | ∆m, where ∆i = ∆j = {A} for some i, j s.t. i 6= j and ∆k = ∅ otherwise (A is any formula), (2) For every connective 2 and every truth value vi an n-admissible introduction rule 2:i, (3) For every quantifier Q and every truth value vi an n-admissible introduction rule Q:i, (4) Weakening rules (identical to the ones tor p-sequent calculi) (5) The cut rule: 〈 Γ i1 | . . . | Γ ii , A | . . . | Γ im 〉m i=1 Γ1 | . . . | Γm cut: where Γl = ⋃ 1≤j≤m Γ j l . 7. Theorem (Soundness and cut-free Completeness) For every p(n)-sequent calculus the following holds: A sequent is p(n)-provable without cut rule(s) iff it is p(n)-valid. Analytic tableaux for many-valued logics have been investigated by Surma [1977] and Carnielli [1987]. Hähnle [1991], based on the aforementioned work, studied the applicability of these systems for automated theorem proving. Hähnle introduced the notation of sets-of-signs which allows a more efficient representation of tableaux and presented streamlined calculi for certain classes of logics. Here, we want to stress the striking similarity between tableaux systems and sequent calculi: In fact, there is an immediate correspondence between cut-free sequent calculus proofs and closed tableaux. Again, there are two dual systems for any logic. 8. Definition A signed formula is an expression of the form {w}A, where w ∈ V . 3 9. Definition A tableau is a downward tree of sets of signed formulas where every set is obtained from a set preceding it in the tree by application of one of the rules of the tableau system: Let R:i be a p(n)-admissible introduction rule for a connective or a quantifier as given in Definitions 3 and 5, where at least one of the ∆j is nonempty. Moreover, let F be the formula being introduced (i.e., F ≡ 2(A1, . . . , An) or F ≡ (Qx)A(x)). The p(n)-tableau rule corresponding to R:i is: Γ, {vi}F〈 Γ, ⋃m k=1∆k 〉 j∈I where ∆k is obtained from ∆k by replacing every formula A ∈ ∆k by {vk}A. A p(n)-analytic tableau is called closed, if every leaf contains formulas {vk}A for all k ∈ {1, . . . ,m} (for k ∈ {i, j}, i 6= j). 10. Theorem Every closed p(n)-tableau with the root ⋃ Γ k corresponds to a cutfree p(n)-sequent calculus proof of Γ1 | . . . | Γm. We finally remark that also resolution calculi can be derived from sequent calculi: The introduction rules for sequents convert into reduction rules that translate finite sets of assignments of truth values to formulas into clause forms. (Clauses are finite sets of assignments of truth values to atomic formulas; cf. Baaz and Fermüller [1992]) References Baaz, M. and C. G. Fermüller. [1992] Resolution for many-valued logics. In Proc. Logic Programming and Automated Reasoning LPAR'92, A. Voronkov, editor, LNAI 624, pp. 107–118, Springer, Berlin. Carnielli, W. A. [1987] Systematization of finite many-valued logics through the method of tableaux. J. Symbolic Logic, 52(2), 473–493. [1991] On sequents and tableaux for many-valued logics. J. Non-Classical Logic, 8(1), 59–76. Hähnle, R. [1991] Uniform notation of tableaux rules for multiple-valued logics. In Proc. IEEE International Symposium on Multiple-valued Logic, pp. 238 – 245. Rousseau, G. [1967] Sequents in many valued logic I. Fund. 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