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Dual tableau for monoidal triangular norm logic MTL
Fuzzy Sets and Systems 162 (2011) 39 – 52 www.elsevier.com/locate/fss
Dual tableau for monoidal triangular norm logic MTL夡 Joanna Goli´nska-Pilareka,b,∗ , Ewa Orłowskab a Institute of Philosophy, Warsaw University, Poland b National Institute of Telecommunications, Warsaw, Poland
Received 30 September 2009; received in revised form 5 September 2010; accepted 16 September 2010 Available online 22 September 2010
Abstract Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system. © 2010 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Monoidal triangular norm logic; Relational logic; Dual tableau system
1. Introduction Monoidal triangular norm logic, MTL, was introduced in [6]. From the perspective of substructural logics it is the logic of full Lambek calculus endowed with the rules of exchange and weakening, FLew , with the additional axiom ( → ) ∨ ( → ) referred to as prelinearity. From the perspective of fuzzy logics it is a logic of left-continuous triangular norms, t-norms for short (see [12]). The classical examples of t-norms which are applied to modelling logical operator of conjunction include Łukasiewicz t-norm x y = max(0, x +y−1), product t-norm defined as multiplication of reals, and Gödel t-norm x y = min(x, y). All those t-norms are continuous. A t-norm is left-continuous iff it has a residuum, →, i.e., the two operators satisfy z ≤ x → y iff x z ≤ y. The residua of the three t-norms mentioned above play the role of implications: Łukasiewicz implication x → y = min(1, 1 − x + y), Gougen implication x → y = 1 if x ≤ y and y/x otherwise, and Gödel implication x → y = 1 if x ≤ y and y otherwise, respectively. Algebraic semantics of logic MTL is provided by the class of MTL-algebras. They are abstract counterparts to the standard structures ([0, 1], ≤, , →), where is a left-continuous t-norm and → is its residuum. A completeness of logic MTL with respect to the semantics determined by those standard structures is presented in [11]. In [18] a Kripke-style semantics for a first-order version of logic MTL is presented, defined along the lines of semantics for substructural logics presented in [22,23]. In that semantics an algebraic structure is assumed in the universes of the models. 夡 Partial support from the Polish Ministry of Science and Higher Education Grant no. N206 399134 is gratefully acknowledged. ∗ Corresponding author at: Institute of Philosophy, Warsaw University, Poland. Tel.: +48 225520140.
E-mail addresses: [email protected] (J. Goli´nska-Pilarek), [email protected] (E. Orłowska). 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.09.007
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In this paper, first, we present MTL-algebras which provide an algebraic semantics of the logic MTL. Next, we present logic MTL with a purely relational semantics developed in [4,28]. Based on this semantics a relational dual tableau for the logic is developed. Relational dual tableaux are based on Rasiowa–Sikorski diagrams for first-order logic [29]. The common language of all relational dual tableaux is a logic of relations. Relational deduction systems in the Rasiowa–Sikorski style have been constructed for a great variety of theories, e.g., intuitionistic, modal, relevant, and multiple-valued logics, as well as for temporal logic, various logics of programs, logics of rough sets, theories of spatial reasoning including region connection calculus, theories of order of magnitude reasoning, and formal concept analysis. A survey of these results is presented in [24]. Specific methodological principles of construction of dual tableaux which make possible such a broad applicability of these systems are: • first, given a theory, a truth preserving translation is defined of the language of the theory into an appropriate relational language (most often a language of binary relations); • second, a dual tableau is constructed for this relational language so that it provides a deduction system for the original theory. This methodology, reflecting the paradigm “Formulas are Relations”, enables us to represent within a uniform formalism the three basic components of formal systems: syntax, semantics, and deduction apparatus. Relational approach enables us to build dual tableaux in a systematic modular way. First, deduction rules are defined for the common relational core of the theories. These rules constitute a basis of all the relational dual tableau proof systems. Next, for any particular theory specific rules are added to the basic set of rules. They reflect the semantic constraints assumed in the models of the theory. As a consequence, we need not implement each deduction system from scratch, we should only extend the basic system with a module corresponding to the specific part of a theory under consideration. Relational dual tableaux are powerful tools which can be used to solve the four major problems that may be formulated in a theory: • • • •
proving general laws holding in the theory; proving that from a finite number of laws of the theory some other law follows; proving that a particular model of the theory obeys some laws; proving that some objects which the theory deals with satisfy a law.
These tasks are represented, respectively, as: the validity problem, the entailment, the model checking problem, and the satisfaction problem in the relational logic. A recent implementation of a proof system for a relational logic is available in [8]. In [7] an implementation of translation procedures from non-classical logics to a relational logic is presented. In [10,2], implementations of relational logics for order of magnitude reasoning are presented. The paper is organized as follows. In Section 2 we present MTL-algebras which provide algebraic semantics of monoidal triangular norm logic MTL. In Section 3 we define the logic MTL, its language and semantics. Relational formalization of logic MTL is presented in Section 4. In Section 5 we present a relational dual tableau that can be used for verification of validity of MTL-formulas. We prove its soundness and completeness in Section 6. In Section 7 we discuss alternative forms of the rules of the dual tableau for MTL. Finally, comments and conclusions are given in Section 8.
2. MTL-algebras A t-norm is a binary operation on the closed real interval : [0, 1]2 → [0, 1], such that for all x, y, z ∈ [0, 1] the following hold: • is associative and commutative; • 1 is the neutral element of ; • If x ≤ y, then x z ≤ y z. Let (xi )i∈J be an indexed family of elements of [0,1]. A t-norm is said to be left-continuous whenever (sup xi ) y = sup(xi y) for every y ∈ [0, 1].
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The abstract structures which capture the properties of left-continuous t-norms are the MTL-algebras. An MTLalgebra is a structure of the form A = (A, ∨, ∧, , →, 0, 1) such that: • (A, ∨, ∧, 0, 1) is a bounded, distributive lattice; • (A, , 1) is a commutative monoid, i.e., for all a, b, c ∈ A the following conditions are satisfied: a (b c) = (a b) c; a b = b a; 1 a = a; df
• → is a residuum of , i.e., a b ≤ c iff a ≤ b → c, where ≤ is the lattice ordering defined by: a ≤ b ⇐⇒ a = a ∧ b (or equivalently b = a ∨ b); • (a → b) ∨ (b → a) = 1. Observe that monoid ( A, , 1) is integral, i.e., its neutral element coincides with the greatest element of the lattice. 3. The logic MTL The language of the logic MTL consists of symbols from the following pairwise disjoint sets: • V—a countable infinite set of propositional variables; • {0, 1}—the set of propositional constants; • {∨, ∧, , →}—the set of propositional operations of disjunction, conjunction, product, and implication, respectively. As usual, the set of MTL-formulas is the smallest set including the set V ∪ {0, 1} and closed with respect to the propositional operations. A classical formalization of MTL is based on its algebraic semantics defined in terms of MTL-algebras. Here we present a Kripke-style semantics obtained from a discrete duality for MTL-algebras developed in [25,27]. This semantics differs from the semantics presented in [18] in that it is a purely relational semantics not assuming any monoid structure in the universes of the models. An MTL-model is a structure M = (U, ≤, R, m) such that U is a non-empty set, ≤ is a reflexive and transitive relation on U , m is a meaning function satisfying the following conditions: • • • •
m( p) ⊆ U , for every propositional variable p; m(1) = U and m(0) = ∅; (her) If x ≤ y and x ∈ m( p), then y ∈ m( p), for all x, y ∈ U and for every propositional variable p; R is a ternary relation on U satisfying the following conditions, for all x, y, z, x , y , z , t, w ∈ U : (MTL1) If (x, y, z) ∈ R, x ≤ x, y ≤ y, and z ≤ z , then (x , y , z ) ∈ R, (MTL2) If (x, y, z) ∈ R, then (y, x, z) ∈ R, (MTL3) If (x, y, z) ∈ R and (z, y , z ) ∈ R, then there exists u ∈ U such that (y, y , u) ∈ R and (x, u, z ) ∈ R, (MTL4) There exists u ∈ U such that (u, x, x) ∈ R, (MTL5) If (x, y, z) ∈ R, then y ≤ z, (MTL6) If (x, y, z) ∈ R and (x, t, w) ∈ R, then y ≤ w or t ≤ z.
Depending on the intended application, the elements of set U are interpreted in computer science as states of a computation or objects of an information system and in philosophical logic as possible worlds. Let M = (U, ≤, R, m) be an MTL-model and let s ∈ U . Satisfaction of an MTL-formula in model M by state s is defined as: • • • • •
M, s p iff s ∈ m( p), for every p ∈ V ∪ {0, 1}; M, s ∨ iff M, s or M, s; M, s ∧ iff M, s and M, s; M, s iff there exist x, y ∈ U such that (x, y, s) ∈ R and M, x and M, y; M, s → iff for all x, y ∈ U , if (s, x, y) ∈ R and M, x, then M, y.
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A formula is said to be true in an MTL-model M iff it is satisfied in M for every s ∈ U . A formula is MTL-valid iff is true in all MTL-models. As shown in [28], algebraic and Kripke-style semantics of MTL are related according to the principle of duality via truth (see [26]), i.e., the same formulas are validated according to the notions of truth provided by the respective semantic structures. The condition (MTL2) reflects commutativity of ; in the presence of (MTL2) the condition (MTL3) corresponds to associativity of ; (MTL4) expresses the condition a ≤ a 1, while (MTL5) expresses the condition 1 a ≤ a, (MTL6) reflects prelinearity. It follows that a Hilbert-style axiomatization of MTL corresponding to its algebraic semantics is complete with respect to the Kripke-style semantics. 4. Relational formalization of logic MTL In this section we define the relational logic RLMTL corresponding to MTL-logic. Formulas of RLMTL , interpreted as ternary relations, will represent formulas of the logic MTL and the accessibility relation from its models. The language of RLMTL consists of the symbols from the following pairwise disjoint sets: • • • • •
OVRLMTL —a countable infinite set of object variables; RVRLMTL —a countable infinite set of ternary relational variables; {≤}—the set consisting of the binary relational constant ≤; {R, 1, 0}—the set of ternary relational constants, where 0 is the empty relation and 1 is the universal relation; {−, ∪, ∩, , →}—the set of relational operations.
The set of ternary relational terms is the smallest set including RVRLMTL ∪ {R, 1, 0} and closed on the relational operations. The set of RLMTL -terms consists of ternary relational terms and the relational constant ≤. RLMTL -formulas are of the form T (x, y, z) or x ≤ y, where T is a ternary relational term and x, y, z are object variables. As usual, we use the same symbols for operations and constants in the language and for the corresponding entities in the models. An RLMTL -model is a structure M = (U, ≤, R, m) such that U is a non-empty set and the following conditions are satisfied: • • • • •
m(P) = X × U × U , where X ⊆ U , for P ∈ RVRLMTL ; m(1) = U 3 and m(0) = ∅; ≤ is a reflexive and transitive relation on U that provides the interpretation of the relational constant ≤; (her ) If t ≤ x and (t, y, z) ∈ m(P), then (x, y, z) ∈ m(P), for all t, x, y, z ∈ U and for every relational variable P; R is a ternary relation on U providing the interpretation of the relational constant R and satisfying the conditions (MTL1), . . . , (MTL6) of MTL-models; • m extends to all the relational terms as follows: m(−S) = U 3 −m(S), m(S ∪ T ) = m(S) ∪ m(T ), m(S ∩ T ) = m(S) ∩ m(T ), m(S T ) = {(x, y, z) ∈ U 3 : ∃t, w ∈ U, (t, w, x) ∈ R& &(t, y, z) ∈ m(S)&(w, y, z) ∈ m(T ))}, m(S → T ) = {(x, y, z) ∈ U 3 : ∀t, w ∈ U, if (x, t, w) ∈ R& &(t, y, z) ∈ m(S), then (w, y, z) ∈ m(T )}.
In analogy to binary right ideal relations, the ternary relations on U which are of the form X × U × U , for some X ⊆ U , will be referred to as ideal relations. Let M be an RLMTL -model. A valuation in M is a function assigning elements of U to object variables. An RLMTL formula T (x, y, z) (resp. x ≤ y) is satisfied in a model M by a valuation v whenever (v(x), v(y), v(z)) ∈ m(T ) (resp. v(x) ≤ v(y)) and it is true in M whenever it is satisfied in M by all the valuations. A formula is RLMTL -valid iff it is true in all RLMTL -models.
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Now, we define a translation function from formulas of the logic MTL into ternary relational terms of RLMTL . Let : V → RVRLMTL be a one-to-one mapping assigning relational variables to propositional variables. Then, we define: • • • • • •
(0) = 0 and (1) = 1; ( p) = ( p), for every p ∈ V; ( ∨ ) = () ∪ (); ( ∧ ) = () ∩ (); ( ) = () (); ( → ) = () → ().
Observe that given an RLMTL -model M = (U, ≤, R, m), the set of ideal relations on U is closed with respect to the relational operations. Therefore, for every MTL-formula and for every RLMTL -model M = (U, ≤, R, m), m(()) is an ideal relation. Proposition 1. For every MTL-model M = (U, ≤, R, m) there is an RLMTL -model M = (U , ≤ , R , m ) such that for every MTL-formula and for all s, t, u ∈ U , M, s if and only if (s, t, u) ∈ m (()). Proof. Let M = (U, ≤, R, m) be an MTL-model. We define an RLMTL -model M = (U , ≤ , R , m ) as follows: • U = U , ≤ =≤, and R = R; • m (P) = m( p) × U 2 , for every P ∈ RVRLMTL ∪ {0, 1}, where p is such that P = ( p); • m extends to all the compound terms as in RLMTL -models. Clearly, M defined above is an RLMTL -model. The required condition can be easily proved by induction on the complexity of formulas. Proposition 2. For every RLMTL -model M = (U , ≤ , R , m ) there is an MTL-model M = (U, ≤, R, m) such that for every MTL-formula and for all s, t, u ∈ U , M, s if and only if (s, t, u) ∈ m (()). Proof. The required MTL-model is constructed as follows: • U = U , ≤=≤ , and R = R ; • m( p) = {x ∈ U : for some y, z ∈ U, (x, y, z) ∈ m (P)}, for every p ∈ V ∪ {0, 1}, where P is such that ( p) = P. Clearly, the model defined above is an MTL-model. The required condition is proved by induction on the complexity of formulas. Propositions 1 and 2 lead to the following theorem: Theorem 1. For every MTL-formula and for all object variables x, y, and z, the following conditions are equivalent: 1. is MTL-valid; 2. ()(x, y, z) is RLMTL -valid. 5. Relational dual tableau for logic MTL In this section we present a dual tableau for the logic RLMTL that can be used for verification of validity of MTLformulas. Relational dual tableaux are determined by the axiomatic sets of formulas and rules which most often apply to finite sets of relational formulas. The axiomatic sets take the place of axioms. The rules are intended to reflect properties of relational operations and constants. There are two groups of rules: decomposition rules and specific rules. Given a formula, the decomposition rules of the system enable us to transform it into simpler formulas, or the specific rules enable us to replace a formula by some other formulas. The rules have the following general form (rule)
(x) 1 (x 1 , u 1 , w1 )| . . . |n (x n , u n , w n )
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where (x) is a finite (possibly empty) set of formulas whose object variables are among the elements of set(x), where x is a finite sequence of object variables and set (x) is a set of elements of sequence x; every j (x j , u j , w j ), 1 ≤ j ≤ n, is a finite non-empty set of formulas, whose object variables are among the elements of set(x j ) ∪ set(u j ) ∪ set(w j ), where x j , u j , w j are finite sequences of object variables such that set(x j ) ⊆ set(x), set(u j ) consists of the variables that may be instantiated to arbitrary object variables when the rule is applied (usually to the object variables that appear in the set to which the rule is being applied), set(w j ) consists of the variables that must be instantiated to pairwise distinct new variables (not appearing in the set to which the rule is being applied) and distinct from any variable of sequence u j . If n > 1, then (rule) is an n-fold branching rule. A rule of the form (rule) is applicable to a finite set X of formulas whenever (x) ⊆ X . As a result of an application of a rule of the form (rule) to set X , we obtain the sets (X \ (x)) ∪ j (x j , u j , w j ), j ∈ {1, . . . , n}. A set to which a rule is applied is called the premise of the rule, and the sets obtained by the application of the rule are called its conclusions. Hence, given n ≥ 1, a rule of the form (rule) has n conclusions. A finite set {1 , . . . , n } of RLMTL -formulas is said to be an RLMTL -set whenever for every RLMTL -model M and for every valuation v in M there exists i ∈ {1, . . . , n} such that i is satisfied by v in M. It follows that the first-order disjunction of all the formulas from an RLMTL -set is valid in the first-order logic. A rule of the form (rule) is RLMTL -correct whenever for every finite set X of RLMTL -formulas, X ∪ (x) is an RLMTL -set if and only if X ∪ j (x j , u j , w j ) is an RLMTL -set, for every j ∈ {1, . . . , n}, i.e., the rule preserves and reflects validity. It follows that ‘,’ (comma) in the rules is interpreted as disjunction and ‘|’ (branching) is interpreted as conjunction. RLMTL -dual tableau includes decomposition rules of the following forms: For all object variables x, y, z, and for all ternary relational terms S and T , −−S(x, y, z) (−) S(x, y, z) −(S ∪ T )(x, y, z) (S ∪ T )(x, y, z) (−∪) (∪) S(x, y, z), T (x, y, z) −S(x, y, z)|−T (x, y, z) (S ∩ T )(x, y, z) −(S ∩ T )(x, y, z) (∩) (−∩) S(x, y, z)|T (x, y, z) −S(x, y, z), −T (x, y, z) (S T )(x, y, z) () R(t, w, x), |S(t, y, z), |T (w, y, z), t and w are any object variables and = (S T )(x, y, z) −(S T )(x, y, z) (−) −R(t, w, x), −S(t, y, z), −T (w, y, z) t and w are new object variables such that t w (S → T )(x, y, z) (→) −R(x, t, w), −S(t, y, z), T (w, y, z) t and w are new object variables such that t w −(S → T )(x, y, z) (− →) R(x, t, w), |S(t, y, z), |−T (w, y, z), t and w are any object variables and = −(S → T )(x, y, z) The specific rules are of the form: For every relational variable P and for all object variables x, y, z, x , y , z , t, and w, P(x, y, z) (ideal) t, w are any object variables P(x, t, w), P(x, y, z) (0) (tran ≤)
0(x, y, z)
x, y, z are any object variables
x≤y z is any object variable x ≤ z, x ≤ y|z ≤ y, x ≤ y
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(rher ) (rMTL1)
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P(x, y, z) t is any object variable t ≤ x, P(x, y, z)|P(t, y, z), P(x, y, z) R(x, y, z) R(x , y , z ), |x ≤ x , |y ≤ y , |z ≤ z, x , y , z are any object variables and = R(x, y, z)
(rMTL2) (rMTL3)
R(x, y, z) R(y, x, z) R(x, y, z)|R(z, y , z )| − R(y, y , u), −R(x, u, z ) x, y, z, y , z are any object variables u is a new object variable such that {u} ∩ {x, y, z, x , y , z } = ∅
(rMTL4)
(rMTL5)
(rMTL6)
−R(u, x, x) x is any object variable, u is a new object variable such that u x y≤z R(x, y, z), y ≤ z x is any object variable y ≤ w, t ≤ z R(x, y, z), y ≤ w, t ≤ z|R(x, t, w), y ≤ w, t ≤ z x is any object variable
A set of RLMTL -formulas is said to be an RLMTL -axiomatic set whenever it includes a subset of either of the following forms: For all object variables x, y, and z and for every relational term S, (Ax1) {x ≤ x}; (Ax2) {1(x, y, z)}; (Ax3) {S(x, y, z), −S(x, y, z)}. Let be an RLMTL -formula. An RLMTL -proof tree for is a tree with the following properties: • the formula is at the root of this tree; • each node except the root is obtained by an application of an RLMTL -rule to its predecessor node; • a node does not have successors whenever its set of formulas is an RLMTL -axiomatic set or none of the rules is applicable to its set of formulas. Observe that the proof trees are constructed in the top-down manner, and hence every node has a single predecessor node. A branch of an RLMTL -proof tree is said to be closed whenever it contains a node with an RLMTL -axiomatic set of formulas. A tree is closed iff all of its branches are closed. An RLMTL -formula is RLMTL -provable whenever there is a closed RLMTL -proof tree for it which is then referred to as its RLMTL -proof . 6. Soundness and completeness In this section we prove soundness and completeness of the relational dual tableau presented in the previous section. Proposition 3. 1. The RLMTL -rules are RLMTL -correct. 2. The RLMTL -axiomatic sets are RLMTL -sets.
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Proof. By way of example, we prove correctness of the rules (), (→), and (rMTL4). () Let X be a finite set of RLMTL -formulas. Clearly, if X ∪ {(S T )(x, y, z)} is an RLMTL -set, then so are X ∪ {R(t, w, x), (S T )(x, y, z)}, X ∪ {S(t, y, z), (S T )(x, y, z)}, and X ∪ {T (w, y, z), (S T )(x, y, z)}. Now, assume that X ∪{R(t, w, x), (ST )(x, y, z)}, X ∪{S(t, y, z), (ST )(x, y, z)}, and X ∪{T (w, y, z), (ST )(x, y, z)} are RLMTL -sets. Suppose X ∪{(ST )(x, y, z)} is not an RLMTL -set. Then there exist an RLMTL -model M = (U, ≤, R, m) and a valuation v in M such that for every ∈ X ∪ {(S T )(x, y, z)}, M, v / . Therefore, for every a, b ∈ U , (a, b, v(x)) ∈ / R or (a, v(y), v(z)) ∈ / m(S) or (b, v(y), v(z) ∈ / m(T ). By the assumption, (v(t), v(w), v(x)) ∈ R, (v(t), v(y), v(z)) ∈ m(S), and (v(w), v(y), v(z)) ∈ m(T ), a contradiction. (→) Let X be a finite set of RLMTL -formulas and let t, w be variables that do not occur in X and such that t w and {t, w}∩{x, y, z} = ∅. Assume that X ∪{(S → T )(x, y, z)} is an RLMTL -set. Suppose that X ∪{−R(x, t, w), −S(t, y, z), T (w, y, z)} is not an RLMTL -set. Then there exist an RLMTL -model M = (U, ≤, R, m) and a valuation v in M such that for every ∈ X ∪ {−R(x, t, w), −S(t, y, z), T (w, y, z)}, M, v / . Thus, (v(x), v(t), v(w)) ∈ R, (v(t), v(y), v(z)) ∈ / m(S → T ). m(S), and (v(w), v(y), v(z)) ∈ / m(T ). Therefore, by the definition of m(S → T ), (v(x), v(y), v(z)) ∈ However, by the assumption, (v(x), v(y), v(z)) ∈ m(S → T ), a contradiction. Now, assume X ∪ {−R(x, t, w), −S(t, y, z), T (w, y, z)} is an RLMTL -set. By the assumption on variables t and w, for every RLMTL -model M = (U, ≤ , R, m) and for every valuation v in M either there is ∈ X such that M, v or for all a, b ∈ U , if (v(x), a, b) ∈ R and (a, v(y), v(z)) ∈ m(S), then (b, v(y), v(z)) ∈ m(T ), in which case (v(x), v(y), v(z)) ∈ m(S → T ). Hence, X ∪ {(S → T )(x, y, z)} is an RLMTL -set. (rMTL4) Let X be a finite set of RLMTL -formulas and let x and u be object variables such that u does not occur in X and u x. Clearly, if X is an RLMTL -set, then so is X ∪ {−R(u, x, x)}. Now, assume X ∪ {−R(u, x, x)} is an RLMTL -set. Then, due to the assumption on the variables x and u, for every RLMTL -model M = (U, ≤, R, m) and for every valuation v in M either there is ∈ X such that M, v or for every a ∈ U , (a, v(x), v(x)) ∈ / R. Observe that due to condition (MTL4), there exists a ∈ U such that (a, v(x), v(x)) ∈ R, hence X is an RLMTL -set. Correctness of the remaining rules can be proved similarly. The easy proof of 2 is omitted. Proposition 3 implies: Theorem 2 (Soundness of RLMTL ). Let be an RLMTL -formula. If is RLMTL -provable, then it is RLMTL -valid. A branch b of an RLMTL -proof tree is said to be complete whenever it is closed or it satisfies the following completion conditions: For all object variables x, y, z, x , y , z , t and w, for all ternary relational terms S and T , and for every relational variable P, Cpl(−) If −−S(x, y, z) ∈ b, then S(x, y, z) ∈ b, obtained by an application of the rule (−); Cpl(∪) (resp. Cpl(−∩)) If (S ∪ T )(x, y, z) ∈ b (resp. −(S ∩ T )(x, y, z) ∈ b), then both S(x, y, z) ∈ b and T (x, y, z) ∈ b (resp. both −S(x, y, z) ∈ b and −T (x, y, z) ∈ b), obtained by an application of the rule (∪) (resp. (−∩)); Cpl(∩) (resp. Cpl(−∪)) If (S ∩T )(x, y, z) ∈ b (resp. −(S ∪T )(x, y, z) ∈ b), then either S(x, y, z) ∈ b or T (x, y, z) ∈ b (resp. either −S(x, y, z) ∈ b or −T (x, y, z) ∈ b), obtained by an application of the rule (∩) (resp. (−∪)); Cpl() If (S T )(x, y, z) ∈ b, then for all object variables t and w either R(t, w, x) ∈ b or S(t, y, z) ∈ b or T (w, y, z) ∈ b, obtained by an application of the rule (); Cpl(−) If −(S T )(x, y, z) ∈ b, then for some object variables t and w, −R(t, w, x) ∈ b, −S(t, y, z) ∈ b, and −T (w, y, z) ∈ b, obtained by an application of the rule (−); Cpl(→) If (S → T )(x, y, z) ∈ b, then for some object variables t and w, −R(x, t, w) ∈ b, −S(t, y, z) ∈ b, and T (w, y, z) ∈ b, obtained by an application of the rule (→); Cpl(− →) If −(S → T )(x, y, z) ∈ b, then for all object variables t and w either R(x, t, w) ∈ b or S(t, y, z) ∈ b or −T (w, y, z) ∈ b, obtained by an application of the rule (− →); Cpl(ideal) If P(x, y, z) ∈ b, then for all object variables t and w, P(x, t, w) ∈ b, obtained by an application of the rule (P); Cpl(0) For all object variables x, y, and z, 0(x, y, z) ∈ b, obtained by an application of the rule (0); Cpl(tran ≤) If x ≤ y ∈ b, then for every object variable z either x ≤ z ∈ b or z ≤ y ∈ b, obtained by an application of the rule (tran ≤);
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Cpl(her ) If P(x, y, z) ∈ b, then for every object variable t either t ≤ x ∈ b or P(t, y, z) ∈ b, obtained by an application of the rule (her ); Cpl(rMTL1) If R(x, y, z) ∈ b, then for all object variables x , y , and z either R(x , y , z ) ∈ b or x ≤ x ∈ b or y ≤ y ∈ b or z ≤ z ∈ b, obtained by an application of the rule (rMTL1); Cpl(rMTL2) If R(x, y, z) ∈ b, then R(y, x, z) ∈ b, obtained by an application of the rule (rMTL2); Cpl(rMTL3) For all object variables x, y, z, y , z , either R(x, y, z) ∈ b or R(z, y , z ) ∈ b or there is an object variable u such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b, obtained by an application of the rule (rMTL3); Cpl(rMTL4) For every object variable x there exists an object variable u such that −R(u, x, x) ∈ b, obtained by an application of the rule (rMTL4); Cpl(rMTL5) If y ≤ z ∈ b, then for every object variable x, R(x, y, z) ∈ b, obtained by an application of the rule (rMTL5); Cpl(rMTL6) If y ≤ w ∈ b and t ≤ z ∈ b, then for every object variable x either R(x, y, z) ∈ b or R(x, t, w) ∈ b, obtained by an application of the rule (rMTL6). An RLMTL -proof tree is said to be complete if and only if all of its branches are complete. A complete non-closed branch of an RLMTL -proof tree is said to be open. Note that every non-closed RLMTL -proof tree can be extended to a complete RLMTL -proof tree, i.e., for every RLMTL -formula there exists a complete RLMTL -proof tree for . Let b be an open branch of an RLMTL -proof tree. We define a branch structure Mb = (U b , ≤b , R b , m b ) as follows: • • • •
U b = OVRLMTL ; / b} and m b (≤) =≤b ; ≤b = {(x, y) ∈ (U b )2 : x ≤ y ∈ b b 3 m (T ) = {(x, y, z) ∈ (U ) : T (x, y, z) ∈ / b}, for every T ∈ RVRLMTL ∪ {R, 1, 0}; m b extends to all the compound relational terms as in the RLMTL -models. Let v b be a valuation in Mb such that v b (x) = x, for every object variable x. The major steps of a completeness proof are the following three propositions.
Proposition 4 (Closed Branch Property). For every branch b of an RLMTL -proof tree, if T (x, y, z) ∈ b and −T (x, y, z) ∈ b, for an atomic ternary term T , then the branch b can be closed. Proof. Let b be a branch of an RLMTL -proof tree. First, observe that all the rules of RLMTL -dual tableau, except the rule (rMTL2), preserve the formulas built with atomic terms or their complements, that is any application of a rule except of (rMTL2) transfers such formulas from the premises to the conclusions. If a formula does not appear in the premise of a rule and appears in some of its conclusions, then clearly the preservation property is satisfied. If both T (x, y, z) and −T (x, y, z), for an atomic ternary term T , are in the branch b, obtained by an application of a rule different from (rMTL2), then eventually both of these formulas appear in a node of branch b. Since the set containing a subset {T (x, y, z), −T (x, y, z)} is RLMTL -axiomatic, branch b is closed. The rule (rMTL2) does not preserve formulas of the form R(x, y, z), but if R(x, y, z) appears in a node of branch b, then all the successors of this node in b include either R(x, y, z), if the rule (rMTL2) has not been applied yet, or R(y, x, z) if the rule (rMTL2) has been applied. Thus, if R(x, y, z) is in the branch b and −R(x, y, z) is in a node, say n, of branch b, then either R(x, y, z) ∈ n and −R(x, y, z) ∈ n or R(y, x, z) ∈ n and −R(x, y, z) ∈ n. In the former case, the branch is closed. In the latter, we can apply rule (rMTL2) to R(y, x, z) in node n. Then we obtain a node which includes R(x, y, z) and −R(x, y, z) and we close the branch. Although Proposition 4 does not concern formulas of the form x ≤ y, it is sufficient for proving the satisfaction in branch model property (see Proposition 6). In the case of formulas of the form x ≤ y the property follows directly from the definition of the interpretation of the constant ≤ in the branch model. Proposition 5 (Branch Model Property). Let Mb be a branch structure determined by an open branch b of an RLMTL proof tree. Then Mb is an RLMTL -model. Proof. It suffices to show that for every relational variable P, m b (P) = X × U × U for some X ⊆ U , the relation ≤b is reflexive and transitive, m b satisfies the heredity condition, and the relation R b satisfies the conditions (MTL1), . . . , (MTL6).
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Let P be a relational variable. Assume that for some x, y, z ∈ U b , (x, y, z) ∈ m b (P). Suppose that for some t, w ∈ U b , (x, t, w) ∈ / m b (P). Then P(x, t, w) ∈ b and by the completion condition Cpl(ideal), P(x, y, z) ∈ b. Thus, / m b (P), a contradiction. Hence, m b (P) is a right ideal relation on U b . by the definition of m b , (x, y, z) ∈ Since for every x, the set {x ≤ x} is an axiomatic set, ≤b is reflexive. Transitivity of ≤b follows from the completion condition Cpl(tran ≤), while the heredity condition follows from the completion condition Cpl(her ). By way of example, we show that R b satisfies the conditions (MTL1), (MTL3), and (MTL4). (MTL1) Let x, y, z, z , y , z ∈ U b . Assume (x, y, z) ∈ R b , (x , x) ∈≤b , (y , y) ∈≤b , and (z, z ) ∈≤b . Suppose / R b . Then R(x , y , z ) ∈ b. Thus, by the completion condition Cpl(rMTL1), either R(x, y, z) ∈ b or (x , y , z ) ∈ x ≤ x ∈ b or y ≤ y ∈ b or z ≤ z ∈ b. Therefore, by the definition of m b , either (x, y, z) ∈ / R b or (x , x) ∈≤ / b or b b / or (z, z ) ∈≤ / , which contradicts the assumption. (y , y) ∈≤ (MTL3) By the completion condition Cpl(rMTL3), for all x, y, z, y , z ∈ U b either R(x, y, z) ∈ b or R(z, y , z ) ∈ b or there exists u ∈ U b such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b. Thus, if (x, y, z) ∈ R b and (z, y , z ) ∈ R b , then there exists u ∈ U b such that both (y, y , u) ∈ R b and (x, u, z ) ∈ R b , hence the condition (MTL3) is satisfied. (MTL4) By the completion condition Cpl(rMTL4), for every x ∈ U b there exists u ∈ U b such that −R(u, x, x) ∈ b. / b. Hence, there exists Thus, by the closed branch property, for every x ∈ U b there exists u ∈ U b such that R(u, x, x) ∈ u ∈ U b such that (u, x, x) ∈ R b . Proposition 6 (Satisfaction in Branch Model Property). For every open branch b of an RLMTL -proof tree and for every RLMTL -formula , if Mb , v b , then ∈ / b. Proof. Let be an RLMTL -formula. The proof is by induction on the complexity of relational terms. If is an atomic formula, then the condition of the proposition holds by the definition. Let = −S(x, y, z) for an atomic term S. If / b. For binary atomic relational Mb , v b −S(x, y, z), then S(x, y, z) ∈ b. By the closed branch property, −S(x, y, z) ∈ terms the proof is similar. Assume that the condition of the proposition holds for relational terms S and T and their complements. By way of example, we show that it holds for S T , S → T , and −(S → T ). Assume Mb , v b (S T )(x, y, z), that is there are object terms t and w such that (t, w, x) ∈ R b and (t, y, z) ∈ m b (S) and (w, y, z) ∈ m b (T ). Suppose (S T )(x, y, z) ∈ b. By the completion condition Cpl(), for all object terms t and w, either R(t, w, x) ∈ b or S(t, y, z) ∈ b or T (w, y, z) ∈ b. By the induction hypothesis, for all object terms t and w, / m b (S) or (w, y, z) ∈ / m b (T ), a contradiction. either (t, w, x) ∈ / R b or (t, y, z) ∈ b b Assume M , v (S → T )(x, y, z), that is (x, y, z) ∈ m b (S → T ). Thus, for all object terms t and w, if (x, t, w) ∈ b R and (t, y, z) ∈ m b (S), then (w, y, z) ∈ m b (T ). Then, by the induction hypothesis, if R(x, t, w) ∈ / b and S(t, y, z) ∈ / b, then T (w, y, z) ∈ / b. Suppose (S → T )(x, y, z) ∈ b. By the completion condition Cpl(→), there are object variables t and w such that −R(x, t, w) ∈ b and −S(t, y, z) ∈ b, and T (w, y, z) ∈ b, a contradiction. Assume that Mb , v b −(S → T )(x, y, z). Then there are t, w ∈ U b such that (x, t, w) ∈ R b , (t, y, z) ∈ m b (S), and (w, y, z) ∈ / m b (T ). Suppose that −(S → T )(x, y, z) ∈ b. By the completion condition Cpl(− →), for all t, w ∈ U b either R(x, t, w) ∈ b or S(t, y, z) ∈ b or −T (w, y, z) ∈ b, that is either −R(t, w, x) ∈ / b or −S(t, y, z) ∈ / b or T (w, y, z) ∈ / b. Therefore, by the induction hypothesis, either (t, w, x) ∈ / R b or (t, y, z) ∈ / m b (S) or (w, y, z) ∈ m b (T ), a contradiction. The remaining cases can be proved in a similar way. The above propositions imply: Theorem 3 (Completeness of RLMTL ). Let be an RLMTL -formula. If is RLMTL -valid, then is RLMTL -provable. Proof. Assume is RLMTL -valid. Suppose there is no any closed RLMTL -proof tree for . Then there exists a complete RLMTL -proof tree for with an open branch b. Since ∈ b, by Proposition 6, is not satisfied by valuation v b in the branch model Mb . By Proposition 5, Mb is an RLMTL -model. Hence, is not RLMTL -valid, a contradiction. Furthermore, by Theorems 1–3, we have: Theorem 4 (Relational soundness and completeness of MTL). Let be an MTL-formula. Then for all object variables x, y, and z the following conditions are equivalent: 1. is MTL-valid; 2. ()(x, y, z) is RLMTL -provable.
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Fig. 1. An RLMTL -proof of the formula ( p → q) ∨ (q → p).
Fig. 2. An RLMTL -proof of the formula ( p q) → p.
Example. Consider the following MTL-formulas: = ( p → q) ∨ (q → p), = ( p q) → p. The translations of and into RLMTL -terms are () = (P → Q) ∪ (Q → P), () = (P Q) → P, where ( p) = P and (q) = Q. MTL-validity of and is equivalent to RLMTL -provability of the formulas ()(x, y, z) and ()(x, y, z), respectively. Figs. 1 and 2 present their RLMTL -proofs, respectively. In the figures in each node of a proof tree we underline the formulas which determine the rule that has been applied during the construction of the tree and we indicate which rule has been applied. If a rule introduces a variable, then we write how the variable has been instantiated. This concerns both the rules which introduce a new or an arbitrary variable. Furthermore, in each node we write only those formulas which are essential for the application of a rule and the succession of these formulas in the node is usually motivated by the reasons of formatting.
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7. Alternative rules Some relational proof systems known in the literature (see [24]) require cut-like rules in order to get a completeness result. Cut rules in relational proof systems have the form 1 (x 1 )| . . . |k (x k ) where {1 (x 1 ), . . . , k (x k )} is an unsatisfiable set of literals and x i is a sequence of object variables, for i = 1, . . . , k. Recall that a relational literal is a formula of the form T (x) or −T (x) where T is an atomic relational term and x is a sequence of object variables. A cut rule of the form T (x)|−T (x) is referred to as a standard cut rule. In Section 5 we presented the rules of relational dual tableau for logic MTL. Observe that the rules (0), (rMTL3), and (rMTL4) have the form of cut rules. In this section we show that the rule (rMTL3) may be replaced with an ‘analytic’ rule together with the standard cut rule (cut) restricted to some literals, without loosing completeness. The alternative form of the rule (rMTL3) is −R(x, y, z), −R(z, y , z ) (r MTL3) −R(y, y , u), −R(x, u, z ), −R(x, y, z), −R(z, y , z ) u is a new object variable (cut)
R(x, y, z)|−R(x, y, z)
x, y, and z are any object variables
The completion conditions corresponding to these rules are: Cpl(r MTL3) If −R(x, y, z) ∈ b and −R(z, y , z ) ∈ b, then there is an object variable u such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b, obtained by an application of the rule (r MTL3); Cpl(cut) For all object variables x, y, and z, R(x, y, z) ∈ b or −R(x, y, z) ∈ b, obtained by an application of the rule (cut). The rules obtained in this way are RLMTL -correct: Proposition 7. The rule (r MTL3) is RLMTL -correct. Proof. Let X be a finite set of RLMTL -formulas and let x, x , y, y , z, z , u be object variables such that u does not occur in X and u ∈ / {x, x , y, y , z, z }. Assume X ∪ {−R(y, y , u), −R(x, u, z ), −R(x, y, z), −R(z, y , z )} is an RLMTL -set. By the assumption on variable u, for every RLMTL -model M = (U, ≤, R, m) and for every valuation v in / R or for every a ∈ U , M, either there exists ∈ X such that M, v or (v(x), v(y), v(z)) ∈ / R or (v(z), v(y ), v(z )) ∈ (v(y), v(y ), a) ∈ / R or (v(x), a, v(z )) ∈ / R. Suppose that X ∪ {−R(x, y, z), −R(z, y , z )} is not an RLMTL -set. Then, / and (v(x), v(y), v(z)) ∈ R and there exist an RLMTL -model M and v in M such that for every ∈ X , M, v / R or (v(x), a, v(z )) ∈ / (v(z), v(y ), v(z )) ∈ R. Therefore, by the assumption, for every a ∈ U , either (v(y), v(y ), a) ∈ R. Hence, by condition (MTL3) and since (v(x), v(y), v(z)) ∈ R and (v(z), v(y ), v(z )) ∈ R, there exists a ∈ U such that (v(y), v(y ), a) ∈ R and (v(x), a, v(z )) ∈ R, a contradiction. The rule (cut) is needed in the proof of completeness. Let Mb = (U b , ≤b , R b , m b ) be a branch structure defined in Section 6. Now, we prove that RLMTL -dual tableau obtained by replacing the rule (rMTL3) with the rules (r MTL3) and (cut) satisfies the branch model property. The remaining parts of completeness proof can be shown as in Section 6. Proposition 8 (Branch Model Property). Let Mb be a branch structure determined by an open branch b of an RLMTL proof tree. Then Mb is an RLMTL -model. Proof. We show that R b satisfies the condition (MTL3). Let x, y, z, z , y , z ∈ U b . Assume that (x, y, z) ∈ R b and / b and R(z, y , z ) ∈ / b. By the completion condition Cpl(cut), −R(x, y, z) ∈ b and (z, y , z ) ∈ R b . Then, R(x, y, z) ∈
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Fig. 3. An alternative RLMTL -proof of the formula ( p q) → p.
−R(z, y , z ) ∈ b. Thus, by the completion condition Cpl(r MTL3), there is u ∈ U b such that −R(y, y , u) ∈ b and / b and R(x, u, z ) ∈ / b, for otherwise b would be closed. −R(x, u, z ) ∈ b. By the closed branch property, R(y, y , u) ∈ b b These yield (y, y , u) ∈ R and (x, u, z ) ∈ R , and hence the condition (MTL3) holds. Example. In the preceding section we presented examples of proofs of MTL-formulas in RLMTL -dual tableau. In Fig. 3 we present an alternative proof of the formula = ( p q) → p with the rules (r MTL3) and (cut) instead of the rule (rMTL3). 8. Conclusion Dual tableaux systems and, in particular, relational dual tableaux are semantics based. The rules must be designed so that they could reflect semantics of the logical operations and constrains posed on the models. Kripke-style semantics is very useful in that respect. There are several correspondences showing how to define a rule for a given constraint. The rules constructed in this paper follow general methods presented in [13,24, Chapter 25]. However, fuzzy logics are usually defined with an algebraic semantics. Establishing discrete duality (see [26]) for fuzzy logics may be useful in developing Kripke-style semantics which in turn could lead to a development of dual tableaux. The discrete duality for logic MTL is presented in [27], it is based on [3]. The Kripke-style semantics obtained as a result of this duality is used in this paper. Based on this semantics we developed dual tableau for logic MTL and we proved its soundness and completeness. Following the developments in [24] we discussed some modifications of the rules which preserve completeness of the dual tableau. Dual tableaux are deduction systems alternative to other formalisms such as tableaux, Gentzen sequent calculi, resolution or Hilbert-style systems. A discussion of relationships between dual tableaux and those systems is presented in [24]. The distinguishing features of the system presented in this paper are: the system operates on relational representations of the MTL-formulas and not on the formulas themselves; the system is a validity checker; the rules of the system preserve and reflect validity of sets of formulas which are their premises and conclusions. An advantage of relational approach to deduction is modularity of construction of proof systems. Relational representation of formulas of non-classical logics enables us to exhibit algebras of binary or ternary relations as a common core of many non-classical logics. Once a proof system for such an algebra is given, a proof system for a particular logic can be obtained from it by adding some rules specific for the logic. Furthermore, the general method of construction of dual tableaux provides a standard and intuitively simple way of proving completeness by defining a model which is built from the syntactic resources of the tree built during the proof search process and falsifies a non-provable formula. Dual tableau approach enables us an almost automatic way of transforming a dual tableau proof into a proof in Gentzen
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sequent calculi. A strategy for application of the rules of MTL-dual tableau leading to a decision procedure is an open problem. Various proof-theoretic results in fuzzy logics can be found in [20,30,5,14,19,21,1,9,15,16], among others. A comprehensive survey can be found in [17]. The dual tableau presented in this paper contributes to this area. The relational approach employed in its construction can be applied to other fuzzy logics, in particular, to the axiomatic extensions of logic MTL presented in [25]. Acknowledgments We thank anonymous referees for helpful remarks and suggestions for improvements in the paper. References [1] M. Baaz, A. Ciabattoni, F. Montagna, Analytic calculi for monoidal t-norm based logic, Fundamenta Informaticae 59 (2004) 315–332. [2] A. Burrieza, M. Ojeda-Aciego, E. Orłowska, An implementation of a dual tableaux system for order-of-magnitude qualitative reasoning, International Journal of Computer Mathematics 86 (10/11) (2009) 1852–1866. [3] L.M. Cabrer, S.A. Celani, Priestley dualities for some lattice-ordered algebraic structures, including MTL, IMTL and MV-algebras, Central European Journal of Mathematics 4 (4) (2006) 600–623. [4] L.M. Cabrer, S.A. Celani, Kripke semantics for monoidal t-norm based logics MTL and IMTL, in: Centre for Logic, Epistemology and the History of Science, vol. 8(6), 2008, pp. 1–3. [5] A. Ciabattoni, C.G. Fermüller, Hypersequents as a uniform framework for Urquhart’s C, MTL and related logics, in: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, Warsaw, Poland, 2001, pp. 227–232. [6] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (3) (2001) 271–288. [7] A. Formisano, E.G. Omodeo, E. Orłowska, A PROLOG tool for relational translation of modal logics: a front-end for relational proof systems, in: B. 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Orłowska, Correspondence results for relational proof systems with application to the Lambek calculus, Studia Logica 71 (3) (2002) 389–414. [14] G. Metcalfe, Proof theory for propositional fuzzy logics, Ph.D. Thesis, King’s College London, 2003. [15] G. Metcalfe, N. Olivetti, D. Gabbay, Analytic proof calculi for product logics, Archive for Mathematical Logic 43 (7) (2004) 859–889. [16] G. Metcalfe, N. Olivetti, D. Gabbay, Sequent and hypersequent calculi for Abelian and Łukasiewicz logics, ACM Transactions on Computational Logic 6 (3) (2005) 578–613. [17] G. Metcalfe, N. Olivetti, D. Gabbay, Proof Theory for Fuzzy Logics, Springer, Heidelberg, 2009. [18] F. Montagna, H. Ono, Kripke semantics, undecidability and standard completeness for Esteva and Godo’s logic MTL∀, Studia Logica 71 (2) (2002) 227–245. [19] F. Montagna, G.M. Pinna, E.B.P. Tiezzi, A tableau calculus for Hájek’s logic BL, Journal of Logic and Computation 13 (2) (2003) 241–259. [20] D. Mundici, N. Olivetti, Resolution and model building in the infinite-valued calculus of Łukasiewicz, Theoretical Computer Science 200 (1–2) (1998) 335–366. [21] N. Olivetti, Tableaux for Łukasiewicz infinite-valued logic, Studia Logica 73 (1) (2003) 81–111. [22] H. Ono, Semantical analysis of predicate logics without the contraction rule, Studia Logica 44 (2) (1985) 187–196. [23] H. Ono, Semantics for substructural logics, in: K. Došen, P. Schroeder-Heister (Eds.), Substructural Logics, Oxford University Press, 1993, pp. 259–291. [24] E. Orłowska, J. Goli´nska-Pilarek, Dual Tableaux: Foundations, Methodology, Case Studies, Trends in Logic 36. Springer, 2011, DOI 10.1007/978-94-007-0005-5 1, a book in print [25] E. Orłowska, A. Radzikowska, Discrete duality for some axiomatic extensions of MTL algebras, in: P. Cintula, Z. Haniková, V. Švejdar (Eds.), Witnessed Years—Essays in Honour of Petr Hájek, vol. 10, College Publications, King’s College London, London, 2009, pp. 329–344. [26] E. Orłowska, I. Rewitzky, Discrete duality and its applications to reasoning with incomplete information, in: Lecture Notes in Computer Science, vol. 4585, 2007, pp. 51–56. [27] E. Orłowska, I. Rewitzky, Algebras of Galois-style connections and their discrete duality, Fuzzy Sets and Systems 161 (9) (2010) 1325–1342. [28] E. Orłowska, A. Radzikowska, I. Rewitzky, Discrete Representability and Discrete Duality, 2010, A draft of the book. [29] H. Rasiowa, R. Sikorski, On Gentzen theorem, Fundamenta Mathematicae 48 (1960) 57–69. [30] H. Wagner, A new resolution calculus for the infinite-valued propositional logic of Łukasiewicz, in: Proceedings of FTP 98, Technical Report E1852-GS-981, TU Wien, Austria, 1998, pp. 234–243.
Dual tableau for monoidal triangular norm logic MTL夡 Joanna Goli´nska-Pilareka,b,∗ , Ewa Orłowskab a Institute of Philosophy, Warsaw University, Poland b National Institute of Telecommunications, Warsaw, Poland
Received 30 September 2009; received in revised form 5 September 2010; accepted 16 September 2010 Available online 22 September 2010
Abstract Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system. © 2010 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Monoidal triangular norm logic; Relational logic; Dual tableau system
1. Introduction Monoidal triangular norm logic, MTL, was introduced in [6]. From the perspective of substructural logics it is the logic of full Lambek calculus endowed with the rules of exchange and weakening, FLew , with the additional axiom ( → ) ∨ ( → ) referred to as prelinearity. From the perspective of fuzzy logics it is a logic of left-continuous triangular norms, t-norms for short (see [12]). The classical examples of t-norms which are applied to modelling logical operator of conjunction include Łukasiewicz t-norm x y = max(0, x +y−1), product t-norm defined as multiplication of reals, and Gödel t-norm x y = min(x, y). All those t-norms are continuous. A t-norm is left-continuous iff it has a residuum, →, i.e., the two operators satisfy z ≤ x → y iff x z ≤ y. The residua of the three t-norms mentioned above play the role of implications: Łukasiewicz implication x → y = min(1, 1 − x + y), Gougen implication x → y = 1 if x ≤ y and y/x otherwise, and Gödel implication x → y = 1 if x ≤ y and y otherwise, respectively. Algebraic semantics of logic MTL is provided by the class of MTL-algebras. They are abstract counterparts to the standard structures ([0, 1], ≤, , →), where is a left-continuous t-norm and → is its residuum. A completeness of logic MTL with respect to the semantics determined by those standard structures is presented in [11]. In [18] a Kripke-style semantics for a first-order version of logic MTL is presented, defined along the lines of semantics for substructural logics presented in [22,23]. In that semantics an algebraic structure is assumed in the universes of the models. 夡 Partial support from the Polish Ministry of Science and Higher Education Grant no. N206 399134 is gratefully acknowledged. ∗ Corresponding author at: Institute of Philosophy, Warsaw University, Poland. Tel.: +48 225520140.
E-mail addresses: [email protected] (J. Goli´nska-Pilarek), [email protected] (E. Orłowska). 0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2010.09.007
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In this paper, first, we present MTL-algebras which provide an algebraic semantics of the logic MTL. Next, we present logic MTL with a purely relational semantics developed in [4,28]. Based on this semantics a relational dual tableau for the logic is developed. Relational dual tableaux are based on Rasiowa–Sikorski diagrams for first-order logic [29]. The common language of all relational dual tableaux is a logic of relations. Relational deduction systems in the Rasiowa–Sikorski style have been constructed for a great variety of theories, e.g., intuitionistic, modal, relevant, and multiple-valued logics, as well as for temporal logic, various logics of programs, logics of rough sets, theories of spatial reasoning including region connection calculus, theories of order of magnitude reasoning, and formal concept analysis. A survey of these results is presented in [24]. Specific methodological principles of construction of dual tableaux which make possible such a broad applicability of these systems are: • first, given a theory, a truth preserving translation is defined of the language of the theory into an appropriate relational language (most often a language of binary relations); • second, a dual tableau is constructed for this relational language so that it provides a deduction system for the original theory. This methodology, reflecting the paradigm “Formulas are Relations”, enables us to represent within a uniform formalism the three basic components of formal systems: syntax, semantics, and deduction apparatus. Relational approach enables us to build dual tableaux in a systematic modular way. First, deduction rules are defined for the common relational core of the theories. These rules constitute a basis of all the relational dual tableau proof systems. Next, for any particular theory specific rules are added to the basic set of rules. They reflect the semantic constraints assumed in the models of the theory. As a consequence, we need not implement each deduction system from scratch, we should only extend the basic system with a module corresponding to the specific part of a theory under consideration. Relational dual tableaux are powerful tools which can be used to solve the four major problems that may be formulated in a theory: • • • •
proving general laws holding in the theory; proving that from a finite number of laws of the theory some other law follows; proving that a particular model of the theory obeys some laws; proving that some objects which the theory deals with satisfy a law.
These tasks are represented, respectively, as: the validity problem, the entailment, the model checking problem, and the satisfaction problem in the relational logic. A recent implementation of a proof system for a relational logic is available in [8]. In [7] an implementation of translation procedures from non-classical logics to a relational logic is presented. In [10,2], implementations of relational logics for order of magnitude reasoning are presented. The paper is organized as follows. In Section 2 we present MTL-algebras which provide algebraic semantics of monoidal triangular norm logic MTL. In Section 3 we define the logic MTL, its language and semantics. Relational formalization of logic MTL is presented in Section 4. In Section 5 we present a relational dual tableau that can be used for verification of validity of MTL-formulas. We prove its soundness and completeness in Section 6. In Section 7 we discuss alternative forms of the rules of the dual tableau for MTL. Finally, comments and conclusions are given in Section 8.
2. MTL-algebras A t-norm is a binary operation on the closed real interval : [0, 1]2 → [0, 1], such that for all x, y, z ∈ [0, 1] the following hold: • is associative and commutative; • 1 is the neutral element of ; • If x ≤ y, then x z ≤ y z. Let (xi )i∈J be an indexed family of elements of [0,1]. A t-norm is said to be left-continuous whenever (sup xi ) y = sup(xi y) for every y ∈ [0, 1].
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The abstract structures which capture the properties of left-continuous t-norms are the MTL-algebras. An MTLalgebra is a structure of the form A = (A, ∨, ∧, , →, 0, 1) such that: • (A, ∨, ∧, 0, 1) is a bounded, distributive lattice; • (A, , 1) is a commutative monoid, i.e., for all a, b, c ∈ A the following conditions are satisfied: a (b c) = (a b) c; a b = b a; 1 a = a; df
• → is a residuum of , i.e., a b ≤ c iff a ≤ b → c, where ≤ is the lattice ordering defined by: a ≤ b ⇐⇒ a = a ∧ b (or equivalently b = a ∨ b); • (a → b) ∨ (b → a) = 1. Observe that monoid ( A, , 1) is integral, i.e., its neutral element coincides with the greatest element of the lattice. 3. The logic MTL The language of the logic MTL consists of symbols from the following pairwise disjoint sets: • V—a countable infinite set of propositional variables; • {0, 1}—the set of propositional constants; • {∨, ∧, , →}—the set of propositional operations of disjunction, conjunction, product, and implication, respectively. As usual, the set of MTL-formulas is the smallest set including the set V ∪ {0, 1} and closed with respect to the propositional operations. A classical formalization of MTL is based on its algebraic semantics defined in terms of MTL-algebras. Here we present a Kripke-style semantics obtained from a discrete duality for MTL-algebras developed in [25,27]. This semantics differs from the semantics presented in [18] in that it is a purely relational semantics not assuming any monoid structure in the universes of the models. An MTL-model is a structure M = (U, ≤, R, m) such that U is a non-empty set, ≤ is a reflexive and transitive relation on U , m is a meaning function satisfying the following conditions: • • • •
m( p) ⊆ U , for every propositional variable p; m(1) = U and m(0) = ∅; (her) If x ≤ y and x ∈ m( p), then y ∈ m( p), for all x, y ∈ U and for every propositional variable p; R is a ternary relation on U satisfying the following conditions, for all x, y, z, x , y , z , t, w ∈ U : (MTL1) If (x, y, z) ∈ R, x ≤ x, y ≤ y, and z ≤ z , then (x , y , z ) ∈ R, (MTL2) If (x, y, z) ∈ R, then (y, x, z) ∈ R, (MTL3) If (x, y, z) ∈ R and (z, y , z ) ∈ R, then there exists u ∈ U such that (y, y , u) ∈ R and (x, u, z ) ∈ R, (MTL4) There exists u ∈ U such that (u, x, x) ∈ R, (MTL5) If (x, y, z) ∈ R, then y ≤ z, (MTL6) If (x, y, z) ∈ R and (x, t, w) ∈ R, then y ≤ w or t ≤ z.
Depending on the intended application, the elements of set U are interpreted in computer science as states of a computation or objects of an information system and in philosophical logic as possible worlds. Let M = (U, ≤, R, m) be an MTL-model and let s ∈ U . Satisfaction of an MTL-formula in model M by state s is defined as: • • • • •
M, s p iff s ∈ m( p), for every p ∈ V ∪ {0, 1}; M, s ∨ iff M, s or M, s; M, s ∧ iff M, s and M, s; M, s iff there exist x, y ∈ U such that (x, y, s) ∈ R and M, x and M, y; M, s → iff for all x, y ∈ U , if (s, x, y) ∈ R and M, x, then M, y.
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A formula is said to be true in an MTL-model M iff it is satisfied in M for every s ∈ U . A formula is MTL-valid iff is true in all MTL-models. As shown in [28], algebraic and Kripke-style semantics of MTL are related according to the principle of duality via truth (see [26]), i.e., the same formulas are validated according to the notions of truth provided by the respective semantic structures. The condition (MTL2) reflects commutativity of ; in the presence of (MTL2) the condition (MTL3) corresponds to associativity of ; (MTL4) expresses the condition a ≤ a 1, while (MTL5) expresses the condition 1 a ≤ a, (MTL6) reflects prelinearity. It follows that a Hilbert-style axiomatization of MTL corresponding to its algebraic semantics is complete with respect to the Kripke-style semantics. 4. Relational formalization of logic MTL In this section we define the relational logic RLMTL corresponding to MTL-logic. Formulas of RLMTL , interpreted as ternary relations, will represent formulas of the logic MTL and the accessibility relation from its models. The language of RLMTL consists of the symbols from the following pairwise disjoint sets: • • • • •
OVRLMTL —a countable infinite set of object variables; RVRLMTL —a countable infinite set of ternary relational variables; {≤}—the set consisting of the binary relational constant ≤; {R, 1, 0}—the set of ternary relational constants, where 0 is the empty relation and 1 is the universal relation; {−, ∪, ∩, , →}—the set of relational operations.
The set of ternary relational terms is the smallest set including RVRLMTL ∪ {R, 1, 0} and closed on the relational operations. The set of RLMTL -terms consists of ternary relational terms and the relational constant ≤. RLMTL -formulas are of the form T (x, y, z) or x ≤ y, where T is a ternary relational term and x, y, z are object variables. As usual, we use the same symbols for operations and constants in the language and for the corresponding entities in the models. An RLMTL -model is a structure M = (U, ≤, R, m) such that U is a non-empty set and the following conditions are satisfied: • • • • •
m(P) = X × U × U , where X ⊆ U , for P ∈ RVRLMTL ; m(1) = U 3 and m(0) = ∅; ≤ is a reflexive and transitive relation on U that provides the interpretation of the relational constant ≤; (her ) If t ≤ x and (t, y, z) ∈ m(P), then (x, y, z) ∈ m(P), for all t, x, y, z ∈ U and for every relational variable P; R is a ternary relation on U providing the interpretation of the relational constant R and satisfying the conditions (MTL1), . . . , (MTL6) of MTL-models; • m extends to all the relational terms as follows: m(−S) = U 3 −m(S), m(S ∪ T ) = m(S) ∪ m(T ), m(S ∩ T ) = m(S) ∩ m(T ), m(S T ) = {(x, y, z) ∈ U 3 : ∃t, w ∈ U, (t, w, x) ∈ R& &(t, y, z) ∈ m(S)&(w, y, z) ∈ m(T ))}, m(S → T ) = {(x, y, z) ∈ U 3 : ∀t, w ∈ U, if (x, t, w) ∈ R& &(t, y, z) ∈ m(S), then (w, y, z) ∈ m(T )}.
In analogy to binary right ideal relations, the ternary relations on U which are of the form X × U × U , for some X ⊆ U , will be referred to as ideal relations. Let M be an RLMTL -model. A valuation in M is a function assigning elements of U to object variables. An RLMTL formula T (x, y, z) (resp. x ≤ y) is satisfied in a model M by a valuation v whenever (v(x), v(y), v(z)) ∈ m(T ) (resp. v(x) ≤ v(y)) and it is true in M whenever it is satisfied in M by all the valuations. A formula is RLMTL -valid iff it is true in all RLMTL -models.
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Now, we define a translation function from formulas of the logic MTL into ternary relational terms of RLMTL . Let : V → RVRLMTL be a one-to-one mapping assigning relational variables to propositional variables. Then, we define: • • • • • •
(0) = 0 and (1) = 1; ( p) = ( p), for every p ∈ V; ( ∨ ) = () ∪ (); ( ∧ ) = () ∩ (); ( ) = () (); ( → ) = () → ().
Observe that given an RLMTL -model M = (U, ≤, R, m), the set of ideal relations on U is closed with respect to the relational operations. Therefore, for every MTL-formula and for every RLMTL -model M = (U, ≤, R, m), m(()) is an ideal relation. Proposition 1. For every MTL-model M = (U, ≤, R, m) there is an RLMTL -model M = (U , ≤ , R , m ) such that for every MTL-formula and for all s, t, u ∈ U , M, s if and only if (s, t, u) ∈ m (()). Proof. Let M = (U, ≤, R, m) be an MTL-model. We define an RLMTL -model M = (U , ≤ , R , m ) as follows: • U = U , ≤ =≤, and R = R; • m (P) = m( p) × U 2 , for every P ∈ RVRLMTL ∪ {0, 1}, where p is such that P = ( p); • m extends to all the compound terms as in RLMTL -models. Clearly, M defined above is an RLMTL -model. The required condition can be easily proved by induction on the complexity of formulas. Proposition 2. For every RLMTL -model M = (U , ≤ , R , m ) there is an MTL-model M = (U, ≤, R, m) such that for every MTL-formula and for all s, t, u ∈ U , M, s if and only if (s, t, u) ∈ m (()). Proof. The required MTL-model is constructed as follows: • U = U , ≤=≤ , and R = R ; • m( p) = {x ∈ U : for some y, z ∈ U, (x, y, z) ∈ m (P)}, for every p ∈ V ∪ {0, 1}, where P is such that ( p) = P. Clearly, the model defined above is an MTL-model. The required condition is proved by induction on the complexity of formulas. Propositions 1 and 2 lead to the following theorem: Theorem 1. For every MTL-formula and for all object variables x, y, and z, the following conditions are equivalent: 1. is MTL-valid; 2. ()(x, y, z) is RLMTL -valid. 5. Relational dual tableau for logic MTL In this section we present a dual tableau for the logic RLMTL that can be used for verification of validity of MTLformulas. Relational dual tableaux are determined by the axiomatic sets of formulas and rules which most often apply to finite sets of relational formulas. The axiomatic sets take the place of axioms. The rules are intended to reflect properties of relational operations and constants. There are two groups of rules: decomposition rules and specific rules. Given a formula, the decomposition rules of the system enable us to transform it into simpler formulas, or the specific rules enable us to replace a formula by some other formulas. The rules have the following general form (rule)
(x) 1 (x 1 , u 1 , w1 )| . . . |n (x n , u n , w n )
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where (x) is a finite (possibly empty) set of formulas whose object variables are among the elements of set(x), where x is a finite sequence of object variables and set (x) is a set of elements of sequence x; every j (x j , u j , w j ), 1 ≤ j ≤ n, is a finite non-empty set of formulas, whose object variables are among the elements of set(x j ) ∪ set(u j ) ∪ set(w j ), where x j , u j , w j are finite sequences of object variables such that set(x j ) ⊆ set(x), set(u j ) consists of the variables that may be instantiated to arbitrary object variables when the rule is applied (usually to the object variables that appear in the set to which the rule is being applied), set(w j ) consists of the variables that must be instantiated to pairwise distinct new variables (not appearing in the set to which the rule is being applied) and distinct from any variable of sequence u j . If n > 1, then (rule) is an n-fold branching rule. A rule of the form (rule) is applicable to a finite set X of formulas whenever (x) ⊆ X . As a result of an application of a rule of the form (rule) to set X , we obtain the sets (X \ (x)) ∪ j (x j , u j , w j ), j ∈ {1, . . . , n}. A set to which a rule is applied is called the premise of the rule, and the sets obtained by the application of the rule are called its conclusions. Hence, given n ≥ 1, a rule of the form (rule) has n conclusions. A finite set {1 , . . . , n } of RLMTL -formulas is said to be an RLMTL -set whenever for every RLMTL -model M and for every valuation v in M there exists i ∈ {1, . . . , n} such that i is satisfied by v in M. It follows that the first-order disjunction of all the formulas from an RLMTL -set is valid in the first-order logic. A rule of the form (rule) is RLMTL -correct whenever for every finite set X of RLMTL -formulas, X ∪ (x) is an RLMTL -set if and only if X ∪ j (x j , u j , w j ) is an RLMTL -set, for every j ∈ {1, . . . , n}, i.e., the rule preserves and reflects validity. It follows that ‘,’ (comma) in the rules is interpreted as disjunction and ‘|’ (branching) is interpreted as conjunction. RLMTL -dual tableau includes decomposition rules of the following forms: For all object variables x, y, z, and for all ternary relational terms S and T , −−S(x, y, z) (−) S(x, y, z) −(S ∪ T )(x, y, z) (S ∪ T )(x, y, z) (−∪) (∪) S(x, y, z), T (x, y, z) −S(x, y, z)|−T (x, y, z) (S ∩ T )(x, y, z) −(S ∩ T )(x, y, z) (∩) (−∩) S(x, y, z)|T (x, y, z) −S(x, y, z), −T (x, y, z) (S T )(x, y, z) () R(t, w, x), |S(t, y, z), |T (w, y, z), t and w are any object variables and = (S T )(x, y, z) −(S T )(x, y, z) (−) −R(t, w, x), −S(t, y, z), −T (w, y, z) t and w are new object variables such that t w (S → T )(x, y, z) (→) −R(x, t, w), −S(t, y, z), T (w, y, z) t and w are new object variables such that t w −(S → T )(x, y, z) (− →) R(x, t, w), |S(t, y, z), |−T (w, y, z), t and w are any object variables and = −(S → T )(x, y, z) The specific rules are of the form: For every relational variable P and for all object variables x, y, z, x , y , z , t, and w, P(x, y, z) (ideal) t, w are any object variables P(x, t, w), P(x, y, z) (0) (tran ≤)
0(x, y, z)
x, y, z are any object variables
x≤y z is any object variable x ≤ z, x ≤ y|z ≤ y, x ≤ y
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(rher ) (rMTL1)
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P(x, y, z) t is any object variable t ≤ x, P(x, y, z)|P(t, y, z), P(x, y, z) R(x, y, z) R(x , y , z ), |x ≤ x , |y ≤ y , |z ≤ z, x , y , z are any object variables and = R(x, y, z)
(rMTL2) (rMTL3)
R(x, y, z) R(y, x, z) R(x, y, z)|R(z, y , z )| − R(y, y , u), −R(x, u, z ) x, y, z, y , z are any object variables u is a new object variable such that {u} ∩ {x, y, z, x , y , z } = ∅
(rMTL4)
(rMTL5)
(rMTL6)
−R(u, x, x) x is any object variable, u is a new object variable such that u x y≤z R(x, y, z), y ≤ z x is any object variable y ≤ w, t ≤ z R(x, y, z), y ≤ w, t ≤ z|R(x, t, w), y ≤ w, t ≤ z x is any object variable
A set of RLMTL -formulas is said to be an RLMTL -axiomatic set whenever it includes a subset of either of the following forms: For all object variables x, y, and z and for every relational term S, (Ax1) {x ≤ x}; (Ax2) {1(x, y, z)}; (Ax3) {S(x, y, z), −S(x, y, z)}. Let be an RLMTL -formula. An RLMTL -proof tree for is a tree with the following properties: • the formula is at the root of this tree; • each node except the root is obtained by an application of an RLMTL -rule to its predecessor node; • a node does not have successors whenever its set of formulas is an RLMTL -axiomatic set or none of the rules is applicable to its set of formulas. Observe that the proof trees are constructed in the top-down manner, and hence every node has a single predecessor node. A branch of an RLMTL -proof tree is said to be closed whenever it contains a node with an RLMTL -axiomatic set of formulas. A tree is closed iff all of its branches are closed. An RLMTL -formula is RLMTL -provable whenever there is a closed RLMTL -proof tree for it which is then referred to as its RLMTL -proof . 6. Soundness and completeness In this section we prove soundness and completeness of the relational dual tableau presented in the previous section. Proposition 3. 1. The RLMTL -rules are RLMTL -correct. 2. The RLMTL -axiomatic sets are RLMTL -sets.
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Proof. By way of example, we prove correctness of the rules (), (→), and (rMTL4). () Let X be a finite set of RLMTL -formulas. Clearly, if X ∪ {(S T )(x, y, z)} is an RLMTL -set, then so are X ∪ {R(t, w, x), (S T )(x, y, z)}, X ∪ {S(t, y, z), (S T )(x, y, z)}, and X ∪ {T (w, y, z), (S T )(x, y, z)}. Now, assume that X ∪{R(t, w, x), (ST )(x, y, z)}, X ∪{S(t, y, z), (ST )(x, y, z)}, and X ∪{T (w, y, z), (ST )(x, y, z)} are RLMTL -sets. Suppose X ∪{(ST )(x, y, z)} is not an RLMTL -set. Then there exist an RLMTL -model M = (U, ≤, R, m) and a valuation v in M such that for every ∈ X ∪ {(S T )(x, y, z)}, M, v / . Therefore, for every a, b ∈ U , (a, b, v(x)) ∈ / R or (a, v(y), v(z)) ∈ / m(S) or (b, v(y), v(z) ∈ / m(T ). By the assumption, (v(t), v(w), v(x)) ∈ R, (v(t), v(y), v(z)) ∈ m(S), and (v(w), v(y), v(z)) ∈ m(T ), a contradiction. (→) Let X be a finite set of RLMTL -formulas and let t, w be variables that do not occur in X and such that t w and {t, w}∩{x, y, z} = ∅. Assume that X ∪{(S → T )(x, y, z)} is an RLMTL -set. Suppose that X ∪{−R(x, t, w), −S(t, y, z), T (w, y, z)} is not an RLMTL -set. Then there exist an RLMTL -model M = (U, ≤, R, m) and a valuation v in M such that for every ∈ X ∪ {−R(x, t, w), −S(t, y, z), T (w, y, z)}, M, v / . Thus, (v(x), v(t), v(w)) ∈ R, (v(t), v(y), v(z)) ∈ / m(S → T ). m(S), and (v(w), v(y), v(z)) ∈ / m(T ). Therefore, by the definition of m(S → T ), (v(x), v(y), v(z)) ∈ However, by the assumption, (v(x), v(y), v(z)) ∈ m(S → T ), a contradiction. Now, assume X ∪ {−R(x, t, w), −S(t, y, z), T (w, y, z)} is an RLMTL -set. By the assumption on variables t and w, for every RLMTL -model M = (U, ≤ , R, m) and for every valuation v in M either there is ∈ X such that M, v or for all a, b ∈ U , if (v(x), a, b) ∈ R and (a, v(y), v(z)) ∈ m(S), then (b, v(y), v(z)) ∈ m(T ), in which case (v(x), v(y), v(z)) ∈ m(S → T ). Hence, X ∪ {(S → T )(x, y, z)} is an RLMTL -set. (rMTL4) Let X be a finite set of RLMTL -formulas and let x and u be object variables such that u does not occur in X and u x. Clearly, if X is an RLMTL -set, then so is X ∪ {−R(u, x, x)}. Now, assume X ∪ {−R(u, x, x)} is an RLMTL -set. Then, due to the assumption on the variables x and u, for every RLMTL -model M = (U, ≤, R, m) and for every valuation v in M either there is ∈ X such that M, v or for every a ∈ U , (a, v(x), v(x)) ∈ / R. Observe that due to condition (MTL4), there exists a ∈ U such that (a, v(x), v(x)) ∈ R, hence X is an RLMTL -set. Correctness of the remaining rules can be proved similarly. The easy proof of 2 is omitted. Proposition 3 implies: Theorem 2 (Soundness of RLMTL ). Let be an RLMTL -formula. If is RLMTL -provable, then it is RLMTL -valid. A branch b of an RLMTL -proof tree is said to be complete whenever it is closed or it satisfies the following completion conditions: For all object variables x, y, z, x , y , z , t and w, for all ternary relational terms S and T , and for every relational variable P, Cpl(−) If −−S(x, y, z) ∈ b, then S(x, y, z) ∈ b, obtained by an application of the rule (−); Cpl(∪) (resp. Cpl(−∩)) If (S ∪ T )(x, y, z) ∈ b (resp. −(S ∩ T )(x, y, z) ∈ b), then both S(x, y, z) ∈ b and T (x, y, z) ∈ b (resp. both −S(x, y, z) ∈ b and −T (x, y, z) ∈ b), obtained by an application of the rule (∪) (resp. (−∩)); Cpl(∩) (resp. Cpl(−∪)) If (S ∩T )(x, y, z) ∈ b (resp. −(S ∪T )(x, y, z) ∈ b), then either S(x, y, z) ∈ b or T (x, y, z) ∈ b (resp. either −S(x, y, z) ∈ b or −T (x, y, z) ∈ b), obtained by an application of the rule (∩) (resp. (−∪)); Cpl() If (S T )(x, y, z) ∈ b, then for all object variables t and w either R(t, w, x) ∈ b or S(t, y, z) ∈ b or T (w, y, z) ∈ b, obtained by an application of the rule (); Cpl(−) If −(S T )(x, y, z) ∈ b, then for some object variables t and w, −R(t, w, x) ∈ b, −S(t, y, z) ∈ b, and −T (w, y, z) ∈ b, obtained by an application of the rule (−); Cpl(→) If (S → T )(x, y, z) ∈ b, then for some object variables t and w, −R(x, t, w) ∈ b, −S(t, y, z) ∈ b, and T (w, y, z) ∈ b, obtained by an application of the rule (→); Cpl(− →) If −(S → T )(x, y, z) ∈ b, then for all object variables t and w either R(x, t, w) ∈ b or S(t, y, z) ∈ b or −T (w, y, z) ∈ b, obtained by an application of the rule (− →); Cpl(ideal) If P(x, y, z) ∈ b, then for all object variables t and w, P(x, t, w) ∈ b, obtained by an application of the rule (P); Cpl(0) For all object variables x, y, and z, 0(x, y, z) ∈ b, obtained by an application of the rule (0); Cpl(tran ≤) If x ≤ y ∈ b, then for every object variable z either x ≤ z ∈ b or z ≤ y ∈ b, obtained by an application of the rule (tran ≤);
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Cpl(her ) If P(x, y, z) ∈ b, then for every object variable t either t ≤ x ∈ b or P(t, y, z) ∈ b, obtained by an application of the rule (her ); Cpl(rMTL1) If R(x, y, z) ∈ b, then for all object variables x , y , and z either R(x , y , z ) ∈ b or x ≤ x ∈ b or y ≤ y ∈ b or z ≤ z ∈ b, obtained by an application of the rule (rMTL1); Cpl(rMTL2) If R(x, y, z) ∈ b, then R(y, x, z) ∈ b, obtained by an application of the rule (rMTL2); Cpl(rMTL3) For all object variables x, y, z, y , z , either R(x, y, z) ∈ b or R(z, y , z ) ∈ b or there is an object variable u such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b, obtained by an application of the rule (rMTL3); Cpl(rMTL4) For every object variable x there exists an object variable u such that −R(u, x, x) ∈ b, obtained by an application of the rule (rMTL4); Cpl(rMTL5) If y ≤ z ∈ b, then for every object variable x, R(x, y, z) ∈ b, obtained by an application of the rule (rMTL5); Cpl(rMTL6) If y ≤ w ∈ b and t ≤ z ∈ b, then for every object variable x either R(x, y, z) ∈ b or R(x, t, w) ∈ b, obtained by an application of the rule (rMTL6). An RLMTL -proof tree is said to be complete if and only if all of its branches are complete. A complete non-closed branch of an RLMTL -proof tree is said to be open. Note that every non-closed RLMTL -proof tree can be extended to a complete RLMTL -proof tree, i.e., for every RLMTL -formula there exists a complete RLMTL -proof tree for . Let b be an open branch of an RLMTL -proof tree. We define a branch structure Mb = (U b , ≤b , R b , m b ) as follows: • • • •
U b = OVRLMTL ; / b} and m b (≤) =≤b ; ≤b = {(x, y) ∈ (U b )2 : x ≤ y ∈ b b 3 m (T ) = {(x, y, z) ∈ (U ) : T (x, y, z) ∈ / b}, for every T ∈ RVRLMTL ∪ {R, 1, 0}; m b extends to all the compound relational terms as in the RLMTL -models. Let v b be a valuation in Mb such that v b (x) = x, for every object variable x. The major steps of a completeness proof are the following three propositions.
Proposition 4 (Closed Branch Property). For every branch b of an RLMTL -proof tree, if T (x, y, z) ∈ b and −T (x, y, z) ∈ b, for an atomic ternary term T , then the branch b can be closed. Proof. Let b be a branch of an RLMTL -proof tree. First, observe that all the rules of RLMTL -dual tableau, except the rule (rMTL2), preserve the formulas built with atomic terms or their complements, that is any application of a rule except of (rMTL2) transfers such formulas from the premises to the conclusions. If a formula does not appear in the premise of a rule and appears in some of its conclusions, then clearly the preservation property is satisfied. If both T (x, y, z) and −T (x, y, z), for an atomic ternary term T , are in the branch b, obtained by an application of a rule different from (rMTL2), then eventually both of these formulas appear in a node of branch b. Since the set containing a subset {T (x, y, z), −T (x, y, z)} is RLMTL -axiomatic, branch b is closed. The rule (rMTL2) does not preserve formulas of the form R(x, y, z), but if R(x, y, z) appears in a node of branch b, then all the successors of this node in b include either R(x, y, z), if the rule (rMTL2) has not been applied yet, or R(y, x, z) if the rule (rMTL2) has been applied. Thus, if R(x, y, z) is in the branch b and −R(x, y, z) is in a node, say n, of branch b, then either R(x, y, z) ∈ n and −R(x, y, z) ∈ n or R(y, x, z) ∈ n and −R(x, y, z) ∈ n. In the former case, the branch is closed. In the latter, we can apply rule (rMTL2) to R(y, x, z) in node n. Then we obtain a node which includes R(x, y, z) and −R(x, y, z) and we close the branch. Although Proposition 4 does not concern formulas of the form x ≤ y, it is sufficient for proving the satisfaction in branch model property (see Proposition 6). In the case of formulas of the form x ≤ y the property follows directly from the definition of the interpretation of the constant ≤ in the branch model. Proposition 5 (Branch Model Property). Let Mb be a branch structure determined by an open branch b of an RLMTL proof tree. Then Mb is an RLMTL -model. Proof. It suffices to show that for every relational variable P, m b (P) = X × U × U for some X ⊆ U , the relation ≤b is reflexive and transitive, m b satisfies the heredity condition, and the relation R b satisfies the conditions (MTL1), . . . , (MTL6).
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Let P be a relational variable. Assume that for some x, y, z ∈ U b , (x, y, z) ∈ m b (P). Suppose that for some t, w ∈ U b , (x, t, w) ∈ / m b (P). Then P(x, t, w) ∈ b and by the completion condition Cpl(ideal), P(x, y, z) ∈ b. Thus, / m b (P), a contradiction. Hence, m b (P) is a right ideal relation on U b . by the definition of m b , (x, y, z) ∈ Since for every x, the set {x ≤ x} is an axiomatic set, ≤b is reflexive. Transitivity of ≤b follows from the completion condition Cpl(tran ≤), while the heredity condition follows from the completion condition Cpl(her ). By way of example, we show that R b satisfies the conditions (MTL1), (MTL3), and (MTL4). (MTL1) Let x, y, z, z , y , z ∈ U b . Assume (x, y, z) ∈ R b , (x , x) ∈≤b , (y , y) ∈≤b , and (z, z ) ∈≤b . Suppose / R b . Then R(x , y , z ) ∈ b. Thus, by the completion condition Cpl(rMTL1), either R(x, y, z) ∈ b or (x , y , z ) ∈ x ≤ x ∈ b or y ≤ y ∈ b or z ≤ z ∈ b. Therefore, by the definition of m b , either (x, y, z) ∈ / R b or (x , x) ∈≤ / b or b b / or (z, z ) ∈≤ / , which contradicts the assumption. (y , y) ∈≤ (MTL3) By the completion condition Cpl(rMTL3), for all x, y, z, y , z ∈ U b either R(x, y, z) ∈ b or R(z, y , z ) ∈ b or there exists u ∈ U b such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b. Thus, if (x, y, z) ∈ R b and (z, y , z ) ∈ R b , then there exists u ∈ U b such that both (y, y , u) ∈ R b and (x, u, z ) ∈ R b , hence the condition (MTL3) is satisfied. (MTL4) By the completion condition Cpl(rMTL4), for every x ∈ U b there exists u ∈ U b such that −R(u, x, x) ∈ b. / b. Hence, there exists Thus, by the closed branch property, for every x ∈ U b there exists u ∈ U b such that R(u, x, x) ∈ u ∈ U b such that (u, x, x) ∈ R b . Proposition 6 (Satisfaction in Branch Model Property). For every open branch b of an RLMTL -proof tree and for every RLMTL -formula , if Mb , v b , then ∈ / b. Proof. Let be an RLMTL -formula. The proof is by induction on the complexity of relational terms. If is an atomic formula, then the condition of the proposition holds by the definition. Let = −S(x, y, z) for an atomic term S. If / b. For binary atomic relational Mb , v b −S(x, y, z), then S(x, y, z) ∈ b. By the closed branch property, −S(x, y, z) ∈ terms the proof is similar. Assume that the condition of the proposition holds for relational terms S and T and their complements. By way of example, we show that it holds for S T , S → T , and −(S → T ). Assume Mb , v b (S T )(x, y, z), that is there are object terms t and w such that (t, w, x) ∈ R b and (t, y, z) ∈ m b (S) and (w, y, z) ∈ m b (T ). Suppose (S T )(x, y, z) ∈ b. By the completion condition Cpl(), for all object terms t and w, either R(t, w, x) ∈ b or S(t, y, z) ∈ b or T (w, y, z) ∈ b. By the induction hypothesis, for all object terms t and w, / m b (S) or (w, y, z) ∈ / m b (T ), a contradiction. either (t, w, x) ∈ / R b or (t, y, z) ∈ b b Assume M , v (S → T )(x, y, z), that is (x, y, z) ∈ m b (S → T ). Thus, for all object terms t and w, if (x, t, w) ∈ b R and (t, y, z) ∈ m b (S), then (w, y, z) ∈ m b (T ). Then, by the induction hypothesis, if R(x, t, w) ∈ / b and S(t, y, z) ∈ / b, then T (w, y, z) ∈ / b. Suppose (S → T )(x, y, z) ∈ b. By the completion condition Cpl(→), there are object variables t and w such that −R(x, t, w) ∈ b and −S(t, y, z) ∈ b, and T (w, y, z) ∈ b, a contradiction. Assume that Mb , v b −(S → T )(x, y, z). Then there are t, w ∈ U b such that (x, t, w) ∈ R b , (t, y, z) ∈ m b (S), and (w, y, z) ∈ / m b (T ). Suppose that −(S → T )(x, y, z) ∈ b. By the completion condition Cpl(− →), for all t, w ∈ U b either R(x, t, w) ∈ b or S(t, y, z) ∈ b or −T (w, y, z) ∈ b, that is either −R(t, w, x) ∈ / b or −S(t, y, z) ∈ / b or T (w, y, z) ∈ / b. Therefore, by the induction hypothesis, either (t, w, x) ∈ / R b or (t, y, z) ∈ / m b (S) or (w, y, z) ∈ m b (T ), a contradiction. The remaining cases can be proved in a similar way. The above propositions imply: Theorem 3 (Completeness of RLMTL ). Let be an RLMTL -formula. If is RLMTL -valid, then is RLMTL -provable. Proof. Assume is RLMTL -valid. Suppose there is no any closed RLMTL -proof tree for . Then there exists a complete RLMTL -proof tree for with an open branch b. Since ∈ b, by Proposition 6, is not satisfied by valuation v b in the branch model Mb . By Proposition 5, Mb is an RLMTL -model. Hence, is not RLMTL -valid, a contradiction. Furthermore, by Theorems 1–3, we have: Theorem 4 (Relational soundness and completeness of MTL). Let be an MTL-formula. Then for all object variables x, y, and z the following conditions are equivalent: 1. is MTL-valid; 2. ()(x, y, z) is RLMTL -provable.
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Fig. 1. An RLMTL -proof of the formula ( p → q) ∨ (q → p).
Fig. 2. An RLMTL -proof of the formula ( p q) → p.
Example. Consider the following MTL-formulas: = ( p → q) ∨ (q → p), = ( p q) → p. The translations of and into RLMTL -terms are () = (P → Q) ∪ (Q → P), () = (P Q) → P, where ( p) = P and (q) = Q. MTL-validity of and is equivalent to RLMTL -provability of the formulas ()(x, y, z) and ()(x, y, z), respectively. Figs. 1 and 2 present their RLMTL -proofs, respectively. In the figures in each node of a proof tree we underline the formulas which determine the rule that has been applied during the construction of the tree and we indicate which rule has been applied. If a rule introduces a variable, then we write how the variable has been instantiated. This concerns both the rules which introduce a new or an arbitrary variable. Furthermore, in each node we write only those formulas which are essential for the application of a rule and the succession of these formulas in the node is usually motivated by the reasons of formatting.
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7. Alternative rules Some relational proof systems known in the literature (see [24]) require cut-like rules in order to get a completeness result. Cut rules in relational proof systems have the form 1 (x 1 )| . . . |k (x k ) where {1 (x 1 ), . . . , k (x k )} is an unsatisfiable set of literals and x i is a sequence of object variables, for i = 1, . . . , k. Recall that a relational literal is a formula of the form T (x) or −T (x) where T is an atomic relational term and x is a sequence of object variables. A cut rule of the form T (x)|−T (x) is referred to as a standard cut rule. In Section 5 we presented the rules of relational dual tableau for logic MTL. Observe that the rules (0), (rMTL3), and (rMTL4) have the form of cut rules. In this section we show that the rule (rMTL3) may be replaced with an ‘analytic’ rule together with the standard cut rule (cut) restricted to some literals, without loosing completeness. The alternative form of the rule (rMTL3) is −R(x, y, z), −R(z, y , z ) (r MTL3) −R(y, y , u), −R(x, u, z ), −R(x, y, z), −R(z, y , z ) u is a new object variable (cut)
R(x, y, z)|−R(x, y, z)
x, y, and z are any object variables
The completion conditions corresponding to these rules are: Cpl(r MTL3) If −R(x, y, z) ∈ b and −R(z, y , z ) ∈ b, then there is an object variable u such that both −R(y, y , u) ∈ b and −R(x, u, z ) ∈ b, obtained by an application of the rule (r MTL3); Cpl(cut) For all object variables x, y, and z, R(x, y, z) ∈ b or −R(x, y, z) ∈ b, obtained by an application of the rule (cut). The rules obtained in this way are RLMTL -correct: Proposition 7. The rule (r MTL3) is RLMTL -correct. Proof. Let X be a finite set of RLMTL -formulas and let x, x , y, y , z, z , u be object variables such that u does not occur in X and u ∈ / {x, x , y, y , z, z }. Assume X ∪ {−R(y, y , u), −R(x, u, z ), −R(x, y, z), −R(z, y , z )} is an RLMTL -set. By the assumption on variable u, for every RLMTL -model M = (U, ≤, R, m) and for every valuation v in / R or for every a ∈ U , M, either there exists ∈ X such that M, v or (v(x), v(y), v(z)) ∈ / R or (v(z), v(y ), v(z )) ∈ (v(y), v(y ), a) ∈ / R or (v(x), a, v(z )) ∈ / R. Suppose that X ∪ {−R(x, y, z), −R(z, y , z )} is not an RLMTL -set. Then, / and (v(x), v(y), v(z)) ∈ R and there exist an RLMTL -model M and v in M such that for every ∈ X , M, v / R or (v(x), a, v(z )) ∈ / (v(z), v(y ), v(z )) ∈ R. Therefore, by the assumption, for every a ∈ U , either (v(y), v(y ), a) ∈ R. Hence, by condition (MTL3) and since (v(x), v(y), v(z)) ∈ R and (v(z), v(y ), v(z )) ∈ R, there exists a ∈ U such that (v(y), v(y ), a) ∈ R and (v(x), a, v(z )) ∈ R, a contradiction. The rule (cut) is needed in the proof of completeness. Let Mb = (U b , ≤b , R b , m b ) be a branch structure defined in Section 6. Now, we prove that RLMTL -dual tableau obtained by replacing the rule (rMTL3) with the rules (r MTL3) and (cut) satisfies the branch model property. The remaining parts of completeness proof can be shown as in Section 6. Proposition 8 (Branch Model Property). Let Mb be a branch structure determined by an open branch b of an RLMTL proof tree. Then Mb is an RLMTL -model. Proof. We show that R b satisfies the condition (MTL3). Let x, y, z, z , y , z ∈ U b . Assume that (x, y, z) ∈ R b and / b and R(z, y , z ) ∈ / b. By the completion condition Cpl(cut), −R(x, y, z) ∈ b and (z, y , z ) ∈ R b . Then, R(x, y, z) ∈
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Fig. 3. An alternative RLMTL -proof of the formula ( p q) → p.
−R(z, y , z ) ∈ b. Thus, by the completion condition Cpl(r MTL3), there is u ∈ U b such that −R(y, y , u) ∈ b and / b and R(x, u, z ) ∈ / b, for otherwise b would be closed. −R(x, u, z ) ∈ b. By the closed branch property, R(y, y , u) ∈ b b These yield (y, y , u) ∈ R and (x, u, z ) ∈ R , and hence the condition (MTL3) holds. Example. In the preceding section we presented examples of proofs of MTL-formulas in RLMTL -dual tableau. In Fig. 3 we present an alternative proof of the formula = ( p q) → p with the rules (r MTL3) and (cut) instead of the rule (rMTL3). 8. Conclusion Dual tableaux systems and, in particular, relational dual tableaux are semantics based. The rules must be designed so that they could reflect semantics of the logical operations and constrains posed on the models. Kripke-style semantics is very useful in that respect. There are several correspondences showing how to define a rule for a given constraint. The rules constructed in this paper follow general methods presented in [13,24, Chapter 25]. However, fuzzy logics are usually defined with an algebraic semantics. Establishing discrete duality (see [26]) for fuzzy logics may be useful in developing Kripke-style semantics which in turn could lead to a development of dual tableaux. The discrete duality for logic MTL is presented in [27], it is based on [3]. The Kripke-style semantics obtained as a result of this duality is used in this paper. Based on this semantics we developed dual tableau for logic MTL and we proved its soundness and completeness. Following the developments in [24] we discussed some modifications of the rules which preserve completeness of the dual tableau. Dual tableaux are deduction systems alternative to other formalisms such as tableaux, Gentzen sequent calculi, resolution or Hilbert-style systems. A discussion of relationships between dual tableaux and those systems is presented in [24]. The distinguishing features of the system presented in this paper are: the system operates on relational representations of the MTL-formulas and not on the formulas themselves; the system is a validity checker; the rules of the system preserve and reflect validity of sets of formulas which are their premises and conclusions. An advantage of relational approach to deduction is modularity of construction of proof systems. Relational representation of formulas of non-classical logics enables us to exhibit algebras of binary or ternary relations as a common core of many non-classical logics. Once a proof system for such an algebra is given, a proof system for a particular logic can be obtained from it by adding some rules specific for the logic. Furthermore, the general method of construction of dual tableaux provides a standard and intuitively simple way of proving completeness by defining a model which is built from the syntactic resources of the tree built during the proof search process and falsifies a non-provable formula. Dual tableau approach enables us an almost automatic way of transforming a dual tableau proof into a proof in Gentzen
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sequent calculi. A strategy for application of the rules of MTL-dual tableau leading to a decision procedure is an open problem. Various proof-theoretic results in fuzzy logics can be found in [20,30,5,14,19,21,1,9,15,16], among others. A comprehensive survey can be found in [17]. The dual tableau presented in this paper contributes to this area. The relational approach employed in its construction can be applied to other fuzzy logics, in particular, to the axiomatic extensions of logic MTL presented in [25]. Acknowledgments We thank anonymous referees for helpful remarks and suggestions for improvements in the paper. References [1] M. Baaz, A. Ciabattoni, F. Montagna, Analytic calculi for monoidal t-norm based logic, Fundamenta Informaticae 59 (2004) 315–332. [2] A. Burrieza, M. Ojeda-Aciego, E. Orłowska, An implementation of a dual tableaux system for order-of-magnitude qualitative reasoning, International Journal of Computer Mathematics 86 (10/11) (2009) 1852–1866. [3] L.M. 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