A note on fuzzy predicate logic Petr H jek 1 Institute of Computer Science, Academy of Sciences of the Czech Republic Pod vod renskou v 2, 182 07 Prague. Abstract. Recent development of mathematical fuzzy logic is briey surveyed. The set of all formulas of predicate logic that are tautologies with respect to all continuous t-norms is shown to be heavily non-recursive ( 2 -hard). 1 Introduction Recently considerable progres has been made in strictly mathematical (formal, symbolic) aspects of fuzzy logic as a logic with a comparative notion of truth, logic of vague propositions that may be more true or less true. My book [4] Metamathematics of fuzzy logic contains a unied theory of logics based on continuous t-norms (as truth functions of conjunction); a basic fuzzy logic is introduced and studied in depth, together with three stronger logics ( Lukasiewicz, Godel and product logic) corresponding to three most important continuous t-norms. Needless to say, several results were obtained by other authors; let we mention at least the books [1, 3]. My [5] is a critical self-review of my book and a survey of further results in mathematical fuzzy logic. The development of mathematical fuzzy logic is expected to help in building a bridge between fuzzy logic in the broad sense (term used by L. Zadeh, the founder of fuzzy set theory) and pure logicians. In the present paper I oer the reader a short survey of some selected fundamental results serving as preliminaries to just one (important) new undecidability result concerning t-norm predicate logics. I hope that the paper helps the reader to get some insight into mathematical fuzzy logic in its present state of development. 2 Preliminaries Our standard set of truth values is [0; 1] { the real unit interval. Recall that a t-norm in a binary operation in [0; 1] which is commutative, associative, nondecreasing in both arguments and satisfying 0x = 0 and 1x = x for each x: We restrict ourselves to continuous t-norms. Each continuous t-norm determines uniquely its residuum ) satisfying, for each x; y; z; the condition x x ) y i x z y: () 1 Partial support by COST Action 15 is acknowledged. 1
Note that x ) y = 1 i x y: The following are three important continuous t-norms and their residua: name x + y x ) y for a > y negation x ) 0 Lukasiewicz max(0; x + y? 1) 1? x + y ( 1? x Godel min(x; y) y 1 for x = 0 product x y y=x 0 for x > 0 For each continuous t-norm ; the structure ([0; 1]; ; ; ); 0; 1); i.e. [0; 1] with its natural ordering, operations ; ) and designated elements 0, 1 is the t-algebra determined by : Note that minimum is denable from ; ); indeed, min(x; y) = x (x ) y): More generally, a BL-algebra is a structure L = (L; ; ; ); 0 L ; 1 L ) where (L; ; 0 L ; 1 L ) is a lattice with the least element 0 L and largest element 1 L ; is commutative and associative, 1 x = x for each x; ) is a residuum of (i.e. the equivalence () above holds for each x:y:z) and x \ y = x (x ) y); (x ) y) [ (y ) x) = 1 L where \; [ are the lattice operation of inmum and supremum. Each t-algebra is a particular BL-algebra; and each BL-algebra L can serve as the algebra of truth functions of a propositional calculus, i.e. we may evaluate propositional atoms p by elements e(p) 2 L and compute the value el(') of each formula ' built up from propositional variables using conjunction & and implication!: we take to be the truth function if & and ) the truth function of! : (Note that :' is dened as '! 0 where 0 is the truth constant with the value 0 L :) A propositional formula ' is an L-tautology if el(') = 1 L for each evaluation e of atoms in L; ' is a t-tautology of it is an L-tautology for each t-algebra L, and ' is a BL-tautology if it is an L-tautology for each BL-algebra L. There is a simple set of 7 (schemas of) axioms of BL { particular BLtautologies, complete (together with the rule of modus ponens) with respect to BL-tautologies. Note a recent result [2] saying that t-tautologies coincide with BL-tautologies; thus those axioms are also complete for t-tautologies. (See Appendix for the axioms.) Extending axioms of BL by simple additions axioms we get axioms of Lukasiewicz logic L (additional axiom: ::' '); Godel logic G (('&') ') and product logic (2 additional axioms) complete for L-tautologies, L being given by Lukasiewicz, Godel and product t-norm respectively. (Notation: L = [0; 1] L ; [0; 1] G; [0; 1] respectively.) See [4] for details.) Additional axioms lead to particular classes of BL-algebras: BL-algebras L such that the additional axiom of Lukasiewicz logic is and L-tautology are called MValgebras; similarly for Godel logic (G-algebras) and product logic (-algebras). One can prove that a formula ' is a [0; 1] L-tautology i it is an L-tautology for each MV-algebra L; similarly for [0; 1] G (G-algebras) and [0; 1] (-algebras). 2
Consider the language of predicate logic with some predicates P 1 ; : : : P n (each with its arity), object variables, connectives &!; truth constant 0 and quantiers 8; 9: Given a BL-algebra L, an L-interpretation of our language is a structure M = (M; (r P ) P predicate ) where M 6= ; and for each predicate P of arity n; r P is an n-ary L-fuzzy relation on M; i.e. r P : M n! L: The truth value k'k L M;v of a formula ' in M for an evaluation v (of object variables by elements of M) given by the truth functions is given by the usual Tarski style conditions, e.g. kp (x; y)k L M;v = r P (v(x); v(y)); k'& k L M;v = k'kl M;v k kl M;v ; k(8x)'kl M;v = inffk'k L M;v jv0 x vg (where v 0 x v means that v 0 (y) = v(y) for all y dierent from x:) This is always dened if L is a t-algebra (all inma and suprema exist). For a general BL-algebra L we call M L-safe if all truth values k'k L are well M;v dened. A formula ' of predicate logic is an L-tautology if k'k L M;v = 1 L for all L-safe M and all v: The notion of a t-tautology (BL-tautology) is obvious. (Let us recall that the condition of safeness is superuous in the case of t-algebras.) Axioms of the basic predicate fuzzy logic BL8 are those of BL plus 5 axiom schemas for quantiers (see Appendix). Deduction rules are modus ponens and generalization. The completeness theorem says that a predicate formula ' is provable in BL8 i it is a BL-tautology. Axioms of Lukasiewicz predicate logic L8 are those of BL8 plus the additional axiom schema of Lukasiewicz propositional logic ::' '): Similarly for G8; 8: We have analogous completeness theorems: L8 proves ' i ' is an L-tautology for each MV-algebra L. Similarly for G8 (G-algebras), 8 (-algebras). For G8 it can be proved that ' is an L-tautology for each G-algebra i it is a [0; 1] G -tautology (where, once more, [0; 1] G is the t-algebra given by Godel t-norm min). But the analogous statement is false both for [0; 1] L (MV-algebras) and [0; 1] (-algebras): whereas MV-tautologies equal L8-provable formulas and hence form a 1 (rec. enumerable) set, [0; 1] L -tautologies from a 2-complete set (rst proved by Ragaz). The set of [0; 1] -tautologies is 2 -hard. Thus neither the set of (predicate) [0; 1] L -tautologies nor the set of (predicate) [0; 1] -tautologies is recursively axiomatizable. A natural question remains if the set of t-tautologies is 1 (recursively axiomatizable). In the next section we show that it is not. It follows that, in contradistinction to the propositional calculus (recall the result of [2] mentioned above), t-tautologies of the predicate calculus form a proper subclass of predicate BL-tautologies. 3 Complexity of predicate t-tautologies We are going to present a recursive reduction of [0; 1] L-tautologies to t-tautologies, i.e. we associate in an eective way with each (closed) formula ' another formula ' ## such that ' is a [0; 1] L -tautology i '## is an L-tautology for each t-algebra 3
L. For this purpose we shall need some known facts on the structure of continuous t-norms. Let us start with an example. Take n + 1 numbers 0 = a 0 < : : : < a n = 1 and decide if on [a i ; a i+1 ] your t-norm should be isomorphic to Lukasiewicz, Godel or product t-norm; choose isomorphisms f i : [0; 1] $ [a i ; a i+1 ]: For x; y from the same interval [a i ; a i+1 ] let x y = f i (f?1 (x) i f?1 (y)) where i is the t-norm you have chosen for the i-th interval; for x; y not from the same interval let x y = min(x; y): This is a continuous t-norm; the corresponding residuum ) looks as follows: for x y; x ) y = 1; for x > y from the same interval x ) y = f i (f?1 (x) ) i f?1 (y)) (where ) i is the residuum of i ); for x > y not from the same interval x ) y = y: Each continuous t-norm is like this, but not necesary with a nite set of \division points" a i ; in general the division points closed nowhere dense subset of [0; 1]: (For example, think of the set f0g[f 1 jn positive naturalg; being isomorphic to Lukasiewicz on [ ; 1 ] for n even and to product for n odd.) We call n 1 n+1 n this representation of the t-norm Mostert-Shields representation. (See [4] for details and references.) One more thing we need is the fact that in Lukasiewicz predicate logic the existential quantier is denable from 8 as in classical logic: (9x)' :(8x):' is a tautology. Thus in L8 each formula is equivalent to a formula not containing the existential quantier. Last preliminary thing: let us say that the rst summand of a continuous t-norm is Lukasiewicz if there is a least positive division point a and on [0; a]; is isomorphic to Lukasiewicz t-norm on [0; 1] as above. Similarly for Godel and product; note that there may be no rst summand, as in the example with 1 n above. The reader easily veries the following Lemma. Let be a continuous t-norm and [0; 1] the corresponding t-algebra; let x 2 [0; 1]; x > 0: (1) (?)(?)x = x i x = 0 or x = 1 or the rst summand [0; a] is Lukasiewicz and 0 < x < a: (2) In all remaining cases (?)x = 0 and hence (?)(?)x = 1: Now we are ready to present our reduction. Denition. For each '; let ' # be the result of replacing, in '; each atom P (t 1 ; : : : ; t n ) by its double negation ::P (t 1 ; : : : ; t n ): Furthermore, let Q be a unary predicate and c a constant not occuring in '; let ' ## be the formula :Q(c) _ ::Q(c) _ ' # : Theorem. Let ' be a formula not containing the existential quantier. Then (i), (ii), (iii) are mutually, equivalent, where (i) ' is a [0; 1] L -tautology; 4
(ii) ' # is an L-tautology for each L given by a continuous t-norm whose rst summand (in the Mostert-Shields representation) is Lukasiewicz; (iii) ' ## is a t-tautology. Proof. First let be a continuous t-norm whose rst summand on [0; a] 2 is isomorphic to Lukasiewicz t-norm (a 1); let L be the BL-algebra on [0; 1] given by : Then [0; a)[f1g is the domain of a BL-subalgebra L 1 of L and the mapping f dened by f(x) = (?)(?)x is a homomorphism of L onto L 1 (identical on [0; a) and mapping the rest to 1). Moreover, for each L-structure M, formula ' and evaluation v of variables, k' # k L M;v = f(k'k L M;v) = k'k L 1 M ; 1 ;v where M 1 results from M by replacing the interpretation r P by the L-interpretation r 0 of ::P; i.e. of each predicate P r 0 (a 1 ; : : : ; a n ) = (?)(?)r(a 1 ; : : : ; a n ): This is shown by induction on the complexity of '; observing that f preserves even innite infs. (Caution: this would fail if we did not assume that the rst factor is Lukasiewicz. If it is not then f(0) = 0 and f(x) = 1 for x > 0 and innite infs are not preserved!) Recall that L 1 is isomorphic to [0; 1] L ; thus if ' is a [0; 1] L-tautology, then ' is an L 1 -tautology and ' # is an L-tautology. Thus (i) implies (ii). Now consider ' ##, i.e. :Q(c)_::Q(c)_' # and let L be given by any continuous t-norm ; let ' be a [0; 1] L-tautology. If the rst factor of is Lukasiewicz then ' # is an L-tautology and so is ' ## : In all other cases for (the rst factor is product or 0 is the inmum if positive idempotents), (8x)(:Q(x) _ ::Q(x)) is an L-tautology and so is ' ## : Thus (ii) implies (iii). Finally let us show that (iii) implies (i). Let ' be such that ' ## is a t- tautology and hence a [0; 1] L -tautology. Let L stand for [0; 1] L ; let M be any L-interpretation of the language of ' and let m 2 M: We may interpret c as m and set the truth value of Q(c) to be 1 ; interpreting Q arbitrarily for other elements of 2 M: Let M 0 be the expanded structure; then kq(c) _ :Q(c)k L = 1 and therefore M0 2 k' # k L = k' # k L M0 M = 1: Moreover, since L = [0; 1] L we get k'# k L M = k'kl M (due to the L-validity of the axiom of double negation). Thus k'k L M = 1 for arbitrarily M, and ' is a [0; 1] L-tautology. This completes the proof. Corollary. The set of all t-tautologies of the predicate calculus is 2 -hard and hence not recursively axiomatizable. Remark. Thus formulas provable in the basic predicate logic BL8 form a proper subset of the set of all t-tautologies of predicate calculus. It would be very interesting to nd a natural formula ' which is a predicate t-tautology but is not an L-tautology for an appropriate BL-algebra L. 5
4 Appendix. Axioms of the basic fuzzy predicate calculus. Axioms for connectives: (A1) ('! )! ((! )! ('! )) (A2) ('& )! ' (A3) ('& )! ( &') (A4) ('&('! ))! ( &(! ')) (A5a) ('! (! ))! (('& )! ) (A5b) (('& )! )! ('! (! )) (A6) (('! )! )! (((! ')! )! ) (A7) 0! ' Axioms for quantiers: (81) (8x)'(x)! '(y) (91) '(y)! (8x)'(x) (82) (8x)(! )! (! (8x) ) (92) (8x)('! )! ((9x)'! ) (83) (8x)(' _ )! ((8x)' _ where y is a constant or a variable substitutable for x in ' and the formula does not contain free occurences of x: References [1] Cignoli R., d'ottaviano I. M. L., Mundici D.: Algebraic Foundations of Many-valued Reasoning, Kluwer (to appear). [2] Cignoli R., Esteva F., Godo L., Torrens A.: Basic fuzzy logic is the logic of continuous t-norms and their residua, submitted. [3] Gottwald S.: Fuzzy sets and fuzzy logic, Viehweg Wiesbaden 1993. [4] H jek P.: Metamathematics of fuzzy logic, Kluwer 1998 [5] H jek P.: Mathematical fuzzy logic { state of art. Proc. Logic Colloquium'98 (Buss at al., ed.) Lect. Notices in Logic, Springer-Verlag, to appear. [6] H jek P.: Basic fuzzy logic and BL-algebras, Soft Computing 2 (1998) 124{128 6