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Computational complexity of t-norm based propositional fuzzy logics with rational truth constants
Fuzzy Sets and Systems 157 (2006) 677 – 682 www.elsevier.com/locate/fss
Computational complexity of t-norm based propositional fuzzy logics with rational truth constants夡 Petr Hájek∗ Institute of Computer Science, Academy of Sciences of the Czech Republic, 182 07 Prague, Czech Republic Received 26 July 2005; received in revised form 5 December 2005; accepted 12 December 2005 Available online 9 January 2006
Abstract If a continuous t-norm on [0, 1] maps pairs of rationals into rationals then the corresponding fuzzy propositional calculus can be extended by rational truth constants and “bookkeeping” axioms for them. (Łukasiewicz t-norm is the classical example.) Computational complexity of such logics is studied. Consequences for fuzzy description logic are formulated. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Rational constants; Computational complexity; Description logic
1. Introduction The reader is assumed to be familiar with continuous t-norms and the corresponding fuzzy propositional logics (see the monograph [6] and/or the recent detailed survey [5]), in particular Łukasiewicz, Gödel and product t-norm (denoted Ł, G, ) and the representation of each continuous t-norm as an ordered sum of copies of these three ones (Mostert-Shields). For each continuous t-norm ∗ the corresponding propositional logic L(∗) has ∗ as the standard truth function of conjunction & and the residuum ⇒ of ∗ as the truth function of implication →. Defined connectives are ¬, ∧, ∨, ≡; 0¯ is the truth constant for 0, 1¯ for 1. A propositional formula is a (standard) ∗-tautology if it has the value 1 for each evaluation of propositional variables by elements of [0, 1], when computed using the truth functions given by ∗. The definition of to be (standardly) ∗-satisfiable is evident. (In this paper we do not work with general semantics given by BL-algebras.) 1 For each ∗, the set of ∗-tautologies is coNP-complete and the set of all ∗-satisfiable formulas in NP-complete (first proved for Łukasiewicz t-norm by Mundici; the final result and references to intermediate results is in [8]). The idea to work explicitly with truth constants r¯ for r ∈ [0, 1] was first elaborated by Pavelka [10] in his logic which is in fact Łukasiewicz logic extended by r for all r ∈ [0, 1]. If turned out to be reasonable to work with r¯ for rational elements of [0, 1], keeping the language countable. (For axioms see below.) Note that Łukasiewicz logic extended by 夡 This work was partially supported by the project ITI, No. 1M0021620808 of MŠMT and partly by the Institutional Research Plan AV0Z10300504. Thanks are due to Z. Haniková for important comments. ∗ Tel.: +42 2 6884 244; fax: +42 2 8585 789. E-mail address: [email protected]. 1 More generally, for each set D ⊆ [0, 1] of designated values we can define D-∗-tautologies and D-∗-satisfiable formulas; ∗-tautologies are then {1}-∗-tautologies etc. For D being the half-open interval (0, 1] we speak on positive ∗-tautologies and positively ∗-satisfiable formulas.
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truth constants has been systematically developed by Novák and his group, see e.g. [9]. The fact that the residuum of Łukasiewicz t-norm is continuous makes the so-called (strong) Pavelka-style completeness theorem possible: truth degree of a formula over a theory equals its provability degree in that theory), see e.g. [9,6]. In the sequel, given a continuous t-norm ∗, we denote by RL(∗) the extension of L(∗) by adding the truth constants r¯ for each rational r ∈ [0, 1] to the language, declaring that r¯ denotes just r. To have it well-behaved, we have to restrict our attention to continuous t-norms such that the set Q ∩ [0, 1] of all rational elements of [0, 1] is closed under ∗ and under ⇒ . We shall call such t-norms r-admissible (admissible w.r.t. rational numbers). Given an axiom system for L(∗), (∗ r-admissible) the corresponding axiom system for RL(∗) results by adding the “bookkeeping” axioms for all rational r, s ∈ [0, 1] : (¯r &¯s ) ≡ r ∗ s, (¯r → s¯ ) ≡ r ⇒ s, Note that for ∗ being Ł (Łukasiewicz t-norm) we have (usual, standard) completeness for tautologies: is a tautology of RL(Ł) iff RL(Ł) (see [6]). Recently the completeness for tautologies was proved for Gödel t-norm [4] and for product t-norm [11]. (See also some results on RL(∗) in [3].) This has increased the interest in t-norm based logics with rational truth constants, different from RL(Ł). In this paper, we investigate the computational complexity of various logics RL(∗) (∗ r-admissible). We show RL(Ł) and RL(G) (Łukasiewicz, Gödel) to have the set of tautologies coNP-complete and the set of satisfiable formulas NPcomplete; for the product t-norm we are only able to show that both the set of tautologies and the set of satisfiable formulas of RL() is in PSPACE (better results being not excluded). We also get results on finite semantic consequence (whenever 1 , . . . , n are true then is also true). Then we discuss r-admissible isomorphic copies of Ł, G, , radmissible t-norms having finitely many components and give an example of an r-admissible ∗ with infinitely many components such that tautologies of RL(∗) are very heavily undecidable (form a non-arithmetical set). Finally we comment on consequences of our results for fuzzy description logic.
2. Łukasiewicz, Gödel and product with rational constants Definition 1. For any r-admissible ∗, TAUT (RL(∗)) denotes the set of all tautologies of RL(∗), similarly for SAT (RL(∗)) and satisfiable formulas; furthermore, SCONS(RL(∗)) is the set of all pairs (, ) such that is a semantic consequence of in the sense of RL(∗). Theorem 1. For Łukasiewicz t-norm Ł, TAUT (RL(Ł)) and SCONS(RL(Ł)) are coNP-complete and SAT (RL(Ł)) is NP-complete. Proof. Clearly the first two sets are coNP-hard and the third is NP-hard since so are the corresponding sets for Łukasiewicz logic without rational constants. To show that TAUT (RL(Ł)) is in coNP (equivalently that the set posSAT (RL(Ł)) of all positively satisfiable formulas of RL(Ł) is in N P ) and that SAT (RL(Ł)) is in NP just inspect [6] 6.2.18–6.2.20 (and 6.1.8) to reduce ∈ SAT (RL(Ł)), ∈ posSAT (RL(Ł)) to a particular mixed integer problem; the only thing to be added is the replacement of e(¯r ) i by r i and e(¯r ) i by r i. Then everything works. ¯ i0 , i0 > 0. For testing (, ) ∈ / SCON S start with e()1, e( → 0) Theorem 2. For Gödel t-norm G, TAUT (RL(G)) and SCONS(RL(G)) is co-NP complete and SAT (RL(G)) is NPcomplete. Proof. Given a formula , let all rational truth constants occurring in be among 0 = r0 < r1 < · · · < rm−1 < rm = 1. Clearly we may assume ri = i/m, i = 0, . . . , m. If contains n propositional variables then for each i = 0, . . . , m − 1 choose n rational elements ai1 , . . . , ain strictly between ri , ri+1 . The formula is in SAT (RL(G)) iff you may guess an evaluation of p1 , . . . , pn taking values in {r0 , . . . , rm } ∪ m−1 i=0 {ai1 , . . . , ain } giving the formula the value 1. is not in TAUT (RL(G)) iff you can guess such an evaluation giving the formula a value less than 1. Similarly for SCONS.
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Now we turn to RL(). We use the result by Canny [2] saying that the existential theory of real numbers can be decided in P SP ACE. In more details, consider the predicate language with predicates =, <, , function symbols +, · (binary), − (unary) and constants 0, 1. Given an open (quantifier free) formula (x, . . . , y), the problem is if is satisfiable in the structure of real numbers, i.e. if (∃x, . . . , y)(x, . . . , y) is true in this structure. By Canny’s result the problem is in PSPACE. Our aim is to reduce (in P time) the tautology and satisfiability problem of RL() to the former problem. This will be done as follows: Observe that if lh() = n (length) then the number of subformulas of is n. For each such subformula of introduce an object variable z and for each such which is not a propositional variable form a definition of z as follows: if is r¯ where r = h/k (h, k integers coded as binary numbers, i.e. sequences of zeros and ones) then def (z ) is k¯ · z = h¯ where for h = 0, 1, h¯ is represented in the theory of real numbers as a sum of powers of 2 (i.e. of (1 + 1)) corresponding to digits 1, where for i > 0, 2i is 2 · 2 · . . . 2, i factors; thus e.g. 1101 becomes 2.2.2 + 2.2 + 1. If is 1 &2 then def (x ) is z = z1 · z2 ; if is 1 → 2 then def (z ) is (z1 z2 & z = 1)∨ (z1 > z2 & z · z1 = z2 ). Finally, let ˆ be {def ()| subformula of , not a variable}. The length of ˆ is quadratic in the length of . Moreover, • ∈ SAT (RL()) iff &(z ˆ = 1) is satisfiable in the reals, • ∈ / TAUT (RL()) iff &(z ˆ < 1) is satisfiable in the reals, ˆ • (, ) ∈ / SCONS(RL()) iff & ˆ &(z = 1)&(z < 1) is satisfiable in the reals. Thus we have the following: Theorem 3. For the product t-norm , the sets TAUT (RL()), SAT (RL()), SCONS(RL()) are in PSPACE. 3. Other r-admissible t-norms Each increasing one–one mapping f of [0, 1] onto itself can be used to transform a given continuous t-norm ∗ to a new continuous t-norm ∗f defined as x ∗f y = f −1 (f (x) ∗ f (y)); thus f becomes an isomorphism of ([0, 1], ∗, ⇒) and ([0, 1], ∗f , ⇒f ) where ⇒f is the residuum of ∗f and satisfies x ⇒f u = f −1 (f (x) ⇒ f (y)). Call f r-admissible if it maps rationals to rationals; if f is an r-admissible mapping and ∗ is an r-admissible t-norm then ∗f is also an r-admissible t-norm (both continuous). Clearly, L(∗) and L(∗f ) have the same tautologies and satisfiables; but this evidently need not be true for RL(∗) and RL(∗f ) since in RL(∗), r¯ &¯s ≡ r ∗ s is a tautology whereas in RL(∗f ) the corresponding tautology is r¯ &¯s ≡ r ∗f s. (Observe that ∗ is the same as ∗f iff ∗ is Gödel t-norm, i.e. minimum.) A particular kind of an r-admissible mapping f is an increasing piecewise linear mapping of [0, 1] onto itself, consisting of finitely many linear pieces, the linear pieces having rational coefficients, e.g. f having a linear piece between (0, 0) and ( 21 , 41 ) and another linear piece between ( 21 , 41 ) and (1, 1). Trivially, both f and f −1 are P (deterministically polynomially computable) for rational arguments. Definition 2. An r-admissible mapping of [0, 1] into itself is strongly r-admissible if both f and f −1 are in P when restricted to rational arguments. Theorem 4. Assume TAUT (RL(∗)) is co-NP complete and SAT (RL(∗)) is NP-complete (e.g. ∗ can be Łukasiewicz). Let f be a strongly r-admissible mapping of [0, 1] onto itself. Then TAUT (RL(∗f )) is co-NP complete and SAT (RL(∗f )) is NP-complete. Proof. This is easily seen from the fact that for each evaluation e of propositional variables and each formula (p1 , . . . , r¯1 , . . .), f (e∗ ((p1 , . . . , r¯1 , . . .)) = e∗ f ((p1 , . . . , f (r1 ), . . .)) e (pi ) = f (e(pi )). Thus (p1 , . . . , r¯1 , . . .) is tautologous/satisfiable is RL(∗) iff so is (p1 , . . . , f (r1 ), . . .) in RL(∗f ). Thus TAUT (RL(∗)) and TAUT (RL(∗f )) are P-reducible to each other and the same for SAT (and for SCON S).
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Similarly we get the following: Theorem 5. Assume that TAUT (RL(∗)) and SAT (RL(∗)) are PSPACE and let f be an r-admissible mapping such that both f and f −1 are PSPACE-computable. Then TAUT (RL(∗f )) and SAT (RL(∗f )) are PSPACE (and similarly for SCONS). Now let us turn to continuous t-norms that are ordinal sums of finitely many components, i.e. ∗ = ∗1 ⊕ · · · ⊕ ∗n where each ∗i is Ł, G or (but there are no two consecutive G’s). Thus there are 0 = d0 < d1 < · · · < dn = 1 such that ∗ restricted to [di , di+1 ] is isomorphic to Ł, G or . The di ’s will be called endpoints of the components. Lemma 1. Let ∗ be an r-admissible t-norm with n components. Then the left endpoint of each Ł-component is rational.
Proof. The left endpoint di corresponds to 0 and Ł is nilpotent; thus take any rational x between the endpoints, then for a sufficiently large natural k, x k = x ∗ · · · ∗ x = di , hence di is rational. Note that the right point may be irrational: let 0 < e < 1, e irrational, let 0 = a0 , a1 , . . . be an increasing sequence of rationals with lim an = 1 and 0 = b0 , b1 , . . . an increasing sequence of rationals with lim bn = e. Let f be a 1-1 increasing mapping of [0, 1] onto [0, e] which maps each [ai , ai+1 ] linearly to [bi , bi+1 ], thus sending all rationals 0 x < 1 on rationals 0 f (x) < e. Use f to induce the continuous t-norm ∗ which is isomorphic to Łukasiewicz on [0, e] and to Gödel on [e, 1]. Clearly the t-norm is r-admissible. We restrict ourselves to r-admissible t-norms with finitely many components, all having rational endpoints. Definition 3. (1) Let e < e be two rationals from [0, 1]. A bijection of [0, 1] to [e, e ] is strongly r-admissible if f maps rational elements of [0, 1] onto rational elements of [e, e ] and both f and f −1 restricted to rational arguments are P-computable. (2) Let ∗ be an r-admissible t-norm with n components, ∗ is strongly r-admissible if each its Ł-component is isomorphic to [0, 1]Ł via a strongly r-admissible bijection and the same for each -component and [0, 1] . (Nothing needs to be assumed on G-components.) Theorem 6. If ∗ is a strongly r-admissible t-norm with finitely many components, having no -component then TAUT (RL(∗)) is co-NP-complete, SAT (RL(∗)) is NP-complete and SCONS(RL(∗)) is co-NP-complete. Proof. The proof is got by analyzing Haniková [8] Section 2. Let 0 = r0 < r1 < · · · < rn = 1 list all rationals occurring in a given formula or being endpoints of components of ∗. For each i = 0, . . . , n introduce variables yi0 , . . . , yim (where m is the length of ) and postulate ri = yi0 < yi1 < · · · < yim = ri+1 . Guess assignment: Choose indeterministically a mapping g assigning to each subformula of a variable yij such that g() = ynm (= rn = 1). Check external: Let ri be left endpoint of a component; recall g(ri ) = yi0 . Further let f (1 ) = u, f (2 ) = v, 0 i n − 1. Check: • if 1 &2 is a subformula of and u yi0 v then f (1 &2 ) = u; • if 1 → 2 is a subformula of and uv then f (1 → 2 ) = 1 = rn ; • if 1 → 2 is a subformula of and v < ri u then f (1 → 2 ) = u. Check internal: For each component [ri , rj ] (i < j ) of ∗ check behavior of g in the segment. Whenever g(1 ) = u, g(2 ) = v with ri u, v rj do the following: check g(1 &2 ) = w ∈ [ri , rj ] and produce the equation w = u ∗ v; if u > v then check g(1 → 2 ) = w ∈ [ri , rj ] and produce the equation w = u ⇒ v. The produced equations together with the inequations yk0 < · · · < ykm for i k < j form the system belonging to [i, j ]. For [i, j ] being a G-component replace w = u ∗ v by w = min(u, v) and w = u ⇒ v by w = v. Check satisfiability of the system by taking an arbitrary (fixed) chain of rational values from [0, 1] whose cardinality is the number of y-variables from [ri , rj ] and guessing evaluation q of these variables in the chain sending yi0 to 0 and yj 0 to 1. Verify validity of the system belonging to [i, j ]. If successful, map [0, 1] to [ri , rj ] by an increasing bijection sending g(yk,0 ) to rk for i k j .
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For [i, j ] being a Ł-component and fi the corresponding strongly r-admissible bijection of [0, 1] onto [ri , rj ] then check the system belonging to [i, j ] extended by equations yk,0 = fi−1 (rk ) for i k j in the sense of Łukasiewicz connectives. This can be done by a reduction to a MIP-problem like above (cf. also [1]). This ends the definition of a non-deterministic polynomial algorithm; it succeeds iff is RL(∗) – satisfiable. This proves SAT (RL(∗)) being in NP (and clearly it is NP-hard). Modifying the algorithm by demanding at the beginning g() < ynm (instead of = ynm ) we get an algorithm succeeding iff is not a RL(∗)-tautology; hence TAUT (RL(∗)) is co-NP-complete. Similarly for SCONS. For t-norms with some -components we get the following: Theorem 7. If ∗ is a strongly r-admissible t-norm with finitely many components then SAT (RL(∗)), TAUT (RL(∗)) and SCONS(RL(∗)) are in PSPACE. Proof. Everything as in the preceding theorem, only add the case of [ri , rj ] being a -component: then take the corresponding system of equalities and inequalities extended by yk , 0 = fk−1 (rk ) for i k j where fi is the strongly r-admissible bijection of [0, 1] to [ri , rj ] from the definition of ∗ being strongly r-admissible) and reduce the satisfiability of this system to the satisfiability of a corresponding open formula in the reals as in the proof concerning RL(). (The case of an Ł-component may be handled as in the previous theorem or also by translating into an open formula in the language of reals.) Details are left to the reader. To close this section we give an example of an r-admissible continuous t-norm with infinitely many components, all being Ł, which is as undecidable as you want. Example 1. Let A be any set of positive natural numbers. Let ∗ be a continuous t-norm whose idempotents are 0 and all n1 , n ∈ A; if n1 < n2 are two neighbor elements of A (i.e. there is no n ∈ A with n1 < n < n2 ) then on [1/n2 , 1/n1 ] ∗ is isomorphic to [0, 1]Ł by the obvious linear bijection of [0, 1] to [1/n2 , 1/n1 ]. Then ∗ is r-admissible; moreover, n ∈ A iff the formula 1/n ∗ 1/n = 1/n is in TAUT (RL(∗)) iff this formula is in SAT (RL(∗)). Therefore A is recursively reducible both to TAUT (RL(∗)) and to SAT (RL(∗)); hence the Turing degrees of the two last sets is bigger than or equal to the Turing degree of A. In particular there is an r-admissible ∗ such that both TAUT (RL(∗)) and SAT (RL(∗)) is not arithmetical. (But recall that TAUT (L(∗)) is co-NP-complete and SAT (L(∗)) is NP-complete by [8].) Admittedly our results isolate “reasonable” continuous t-norms for which adding rational constants either keeps the computational complexity equal to the complexity of the logic without those constants or shows a reasonable upper bound (PSPACE). Several problems remain open.
4. Consequences for fuzzy description logic Description logics and their fuzzy versions have become a topic of increased interest due to possible applications in semantic web. First papers on fuzzy description logic worked with an oversimplified “minimalistic” fuzzy logic; in [7] I show how to reduce the question of satisfiability and validity of concepts (which is in fact a question of satisfiability/validity of a particular kind of formulas of first-order predicate logic) to satisfiability/validity of formulas of the corresponding propositional logic—uniformly for all logics given by continuous t-norms; the reduction gives then decidability of our questions. The paper [7] deals with a fuzzy variant of the description logic ALC over our t-norm logic L(∗) (∗ any continuous t-norm). Rational truth constants are not used there. But their use is natural in fuzzy description logic, in particular for giving lower and upper bounds of truth degrees of concepts ( → r¯ , r¯ → ). Here we just survey how to read the paper [7] using RL(∗) and RL(∗)∀ to get decidability for this generalized approach. Similarly as in [7] we disregard any particular axioms (facts and terminological axioms); but admitting them is an obvious variant of what is presented below.
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The language consists of some atomic concepts Ai (unary predicates), role names Rj (binary predicates) and some object constants (finitely many symbols). Concepts are built from atomic concepts and from rational truth constants r¯ using connectives &, → (and defined connectives ∧, ∨, ¬, ≡) and quantifier constructs ∀R.C, ∃R.C. Instances C(t) of concepts (t a variable or a constant) are defined in the obvious way, recalling that (∀R.C)(t) means (∀x)(R(t, x) → C(x)) and (∃R.C)(t) means (∃x)(R(t, x)&C(x)). One works with witnessed models of L(∗)∀ – and now of RL(∗)∀. A model M is witnessed if for each formula (∀x)(x, y) and each tuple b of elements of M there is an a ∈ M with (∀x)(x, b)∗M = (a, b)∗M and similarly for (∃x)(x, y). Clearly each finite model is witnessed. The algorithm in [7] assigns to each instance C(a) of a concept a finite theory T (C(a)) over predicate logic (now: RL(∗)∀) that introduces witnessing constants and corresponding witnessing axioms for relevant quantified formulas determined by C(a). One converts T (C(a)) into a finite propositional theory by replacing each atomic formula of the form A(b) or R(b, c) and each instance of a quantified concept (∀R.C)(b) or (∃R.C)(b) by a propositional variable. The resulting theory is called prop(T (C(a))); the propositional variable assigned to C(a) is denoted pc(a) . 2 Now one proves exactly as in [7] that for any continuous t-norm ∗, each propositional RL(∗)-model e of prop (T (C(a))) determines a finite RL(∗)∀-model M of T (C(a)) such that e∗ (pc(a)) = C(a)∗M and, conversely, each witnessed RL(∗)∀-model M of T (C(a)) determines a RL(∗)-model e of prop(T (C(a)) such that e∗ (pc(a)) = C(a)∗M . Since the size of prop(T (C(a))) may be exponential in the size of C(a), the results on the complexity of RL(∗) do not immediately give results on the complexity of our description logic – denote it by R-ALC(*); but we do have the following (cf. Theorems 2–4 of [7]): Theorem 8. For any (r-admissible) continuous t-norm ∗ and any R-ALC∗-concept C, the following are equivalent: (i) C is satisfiable by a witnessed RL(∗)∀-model, (ii) C is satisfiable by a finite RL(∗)∀-model, (iii) prop(T (C(a))) ∪ {pC(a) } is RL(∗)-satisfiable. Theorem 9. For any (r-admissible) continuous t-norm ∗ and any R-ALC∗-concept C, the following are equivalent: (i) C is valid in all witnessed RL(∗)∀-models, (ii) C is valid in all finite RL(∗)∀-models, (iii) prop(T (C(a))) RL(∗)-entails pC(a) . Corollary 1. For any strongly r-admissible continuous t-norm ∗, the problem of witnessed satisfiability of R-ALC∗concepts is decidable and so is the problem of their validity. References [1] M. Baaz, P. Hájek, F. Montagna, H. Veith, Complexity of t-tautologies, -Ann. Pure Appl. Logic 113 (2002) 3–11. [2] J. Canny, Some algebraic and geometric computations in PSPACE, In Proc. 20th Annu. ACM Symp. on Theory of Computing, 1988, pp. 460–467. [3] P. Cintula, From Fuzzy Logic to Fuzzy Mathematics. Thesis, Czech Technical University, Prague, 2005. [4] F. Esteva, L. Godo, C. Noguera, On rational weak nilpotent minimum logics, J. Multiple-valued Logic Soft Comput. (short version in Proc. ESTYLF 2004, Jaén, Spain), to appear. [5] S. Gottwald, P. Hájek, Triangular norm based mathematical fuzzy logics, In: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, Elsevier, Amsterdam, 2005, pp. 275–299. [6] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [7] P. Hájek, Making fuzzy description logic more general, Fuzzy Sets and Systems 154 (2005) 1–15. [8] Z. Haniková, A note on the complexity of propositional tautologies of individual t-algebras, Neural Network World 5 (2002) 453–460. [9] V. Novák, I. Perfilieva, J. Moˇckoˇr, Mathematical Principles of Fuzzy Logic, Kluwer, Dordrecht, 2000. [10] J. Pavelka, On fuzzy logic I, II, III. Z. Math. Logik Grundlagen Math. 25 (1979) 45–52, 119–134, 447–464. [11] P. Savický, R. Cignoli, F. Esteva, L. Godo, C. Noguera, On product logic with truth constants. J. Logic Comput., to appear.
2 In [7], when defining prop(T (C(a))), the propositional variables p R(b,c) assigned to atomic formulas R(b, c) are not explicitly introduced by an overlook, but are clear from the context.
Computational complexity of t-norm based propositional fuzzy logics with rational truth constants夡 Petr Hájek∗ Institute of Computer Science, Academy of Sciences of the Czech Republic, 182 07 Prague, Czech Republic Received 26 July 2005; received in revised form 5 December 2005; accepted 12 December 2005 Available online 9 January 2006
Abstract If a continuous t-norm on [0, 1] maps pairs of rationals into rationals then the corresponding fuzzy propositional calculus can be extended by rational truth constants and “bookkeeping” axioms for them. (Łukasiewicz t-norm is the classical example.) Computational complexity of such logics is studied. Consequences for fuzzy description logic are formulated. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy logic; Rational constants; Computational complexity; Description logic
1. Introduction The reader is assumed to be familiar with continuous t-norms and the corresponding fuzzy propositional logics (see the monograph [6] and/or the recent detailed survey [5]), in particular Łukasiewicz, Gödel and product t-norm (denoted Ł, G, ) and the representation of each continuous t-norm as an ordered sum of copies of these three ones (Mostert-Shields). For each continuous t-norm ∗ the corresponding propositional logic L(∗) has ∗ as the standard truth function of conjunction & and the residuum ⇒ of ∗ as the truth function of implication →. Defined connectives are ¬, ∧, ∨, ≡; 0¯ is the truth constant for 0, 1¯ for 1. A propositional formula is a (standard) ∗-tautology if it has the value 1 for each evaluation of propositional variables by elements of [0, 1], when computed using the truth functions given by ∗. The definition of to be (standardly) ∗-satisfiable is evident. (In this paper we do not work with general semantics given by BL-algebras.) 1 For each ∗, the set of ∗-tautologies is coNP-complete and the set of all ∗-satisfiable formulas in NP-complete (first proved for Łukasiewicz t-norm by Mundici; the final result and references to intermediate results is in [8]). The idea to work explicitly with truth constants r¯ for r ∈ [0, 1] was first elaborated by Pavelka [10] in his logic which is in fact Łukasiewicz logic extended by r for all r ∈ [0, 1]. If turned out to be reasonable to work with r¯ for rational elements of [0, 1], keeping the language countable. (For axioms see below.) Note that Łukasiewicz logic extended by 夡 This work was partially supported by the project ITI, No. 1M0021620808 of MŠMT and partly by the Institutional Research Plan AV0Z10300504. Thanks are due to Z. Haniková for important comments. ∗ Tel.: +42 2 6884 244; fax: +42 2 8585 789. E-mail address: [email protected]. 1 More generally, for each set D ⊆ [0, 1] of designated values we can define D-∗-tautologies and D-∗-satisfiable formulas; ∗-tautologies are then {1}-∗-tautologies etc. For D being the half-open interval (0, 1] we speak on positive ∗-tautologies and positively ∗-satisfiable formulas.
0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.12.008
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truth constants has been systematically developed by Novák and his group, see e.g. [9]. The fact that the residuum of Łukasiewicz t-norm is continuous makes the so-called (strong) Pavelka-style completeness theorem possible: truth degree of a formula over a theory equals its provability degree in that theory), see e.g. [9,6]. In the sequel, given a continuous t-norm ∗, we denote by RL(∗) the extension of L(∗) by adding the truth constants r¯ for each rational r ∈ [0, 1] to the language, declaring that r¯ denotes just r. To have it well-behaved, we have to restrict our attention to continuous t-norms such that the set Q ∩ [0, 1] of all rational elements of [0, 1] is closed under ∗ and under ⇒ . We shall call such t-norms r-admissible (admissible w.r.t. rational numbers). Given an axiom system for L(∗), (∗ r-admissible) the corresponding axiom system for RL(∗) results by adding the “bookkeeping” axioms for all rational r, s ∈ [0, 1] : (¯r &¯s ) ≡ r ∗ s, (¯r → s¯ ) ≡ r ⇒ s, Note that for ∗ being Ł (Łukasiewicz t-norm) we have (usual, standard) completeness for tautologies: is a tautology of RL(Ł) iff RL(Ł) (see [6]). Recently the completeness for tautologies was proved for Gödel t-norm [4] and for product t-norm [11]. (See also some results on RL(∗) in [3].) This has increased the interest in t-norm based logics with rational truth constants, different from RL(Ł). In this paper, we investigate the computational complexity of various logics RL(∗) (∗ r-admissible). We show RL(Ł) and RL(G) (Łukasiewicz, Gödel) to have the set of tautologies coNP-complete and the set of satisfiable formulas NPcomplete; for the product t-norm we are only able to show that both the set of tautologies and the set of satisfiable formulas of RL() is in PSPACE (better results being not excluded). We also get results on finite semantic consequence (whenever 1 , . . . , n are true then is also true). Then we discuss r-admissible isomorphic copies of Ł, G, , radmissible t-norms having finitely many components and give an example of an r-admissible ∗ with infinitely many components such that tautologies of RL(∗) are very heavily undecidable (form a non-arithmetical set). Finally we comment on consequences of our results for fuzzy description logic.
2. Łukasiewicz, Gödel and product with rational constants Definition 1. For any r-admissible ∗, TAUT (RL(∗)) denotes the set of all tautologies of RL(∗), similarly for SAT (RL(∗)) and satisfiable formulas; furthermore, SCONS(RL(∗)) is the set of all pairs (, ) such that is a semantic consequence of in the sense of RL(∗). Theorem 1. For Łukasiewicz t-norm Ł, TAUT (RL(Ł)) and SCONS(RL(Ł)) are coNP-complete and SAT (RL(Ł)) is NP-complete. Proof. Clearly the first two sets are coNP-hard and the third is NP-hard since so are the corresponding sets for Łukasiewicz logic without rational constants. To show that TAUT (RL(Ł)) is in coNP (equivalently that the set posSAT (RL(Ł)) of all positively satisfiable formulas of RL(Ł) is in N P ) and that SAT (RL(Ł)) is in NP just inspect [6] 6.2.18–6.2.20 (and 6.1.8) to reduce ∈ SAT (RL(Ł)), ∈ posSAT (RL(Ł)) to a particular mixed integer problem; the only thing to be added is the replacement of e(¯r ) i by r i and e(¯r ) i by r i. Then everything works. ¯ i0 , i0 > 0. For testing (, ) ∈ / SCON S start with e()1, e( → 0) Theorem 2. For Gödel t-norm G, TAUT (RL(G)) and SCONS(RL(G)) is co-NP complete and SAT (RL(G)) is NPcomplete. Proof. Given a formula , let all rational truth constants occurring in be among 0 = r0 < r1 < · · · < rm−1 < rm = 1. Clearly we may assume ri = i/m, i = 0, . . . , m. If contains n propositional variables then for each i = 0, . . . , m − 1 choose n rational elements ai1 , . . . , ain strictly between ri , ri+1 . The formula is in SAT (RL(G)) iff you may guess an evaluation of p1 , . . . , pn taking values in {r0 , . . . , rm } ∪ m−1 i=0 {ai1 , . . . , ain } giving the formula the value 1. is not in TAUT (RL(G)) iff you can guess such an evaluation giving the formula a value less than 1. Similarly for SCONS.
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Now we turn to RL(). We use the result by Canny [2] saying that the existential theory of real numbers can be decided in P SP ACE. In more details, consider the predicate language with predicates =, <, , function symbols +, · (binary), − (unary) and constants 0, 1. Given an open (quantifier free) formula (x, . . . , y), the problem is if is satisfiable in the structure of real numbers, i.e. if (∃x, . . . , y)(x, . . . , y) is true in this structure. By Canny’s result the problem is in PSPACE. Our aim is to reduce (in P time) the tautology and satisfiability problem of RL() to the former problem. This will be done as follows: Observe that if lh() = n (length) then the number of subformulas of is n. For each such subformula of introduce an object variable z and for each such which is not a propositional variable form a definition of z as follows: if is r¯ where r = h/k (h, k integers coded as binary numbers, i.e. sequences of zeros and ones) then def (z ) is k¯ · z = h¯ where for h = 0, 1, h¯ is represented in the theory of real numbers as a sum of powers of 2 (i.e. of (1 + 1)) corresponding to digits 1, where for i > 0, 2i is 2 · 2 · . . . 2, i factors; thus e.g. 1101 becomes 2.2.2 + 2.2 + 1. If is 1 &2 then def (x ) is z = z1 · z2 ; if is 1 → 2 then def (z ) is (z1 z2 & z = 1)∨ (z1 > z2 & z · z1 = z2 ). Finally, let ˆ be {def ()| subformula of , not a variable}. The length of ˆ is quadratic in the length of . Moreover, • ∈ SAT (RL()) iff &(z ˆ = 1) is satisfiable in the reals, • ∈ / TAUT (RL()) iff &(z ˆ < 1) is satisfiable in the reals, ˆ • (, ) ∈ / SCONS(RL()) iff & ˆ &(z = 1)&(z < 1) is satisfiable in the reals. Thus we have the following: Theorem 3. For the product t-norm , the sets TAUT (RL()), SAT (RL()), SCONS(RL()) are in PSPACE. 3. Other r-admissible t-norms Each increasing one–one mapping f of [0, 1] onto itself can be used to transform a given continuous t-norm ∗ to a new continuous t-norm ∗f defined as x ∗f y = f −1 (f (x) ∗ f (y)); thus f becomes an isomorphism of ([0, 1], ∗, ⇒) and ([0, 1], ∗f , ⇒f ) where ⇒f is the residuum of ∗f and satisfies x ⇒f u = f −1 (f (x) ⇒ f (y)). Call f r-admissible if it maps rationals to rationals; if f is an r-admissible mapping and ∗ is an r-admissible t-norm then ∗f is also an r-admissible t-norm (both continuous). Clearly, L(∗) and L(∗f ) have the same tautologies and satisfiables; but this evidently need not be true for RL(∗) and RL(∗f ) since in RL(∗), r¯ &¯s ≡ r ∗ s is a tautology whereas in RL(∗f ) the corresponding tautology is r¯ &¯s ≡ r ∗f s. (Observe that ∗ is the same as ∗f iff ∗ is Gödel t-norm, i.e. minimum.) A particular kind of an r-admissible mapping f is an increasing piecewise linear mapping of [0, 1] onto itself, consisting of finitely many linear pieces, the linear pieces having rational coefficients, e.g. f having a linear piece between (0, 0) and ( 21 , 41 ) and another linear piece between ( 21 , 41 ) and (1, 1). Trivially, both f and f −1 are P (deterministically polynomially computable) for rational arguments. Definition 2. An r-admissible mapping of [0, 1] into itself is strongly r-admissible if both f and f −1 are in P when restricted to rational arguments. Theorem 4. Assume TAUT (RL(∗)) is co-NP complete and SAT (RL(∗)) is NP-complete (e.g. ∗ can be Łukasiewicz). Let f be a strongly r-admissible mapping of [0, 1] onto itself. Then TAUT (RL(∗f )) is co-NP complete and SAT (RL(∗f )) is NP-complete. Proof. This is easily seen from the fact that for each evaluation e of propositional variables and each formula (p1 , . . . , r¯1 , . . .), f (e∗ ((p1 , . . . , r¯1 , . . .)) = e∗ f ((p1 , . . . , f (r1 ), . . .)) e (pi ) = f (e(pi )). Thus (p1 , . . . , r¯1 , . . .) is tautologous/satisfiable is RL(∗) iff so is (p1 , . . . , f (r1 ), . . .) in RL(∗f ). Thus TAUT (RL(∗)) and TAUT (RL(∗f )) are P-reducible to each other and the same for SAT (and for SCON S).
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Similarly we get the following: Theorem 5. Assume that TAUT (RL(∗)) and SAT (RL(∗)) are PSPACE and let f be an r-admissible mapping such that both f and f −1 are PSPACE-computable. Then TAUT (RL(∗f )) and SAT (RL(∗f )) are PSPACE (and similarly for SCONS). Now let us turn to continuous t-norms that are ordinal sums of finitely many components, i.e. ∗ = ∗1 ⊕ · · · ⊕ ∗n where each ∗i is Ł, G or (but there are no two consecutive G’s). Thus there are 0 = d0 < d1 < · · · < dn = 1 such that ∗ restricted to [di , di+1 ] is isomorphic to Ł, G or . The di ’s will be called endpoints of the components. Lemma 1. Let ∗ be an r-admissible t-norm with n components. Then the left endpoint of each Ł-component is rational.
Proof. The left endpoint di corresponds to 0 and Ł is nilpotent; thus take any rational x between the endpoints, then for a sufficiently large natural k, x k = x ∗ · · · ∗ x = di , hence di is rational. Note that the right point may be irrational: let 0 < e < 1, e irrational, let 0 = a0 , a1 , . . . be an increasing sequence of rationals with lim an = 1 and 0 = b0 , b1 , . . . an increasing sequence of rationals with lim bn = e. Let f be a 1-1 increasing mapping of [0, 1] onto [0, e] which maps each [ai , ai+1 ] linearly to [bi , bi+1 ], thus sending all rationals 0 x < 1 on rationals 0 f (x) < e. Use f to induce the continuous t-norm ∗ which is isomorphic to Łukasiewicz on [0, e] and to Gödel on [e, 1]. Clearly the t-norm is r-admissible. We restrict ourselves to r-admissible t-norms with finitely many components, all having rational endpoints. Definition 3. (1) Let e < e be two rationals from [0, 1]. A bijection of [0, 1] to [e, e ] is strongly r-admissible if f maps rational elements of [0, 1] onto rational elements of [e, e ] and both f and f −1 restricted to rational arguments are P-computable. (2) Let ∗ be an r-admissible t-norm with n components, ∗ is strongly r-admissible if each its Ł-component is isomorphic to [0, 1]Ł via a strongly r-admissible bijection and the same for each -component and [0, 1] . (Nothing needs to be assumed on G-components.) Theorem 6. If ∗ is a strongly r-admissible t-norm with finitely many components, having no -component then TAUT (RL(∗)) is co-NP-complete, SAT (RL(∗)) is NP-complete and SCONS(RL(∗)) is co-NP-complete. Proof. The proof is got by analyzing Haniková [8] Section 2. Let 0 = r0 < r1 < · · · < rn = 1 list all rationals occurring in a given formula or being endpoints of components of ∗. For each i = 0, . . . , n introduce variables yi0 , . . . , yim (where m is the length of ) and postulate ri = yi0 < yi1 < · · · < yim = ri+1 . Guess assignment: Choose indeterministically a mapping g assigning to each subformula of a variable yij such that g() = ynm (= rn = 1). Check external: Let ri be left endpoint of a component; recall g(ri ) = yi0 . Further let f (1 ) = u, f (2 ) = v, 0 i n − 1. Check: • if 1 &2 is a subformula of and u yi0 v then f (1 &2 ) = u; • if 1 → 2 is a subformula of and uv then f (1 → 2 ) = 1 = rn ; • if 1 → 2 is a subformula of and v < ri u then f (1 → 2 ) = u. Check internal: For each component [ri , rj ] (i < j ) of ∗ check behavior of g in the segment. Whenever g(1 ) = u, g(2 ) = v with ri u, v rj do the following: check g(1 &2 ) = w ∈ [ri , rj ] and produce the equation w = u ∗ v; if u > v then check g(1 → 2 ) = w ∈ [ri , rj ] and produce the equation w = u ⇒ v. The produced equations together with the inequations yk0 < · · · < ykm for i k < j form the system belonging to [i, j ]. For [i, j ] being a G-component replace w = u ∗ v by w = min(u, v) and w = u ⇒ v by w = v. Check satisfiability of the system by taking an arbitrary (fixed) chain of rational values from [0, 1] whose cardinality is the number of y-variables from [ri , rj ] and guessing evaluation q of these variables in the chain sending yi0 to 0 and yj 0 to 1. Verify validity of the system belonging to [i, j ]. If successful, map [0, 1] to [ri , rj ] by an increasing bijection sending g(yk,0 ) to rk for i k j .
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For [i, j ] being a Ł-component and fi the corresponding strongly r-admissible bijection of [0, 1] onto [ri , rj ] then check the system belonging to [i, j ] extended by equations yk,0 = fi−1 (rk ) for i k j in the sense of Łukasiewicz connectives. This can be done by a reduction to a MIP-problem like above (cf. also [1]). This ends the definition of a non-deterministic polynomial algorithm; it succeeds iff is RL(∗) – satisfiable. This proves SAT (RL(∗)) being in NP (and clearly it is NP-hard). Modifying the algorithm by demanding at the beginning g() < ynm (instead of = ynm ) we get an algorithm succeeding iff is not a RL(∗)-tautology; hence TAUT (RL(∗)) is co-NP-complete. Similarly for SCONS. For t-norms with some -components we get the following: Theorem 7. If ∗ is a strongly r-admissible t-norm with finitely many components then SAT (RL(∗)), TAUT (RL(∗)) and SCONS(RL(∗)) are in PSPACE. Proof. Everything as in the preceding theorem, only add the case of [ri , rj ] being a -component: then take the corresponding system of equalities and inequalities extended by yk , 0 = fk−1 (rk ) for i k j where fi is the strongly r-admissible bijection of [0, 1] to [ri , rj ] from the definition of ∗ being strongly r-admissible) and reduce the satisfiability of this system to the satisfiability of a corresponding open formula in the reals as in the proof concerning RL(). (The case of an Ł-component may be handled as in the previous theorem or also by translating into an open formula in the language of reals.) Details are left to the reader. To close this section we give an example of an r-admissible continuous t-norm with infinitely many components, all being Ł, which is as undecidable as you want. Example 1. Let A be any set of positive natural numbers. Let ∗ be a continuous t-norm whose idempotents are 0 and all n1 , n ∈ A; if n1 < n2 are two neighbor elements of A (i.e. there is no n ∈ A with n1 < n < n2 ) then on [1/n2 , 1/n1 ] ∗ is isomorphic to [0, 1]Ł by the obvious linear bijection of [0, 1] to [1/n2 , 1/n1 ]. Then ∗ is r-admissible; moreover, n ∈ A iff the formula 1/n ∗ 1/n = 1/n is in TAUT (RL(∗)) iff this formula is in SAT (RL(∗)). Therefore A is recursively reducible both to TAUT (RL(∗)) and to SAT (RL(∗)); hence the Turing degrees of the two last sets is bigger than or equal to the Turing degree of A. In particular there is an r-admissible ∗ such that both TAUT (RL(∗)) and SAT (RL(∗)) is not arithmetical. (But recall that TAUT (L(∗)) is co-NP-complete and SAT (L(∗)) is NP-complete by [8].) Admittedly our results isolate “reasonable” continuous t-norms for which adding rational constants either keeps the computational complexity equal to the complexity of the logic without those constants or shows a reasonable upper bound (PSPACE). Several problems remain open.
4. Consequences for fuzzy description logic Description logics and their fuzzy versions have become a topic of increased interest due to possible applications in semantic web. First papers on fuzzy description logic worked with an oversimplified “minimalistic” fuzzy logic; in [7] I show how to reduce the question of satisfiability and validity of concepts (which is in fact a question of satisfiability/validity of a particular kind of formulas of first-order predicate logic) to satisfiability/validity of formulas of the corresponding propositional logic—uniformly for all logics given by continuous t-norms; the reduction gives then decidability of our questions. The paper [7] deals with a fuzzy variant of the description logic ALC over our t-norm logic L(∗) (∗ any continuous t-norm). Rational truth constants are not used there. But their use is natural in fuzzy description logic, in particular for giving lower and upper bounds of truth degrees of concepts ( → r¯ , r¯ → ). Here we just survey how to read the paper [7] using RL(∗) and RL(∗)∀ to get decidability for this generalized approach. Similarly as in [7] we disregard any particular axioms (facts and terminological axioms); but admitting them is an obvious variant of what is presented below.
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The language consists of some atomic concepts Ai (unary predicates), role names Rj (binary predicates) and some object constants (finitely many symbols). Concepts are built from atomic concepts and from rational truth constants r¯ using connectives &, → (and defined connectives ∧, ∨, ¬, ≡) and quantifier constructs ∀R.C, ∃R.C. Instances C(t) of concepts (t a variable or a constant) are defined in the obvious way, recalling that (∀R.C)(t) means (∀x)(R(t, x) → C(x)) and (∃R.C)(t) means (∃x)(R(t, x)&C(x)). One works with witnessed models of L(∗)∀ – and now of RL(∗)∀. A model M is witnessed if for each formula (∀x)(x, y) and each tuple b of elements of M there is an a ∈ M with (∀x)(x, b)∗M = (a, b)∗M and similarly for (∃x)(x, y). Clearly each finite model is witnessed. The algorithm in [7] assigns to each instance C(a) of a concept a finite theory T (C(a)) over predicate logic (now: RL(∗)∀) that introduces witnessing constants and corresponding witnessing axioms for relevant quantified formulas determined by C(a). One converts T (C(a)) into a finite propositional theory by replacing each atomic formula of the form A(b) or R(b, c) and each instance of a quantified concept (∀R.C)(b) or (∃R.C)(b) by a propositional variable. The resulting theory is called prop(T (C(a))); the propositional variable assigned to C(a) is denoted pc(a) . 2 Now one proves exactly as in [7] that for any continuous t-norm ∗, each propositional RL(∗)-model e of prop (T (C(a))) determines a finite RL(∗)∀-model M of T (C(a)) such that e∗ (pc(a)) = C(a)∗M and, conversely, each witnessed RL(∗)∀-model M of T (C(a)) determines a RL(∗)-model e of prop(T (C(a)) such that e∗ (pc(a)) = C(a)∗M . Since the size of prop(T (C(a))) may be exponential in the size of C(a), the results on the complexity of RL(∗) do not immediately give results on the complexity of our description logic – denote it by R-ALC(*); but we do have the following (cf. Theorems 2–4 of [7]): Theorem 8. For any (r-admissible) continuous t-norm ∗ and any R-ALC∗-concept C, the following are equivalent: (i) C is satisfiable by a witnessed RL(∗)∀-model, (ii) C is satisfiable by a finite RL(∗)∀-model, (iii) prop(T (C(a))) ∪ {pC(a) } is RL(∗)-satisfiable. Theorem 9. For any (r-admissible) continuous t-norm ∗ and any R-ALC∗-concept C, the following are equivalent: (i) C is valid in all witnessed RL(∗)∀-models, (ii) C is valid in all finite RL(∗)∀-models, (iii) prop(T (C(a))) RL(∗)-entails pC(a) . Corollary 1. For any strongly r-admissible continuous t-norm ∗, the problem of witnessed satisfiability of R-ALC∗concepts is decidable and so is the problem of their validity. References [1] M. Baaz, P. Hájek, F. Montagna, H. Veith, Complexity of t-tautologies, -Ann. Pure Appl. Logic 113 (2002) 3–11. [2] J. Canny, Some algebraic and geometric computations in PSPACE, In Proc. 20th Annu. ACM Symp. on Theory of Computing, 1988, pp. 460–467. [3] P. Cintula, From Fuzzy Logic to Fuzzy Mathematics. Thesis, Czech Technical University, Prague, 2005. [4] F. Esteva, L. Godo, C. Noguera, On rational weak nilpotent minimum logics, J. Multiple-valued Logic Soft Comput. (short version in Proc. ESTYLF 2004, Jaén, Spain), to appear. [5] S. Gottwald, P. Hájek, Triangular norm based mathematical fuzzy logics, In: Logical, Algebraic, Analytic and Probabilistic Aspects of Triangular Norms, Elsevier, Amsterdam, 2005, pp. 275–299. [6] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [7] P. Hájek, Making fuzzy description logic more general, Fuzzy Sets and Systems 154 (2005) 1–15. [8] Z. Haniková, A note on the complexity of propositional tautologies of individual t-algebras, Neural Network World 5 (2002) 453–460. [9] V. Novák, I. Perfilieva, J. Moˇckoˇr, Mathematical Principles of Fuzzy Logic, Kluwer, Dordrecht, 2000. [10] J. Pavelka, On fuzzy logic I, II, III. Z. Math. Logik Grundlagen Math. 25 (1979) 45–52, 119–134, 447–464. [11] P. Savický, R. Cignoli, F. Esteva, L. Godo, C. Noguera, On product logic with truth constants. J. Logic Comput., to appear.
2 In [7], when defining prop(T (C(a))), the propositional variables p R(b,c) assigned to atomic formulas R(b, c) are not explicitly introduced by an overlook, but are clear from the context.