Gödel FL_0 with Greatest Fixed-Point Semantics

Gödel FL

with Greatest Fixed-Point Semantics

Stefan Borgwardt¹, José A. Leyva Galano¹, and Rafael Peñaloza^1,2

1 Theoretical Computer Science, TU Dresden, Germany

2 Center for Advancing Electronics Dresden

{stefborg,penaloza}@tcs.inf.tu-dresden.de jleyva1@gmail.com

Abstract. We study the fuzzy extension ofFL0 with semantics based on the Gödel t-norm. We show that gfp-subsumption w.r.t. a finite set of primitive definitions can be characterized by a relation on weighted automata, and use this result to provide tight complexity bounds for reasoning in this logic.

1 Introduction

Fuzzy Description Logics (DLs) have been introduced as extensions of classical DLs [2] capable of representing and reasoning with vague or imprecise knowledge.

The main idea behind these logics is to allow for a set of truth degrees, beyond the standardtrueandfalse. The area of fuzzy DLs recently experienced a shift, when it was shown that reasoning in these logics easily becomes undecidable [3,6,8].

To guarantee decidability in fuzzy DLs, one can (i) restrict the semantics to consider finitely many truth degrees [7]; (ii) allow only acyclic or unfoldable ontologies [4,18]; or (iii) restrict to Zadeh or Gödel semantics [5,15,16,17].

In the cases where the Gödel t-norm is used, the complexity of reasoning is typically the same as for its classical version, as shown for EL, which is poly- nomial [15,16], andALC,ExpTime-complete [5]. This latter result immediately implies that reasoning inG-FL0with general TBoxes is alsoExpTime-complete.

On the other hand, if TBoxes are restricted to contain only (primitive) defini- tions, then deciding subsumption in classicalFL0under the greatest fixed-point semantics is known to be in PSpace [1]. We show that the same complexity bound holds for the Gödel extension of this logic.

To prove this complexity result, we characterize the greatest fixed-point se- mantics ofG-FL₀by means of weighted automata. We then show that reasoning with these automata can be reduced to a linear number of inclusion tests between unweighted automata, which can be solved using only polynomial space [10].

?Partially supported by the DFG under grant BA 1122/17-1, in the research train- ing group 1763 (QuantLA), and the Cluster of Excellence ‘Center for Advancing Electronics Dresden’.

2 Preliminaries

We first introduce some basic notions of lattice theory. For a more comprehensive overview on the topic, refer to [11]. Afterwards, we introduce fuzzy logics based on Gödel semantics, which are studied in more detail in [9,12,14].

Alattice is an algebraic structure(L,∨,∧)with two commutative, associa- tive and idempotent binary operations ∨ (supremum) and ∧ (infimum) that distribute over each other. It iscompleteif suprema and infima of arbitrary sub- setsS ⊆L, denoted byW

x∈SxandV

x∈Sxrespectively, exist. In this case, the lattice is bounded by the greatest element 1 := W

x∈Lxand the least element 0 := V

x∈Lx. Lattices induce a natural partial ordering on the elements of L wherex≤y iffx∧y=x.

Example 1. One common complete lattice used in fuzzy logics (see e.g. [9,12]) is the interval[0,1]with the usual order on the real numbers. Further complete lattices relevant for this paper can be constructed as follows. Given a complete lattice Land a set S, the setL^S of all functionsf:S →L is also a complete lattice, if infimum and supremum are defined component-wise. More precisely, for any twof1, f2∈L^S, we definef1∨f2for allx∈Sas(f1∨f2)(x) :=f1(x)∨f2(x).

If we similarly define the infimum, we obtain a lattice with the order f1 ≤ f2

ifff1(x)≤f2(x)holds for allx∈S. It is easy to verify that infinite infima and suprema can then also be computed component-wise.

We are particularly interested in operators on complete lattices L and their properties.

Definition 2 (fixed-point). Let L be a complete lattice. A fixed-point of an operator T:L→L is an element x∈L such thatT(x) =x. It is the greatest fixed-point ofT if for any fixed-point y ofT we have y≤x.

The operatorT is monotone if for allx, y∈L,x≤y impliesT(x)≤T(y).

It is downward ω-continuousif for every decreasing chain x₀ ≥x₁≥x₂≥. . . inL we haveT(V

i≥0x_i) =V

i≥0T(x_i).

If it exists, the greatest fixed-point ofT is unique and denoted bygfp(T).

It is easy to verify that every downwardω-continuous operator is also mono- tone. By a fundamental result from [20], every monotone operatorThas a great- est fixed-point. If T is downwardω-continuous, then gfp(T)corresponds to the infimum of the decreasing chain1≥T(1)≥T(T(1))≥ · · · ≥Tⁱ(1)≥. . . [13].

Proposition 3. If Lis a complete lattice andT a downward ω-continuous op- erator on L, thengfp(T) =V

i≥0Tⁱ(1).

Our fuzzy DL is based on the well-known Gödel semantics for fuzzy logics, which is one of the main t-norm-based semantics used in Mathematical Fuzzy Logic [9,12]. This semantics is based on the standard interval[0,1]. The Gödel t-norm is the binary minimum operator on this set. For consistency, we use the lattice-theoretic notation ∧ instead of min. Two important properties of

this operator are that it preserves arbitrary infima and suprema on [0,1], i.e.

V

i∈I(x_i∧x) = V

i∈Ix_i

∧xand W

i∈I(x_i∧x) = W

i∈Ix_i

∧xfor any index set I and elements x, x_i ∈ [0,1] for all i ∈ I. In particular, this means that the Gödel t-norm is monotone in both arguments. The residuum of the Gödel t-norm is the binary operator⇒on[0,1]defined for all x, y∈[0,1]by

x⇒y:=

(1 ifx≤y, y otherwise.

It is a fundamental property of a t-norm and its residuum that for all values x, y, z ∈[0,1]we have x∧y ≤z iff y ≤x⇒z. As with the Gödel t-norm, its residuum preserves arbitrary infima in its second component. However, in the first component the order on[0,1]is reversed.

Proposition 4. For any index setI and valuesx, xi∈[0,1],i∈I, we have x⇒ ^

i∈I

xi

=^

i∈I

(x⇒xi) and _

i∈I

xi

⇒x=^

i∈I

(xi⇒x).

This shows that the residuum is monotone in the second argument and antitone in the first argument. The following reformulation of nested residua in terms of infima will also prove useful.

Proposition 5. For all valuesx, x1, . . . , xn∈[0,1], we have (x₁∧ · · · ∧x_n)⇒x

= x₁⇒. . .(x_n⇒x). . . .

Proof. Both values are eitherxor 1, and they are1 iff one of the operandsxi, 1≤i≤n, is smaller than or equal tox. ut

3 Fuzzy FL

The fuzzy description logic G-FL0 has the same syntax as classical FL0. The difference lies in the interpretation ofG-FL0-concepts.

Definition 6 (syntax).LetNCandNR be two non-empty, disjoint sets of con- cept namesand role names, respectively.Conceptsare built from concept names using the constructors>(top),CuD (conjunction), and∀r.C (value restriction for a role name r).

A(primitive concept) definitionis of the form hAvC≥pi, whereA∈NC, C is a concept, and p ∈ [0,1]. A TBox is a finite set of definitions. Given a TBox T, a concept name is defined if it appears on the left-hand side of a definition inT, and primitiveotherwise.

We use the expression∀w.Cwithw=r₁r₂. . . r_n ∈N^∗_Rto abbreviate the concept

∀r₁.∀r₂. . . .∀r_n.C. We also allow w = ε, in which case∀w.C is simply C. We denote the set of concept names occurring in the TBox T by N^T_C, the set of defined concept names in N^T_C by N^T_D, and the set of primitive concept names inN^T_C byN^T_P. Likewise, we collect all role names occurring inT into the setN^T_R.

Definition 7 (semantics). An interpretation is a pair I = (∆^I,·^I), where

∆^I is a non-empty set, called the domain of I, and the interpretation func- tion ·^I maps every concept name A to a fuzzy set A^I: ∆^I → [0,1] and every role name r to a fuzzy binary relation r^I: ∆^I×∆^I → [0,1]. This function is extended to concepts by setting>^I(x) := 1,(CuD)^I(x) :=C^I(x)∧D^I(x), and (∀r.C)^I(x) :=V

y∈∆^I(r^I(x, y)⇒C^I(y))for allx∈∆^I.

The interpretationI satisfies(or is a model of ) the definitionhAvC ≥pi if A^I(x) ⇒ C^I(x) ≥p holds for all x∈ ∆^I. It satisfies (or is a model of ) a TBox if it satisfies all its definitions.

For an interpretationI= (∆,·^I),w=r1r2. . . rn∈N^∗_R, and elementsx0, xn∈∆, we setw^I(x0, xn) :=W

x1,...,xn−1∈∆(r^I₁(x0, x1)∧ · · · ∧r^I_n(x_n−1, xn)), and can thus treat∀w.C like an ordinary value restriction with

(∀w.C)^I(x0) := ^

xn∈∆

(w^I(x0, xn)⇒C^I(xn))

= ^

x₁,...,x_n∈∆

r₁^I(x₀, x₁)∧ · · · ∧r^I_n(x_n−1, x_n)

⇒C^I(x_n)

= ^

x₁,...,x_n∈∆

r₁^I(x₀, x₁)⇒. . .(r^I_n(x_n−1, x_n)⇒C^I(x_n)). . .

= (∀r1. . . .∀rn.C)^I(x0) for allx0∈∆ (see Propositions 4 and 5).

It is convenient to consider TBoxes in normal form. The TBoxT is innormal form if all definitions in T are of the formhAv ∀w.B≥pi, whereA, B ∈NC, w ∈ N^∗_R, and p ∈ [0,1], and there are no two definitions hA v ∀w.B ≥ pi, hAv ∀w.B≥p⁰iwithp6=p⁰. Every TBox can be transformed into an equivalent TBox in normal form, as follows. First, we distribute the value restrictions over the conjunctions.

Lemma 8. For everyr∈NR, conceptsC, D, and interpretation I= (∆,·^I), it holds that (∀r.(CuD))^I= (∀r.Cu ∀r.D)^I.

Thus, we can equivalently write the right-hand sides of the definitions inT in the form∀w1.B1u · · · u ∀wn.Bn, wherewi ∈N^∗_RandBi∈NC∪ {>},1≤i≤n. Since

∀r.> is equivalent to >, we can remove all conjuncts of the form ∀w.> from this representation. After this transformation, all the definitions in the TBox are of the formhAv ∀w1.B1u · · · u ∀wn.Bn ≥piwith Bi ∈NC,1≤i≤n, or hAv > ≥pi. The latter axioms are tautologies, and can hence be removed from the TBox without affecting the semantics.

It follows from Proposition 4 that an interpretationI satisfies the definition hAv ∀w1.B₁u · · · u ∀wn.B_n ≥piiff it satisfies all the axiomshAv ∀wi.B_i≥pi, 1≤i≤n. Thus, the former axiom can be equivalently replaced by the latter set of axioms.

After these simplification steps, the TBox contains only axioms of the form hA v ∀w.B ≥ pi with A, B ∈ NC, satisfying the first condition of the defi- nition of normal form. Suppose now that T contains two axioms of the form

hAv ∀w.B≥piandhAv ∀w.B≥p⁰iwithp > p⁰. ThenT is equivalent to the TBoxT \ {hAv ∀w.B≥p⁰i}; which means that this axiom can be removed. It is clear that all of these transformations can be done in polynomial time in the size of the original TBox.

Concept definitions can be seen as a restriction of the interpretation of the defined concepts, depending on the interpretation of the primitive concepts. We use this intuition and considergreatest fixed-point semantics, as described next.

A primitive interpretation is a pair J = (∆,·^J) as in Definition 7, except that ·^J is only defined for role names and theprimitive concept names in N^T_P. Given such aJ, we use functionsf ∈([0,1]^∆)^NTD to describe the interpretation of the remaining (defined) concept names. Recall from Example 1 that these functions form a complete lattice. In the following, we use the abbreviation L^T_J := ([0,1]^∆)^NTD for this lattice. Given a primitive interpretation J and a functionf ∈L^T_J, theinduced interpretation I_J,f has the same domain asJ and extends the interpretation function ofJ to the defined concept names A∈N^T_D by taking A^IJ,f :=f(A). The interpretation of the remaining concept names, i.e. those that do not occur inT, is fixed to0.

We can describe the effect that the axioms inT have onL^T_J by the operator T_J^T:L^T_J →L^T_J, which is defined as follows for all f ∈L^T_J,A∈N^T_D, and x∈∆:

T_J^T(f)(A)(x) := ^

hAvC≥pi∈T

(p⇒C^IJ,f(x)).

This operator computes new values of the defined concept names according to the old interpretationI_J,f and their definitions inT.

We are interested in using the greatest fixed-point ofT_J^T, for some primitive interpretationJ, to define a new semantics for TBoxesT inG-FL0. Before being able to do this, we have to ensure that such a fixed-point always exists.

Lemma 9. Given a TBox T and a primitive interpretation J = (∆,·^J), the operator T_J^T on L^T_J is downward ω-continuous.

Proof. Consider a decreasing chainf0≥f1 ≥f2≥. . . of functions inL^T_J. We use the abbreviationsf :=V

i≥0fi,I:=IJ,f, andIi:=IJ,fi for alli≥0, and have to show thatT_J^T(f) =V

i≥0T_J^T(fi)holds.

First, we prove by induction on the structure ofCthatC^I =V

i≥0C^Ii holds for all conceptsC built from N^T_R andN^T_C, whereV

is defined as usual over the complete lattice[0,1]^∆.

ForA∈N^T_P, by the definition ofIJ,f andIJ,fi we haveA^I=A^J =A^Ii for alli≥0, and thusA^I=A^J =V

i≥0A^Ii. ForA∈N^T_D, we have A^I =f(A) = ^

i≥0

fi

(A) =^

i≥0

fi(A) =^

i≥0

A^Ii

by the definition of I_J,f and I_J,f_i and the component-wise ordering on the complete latticeL^T_J.

For concepts of the formCuD, by the induction hypothesis and associativity of∧we have

(CuD)^I=C^I∧D^I = ^

i≥0

C^Ii

∧ ^

i≥0

D^Ii

=^

i≥0

(C^Ii∧D^Ii) =^

i≥0

(CuD)^Ii.

Consider now a value restriction∀r.C. Using Proposition 4 we get for allx∈∆, (∀r.C)^I(x) = ^

y∈∆

(r^I(x, y)⇒C^I(y)) = ^

y∈∆

r^I(x, y)⇒ ^

i≥0

C^Ii(y)

= ^

y∈∆

^

i≥0

(r^Ii(x, y)⇒C^Ii(y)) = ^

i≥0

(∀r.C)^Ii(x) = ^

i≥0

(∀r.C)^Ii (x)

by the induction hypothesis, and the component-wise ordering on[0,1]^∆. Using this, we can now prove the actual claim of the lemma. For allA∈N^T_D and allx∈∆, we get, using again Proposition 4 and the previous claim

T_J^T(f)(A)(x) = ^

hAvC≥pi∈T

(p⇒C^I(x)) = ^

hAvC≥pi∈T

p⇒ ^

i≥0

C^Ii(x)

= ^

hAvC≥pi∈T

^

i≥0

(p⇒C^Ii(x)) = ^

i≥0

T_J^T(fi) (A)(x)

by the definition ofT_J^T, and the component-wise ordering onL^T_J. ut By Proposition 3, we know that gfp(T_J^T) exists and is equal to V

i≥0(T_J^T)ⁱ(1), where1is the greatest element of the latticeL^T_J that maps all defined concept names to >^J. In the following, we denote by gfp_T(J) the interpretationIJ,f

for f := gfp(T_J^T). Note that I := gfp_T(J) is actually a model of T since for every hAvC≥pi ∈ T and everyx∈∆ we have

A^I(x) =f(A)(x) =T_J^T(f)(A)(x) = ^

hAvC⁰≥p⁰i∈T

(p⁰ ⇒C^0I(x))≤p⇒C^I(x),

and thusp∧A^I(x)≤C^I(x), which is equivalent top≤A^I(x)⇒C^I(x).

We can now define the reasoning problem inG-FL₀that we want to solve.

Definition 10 (gfp-subsumption). An interpretationI is a gfp-model of a TBox T if there is a primitive interpretation J such thatI =gfp_T(J). Given A, B ∈ NC and p ∈ [0,1], we say that A is gfp-subsumed by B to degree p w.r.t. T, writtenT |=gfphAvB ≥pi, if for every gfp-modelI of T and every x∈∆^I we haveA^I(x)⇒B^I(x)≥p.

Let nowT be a TBox andT⁰ the result of transformingT into normal form as described before. It is easy to verify that the operatorsT_J^T andT_J^T0coincide, and therefore the gfp-models ofT are the same as those ofT⁰. To solve the problem of deciding gfp-subsumptions, it thus suffices to consider TBoxes in normal form.

4 Characterizing Subsumption using Finite Automata

To decide gfp-subsumption between concept names, we employ an automata- based approach following the ideas from [1]. In contrast to that paper, however, we use aweighted automata model.

Definition 11 (WWA).Aweighted automaton with word transitions (WWA) is a tupleA= (Σ, Q, q₀,wt, q_f), whereΣ is a finite alphabet of input symbols, Q is a finite set of states,q0 ∈Qis the initial state,wt:Q×Σ^∗×Q→[0,1]

is the transition weight function with the property that its support supp(wt) :={(q, w, q⁰)∈Q×Σ^∗×Q|wt(q, w, q⁰)>0}

is finite, andqf ∈Qis the final state.

A finite path in A is a sequence π = q0w1q1w2. . . wnqn, where qi ∈ Q and wi ∈ Σ^∗ for all i ∈ {1, . . . , n}, and qn = qf. Its label is the finite word

`(π) :=w1w2. . . wn. Theweightofπis defined aswt(π) :=Vn

i=1wt(q_i−1, wi, qi).

The set of all finite paths with label w inA is denoted bypaths(A, w). The be- havior kAk:Σ^∗ → [0,1] of A is defined as follows for every word w ∈ Σ^∗: kAk(w) :=W

π∈paths(A,w)wt(π).

If the image of the transition weight function is included in{0,1}, then we have a classical finite automaton with word transitions (WA). In this case,wtis usually described as a subset of Q×Σ^∗×Q and the behavior is characterized by the set L(A), called the language of A, of all words for which the behavior is 1.

Theinclusion problem for WA is to decide, given two such automataAandA⁰, whetherL(A)⊆L(A⁰). This problem is known to bePSpace-complete [10].

Our goal is to describe the restrictions imposed by aG-FL0 TBoxT using a WWA. For the rest of this paper, we assume w.l.o.g. thatT is in normal form.

Definition 12 (automata A^T_A,B). For concept names A, B ∈N^T_C, the WWA A^T_A,B= (NR,N^T_C, A,wt_T, B)is defined by the transition weight function

wt_T(A⁰, w, B⁰) :=

(p if hA⁰v ∀w.B⁰ ≥pi ∈ T, 0 otherwise.

Notice that for a given TBox T and concept names A, A⁰, B, B⁰ ∈ N^T_C, the automataA^T_A,BandA^T_A0,B⁰ differ only on the initial and final states they define;

their sets of states and transition weight function are identical. Since T is in normal form, for any two concept names A⁰, B⁰ ∈N^T_C and w∈N^∗_R, there is at most one axiomhA⁰v ∀w.B⁰≥piinT, and hence the transition weight function is well-defined. This function has finite support sinceT is finite.

We now characterize the gfp-models ofT by properties of the automataA^T_A,B. Lemma 13. For every gfp-model I= (∆,·^I)ofT,x∈∆, andA∈N^T_C,

A^I(x) = ^

B∈N^T_P

^

w∈N^∗_R

kA^T_A,Bk(w)⇒(∀w.B)^I(x) .

Proof. If A is primitive, then the empty path π = A ∈ paths(A^T_A,A, ε) has weightwtT(π) = 1, and hencekA^T_A,Ak(ε) = 1. We also have(∀ε.A)^I(x) =A^I(x);

thus, A^I(x) = (1⇒A^I(x))≥V

B∈N^T_P

V

w∈N^∗_R kA^T_A,Bk(w)⇒(∀w.B)^I(x) . Let now B ∈N^T_P andw ∈N^∗_R such thatA6=B or w6=ε. SinceA is primitive, by Definition 12 any finite pathπ in A^T_A,B with`(π) =wmust have weight 0; i.e.

kA^T_A,Bk(w) = 0, and thus 0 ⇒(∀w.B)^I(x) = 1≥A^I(x). This shows that the whole infimum is equal toA^I(x).

Consider now the case thatA∈N^T_D. Since I is a gfp-model ofT, there is a primitive interpretationJ such thatI =gfp_T(J); let f :=gfp(T_J^T). Thus, we have A^I=f(A) =T_J^T(f)(A) =V

i≥0(T_J^T)ⁱ(1)(A)for allA∈N^T_D.

[≤] For the≤-direction, by Proposition 4 it suffices to show that for allx∈∆, A∈N^T_D, B∈N^T_P, and all finite non-empty pathsπin A^T_A,B it holds that

A^I(x)≤wt_T(π)⇒(∀w.B)^I(x), (1) where w :=`(π). This obviously holds for wt_T(π) = 0, and thus it remains to show this for paths with positive weight. Let π = Aw1A1w2. . . wnAn, where Ai ∈N^T_C andwi ∈ N^∗_R for all i∈ {1, . . . , n} and An =B is the only primitive concept name in this path. We prove (1) by induction onn. Forn= 1, we have π=Aw1Bandwt_T(A, w1, B) =wt_T(π)>0, and thusT contains the definition hAv ∀w1.B≥pi, withp:=wtT(A, w1, B). By the definition ofT_J^T, we obtain

A^I(x) =T_J^T(f)(A)(x)≤p⇒(∀w1.B)^I(x) =wt_T(π)⇒(∀w.B)^I(x).

For n > 1, consider the subpath π⁰ = A₁w₂. . . w_nB in A^T_A

1,B with the label

`(π⁰) = w⁰ := w₂. . . w_n. For all y ∈ ∆, the induction hypothesis yields that A^I₁(y) ≤ wt_T(π⁰) ⇒ (∀w⁰.B)^I(y). Again, p:= wt_T(A, w1, A1) ≥ wt_T(π) > 0, and thusT contains the definitionhAv ∀w1.A1≥pi. By the definitions ofT_J^T, wt_T(π),w^I, and Propositions 4 and 5, we have

A^I(x) =T_J^T(f)(A)(x)

≤p⇒(∀w1.A1)^I(x)

= ^

y∈∆

p⇒(w^I₁(x, y)⇒A^I₁(y))

≤ ^

y∈∆

p⇒

w^I₁(x, y)⇒ wt_T(π⁰)⇒(∀w⁰.B)^I(y)

= p∧wt_T(π⁰)

⇒ ^

y∈∆

w₁^I(x, y)⇒(∀w⁰.B)^I(y)

=wtT(π)⇒(∀w.B)^I(x)

[≥] For the≥-direction, we show by induction onithat for allx∈∆,A∈N^T_D, andi≥0, it holds that

(T_J^T)ⁱ(1)(A)(x)≥ ^

B∈N^T_P

^

w∈N^∗_R

kA^T_A,Bk(w)⇒(∀w.B)^I(x)

. (2)

Fori= 0, we have(T_J^T)⁰(1)(A)(x) =1(A)(x) = 1, which obviously satisfies (2).

Fori >0, by Proposition 4 we obtain

(T_J^T)ⁱ(1)(A)(x) =T_J^T((T_J^T)ⁱ⁻¹(1))(A)(x)

= ^

hAv∀w⁰.A⁰≥pi∈T

(p⇒(∀w⁰.A⁰)^Ii−1(x)), (3)

whereI_i−1:=I_J,(TT

J)ⁱ⁻¹(1). Consider now any definitionhAv ∀w⁰.A⁰≥pi ∈ T. Thenπ⁰ =Aw⁰A⁰ is a finite path inA^T_A,A0 with label w⁰ and weightp.

IfA⁰ is a primitive concept name, then we have

p⇒(∀w⁰.A⁰)^Ii−1(x) =wt_T(π⁰)⇒(∀w⁰.A⁰)^I(x)≥ kA^T_A,A0k(w⁰)⇒(∀w⁰.A⁰)^I(x) by the definition of kA^T_A,A0k(w⁰) and the fact that the interpretation of∀w⁰.A⁰ under I_i−1 andI only depends onJ. IfA⁰ is defined, then we similarly get

p⇒(∀w⁰.A⁰)^Ii−1(x)

= ^

y∈∆

p⇒ w^0J(x, y)⇒A^0Ii−1(y)

≥ ^

y∈∆

^

B∈N^T_P

^

w∈N^∗_R

p⇒ w^0I(x, y)⇒(kA^T_A0,Bk(w)⇒(∀w.B)^I(y))

= ^

B∈N^T_P

^

w∈N^∗_R

p∧ kA^T_A0,Bk(w)

⇒ ^

y∈∆

w^0I(x, y)⇒(∀w.B)^I(y)

= ^

B∈N^T_P

^

w∈N^∗_R

_

π∈paths(A^T_A0,B,w)

(wt_T(π⁰)∧wt_T(π))

⇒(∀w⁰w.B)^I(x)

≥ ^

B∈N^T_P

^

w∈N^∗_R

kA^T_A,Bk(w⁰w)⇒(∀w⁰w.B)^I(x)

by the induction hypothesis, Propositions 4 and 5, and the definition ofkA^T_A,Bk.

In both cases,p⇒(∀w⁰.A⁰)^Ii−1(x)is an upper bound for the infimum on the right-hand side of (2), and thus by (3) the same is true for(T_J^T)ⁱ(1)(A)(x). ut This allows us to prove gfp-subsumptions by comparing the behavior of WWA.

Lemma 14. Let A, B∈N^T_C andp∈[0,1]. Then T |=gfphAvB≥piiff for all C∈N^T_P andw∈N^∗_R it holds thatp∧ kA^T_B,Ck(w)≤ kA^T_A,Ck(w).

Proof. Assume that there exist C ∈ N^T_P and w = r1. . . rn ∈ N^∗_R such that p∧kA^T_B,Ck(w)>kA^T_A,Ck(w). We define the primitive interpretationJ = (∆,·^J) where ∆ := {x0, . . . , xn}, and for all D ∈ N^T_P and r ∈ NR, the interpretation function is given by

D^J(x) :=

(kA^T_A,Ck(w) ifD=C andx=xn,

1 otherwise; and

r^J(x, y) :=

(1 ifx=x_i−1,y=x_i, andr=r_i for somei∈ {1, . . . , n}, 0 otherwise.

Consider now the gfp-model I :=gfp_T(J) of T. By construction, for all pairs (w⁰, D)∈ N^∗_R×N^T_P \ {(w, C)} we have (∀w⁰.D)^I(x0) = 1. Moreover, we know that (∀w.C)^I(x0) is equal to kA^T_A,Ck(w), and thus strictly smaller than pand kA^T_B,Ck(w). By Lemma 13, all this implies that

A^I(x₀) =kA^T_A,Ck(w)⇒(∀w.C)^I(x₀) = 1and B^I(x₀) =kA^T_B,Ck(w)⇒(∀w.C)^I(x₀) = (∀w.C)^I(x₀).

ThusA^I(x0)⇒B^I(x0) = (∀w.C)^I(x0)< p, andT 6|=gfphAvB≥pi.

Conversely, assume that there are a primitive interpretationJ = (∆,·^J)and an elementx∈∆such thatA^I(x)⇒B^I(x)< p, whereI :=gfp_T(J). Thus, we havep∧A^I(x)> B^I(x), which implies by Lemma 13 the existence of aC∈N^T_P and aw∈N^∗_Rwithp∧A^I(x)>kA^T_B,Ck(w)⇒(∀w.C)^I(x). Again by Lemma 13, this implies that

p∧ kA^T_B,Ck(w)> A^I(x)⇒(∀w.C)^I(x)

≥ kA^T_A,Ck(w)⇒(∀w.C)^I(x)

⇒(∀w.C)^I(x).

In particular, the latter value cannot be 1, and thus it is equal to(∀w.C)^I(x).

But this can only be the case if kA^T_A,Ck(w)≤ (∀w.C)^I(x). To summarize, we obtainp∧ kA^T_B,Ck(w)>(∀w.C)^I(x)≥ kA^T_A,Ck(w), as desired. ut Denote by VT := {0,1} ∪ {p ∈ [0,1] | hA v ∀w.B ≥ pi ∈ T } the set of all values appearing in T, together with 0 and 1. Since wt_T has finite sup- port and takes only values from VT, p∧ kA^T_B,Ck(w) > kA^T_A,Ck(w) holds iff p⁰∧ kA^T_B,Ck(w)>kA^T_A,Ck(w), wherep⁰ is the smallest element ofVT such that p⁰≥p. This shows that it suffices to be able to check gfp-subsumptions for the values in VT. We now show how to do this by simulating A^T_B,C and A^T_A,C by polynomially manyunweighted automata.

Definition 15 (automata A_≥p). Given a WWAA= (Σ, Q, q0,wt, qf) and a value p ∈ [0,1], the WA A_≥p = (Σ, Q, q0,wt_≥p, qf) is given by the transition relation wt_≥p:={(q, w, q⁰)∈Q×Σ^∗×Q|wt(q, w, q⁰)≥p}.

The language of this automaton has an obvious relation to the behavior of the original WWA.

Lemma 16. Let Abe a WWA over the alphabetΣandp∈[0,1]. Then we have L(A_≥p) ={w∈Σ^∗| kAk(w)≥p}.

Proof. We havew∈L(A≥p)iff there is a finite pathπ=q₀w₁q₁. . . w_nq_n in A with labelwsuch thatwt(q_i−1, w_i, q_i)≥pholds for alli∈ {1, . . . , n}. The latter condition is equivalent to the fact that wt(π) ≥p. Thus, w∈ L(A_≥p) implies thatkAk(w)≥p. Conversely, sincewthas finite support, there are only finitely many possible weights for any finite path inA, and thuskAk(w)≥palso implies that there exists aπ∈paths(A, w)withwt(π)≥p, and thusw∈L(A_≥p). ut

We thus obtain the following characterization of gfp-subsumption.

Lemma 17. Let A, B∈N^T_C andp∈ V_T. ThenT |=gfp hAvB ≥piiff for all C∈N^T_P andp⁰∈ V_T with p⁰≤pit holds that L((A^T_B,C)_≥p⁰)⊆L((A^T_A,C)_≥p⁰).

Proof. Assume that we have T |=gfp hA v B ≥ pi and consider any C ∈ N^T_P, w∈N^∗_R, and p⁰ ∈ VT ∩[0, p]with w∈L((A^T_B,C)≥p⁰). By Lemma 16, we obtain kA^T_B,Ck(w)≥p⁰, and by Lemma 14 we havekA^T_A,Ck(w)≥p∧ kA^T_B,Ck(w)≥p⁰. Thus,w∈L((A^T_A,C)_≥p⁰).

Conversely, assume that T |=gfp hA v B ≥ pi does not hold. Then by Lemma 14 there areC∈N^T_P andw∈N^∗_Rsuch thatp∧kA^T_B,Ck(w)>kA^T_A,Ck(w).

For the value p⁰ :=p∧ kA^T_B,Ck(w)∈ V_T ∩[0, p], we havekA^T_B,Ck(w)≥p⁰, but kA^T_A,Ck(w)< p⁰, and thusL((A^T_B,C)_≥p⁰)*L((A^T_A,C)_≥p⁰)by Lemma 16. ut A direct consequence of this lemma is that gfp-subsumption between concept names inG-FL0 remains in the same complexity class as for classicalFL0. Theorem 18. Deciding gfp-subsumption between concept names in G-FL0 is PSpace-complete.

Proof. By the reductions above, it suffices to decide the language inclusions L((A^T_B,C)_≥p)⊆ L((A^T_A,C)_≥p) for all C ∈ N^T_P andp ∈ V_T. These polynomially many inclusion tests for WA can be done in polynomial space [10]. The problem is PSpace-hard since gfp-subsumption in classicalFL0is alreadyPSpace-hard [1].

This is a special case of our problem where the input TBox is restricted to the values0and1, and therefore all relevant WWA are already WA. ut

5 Conclusions

We have studied the complexity of reasoning in G-FL₀ w.r.t. primitive concept definitions under greatest fixed-point semantics. More precisely, we have shown that gfp-subsumption between concept names can be reduced to a comparison of the behavior of weighted automata with word transitions. Moreover, the latter can be solved by a polynomial number of inclusion tests onunweightedautomata.

Overall, this shows that gfp-subsumption isPSpace-complete for this logic, just as in the classical case.

This complexity result is consistent with previous work on extensions of de- scription logics with Gödel semantics. Indeed, such extensions ofEL[15,16] and ALC [5] have been shown to preserve the complexity of their classical counter- part. Since reasoning in classical FL0 and in G-ALC w.r.t. general TBoxes is in both cases ExpTime-complete, so is deciding subsumption in G-FL0 w.r.t.

general TBoxes.

We expect our results to generalize easily to any other set of truth degrees that form a total order. However, the arguments used in this paper fail for arbitrary lattices, where incomparable truth degrees might exist [7,19]. Studying these two cases in detail is a task for future work. We also plan to consider fuzzy extensions ofFL0with semantics based on non-idempotent t-norms, such as the Łukasiewicz or product t-norms [12].

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