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 nite set of primitive denitions 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 standard true and false. 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 nitely 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 also ExpTime-complete.

On the other hand, if TBoxes are restricted to contain only (primitive) deni- tions, then deciding subsumption in classicalFL₀under the greatest xed-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 xed-point se- mantics of G-FL0 by means of weighted automata over lattices. We then show that reasoning with these automata can be reduced to a linear number of in- clusion 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 rst 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].

A lattice is an algebraic structure(L,∨,∧)with two commutative, associa- tive and idempotent binary operations ∨ (supremum) and ∧ (inmum) that distribute over each other. It is complete if suprema and inma 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 ix∧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 inmum and supremum are dened component-wise. More precisely, for any twof1, f2∈L^S, we denef1∨f2for allx∈Sas(f1∨f2)(x) :=f1(x)∨f2(x). If we similarly dene the inmum, we obtain a lattice with the order f1 ≤ f2

if1(x)≤f2(x)holds for allx∈S. It is easy to verify that innite inma and suprema can then also be computed component-wise.

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

Denition 2 (xed-point). Let L be a complete lattice. A xed-point of an operator T:L→L is an element x∈L such thatT(x) =x. It is the greatest xed-point ofT if for any xed-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-continuous if for every decreasing chainx₀ ≥x₁≥x₂≥. . . inL we haveT(V

i≥0x_i) =V

i≥0T(x_i).

If it exists, the greatest xed-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 xed-point. If T is downwardω-continuous, then gfp(T)corresponds to the inmum 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 inma 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]dened 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 i y ≤x⇒z. As with the Gödel t-norm, its residuum preserves arbitrary inma in its second component. However, in the rst 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 rst argument. The following reformulation of nested residua in terms of inma will also prove useful.

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

= x₁⇒. . .(x_n⇒x). . . .

Proof. Both values are eitherxor 1, and they are1 i 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 dierence lies in the interpretation ofG-FL0-concepts.

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

A (primitive concept) denition is of the formhAvC≥pi, whereA∈NC, C is a concept, and p ∈ [0,1]. A TBox is a nite set of denitions. Given a TBox T, a concept name is dened if it appears on the left-hand side of a denition inT, and primitive otherwise.

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 dened 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.

Denition 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 satises (or is a model of) the denitionhAvC ≥pi if A^I(x) ⇒ C^I(x) ≥p holds for all x∈ ∆^I. It satises (or is a model of) a TBox if it satises all its denitions.

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 in normal form if all denitions inT are of the formhAv ∀w.B≥pi, whereA, B ∈NC, w ∈ N^∗_R, and p ∈ [0,1], and there are no two denitions 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 denitions 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 denitions 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 aecting the semantics.

It follows from Proposition 4 that an interpretationI satises the denition hAv ∀w1.B₁u · · · u ∀wn.B_n ≥pii it satises 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 simplication steps, the TBox contains only axioms of the form hA v ∀w.B ≥ pi with A, B ∈ NC, satisfying the rst condition of the de- 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 denitions can be seen as a restriction of the interpretation of the dened concepts, depending on the interpretation of the primitive concepts. We use this intuition and consider greatest xed-point semantics, as described next.

A primitive interpretation is a pair J = (∆,·^J) as in Denition 7, except that ·^J is only dened for role names and the primitive concept names inN^T_P. Given such aJ, we use functionsf ∈([0,1]^∆)^NTD to describe the interpretation of the remaining (dened) 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 function f ∈ L^T_J, the induced interpretation I_J,f has the same domain as J and extends the interpretation function of J to the dened concepts 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 in T, is xed to0.

We can describe the eect that the axioms inT have onL^T_J by the operator T_J^T:L^T_J →L^T_J, which is dened 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 dened concept names according to the old interpretationI_J,f and their denitions inT.

We are interested in using the greatest xed-point ofT_J^T, for some primitive interpretationJ, to dene a new semantics for TBoxesT inG-FL0. Before being able to do this, we have to ensure that such a xed-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 dened as usual over the complete lattice[0,1]^∆.

ForA∈N^T_P, by the denition 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 denition 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 denition 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 dened 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 dene the reasoning problem inG-FL₀that we want to solve.

Denition 10 (gfp-subsumption). An interpretation I 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 suces 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 a weighted automata model.

Denition 11 (WWA). A weighted automaton with word transitions (WWA) is a tupleA= (Σ, Q, q₀,wt, q_f), whereΣ is a nite alphabet of input symbols, Q is a nite 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 nite, andqf ∈Qis the nal state.

A nite 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 nite word

`(π) :=w1w2. . . wn. The weight ofπis dened aswt(π) :=Vn

i=1wt(q_i−1, wi, qi). The set of all nite paths with label w inA is denoted bypaths(A, w). The be- havior kAk:Σ^∗ → [0,1] of A is dened 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 nite 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. The inclusion problem for WA is to decide, given two such automataAandA⁰, whetherL(A)⊆L(A⁰). This problem is known to be PSpace-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.

Denition 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 dened 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 dier only on the initial and nal states they dene;

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-dened. This function has nite support sinceT is nite.

We now characterize the gfp-models ofT by properties of the automataA^T_A,B. Lemma 13. For every gfp-modelI= (∆,·^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 Denition 12 any nite 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 inmum 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 suces to show that for allx∈∆, A∈N^T_D, B∈N^T_P, and all nite 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 denition hAv ∀w1.B≥pi, withp:=wtT(A, w1, B). By the denition 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 denitionhAv ∀w1.A1≥pi. By the denitions 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 satises (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 denitionhAv ∀w⁰.A⁰≥pi ∈ T. Thenπ⁰ =Aw⁰A⁰ is a nite 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 denition 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 dened, 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 denition ofkA^T_A,Bk. In both cases,p⇒(∀w⁰.A⁰)^Ii−1(x)is an upper bound for the inmum 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. LetA, B∈N^T_C andp∈[0,1]. Then T |=gfphAvB≥pii 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 dene 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 nite sup- port and takes only values from VT, p∧ kA^T_B,Ck(w) > kA^T_A,Ck(w) holds i 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 suces 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 many unweighted automata.

Denition 15 (automata A_≥p). Given a WWA A= (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. LetAbe a WWA over the alphabetΣandp∈[0,1]. Then we have L(A_≥p) ={w∈Σ^∗| kAk(w)≥p}.

Proof. We have w∈L(A≥p)i there is a nite 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 nite support, there are only nitely many possible weights for any nite 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 ≥pii 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 suces 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 already PSpace-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 denitions under greatest xed-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 on unweighted automata.

Overall, this shows that gfp-subsumption is PSpace-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

Šukasiewicz or product t-norms [12].

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