Stable Models of Fuzzy Propositional Formulas

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Joohyung Lee and Yi Wang

School of Computing, Informatics, and Decision Systems Engineering Arizona State University, Tempe, USA

{joolee,ywang485}@asu.edu

Abstract. We introduce the stable model semantics for fuzzy propositional formulas, which generalizes both fuzzy propositional logic and the stable model semantics of classical propositional formulas. Combining the advantages of both formalisms, the introduced language allows highly configurable default reasoning involving fuzzy truth values. We show that several properties of Boolean stable models are naturally extended to this formalism, and discuss how it is related to other approaches to combining fuzzy logic and the stable model semantics.

1 Introduction

Answer set programming (ASP) [1] is a widely applied declarative programming paradigm for the design and implementation of knowledge intensive applications. One of the at- tractive features of ASP is its capability to model the nonmonotonic aspect of knowledge. However, as its mathematical basis, the stable model semantics, is restricted to Boolean values, it is too rigid to represent imprecise and vague information. Fuzzy logic [2], as a form of many-valued logic, can handle vague information by interpreting propositions with a truth degree in the interval of real numbers[0,1]. The availability of various fuzzy operators gives the user great flexibility in combining truth degrees.

However, the semantics of fuzzy logic is monotonic, and is not flexible enough to handle default reasoning allowed in answer set programming.

Both the stable model semantics and fuzzy logic are generalizations of classical propositional logic in different ways. While they do not subsume each other, it is clear that many real-world problems require both their strengths. This led to the body of work on combining fuzzy logic and the stable model semantics, known as fuzzy answer set programming (e.g., [3, 4]). However, their syntax is restricted to rules, and does not allow connectives nested arbitrarily as in fuzzy logic.

Unlike existing work on fuzzy answer set semantics, in this paper, we extend the general stable model semantics from [5] to many-valued propositional formulas. The syntax of this language is the same as the syntax of fuzzy propositional logic. The semantics, on the other hand, definesstablemodels instead of models. The language is a proper generalization of both fuzzy propositional logic and the stable model semantics for Boolean propositional formulas. This generalization is not simply a pure theoretical pursuit, but has practical use in conveniently modeling defaults involving fuzzy truth values in dynamic domains. For example, consider modeling dynamics oftrustin social network. People trust each other in different degrees under some normal assumptions.

If person Atrusts another personB, thenA tends to trust personC whom B trusts

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to a degree which is positively correlated to the degree to which Atrusts B and the degree to whichB trustsC. If nothing happens, the trust degrees would not change.

But there may be less trust between two people when a conflict arises between them.

Modeling such a domain requires expressing defaults involving fuzzy truth values. We demonstrate that such examples can be conveniently modelled in our proposed language by taking advantage of its generality over the existing approaches to fuzzy ASP.

The paper is organized as follows. Section 2 reviews the syntax and the semantics of fuzzy propositional logic we discuss in the paper, as well as the stable model semantics of classical propositional formulas. Section 3 presents the stable model semantics of fuzzy propositional formulas along with examples, followed by Section 4 that formal- izes the trust example above in the proposed language. Section 5 shows how the fuzzy stable model semantics is related to the Boolean stable model semantics, and Section 6 shows how our fuzzy stable model semantics is related to other approaches to fuzzy ASP. Section 7 shows that several well-known properties of the Boolean stable model semantics can be easily extended to our fuzzy stable model semantics.

2 Preliminaries

2.1 Review: Stable Models of Classical Propositional Formulas

We review the definition of a stable model from [5] by limiting attention to the syntax of propositional formulas. Instead of defining stable models in terms of second-order logic as in [5] , we express the same concept using auxiliary atoms that do not belong to the original signature. This slight reformulation will simplify our efforts in extending the stable model semantics to fuzzy propositional formulas without resorting to “second- order fuzzy logic.”

Letσbe a classical propositional signature,p= (p₁, . . . , p_n)be a list of distinct atoms belonging toσ, and letq= (q₁, . . . , q_n)be a list of new propositional atoms not inσ.

For two interpretationsIandJofσ,I∪J_q^pdenotes the interpretation ofσ∪qthat agrees withIandJ on all atoms not inp∪q, and

– for eachp∈p,(I∪J_q^p)(p) =I(p);

– for eachq∈q,(I∪J_q^p)(q) =J(p).¹

For any classical propositional formulaFof signatureσ,F^∗(q)is a classical propositional formula of signatureσ∪qthat is defined recursively as follows:

– p^∗_i =qifor eachpi∈p;

– F^∗=Ffor any atomF 6∈p;

– ⊥^∗=⊥; >^∗=>;

– (¬F)^∗=¬F;

– (F∧G)^∗=F^∗∧G^∗; (F∨G)^∗=F^∗∨G^∗; – (F →G)^∗= (F^∗→G^∗)∧(F →G).

1I(p)denotes the truth value ofpunderI. We identify a list with a set if there is no confusion.

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LetIandJbe two interpretations ofσ, and letpbe a subset ofσ. We sayJ ≤^pI if

– JandIagree on all atoms not inp, and – for allp∈p, ifJ |=p, thenI|=p.

We sayJ <^pIifJ ≤^pIandJ 6=I.

Definition 1. An interpretationI is astable modelofF relative top(denotedI |= SM[F;p])

– ifI|=F, and

– there is no interpretationJsuch thatJ <^pIandI∪J_q^p|=F^∗(q).

Example 1. Consider a logic program

p←notq, q←notp

which is understood as an alternative notation for propositional formula F1= (¬q→p)∧(¬p→q).

F₁^∗(u, v)is

(¬q→u)∧(¬q→p)∧(¬p→v)∧(¬p→q).

We check thatI₁ ={p}(that is,pisTRUEandqisFALSE)²is a stable model ofF₁ (relative to{p, q}):I₁satisfiesF₁, and∅is the only interpretationJsuch thatJ <^pqI₁. However,I1∪J_uv^pq={p}does not satisfyF₁^∗(u, v)because it does not satisfy the first conjunctive term ofF₁^∗(u, v).

Similarly, we can check that{q}is another stable model ofF1.

2.2 Review: Fuzzy Logic

Letσbe a fuzzy propositional signature, which is a set of symbols calledfuzzy atoms.

In addition, we assume the presence of a setCof fuzzy conjunction symbols, a setD of fuzzy disjunction symbols, a setNof fuzzy negation symbols, and a setIof fuzzy implication symbols.

Afuzzy (propositional) formulaofσis defined recursively as follows.

– every fuzzy atomp∈σis a fuzzy formula;

– every numeric constantcwherecis a real number in[0,1]is a formula;

– ifF is a formula, then¬F is a formula, where¬ ∈N;

– ifF andGare formulas, thenF ⊗G,F ⊕GandF → Gare formulas, where

⊗ ∈C,⊕ ∈D, and→ ∈I.

2We identify a propositional interpretation with the set of atoms that are true in it.

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The models of a fuzzy formula are defined as follows [2]. Thefuzzy truth valuesare the real numbers in the range[0,1]. Afuzzy interpretationIofσis a mapping fromσ into[0,1].

The fuzzy operators are functions mapping one or two truth values into a truth value.

Among the operators,¬denotes a function from[0,1]into[0,1];⊗,⊕, and→denote functions from[0,1]×[0,1]into[0,1]. The actual mapping performed by each operator can be defined in many different ways, but all of them satisfy the following conditions, which imply that they are generalizations of the corresponding classical propositional connectives:³

– a fuzzy negation¬is decreasing, and satisfies¬(0) = 1and¬(1) = 0;

– a fuzzy conjunction⊗is increasing, commutative, associative, and⊗(1, x) =xfor allx∈[0,1];

– a fuzzy disjunction⊕is increasing, commutative, associative, and⊕(0, x) =xfor allx∈[0,1];

– a fuzzy implication→is decreasing in its first argument and increasing in its second argument; and→(1, x) =xand→(0,0) = 1for allx∈[0,1].

Figure 1 lists some specific fuzzy operators that we use in this paper.

Symbol Name Definition

⊗l Łukasiewicz t-norm ⊗l(x, y) =max(x+y−1,0)

⊕l Łukasiewicz t-conorm ⊕l(x, y) =min(x+y,1)

⊗m minimum t-norm ⊗m(x, y) =min(x, y)

⊕m maximum t-conorm ⊕m(x, y) =max(x, y)

⊗p product t-norm ⊗p(x, y) =x·y

⊕p product t-conorm ⊕p(x, y) =x+y−x·y

¬s standard negator ¬s(x) = 1−x

→r the residual implicator of⊕m →r(x, y) =

(1 ifx≤y y otherwise

→s the S-implicator induced by¬sand⊕m→s(x, y) =max(1−x, y) Fig. 1.Some t-norms, t-conorms, negator, and implicators

Thetruth valueof a formulaF underI, denotedF^I, is defined recursively as follows:

– for any atomp∈σ, p^I =I(p);

– for any numeric constantc,c^I =c;

– (¬F)^I =¬(F^I);

– (F⊗G)Î =⊗(FÎ, GÎ); (F⊕G)Î =⊕(FÎ, GÎ); (F →G)Î =→(FÎ, GÎ).

3We say that a function f of arityn is increasing in its i-th argument(1 ≤ i ≤ n) if f(arg1, . . . , argi, . . . , argn) ≤ f(arg1, . . . , arg_i⁰, . . . , argn) for all arguments such that argi≤argi⁰;fis said to beincreasingif it is increasing in all its arguments. The definition of decreasingis similar.

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(For simplicity, we identify the symbols for the fuzzy operators with the truth value functions represented by them.)

Definition 2. We say that a fuzzy interpretationI satisfies a fuzzy formulaF w.r.t. a thresholdy ∈ [0,1]ifF^I ≥ y, and denote it byI |=_y F. We callIafuzzyy-model ofF.

We often omit the thresholdywhen it is1.

3 Definition and Examples

We extend the notion ofJ <^p I in Section 2.1 as follows. For any two fuzzy inter- pretationsJ andI of the same signatureσ and any subsetpof σ, we sayJ ≤^p I if

– JandIagree on all fuzzy atoms not inp, and – for allp∈p,p^J≤p^I.

We sayJ <^pIifJ ≤^pIandJ 6=I.

As before, we assume a listqof new, distinct fuzzy atoms, and defineI∪J_q^pin the same way. That is,I∪J_q^pdenotes the interpretation ofσ∪qthat agrees withIandJ on all atoms not inp∪q, and

– for eachp∈p,(I∪J_q^p)(p) =I(p);

– for eachq∈q,(I∪J_q^p)(q) =J(p).

The definition ofF^∗is also extended in a straightforward way: For any fuzzy for- mulaFof signatureσ,F^∗(q)is defined as follows.

– p^∗_i =q_ifor eachp_i∈p;

– F^∗=Ffor any atomF 6∈p;

– c^∗=cfor any numeric constantc;

– (¬F)^∗=¬F;

– (F⊗G)^∗=F^∗⊗G^∗; (F⊕G)^∗=F^∗⊕G^∗; – (F →G)^∗= (F^∗→G^∗)⊗m(F →G).

Definition 3. An interpretationIis ay-stable model ofF relative top(denotedI|=_y SM[F;p]) if

– I|=yF, and

– there is no interpretationJsuch thatJ <^pIandI∪J_q^p|=_yF^∗(q).

We often omit the thresholdywhen it is1, and omitpif it contains all atoms inσ.

Clearly, whenpis empty, Definition 3 reduces to the definition of a fuzzy model in Definition 2 because there is noJsuch thatJ <^∅I.

Also, Definition 3 is very similar to the definition of a stable model for classical propositional formulas in Definition 1. The main difference is that simply in the latter, atoms may have various degrees of truth, and accordingly the notion ofJ <^pIis more general. The precise relationship between the definitions is discussed in Section 5.

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Example 2. Consider the formulaF =¬_sp→_rqand the interpretationI={(p,0),(q,0.6)}.

F^∗(u, v)is

((¬sp)^∗→rq^∗)⊗m(¬sp→rq) = (¬sp→rv)⊗m(¬sp→rq).

I|=0.6SM[F; p, q]. First, it is easy to see thatI|=0.6F, as

FÎ =→_r((¬_sp)Î, qÎ) =→_r(1−pÎ, qÎ) =→_r(1,0.6) = 0.6.

Suppose there existsJ <^pqIsuch thatI∪J_uv^pq|=0.6F, i.e.,

F^∗(u, v)Î∪Jûv^pq =min →r(¬s(pÎ), v^Jûv^pq),→r(¬s(pÎ), qÎ)

=min →r(1, v^J^uv^pq),0.6

=min v^J^uv^pq,0.6

≥0.6.

Sov^Jûv^pq ≥0.6. Sincev^Jûv^pq ≤ qÎ = 0.6, we conclude thatv^Jûv^pq = 0.6. However, this contradicts the assumption thatJ <^pq I. Therefore, suchJ does not exist, andI is a 0.6-stable model ofF.

Example 3. pand¬s¬sphave the same fuzzy models, but their stable models are different. This is similar to the fact thatpand¬¬phave different stable models according to the semantics from [5].

Clearly, any interpretationI = {(p, y)}, wherey is any positive real number in [0,1], is ay-stable model ofprelative to{p}. On the other hand,I={(p, y)}is not a y-stable model ofF =¬s¬sprelative to{p}. FormulaF^∗(q)is¬s¬sF, and although I|=y F, we haveI∪J_q^p |=yF^∗(q)regardless of anyJ.

Example 4. LetF1=p→spandF2=¬sp⊕mp. Their fuzzy models are the same, but their stable models are not. This is similar to the relation betweenp→pand¬p∨pin the Boolean stable model semantics. Indeed, observe thatF₁^∗(q) = (p→sp)⊗m(q→sq) andF₂^∗(q) =¬sp⊕mq.

The interpretationI ={(p,1)}is not a1-stable model ofF₁relative top, as witnessed byJ ={(p,0)}. However,Iis a1-stable model ofF₂relative top: for anyJ_q^p,

F₂^∗(q)^I∪J^q^p=max

1−p^I, q^J^q^p

=max 0, q^J^q^p

=q^J^q^p.

So, forI∪J_q^pto satisfyF₂^∗(q)to degree1,q^J^q^pshould be1, or equivalently,p^Jshould be1. Consequently, it is not possible to haveJ <^pI.

The following example illustrates how the commonsense law of inertia involving fuzzy truth values can be represented.

Example 5. Letσbe{p,∼p, q,∼q}⁴and letF beF₁⊗mF₂, whereF₁represents that pand∼pare complementary, i.e., the sum of their truth values is1:

F1=¬s(p⊗l∼p)⊗m¬s¬s(p⊕l∼p).

4Note that∼is not a connective; it is just a part of the symbol representing an atom.

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F₂represents that by defaultphas the truth value ofq, and∼phas the truth value of∼q:

F₂= ((q⊗m¬s¬sp)→rp)⊗m((∼q⊗m¬s¬s∼p)→r∼p).

Letp={p,∼p}andu={u,∼u}.F^∗(u)is

¬s(p⊗l∼p)⊗m¬s¬s(p⊕l∼p)

⊗m((q⊗m¬s¬sp)→ru)⊗m((q⊗m¬s¬sp)→rp)

⊗m((∼q⊗m¬s¬s∼p)→r∼u)⊗m((∼q⊗m¬s¬s∼p)→r∼p).

One can check that the interpretation I1 = {(p, x),(∼p,1−x),(q, x),(∼q,1−x)}

(xis any value in[0,1]) is a1-stable model ofFrelative to(p,∼p); The interpretation I2={(p, y),(∼p,1−y),(q, x),(∼q,1−x)}, wherey > x, is not. Similarly, ify < x, I2is not a1-stable model ofF relative to(p,∼p).

On the other hand, if we conjoinFwithy→rpto yieldF⊗m(y→rp), then the default behavior is overridden, andI2is a1-stable model ofF⊗m(y→rp)relative to (p,∼p).

This behavior is useful in expressing the commonsense law of inertia involving fuzzy values. Supposeqrepresents some fluent at timet, andprepresents the fluent at timet+ 1. ThenF states that, “by default, the fluent retains the previous value.” The default value is overridden if there is an action that setspto a different value.

4 Further Examples

The trust example in the introduction can be formalized in the fuzzy stable model semantics as follows. Belowx,y,zare schematic variables ranging over people, andtis a schematic variable ranging over time steps.Trust(x, y, t)is a fuzzy atom representing that “xtrustsyat timet.” Similarly,Distrust(x, y, t)is a fuzzy atom representing that

“xdistrustsyat timet.”

The trust relation is reflexive:

F1=Trust(x, x, t).

The trust and distrust degrees are complementary, i.e., their sum is1 (similar to Example 5):

F₂=¬_s(Trust(x, y, t)⊗_lDistrust(x, y, t)), F3=¬s¬s(Trust(x, y, t)⊕lDistrust(x, y, t)).

Initially, ifxtrustsy to degree d1andy trusts z to degreed2, thenxtrusts zto degreed1×d2; further the initial distrust degree is1minus the initial trust degree.

F₄=Trust(x, y,0)⊗pTrust(y, z,0)→rTrust(x, z,0), F₅=¬_sTrust(x, y,0)→_rDistrust(x, y,0).

The inertia assumption (similar to Example 5):

F₆=Trust(x, y, t)⊗_m¬_s¬_sTrust(x, y, t+1)→_rTrust(x, y, t+1),

F7=Distrust(x, y, t)⊗m¬s¬sDistrust(x, y, t+1)→rDistrust(x, y, t+1).

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A conflict increases the distrust degree by the conflict degree:

F8=Conflict(x, y, t)⊕lDistrust(x, y, t)→rDistrust(x, y, t+1), F9=¬s(Conflict(x, y, t)⊕lDistrust(x, y, t))→rTrust(x, y, t+1).

LetF_{T W} be F₁⊗mF₂ ⊗m· · · ⊗mF₉. Suppose we have the formulaF_{F act} = Fact1⊗_mFact2that gives the initial trust degree.

Fact1= 0.8→rTrust(Alice,Bob,0), Fact2= 0.7→rTrust(Bob,Carol,0).

Although there is no fact about how much AlicetrustsCarol, any1-stable model of F_{T W} ⊗m F_{F act} assigns value 0.56to the atom Trust(Alice,Carol,0). On the other hand, the 1-stable model assigns value 0 to Trust(Alice,David,0) due to the closed world assumption under the stable model semantics.

When we conjoinF_{T W} ⊗F_{F act}with0.2→Conflict(Alice,Carol,0), the1-stable model of F_{T W} ⊗_mF_{F act}⊗_m(0.2 → Conflict(Alice,Carol,0)), manifests that the trust degree betweenAliceandCaroldecreases to0.36at time1. More generally, if we have more actions that change the trust degree in various ways, by specifying the entire history of actions, we can determine the evolution of the trust distribution among all the participants. Useful decisions can be made based on this information. For example, Alicemay decide not to share her personal pictures to those whom she trusts less than degree0.48.

Note that this example, like Example 5, uses nested connectives, such as¬s¬s, that are not available in previous fuzzy ASP semantics, such as [3, 4].

5 Relation to Boolean-Valued Stable Models

The Boolean stable model semantics in Section 2.1 can be embedded into the fuzzy stable model semantics as follows:

For any classical propositional formulaF, defineF^fuzzy to be the fuzzy propositional formula obtained fromFby replacing⊥with0,>with1,¬with¬s,∧with⊗m,

∨with⊕_m, and→with→_s. We identify the signature ofF^{f uzzy} with the signature ofF. Also, for any interpretationI, we define the corresponding fuzzy interpretation I^{f uzzy}as

– I^fuzzy(p) = 1ifI(p) =TRUE; – I^fuzzy(p) = 0otherwise.

The following theorem tells us that the Boolean-valued stable model semantics can be viewed as a special case of the fuzzy stable model semantics.

Theorem 1 For any classical propositional formulaFand any classical propositional interpretationI,Iis a stable model ofF relative topiffI^fuzzyis a1-stable model of F^{f uzzy}relative top.

Example 6. Let F be the classical propositional formula ¬p → q. F has only one stable modelI={q}. ClearlyI^fuzzy={(p,0),(q,1)}is a1-stable model ofF^fuzzy=

¬sp→sq.

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Theorem 1 does not hold for an arbitrary choice of operators, as illustrated by the following example.

Example 7. LetF be the classical propositional formulap∨p. Classical interpretation I ={p}is a stable model ofF. However,I^fuzzy = {(p,1)}is not a stable model of F⁰=p⊕lpbecause there isJ ={(p,0.5)}such thatI∪J_q^p |=1q⊕lq.

However, one direction of Theorem 1 holds for arbitrary choice of fuzzy operators.

Theorem 2 For any classical propositional formulaF, letF₁^fuzzy be the formula obtained fromF by replacing⊥with0,>with1,¬with any fuzzy negation symbol,∧ with any fuzzy conjunction symbol,∨with any fuzzy disjunction symbol, and→with any fuzzy implication symbol. For any classical propositional interpretationI, ifI^fuzzy is a1-stable model ofF₁^{f uzzy} relative top, thenIis a stable model ofFrelative top.

6 Relation to Other Approaches to Fuzzy ASP

6.1 Relation to Stable Models of Normal FASP Programs

A normal FASP program is a finite set of rules of the form a ← b1⊗. . .⊗bm⊗ ¬bm+1⊗. . .⊗ ¬bn,

wheren≥m≥0,a, b₁, . . . , b_nare fuzzy atoms or numeric constants in[0,1], and⊗ is any fuzzy conjunction. We identify the rule with the fuzzy implication

b₁⊗. . .⊗b_m⊗ ¬_sb_m+1⊗. . .⊗ ¬_sb_n →_r a.

We say that a fuzzy interpretationIof signatureσsatisfiesa ruleRifR^I = 1.I satisfiesan FASP program Π if I |= Rfor every ruleR inΠ. According to [3], an interpretationI is afuzzy answer setof a normal FASP programΠ ifI satisfies Π, and no interpretationJ such thatJ <^σ I satisfies the reduct of Π w.r.t. I, which is the program obtained fromΠ by replacing each negative literal¬b with the constant for1−b^I.

Theorem 3 For any normal FASP programΠ ={r1, . . . , r_n}, letF be the formula r₁⊗m. . .⊗mr_n. An interpretationIis a fuzzy answer set ofΠ in the sense of [3] if and only ifIis a1-stable model ofF.

Example 8. LetΠbe the following program

p← ¬q, q← ¬p.

The answer sets ofΠ according to [3] are{(p, x),(q,1−x)}, wherexis any value in [0,1]: the corresponding fuzzy formulaF is(¬sq→rp)⊗m(¬sp→rq); F^∗(u, v)is

F⊗m((¬sq→ru)⊗m(¬sp→rv)).

One can check that the1-stable models ofF are also{(p, x),(q,1−x)}, wherex∈ [0,1].

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6.2 Relation to Fuzzy Equilibrium Logic

Like our fuzzy stable model semantics, fuzzy equilibrium logic [6] generalizes fuzzy ASP programs to arbitrary propositional formulas, but its definition is highly complex.

Nonetheless we show that if we disregard strong negation considered there, fuzzy equilibrium logic is essentially equivalent to the fuzzy stable model semantics where the threshold is set to1and all atoms are subject to minimization.⁵

We review the definition of fuzzy equilibrium logic in the absence of strong negation. For any fuzzy propositional signatureσ, a (fuzzy N5)valuationis a mapping from {h, t} ×σto subintervals of[0,1]such thatV(t, a)⊆V(h, a)for each atoma∈σ. For V(w, a) = [u, v], wherew∈ {h, t}, we writeV⁻(w, a)to denote the lower boundu andV⁺(w, a)to denote the upper boundv. The truth value of a formula underV is defined as follows.

– V(w, c) = [c, c]for any numeric constantc;

– V(w, F ⊗G) = [V⁻(w, F)⊗V⁻(w, G), V⁺(w, F)⊗V⁺(w, G)];⁶ – V(w, F ⊕G) = [V⁻(w, F)⊕V⁻(w, G), V⁺(w, F)⊕V⁺(w, G)];

– V(h,¬F) = [1−V⁻(t, F), 1−V⁻(h, F)];

– V(t,¬F) = [1−V⁻(t, F), 1−V⁻(t, F)];

– V(h, F →G) = [min(V⁻(h, F)→V⁻(h, G), V⁻(t, F)→V⁻(t, G)),

V⁻(h, F)→V⁺(h, G)];

– V(t, F →G) = [V⁻(t, F)→V⁻(t, G), V⁻(t, F)→V⁺(t, G)].

A valuationV is a (fuzzy N5) model of a formulaF if V⁻(h, F) = 1, which impliesV⁺(h, F) =V⁻(t, F) =V⁺(t, F) = 1. For two valuationsV andV⁰, we say V⁰ V ifV⁰(t, a) =V(t, a)andV(h, a)⊆V⁰(h, a)for all atomsa. We sayV⁰ ≺V ifV⁰ V andV⁰6=V. We say that a modelV ofFish-minimalif there is no model V⁰ofFsuch thatV⁰≺V. An h-minimal fuzzy N5 modelV ofFis afuzzy equilibrium modelofFifV(h, a) =V(t, a)for all atomsa.

For two fuzzy interpretationsI,J of signatureσsuch thatJ ≤^σ I, define the N5 fuzzy valuationV_J,I asV_J,I(h, a) =

a^J,1

,V_J,I(t, a) = a^I,1

for all atomsainσ.

SinceJ ≤^σI, we havea^J ≤a^I, andV_J,I(t, a)⊆V_J,I(h, a)for all atomsa.

As in [6], we assume that the fuzzy negation¬is¬s.

The following proposition relates the notions used in the fuzzy equilibrium models and the fuzzy stable models.

Proposition 1 (a) I|=₁Fif and only ifV_I,Iis a model ofF.

(b) Forp=σ,I∪J_q^p|=1F^∗(q)if and only ifVJ,I is a model ofF.

(c) For two interpretationsIandJ, we haveV_J,I ≺V_I,Iif and only ifJ < I.

The theorem below shows that the fuzzy stable model semantics can be reduced to fuzzy equilibrium logic semantics.

Theorem 4 For any fuzzy formulaF(that contains no strong negation) and any fuzzy interpretation I,I is a1-stable model ofF if and only ifVI,I is a fuzzy equilibrium model ofF.

5Strong negation can be simulated in our semantics using new atoms as illustrated in Example 5.

6For readability, we write the infix notation(xy)in place of(x, y).

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Next we show the other direction, i.e., reducing fuzzy equilibrium logic to the fuzzy stable model semantics. For any valuation V, define the fuzzy interpretationI_V as a^I^V =V⁻(h, a)for all atomsa.

Theorem 5 For any fuzzy formulaF (that contains no strong negation) and any valu- ationV, we have thatV is an equilibrium model ofFif and only if

(i) V⁺(h, a) =V⁺(t, a) = 1for all atomsa, and (ii) I_V is a1-stable model ofFrelative toσ.

Theorem 5 tells us that in the absence of strong negation, the upper bounds of both worlds in any equilibrium model are always1.

7 Properties of Fuzzy Stable Models

In this section, we show that several well-known properties of the Boolean stable model semantics can be naturally extended to the fuzzy stable model semantics.

7.1 Alternative Definition ofF^∗

Proposition 2 For any fuzzy formulaFand any fuzzy interpretationsI,JwithJ ≤^pI, – I∪J_q^p|=_y ¬F^∗(q)⊗m¬F iff I∪J_q^p|=_y¬F.

– I∪J_q^p|=_y (F^∗⊗G^∗)(q)⊗m(F⊗G) iff I∪J_q^p|=_y(F^∗⊗G^∗)(q).

– I∪J_q^p|=_y (F^∗⊕G^∗)(q)⊗_m(F⊕G) iff I∪J_q^p|=_y(F^∗⊕G^∗)(q).

This proposition tells us thatF^∗in Section 3 can be equivalently defined by treating the fuzzy operators in the uniform way:

– (¬F)^∗=¬F^∗⊗m¬F;

– (FG)^∗= (F^∗G^∗)⊗m(FG)for any binary operator.

7.2 Theorem on Constraints

In answer set programming, constraints—rules with⊥in the head—play an important role in view of the fact that adding a constraint eliminates the stable models that “vi- olate” the constraint. The following theorem is the counterpart of Theorem 3 from [5]

for fuzzy propositional formulas.

Theorem 6 For any fuzzy formulasFandG,Iis a1-stable model ofF⊗ ¬G(relative top) if and only ifIis a1-stable model ofF (relative top) andI|=₁¬G.

Example 9. Consider F = (¬sp →r q)⊗m(¬sq →r p)⊗m¬sp. Formula F has only one 1-stable model I = {(p,0),(q,1)}, which is the only 1-stable model of (¬sp→rq)⊗m(¬sq→rp)that satisfies¬spto degree1.

If we consider a more generaly-stable model, then only one direction holds.

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Theorem 7 For any fuzzy formulas F and G, if I is a y-stable model ofF ⊗ ¬G (relative top), thenIis ay-stable model ofF(relative top) andI|=_y ¬G.

Example 10. The other direction, that is, “ifIis ay-stable model ofFandI|=y¬G, thenIis ay-stable model ofF⊗ ¬G,” does not hold in general. For example, consider F =G=pand⊗to be⊗l, and interpretationI={(p,0.4)}. ClearlyIis a0.4-stable model ofpandI |=0.4¬p, butIis not a0.4-stable model ofp⊗l¬p. In fact,Iis not even a0.4-model of the formula.

7.3 Theorem on Choice Formulas

In the Boolean stable model semantics, formulas of the formp∨ ¬pare calledchoice formulas, and adding them to the program makes atoms p exempt from minimization. Choice formulas have been shown to be useful in composing a program in the

“Generate-and-Test” method. This section shows their counterpart in the fuzzy stable model semantics.

For any fuzzy atomp,Choice(p)stands forp⊕l¬sp. For any listp= (p1, . . . pn) of fuzzy atoms,Choice(p)stands for

Choice(p₁)⊗. . .⊗Choice(p_n), where⊗is any fuzzy conjunction.

The following proposition tells that choice formulas are tautological.

Proposition 3 For any fuzzy interpretationIand any listpof fuzzy atoms,I|=₁Choice(p).

Theorem 8 is an extension of Theorem 2 from [5].

Theorem 8 (a) IfI is ay-stable model ofF relative top∪q, thenI is ay-stable model ofF relative top.

(b) Iis a1-stable model ofFrelative topiffIis a1-stable model ofF⊗Choice(q) relative top∪q.

Theorem 8 (b) does not hold for arbitrary thresholdy(i.e., if “1−” is replaced with

“y−”). For example, considerF =¬s¬sqandI={(q,0.5)}. ClearlyIis a0.5-model ofF, and thusIis a0.5-stable model ofF relative to∅. However,Iis not a0.5-stable model ofF⊗_mChoice(q) =¬_s¬_sq⊗_m(q⊕_l¬_sq)relative to∅ ∪ {q}, as witnessed byJ ={(q,0)}.

Since the1-stable models ofF relative to∅ are the models ofF, it follows from Theorem 8 (b) that the1-stable modelsofF⊗Choice(σ)relative toσare exactly the 1-modelsofF.

Corollary 1 LetF be a formula of a finite signatureσ.I is a1-model ofF relative toσiffIis a1-stable model ofF⊗Choice(σ).

Example 11. Consider the formulaF =¬sp→rq. Although any interpretationIthat satisfies1−p^I ≤q^I is a1-model ofF, among them only{(p,0),(q,1)}is a1-stable

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model ofF. However, we check that all1-models ofF are exactly the1-stable models ofG=F⊗_mChoice(p)⊗_mChoice(q): G^∗(u, v)is

(¬sp→rq)⊗m(¬sp→rv)⊗m(u⊕l¬sp)⊗m(v⊕l¬sq) and forK=I∪J_uv^pq,

G^∗(u, v)^K = 1⊗m((1−p^K)→rv^K)⊗m(u^K⊕l(1−p^K))⊗m(v^K⊕l(1−q^K)).

So, forKto satisfyG^∗(u, v)to degree1,u^Kshould be at leastp^Kandv^Kshould be at leastq^K. So there does not existJ <^pq Isuch thatI∪J_uv^pq |=1G^∗(u, v), from which it follows thatIis a1-stable model ofG.

8 Conclusion

We introduced a general stable model semantics for fuzzy propositional formulas, which generalizes both the Boolean stable model semantics and fuzzy propositional logic. The syntax is the same as the syntax of fuzzy propositional logic, but the semantics defines stable modelsinstead ofmodels. The formalism allows highly configurable default reasoning involving fuzzy truth values. Our semantics, when we restrict threshold to be1 and assume all atoms to be subject to minimization, is equivalent to fuzzy equilibrium logic in the absence of strong negation, but is much more simpler. To the best of our knowledge, our representation of commonsense law of inertia involving fuzzy values is new. The representation uses nested fuzzy operators, which are not available in earlier fuzzy ASP semantics for a restricted syntax.

We showed that several traditional results in answer set programming can be naturally extended to this formalism, and expect that more results can be carried over. Future work includes implementing this language using mixed integer programming solvers or bilevel programming solvers [7].

Acknowledgements We are grateful to Joseph Babb, Michael Bartholomew, Enrico Marchioni, and the anonymous referees for their useful comments and discussions related to this paper. This work was partially supported by the National Science Founda- tion under Grant IIS-1319794 and by the South Korea IT R&D program MKE/KIAT 2010-TD-300404-001.

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