Solving Fuzzy Answer Set Programs in Product Logic

Ivor Uhliarik

Department of Applied Informatics, Comenius University, Mlynsk

a dolina, 842 48 Bratislava, Slovak Republic

Keywords:

Fuzzy ASP, Fuzzy Logic, Answer Set Programming, Product Logic, Stable Models, DPLL.

Abstract:

In recent years, foundations have been laid for a turn in logic programming paradigms in continuous domains.

Fuzzy answer set programming (FASP) has emerged as a combination of a tool for non-monotonic reasoning

and solving combinatorial problems (ASP) and a knowledge representation formalism that allows for modeling

partial truth (fuzzy logic). There have been various attempts at designing a solver for FASP, but they either

make use of transformations into optimization programs with scaling problems, operate only on ﬁnite-valued

Łukasiewicz logic, or yield only approximate answer sets. Moreover, there has been no research focused on

the product logic semantics in FASP. In this work we investigate the methods used in state-of-the-art classical

ASP solvers with the aim of designing a FASP solver for product propositional logic. In particular, we base

our approach on the conversion into fuzzy SAT (satisﬁability problem) and the fuzzy generalization of the

DPLL algorithm. Since both Łukasiewicz and (extended) G

odel logic can be embedded into product logic, the

resulting system should be able to operate on all three logics uniformly.

1 INTRODUCTION

Answer set programming (ASP) is a well-known

and popular logic programming paradigm based on

the stable model semantics (Gelfond and Lifschitz,

1988). The solid theory behind ASP (Lifschitz, 1999)

has allowed a range of effective solvers to become

well-established, such as the Potassco suite (Geb-

ser et al., 2012), smodels (Simons et al., 2002), or

DLV (Leone et al., 2006).

The main use of ASP is in modeling and solv-

ing combinatorial search problems. However, to for-

malize a problem requiring several grades of truth, or

solve a continuous optimization problem, it is desir-

able to use a system operating with real values.

One of the ideas to extend the capabilities of ASP

to continuous domains is the combination of fuzzy

logic and ASP (Van Nieuwenborgh et al., 2007b).

Thus, fuzzy answer set programming (FASP) was

born, where atoms may be assigned graded levels of

truth. The area is rather new; the most recent solver

was developed by (Mushthofa et al., 2015a) and

a real-world application was shown in (Mushthofa

et al., 2016) which focused on modeling biological

networks.

Despite numerous proposals of FASP solvers, the

design and implementation of available tools are

still far from reaching the maturity of classical ASP

solvers. One approach of solving FASP relies on

the reduction of programs into fuzzy SAT (Janssen

et al., 2011) using fuzzy extensions of Clark’s com-

pletion (Clark, 1978) and loop formulas (Lin and

Zhao, 2004). However, the deﬁnition of loop formu-

las in FASP relies on the properties of Łukasiewicz

logic and, to our knowledge, no implementation of

this approach is available.

Another approach (Blondeel et al., 2012) ana-

lyzes the complexity of inference in FASP and pro-

poses a reduction of FASP programs to bilevel linear

programming problems. The approach is limited to

Łukasiewicz logic and has been implemented in (Al-

viano and Pealoza, 2013).

There also exists a group of solver proposals that

focus on searching ﬁnite many-valued domains, such

as (Van Nieuwenborgh et al., 2007a) or (Mushthofa

et al., 2014), reﬁned in (Mushthofa et al., 2015b),

where only the latter two support disjunctive FASP

programs and have available implementations.

A method has been proposed that ﬁnds approxi-

mations of fuzzy answer sets (Alviano and Pealoza,

2013), which has also been implemented, but it oper-

ates only on normal programs and yields exact results

only in the case of positive and stratiﬁed programs.

Note that to date, this is the only implemented ap-

proach where any of Łukasiewicz, G

odel, and product

t-norms may be used.

Uhliarik I.

Solving Fuzzy Answer Set Programs in Product Logic.

DOI: 10.5220/0006518303670372

In Proceedings of the 9th International Joint Conference on Computational Intelligence (IJCCI 2017), pages 367-372

ISBN: 978-989-758-274-5

It is clear that the furthest developed proposals are

those based on Łukasiewicz logic. Overall, to the best

of our knowledge, there is no ad-hoc design or imple-

mentation of an exact FASP solver for programs with

odel or product t-norms, i.e. the only existing solver

is limited to stratiﬁed negation and relies on black-box

optimization techniques.

A promising way to ﬁll this gap is to adopt the

mentioned approach of reducing the FASP program

into a fuzzy SAT problem (Janssen et al., 2011). How-

ever, we focus on solving programs with product t-

norms, as both Łukasiewicz and (extended) G

odel

logics have a faithful interpretation in (extended)

product logic (Baaz et al., 1998). Hence, the solver

would be able to uniformly process all three seman-

tics. The steps in the reduction of the FASP pro-

gram need to be generalized to comply with the prop-

erties of product logic. Having available the result

of the reduction, we do not rely on the potential ex-

istence of a fuzzy SAT solver—instead, the fuzzy

theory is then translated into clausal form accord-

ing to (Guller, 2013) and we make use of the Davis-

Putnam-Logemann-Loveland (DPLL) procedure for

propositional product logic (also suggested by Guller)

with modiﬁcations.

Note that this is a work in progress: we describe

our aim and the key concepts and identify the prob-

lems that need to be solved, but we omit the proofs

and implementation details.

2 FUZZY ANSWER SET

PROGRAMMING

In this section we describe the syntax and semantics

of fuzzy answer set programs. We focus on prod-

uct logic semantics, but otherwise we use the deﬁ-

nitions from (Janssen et al., 2012) and the notation

from (Guller, 2013).

2.1 Syntax

Let L be a lattice. In this work we will focus on the

lattice L =

[0, 1],≤

. Let PropAtom be the set of

propositional atoms of product logic over L.

Next, let C = {c|c ∈ C} be a set of truth constants

where {0, 1} ⊆ C ⊆ [0, 1] and C is a countable set;

0, 1 ∈ PropAtom are true and false in product logic,

respectively. A (classical) literal is either a constant

symbol c ∈ C, an atom a or a classical negation literal

¬a. An extended literal is either a classical literal a or

a default negation literal not a.

FASP rules are expressions of the form

r ≡ a ← f (b

, ..., b

, ..., c

)

where a, b

, c

∈ PropAtom for all 1 ≤ i ≤ n, 1 ≤ j ≤ m,

f is a function symbol representing a total mapping

n+m

→ L increasing in its n ﬁrst and decreasing in

its m last arguments. Thus, the atom a is either a

constant or a classical literal, the atoms b

, ..., b

are

classical literals, and the atoms c

, ..., c

are default

negation literals. The head/body of the rule r, r

(also H(r) / B(r)) is the left-hand/right-hand side of

the rule. The Herbrand base of a rule r, B

, is the set

of atoms occurring in r. A rule of the aforementioned

form is called

• a constraint if a ∈ C,

• a fact if all b

, c

∈ C for 1 ≤ i ≤ n, 1 ≤ j ≤ m,

• positive if m = 0 or c

∈ C for 1 ≤ j ≤ m

• simple if it is positive and not a constraint.

A FASP program is a ﬁnite set of FASP rules.

Given a FASP program P, the Herbrand base B

|r ∈ P}. A program is called

• constraint-free if it does not contain constraints,

• positive if all rules occurring in it are positive,

• simple if all rules occurring in it are simple.

2.2 Semantics

We interpret product logic by the Π-algebra

Π = ([0, 1], ≤,∨

∨

∨,∧

∧

∧, ·,⇒

⇒

⇒,∼

∼

∼, 0, 1)

where ∨

∨

∨ is the supremum and ∧

∧

∧ the inﬁmum operator

on [0, 1]; · is the standard multiplication of reals;

a⇒

⇒

⇒b =

(

1 if a ≤ b,

else;

∼

∼ a =



1 if a = 0,

0 else.

Hence, the mapping f in the rule

a ← f (b

, ..., b

, ..., c

)

constructs a body expression recursively:

• a constant c ∈ C and an extended literal are body

expressions,

• if α and β is a body expression, then α  β is also

a body expression for  ∈ {∧, &, ∨, →, ↔}

where ∧, ∨, →, ↔ are standard propositional connec-

tives and & is strong conjunction.

An interpretation of a FASP program P is a

→ L mapping I = {a

, ..., a

} deﬁned as I(a

) =

if 1 ≤ i ≤ n and I(a) = 0 otherwise. We extend I to

constants and expressions as follows:

If the rule has no negative part or consists of constants.

• I(c) = c if c ∈ C

• I(not α) =∼

∼

∼ I(α)

• I(α& β) = I(α)· I(β)

• I(α  β) = I(α)



I(β) for  ∈ {∧, ∨, →}

• I(α ↔ β) = I(α)⇒

⇒

⇒I(β)· I(β)⇒

⇒

⇒I(α)

for expressions α and β. Recall that Π is a complete

linearly ordered lattice algebra; ∨

∨

∨, ∧

∧

∧ is commutative,

associative, idempotent, monotone; 0, 1 is its neutral

element; · is commutative, associative, monotone; 1

is its neutral element; the residuum operator ⇒

⇒

⇒ of ·

satisﬁes the condition of residuation:

for all a, b, c ∈ Π, a·b ≤ c ⇐⇒ a ≤ b⇒

⇒

⇒c; (1)

A rule (r : a ← α) ∈ P is satisﬁed by an inter-

pretation I of P iff I(a) ≥ I(α)

. An interpretation

I of program P is a model of P iff every rule r ∈ P

is satisﬁed by I. For interpretations I and J of P we

deﬁne I ⊆ J iff ∀a ∈ B

: I(a) ≤ J(a) and I ⊂ J iff

(∀a ∈ B

: I(a) ≤ J(a)) ∧(I 6= J). Given this ordering

on interpretations, we say a model I of P is minimal

iff no model J exists such that J ⊂ I.

Let P be a positive FASP program. An interpre-

tation A of P is called the answer set of P iff A is

the minimal model of P. For non-positive programs

we use the fuzzy generalization of the GL reduct.

For a non-positive program P, the reduct of a rule

(r : a ← f (b

, ..., b

, ..., c

)) ∈ P w.r.t. an interpre-

tation I of P is the positive rule r

deﬁned as

= r : a ← f (b

, ..., b

;I(c

), ..., I(c

))

i.e. the occurrences of default negation literals not a

are replaced by the constants I(not a) ∈ L. The reduct

of P is the set of rules P

deﬁned as P

= {r

|r ∈ P}.

An interpretation A of a program P is an answer set

of P iff A is the answer set of P

A simple FASP program has exactly one fuzzy an-

swer set. A positive FASP program may have no, one,

or several fuzzy answer sets.

3 SOLVING FASP

In the introduction we have brieﬂy covered a number

of approaches to solving FASP programs. We have

suggested an approach that would make use of the

reduction of a FASP program into a fuzzy SAT in-

stance (Janssen et al., 2011), but (1) we aim to cover

product (instead of Łukasiewicz) propositional logic

semantics, as there exist embeddings of Łukasiewicz

and extended G

odel logics in product logic (Baaz

This is because we can regard rules as residual impli-

cators.

et al., 1998); (2) we do not rely on the existence of a

potential fuzzy SAT solver—instead, we modify and

integrate the proposed DPLL procedure for product

logic (Guller, 2013); and (3) we study how the em-

beddings could be integrated to lay foundations for a

uniform solver.

The pipeline of the full system shall consist of:

1. reduction of the input FASP program into product

propositional fuzzy logic theory,

2. translation of the theory into clausal form,

3. performing the DPLL procedure for product logic

to ﬁnd a valuation of the atoms which corresponds

to an answer set of the program.

In this section we describe the steps above and iden-

tify the problems that need to be solved in order for

such system to be integrated.

3.1 Reducing FASP to Fuzzy SAT

A theory has been developed (Janssen et al., 2011)

with the idea to translate FASP programs into proposi-

tional fuzzy logic theories whose models correspond

to the answer sets of the original programs. This ap-

proach assumes the availability of a fuzzy SAT solver.

The work generalizes the popular methods used in

classical ASP solvers to the fuzzy case, speciﬁcally

those introduced in (Lin and Zhao, 2004), known as

the ASSAT method. In particular, the contributions of

the approach are the following:

1. The deﬁnition of the completion of a FASP pro-

gram, where the authors also show that the answer

sets of FASP programs without loops are exactly

the models of its completion.

2. The generalization of loop formulas from (Lin

and Zhao, 2004) allowing for the computation

of answer sets of arbitrary FASP programs. The

authors also generalize the ASSAT procedure to

overcome the problem with an exponential num-

ber of loops.

While loop formulas in FASP are formulated us-

ing the unfounded-set semantics, the generalization of

the ASSAT procedure is based on the ﬁxpoint seman-

tics. Hence, the paper also shows that these two coin-

cide in FASP.

The approach in (Janssen et al., 2011) introduces a

ﬂexible framework, as in theory, the only limitations

imposed on the input FASP program P are that (1)

P has to be normal (disjunction cannot appear in the

head of any rule), and (2) the only connectives occur-

ring in P are t-norms. However, in the deﬁnition of

the loop formula in the fuzzy case, an equivalence is

used that is preserved in Łukasiewicz logic, but not in

odel or product logic:

Deﬁnition 1. (Janssen et al., 2012)

(Loop Formula). Let P be a FASP program and

L = {l

, ..., l

} a loop of P. Suppose that R

−

(L) =

, ..., r

}. Then the loop formula induced by loop

L, denoted by LF(L, P), is the following fuzzy logic

formula:

I (max(l

, ..., l

), max((r

)

, ..., (r

)

)) (2)

where I is an arbitrary residual implicator. If

−

(L) =

0, then loop formula becomes

I (max(l

, ..., l

), 0)

Proposition 1. (Janssen et al., 2012)

The loop formula proposed for boolean answer set

programs is of the form

¬(

)

∨ ... ∨

)

) ⇒ (¬l

∨ ... ∨ ¬l

) (3)

which is equivalent to

∨ ... ∨ l

) ⇒ (

)

∨ ... ∨

)

) (4)

This equivalence is preserved in Łukasiewicz logic,

but not in G

odel or product logic.

To extend the deﬁnition to be fully ﬂexible, further

research into generalizing the notion of loop formulas

is required.

3.2 Solving Fuzzy SAT

The reduction approach implies the necessity to

use a solver for the fuzzy SAT problem. For

each logic, however, only few such solvers ex-

ist. According to (Janssen et al., 2012), for

odel logic we can use boolean SAT solvers, for

Łukasiewicz logic we can use mixed integer program-

ming (MIP) (H

ahnle, 1994), and for product logic the

bounded mixed integer quadratically constrained pro-

gramming (bMICQP) used for fuzzy description log-

ics (Bobillo and Straccia, 2007). According to (Al-

viano and Pealoza, 2013), these approaches introduce

many auxiliary variables that may negatively affect

the performance. Moreover, the optimization pro-

grams run as black boxes without any ad-hoc opti-

mizations suitable for the given propositional logic.

Another numerical approach to solving fuzzy SAT

is based on the state-of-the-art black-box optimiza-

tion algorithm on a continuous domain, Covariance

Matrix Adaptation Evolution Strategy (CMA-ES) and

its extension resulting from landscape analysis (Brys

et al., 2013). The approach focuses on Łukasiewicz

logic, but the paper also benchmarks the case for

product logic semantics. The downside of this ap-

proach is the stochastic nature of the algorithm that is

prone to converging to local optima (Hansen, 2006).

Formula (2) in deﬁnition 1 is the fuzzy generalization

of formula (4) in proposition 1.

3.2.1 DPLL-based SAT Solver

A different approach to solving the fuzzy SAT prob-

lem (Guller, 2013) proposes a variant of the DPLL

procedure operating over order product clausal the-

ories. The paper introduces the transformation of a

product propositional theory to order clausal form, es-

tablishes the inference (branching) rules and proposes

a method to compute the valuation of atoms. The pro-

cedure is proved to be refutation sound and complete

for ﬁnite order clausal theories.

The problem with this approach is that it lacks

the notion of intermediate constants, i.e. the terms

of the input theory are limited to propositional atoms,

the constants 0, 1, and expressions built from these

terms using logical connectives—there is no support

for intermediate truth constants

which are present in

most practical applications of FASP. To incorporate

this notion into the fuzzy DPLL procedure, we would

need to modify the theory (branching rules, valuation,

and associated proofs) and study the properties under

which the modiﬁcations are admissible.

Another calculus used for automated deduction

in (G

odel) fuzzy logics is hyperresolution (Guller,

2017). Although the work is concerned with the

odel t-norm, the paper extends the translation of a

fuzzy propositional theory to clausal form and the hy-

perresolution calculus by incorporating intermediate

truth constants, therefore the notions serve as motiva-

tion for the enhancement of the DPLL procedure for

the product case.

4 EMBEDDINGS

A key concept that is to be explored is the embedding

of Łukasiewicz and (extended) G

odel logics in (ex-

tended) product logic driven by the motivation that

once a solver for extended product FASP is imple-

mented, we would be able to uniformly process all

three popular semantics. In this section we deﬁne the

extension of product propositional logic, extend the

axioms of product logic, and cite the theorem stating

that Łukasiewicz logic is a sublogic of extended prod-

uct logic.

4.1 Extended Product Logic

The extended product logic is interpreted by the stan-

dard Π-algebra augmented by the operators P

P, ≺

≺

≺, ∆

∆

for the connectives P, ≺, ∆, respectively.

∆

= ([0, 1], ≤,∨

∨

∨,∧

∧

∧, ·,⇒

⇒

⇒,∼

∼

∼,P

P,≺

≺

≺,∆

∆

∆, 0, 1)

Constants in the open interval (0,1).

where ∨

∨

∨ is the supremum and ∧

∧

∧ the inﬁmum operator

on [0, 1];

x⇒

⇒

⇒y =



1 if x ≤ y,

else;

∼

∼ x =



1 if x = 0,

0 else;

P y =



1 if x = y,

0 else;

x≺

≺

≺y =



1 if x < y,

0 else;

∆

∆x =



1 if x = 1,

0 else.

Similarly to section 2.2, recall that Π is a complete

linearly ordered lattice algebra; ∨

∨

∨, ∧

∧

∧ is commutative,

associative, idempotent, monotone; 0, 1 is its neutral

element; · is commutative, associative, monotone; 1

is its neutral element; the residuum operator ⇒

⇒

⇒ of ·

satisﬁes the residuation principle. G

odel negation ∼

∼

satisﬁes the condition:

for all x ∈ Π

∆

, ∼

∼

∼ x = x⇒

⇒

⇒0;

∆

∆ satisﬁes the condition:

for all x ∈ Π

∆

, ∆

∆

∆x = xP

P 1.

4.2 Embeddings

In this section, we shall use the notation from (H

ajek,

2001). We extend the axioms of product logic by the

following axioms:

∆ϕ ∨ ¬∆ϕ (∆1)

∆(ϕ ∨ ψ) → (∆ϕ ∨ ∆ψ) (∆2)

∆ϕ → ϕ (∆3)

∆ϕ → ∆∆ϕ (∆4)

∆(ϕ → ψ) → (∆ϕ → ∆ψ) (∆5)

(Baaz et al., 1998) deﬁne how Łukasiewicz logic

can be embedded in this extended product logic, i.e.

how the Łukasiewicz t-norm can be isomorphically

transformed to restricted product on [a, 1] for arbitrary

ﬁxed 0 < a < 1. For each formula ϕ with proposi-

tional variables in {p

, ..., p

} a translation ϕ

∗

is de-

ﬁned using one new propositional variable p

. Let:

∗

≡ p

∗

≡ p

∨ p

(ϕ ∧ ψ)

∗

≡ p

∨ (ϕ

∗

∧ ψ

∗

)

(ϕ ∧ ψ)

∗

≡ ϕ

∗

→ ψ

∗

(¬ϕ)

∗

≡ ϕ

∗

→ p

for i ∈ {1, ..., n}.

Then, the following theorem holds:

We assume a decreasing operator precedence: ∼

∼

∼, ∆

∆

∆, ·,

P, ≺

≺

≺, ∧

∧

∧, ∨

∨

∨, ⇒

⇒

⇒.

Theorem 1. (Baaz et al., 1998)

Let ϕ

†

denote the formula ¬¬p

→ ϕ

∗

. For each

formula ϕ not containing p

, ϕ is 1-tautology of

Łukasiewicz logic iff ϕ

†

is a 1-tautology of product

logic.

As such, Łukasiewicz logic has a faithful interpre-

tation in product logic. Similarly, it is also shown how

to embed G

odel (extended by ∆) logic into product

logic (Baaz et al., 1998).

5 CONCLUSIONS

Our solution is based on product propositional logic

extended by the operation ∆. As we have described

in section 4.2, we have that Łukasiewicz and G

odel

logic are both sublogics of the extended product logic

∆

. Given this, we shall deﬁne Π

∆

-FASP programs

and corresponding answer set semantics. With the

aim of developing a FASP solver based on this logic,

the ﬁnal system should be able to uniformly handle

Łukasiewicz, G

odel, and product logics.

As shown in (Guller, 2017), another non-trivial

problem is that of the incorporation of intermediate

truth constants belonging to a countable subset of the

open interval (0, 1), the possibility to use them in the

bodies and heads of rules of product FASP programs,

and their handling in the computation of answer sets.

The occurrence of these truth constants in fact or con-

straint rules is a trivial matter to a FASP solver (as

such, it is merely a condition in the problem of opti-

mization). However, the properties of product logic

will have to be reviewed for possible violations and

required modiﬁcations and proofs.

We shall base the product fuzzy answer set pro-

gramming solver on the reduction of a program to

a fuzzy theory as described in (Janssen et al., 2011)

and formulate the notion of loop formulas for prod-

uct logic. Next, we transform the theory into clausal

form as proposed in (Davis et al., 1962) and ﬁnd

a model satisfying the fuzzy theory using the well-

known DPLL procedure extended with intermediate

constants. That is, we formalize the fuzzy generaliza-

tion of DPLL for product logic and enhance it to allow

for ad-hoc computation of stable models in product

FASP.

ACKNOWLEDGEMENTS

The research reported in this paper was supported by

the grant UK/367/2017.

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