# A Glimpse of Answer Set Programming.

**ABSTRACT** sentation and reasoning. To solve a problem, a programmer designs a logic program so that models of the program determine solutions to the problem. ASP has been identi ed in the late 1990s as a subarea of logic programming and is becoming one of the fastest growing elds in knowledge representation and declarative programming. Major advantages of ASP are (1) its simplicity, (2) its ability to model e ectively incomplete speci cations and closure constraints, and (3) its relation to constraint satisfaction and propositional satis ability, which allows one to take advantage of advances in these areas when designing solvers for ASP systems.

**0**Bookmarks

**·**

**173**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**This document provides an introduction to the Answer Set Programming (ASP) tools gringo, clasp, clingo, and iclingo, developed at the University of Potsdam. The first tool, gringo, is a grounder capable of translating logic pro- grams provided by users into equivalent propositional logic programs. The answer sets of such programs can be computed by clasp, which is a solver. The third tool, clingo, integrates the functionalities of gringo and clasp, thus, acting as a monolithic solver for user programs. Finally, iclingo extends clingo by an incremental mode that incorporates both grounding and solving. For one, this document aims at enabling ASP novices to make use of the aforementioned tools. For another, it provides a reference of their features that ASP adepts might be tempted to exploit. - SourceAvailable from: psu.edu
- SourceAvailable from: Ferda Nur Alpaslan[Show abstract] [Hide abstract]

**ABSTRACT:**Model generation theorem provers have the capability of pro-ducing a model when the first-order input theory is satisfiable. Because grounding step may generate huge propositional instances of the program it hardens the search process of answer set solvers. We propose the use of model generation theorem provers as computational engines for Answer Set Programming (ASP). It can be seen as lifting of SAT-based ASP to the first-order level for tight programs to eliminate the grounding step of ASP or do it more intelligently.01/2007;

Page 1

A Glimpse of Answer Set Programming

Christian Anger and Kathrin Konczak and Thomas Linke and Torsten Schaub

Answer Set Programming (ASP) is a declarative paradigm for solving search problems appearing in knowledge repre-

sentation and reasoning. To solve a problem, a programmer designs a logic program so that models of the program

determine solutions to the problem. ASP has been identified in the late 1990s as a subarea of logic programming and is

becoming one of the fastest growing fields in knowledge representation and declarative programming. Major advantages

of ASP are (1) its simplicity, (2) its ability to model effectively incomplete specifications and closure constraints, and

(3) its relation to constraint satisfaction and propositional satisfiability, which allows one to take advantage of advances

in these areas when designing solvers for ASP systems.

1Introduction

Answer Set Programming (ASP) emerged in the late 1990s

as a new logic programming paradigm [22, 34, 35, 28], ha-

ving its roots in nonmonotonic reasoning, deductive databa-

ses and logic programming with negation as failure. Since its

inception, it has been regarded as the computational embo-

diment of nonmonotonic reasoning and a primary candidate

for an effective knowledge representation tool. This view has

been boosted by the emergence of highly efficient solvers

for ASP [50, 17]. It now seems hard to dispute that ASP

brought new life to logic programming and nonmonotonic

reasoning research and has become a major driving force for

these two fields, helping to dispel gloomy prophecies of their

impending demise.

The basic idea of ASP is to represent a given compu-

tational problem by a logic program whose answer sets cor-

respond to solutions, and then use an answer set solver for

finding answer sets of the program. This approach is close-

ly related to the one pursued in propositional satisfiability

checking (SAT), where problems are encoded as propositio-

nal theories whose models represent the solutions to the gi-

ven problem. Even though syntactically, ASP programs look

like Prolog programs, they are treated by rather different

computational mechanisms. Indeed, the usage of model ge-

neration instead of query evaluation can be seen as a recent

trend in the encompassing field of knowledge representation

and reasoning. ASP is particularly suited for solving difficult

combinatorial search problems. Among these, we find appli-

cations to plan generation, product configuration, diagnosis,

and graph-theoretical problems.

2 Background

We restrict the formal development of ASP to propositional

(normal) logic programs consisting of rules of the form

p0 ← p1,...,pm,not pm+1,...,not pn ,

(1)

where n ≥ m ≥ 0, and each pi (0 ≤ i ≤ n) is an atom.

Given such a rule r, we let head(r) denote the head, p0, of r

and body(r) the body, {p1,...,pm, not pm+1,...,not pn},

of r. Also, let body+(r) = {p1,...,pm} and body−(r) =

{pm+1,...,pn}. The intuitive reading of such a rule is that

of a constraint on an answer set: If all atoms in body+(r)

are included in the set and no atom in body−(r) is in it, then

head(r) must be included in the answer set.

Answer sets as such are defined via a reduction to negation-

as-failure-free programs: A logic program is called basic if

body−(r) = ∅ for all its rules. A set of atoms X is closed

under a basic program Π if for any r ∈ Π, head(r) ∈ X

whenever body+(r) ⊆ X. The smallest set of atoms which

is closed under a basic program Π is denoted by Cn(Π) and

constitutes the answer set of Π.

For the general case, we need the concept of a reduct of

a program Π relative to a set X of atoms:

ΠX= {head(r) ← body+(r) | r ∈ Π,body−(r) ∩ X = ∅}.

With these formalities at hand, we can define answer set

semantics for logic programs: A set X of atoms is an answer

set of a program Π if Cn(ΠX) = X. This definition is due

to [22], where the term stable model is used; the idea traces

back to [45]. In fact, one may regard an answer set as a

model of a program Π that is somehow “stable” under Π.

In other words, an answer set is closed under the rules of Π,

and it is “grounded in Π”, that is, each of its atoms has a

derivation using “applicable” rules from Π.

For illustration, consider the program

Π = {p ← p, q ← not p} .

Among the four candidate sets, we find a single answer set,

{q}, as can be verified by means of the following table:

X

ΠX

∅

q

←

{p}

{q}

q

←

{p,q}

Cn(ΠX)

{q}

p

←

p

p

p

←

←

p

p

∅

{q}

p

←

p

∅

Seite 1

Page 2

A noteworthy fact is that posing the query p or q to Π in a

Prolog system leads to non-terminating situation due to its

top-down approach.

Analogously, we may check that the program

Π1 = {p ← not q, q ← not p}

has the two answer sets {p} and {q}.

X

∅

ΠX

p

q

p

q

Cn(ΠX)

{p,q}←

←

←

←

{p}

{q}

{p,q}

{p}

{q}

∅

The two rules in Π1 are mutually exclusive and capture an

indefinite situation: p can be added unless q has been added

and vice versa. We show in the next section how this can be

exploited in modeling problems in ASP.

Unlike the previous examples,

Π2 = {p ← not p}

admits no answer set. Interestingly, Π2 offers a straightfor-

ward way to model integrity constraints of form

← p1,...,pm,not pm+1,...,not pn .

(2)

This can be done by introducing a new atom f and replacing

(2) by rule f ← not f,p1,...,pm,not pm+1,...,not pn .

Whenever the integrity constraint in (2) is violated by a can-

didate set this set is eliminated by the effect observed on

program Π2. The usefulness of integrity constraints can be

observed by adding ← p to program Π1. In fact, the integri-

ty constraints ← p eliminates the original answer set {p} of

Π1, so that Π1∪ {← p} yields a single answer set {q}.

Although general ASP is principally computationally com-

plete, that is, it can simulate arbitrary Turing machines [6],

one usually deals with decidable fragments. Commonly, we

consider rules with variables that are taken as abbreviations

for all ground instances over a finite set of constants. The

propositional fragment of ASP defined above allows for en-

coding all decision and search problems within NP [48, 33].

3Modeling

The basic approach to writing programs in ASP follows a

“generate-and-test” strategy. First, one writes a group of

rules whose answer sets would correspond to candidate solu-

tions. Then, one adds a second group of rules, mainly consi-

sting of integrity constraints, that eliminates candidates re-

presenting invalid solutions.

As an example consider the well-known n-queens pro-

blem. The goal is to place n queens on an n×n chessboard

so that no two queens appear on the same row, column, or

diagonal. Let us represent the positioning of the queens by

atoms of the form q(i,j), where 1 ≤ i,j ≤ n. That is, an

atom q(i,j) represents that a queen is at position (i,j).

Let us first give a “generator”. To this end, we build

upon the non-deterministic behavior observed on program

Π1. Also, we introduce an auxiliary atom q?(i,j) indicating

that there is no queen at (i,j). Finally, we need a “domain

predicate” d indicating the dimension of the chessboard. Ac-

cordingly, we obtain:

q(X,Y )

q?(X,Y )

←

←

d(X),d(Y ),not q?(X,Y )

(3)

d(X),d(Y ),not q(X,Y )

(4)

So, taking a 1×1 chessboard by adding d(1), we obtain from

(3) and (4) two answer sets, {q(1,1),d(1)} and {q?(1,1),d(1)},

just as with program Π1. Accordingly, we obtain by adding

d(1) and d(2) a 2×2 chessboard, generating 16 answer sets,

among which we find:

{q?(1,1),q?(1,2),q?(2,1),q?(2,2),d(1),d(2)}

(5)

{q(1,1),q?(1,2),q?(2,1),q?(2,2),d(1),d(2)}

q

(6)

{q(1,1),q?(1,2),q?(2,1),q(2,2),d(1),d(2)}

q

q

(7)

{q(1,1),q(1,2),q(2,1),q(2,2),d(1),d(2)}

q

q

q

q

(8)

Now, the “tester” must eliminate candidate sets in which

queens are positioned on the same row, column, or diago-

nal. This can be done by means of the following integrity

constraints:

←

q(X,Y ),q(X?,Y ),X??= X,

d(X),d(Y ),d(X?),d(Y?)

q(X,Y ),q(X,Y?),Y??= Y,

d(X),d(Y ),d(X?),d(Y?)

q(X,Y ),q(X?,Y?),|X − X?| = |Y − Y?|,

X??= X,Y??= Y,d(X),d(Y ),d(X?),d(Y?)

(9)

←

(10)

←

(11)

In fact, all rules rule out candidate set (8). Rule (11) eli-

minates set (7). However, none of them accounts for the

requirement that n queens must be put on the board. This

can be achieved by the following pair of rules.

←

←

d(X),hasq(X)

hasq(X)d(X),d(Y ),q(X,Y )

The two latter rules eliminate candidate sets (5) and (6).

4 Language Extensions

Classical negation. Normal logic programs provide negati-

ve information implicitly through the closed world assump-

tion [44]. Consider the following rule: cross ← not train.

If train is not derivable, the atom cross becomes true. But

this may lead to a disaster because you have no explicit infor-

mation that there really is no train. An alternative would be

to use an explicit negation operator ¬. Then we can express

the previous rule as follows: cross ← ¬train.

Seite 2

Page 3

An atom p or a negated atom ¬p is called a literal.

Logic programs with literals are called extended logic pro-

grams [22]. An extended logic program is contradictory [5]

if complementary literals, e.g. train and ¬train, are deri-

vable. In that case, one obtains exactly one answer set, viz

the set of all literals. If a program is not contradictory, the

definition of answer sets of extended logic programs carries

over from normal programs.

Classical negation can be eliminated by a polynomial

transformation, replacing each negated atom ¬p by a new

atom p?and adding the rules,

q ← p,p?

and

q?← p,p?,

(12)

for each atom p and q. The rules in (12) generate the set of

all literals in case of contradictory programs. For preserving

only consistent answer sets, we may add constraints ← p,p?

for each atom p, instead of the rules in (12).

Disjunctive logic programs extend normal programs by dis-

junctive information in the head of a rule [22]. More precisely,

the head of a rule is a disjunction q0;...;qk for atoms qi,

where 0 ≤ i ≤ k. E.g. p;q expresses that “p is true or q

is true”. Letting head(r) = {q0,...,qk}, a set of atoms

X is closed under a basic program Π if for any r ∈ Π,

head(r) ∩ X ?= ∅ whenever body+(r) ⊆ X. The definition

of ΠXcarries over from normal programs. An answer set X

of a disjunctive logic program is a ⊆- minimal set of atoms

being closed under ΠX. For example, the disjunctive logic

program Π = {p;q ←} has the answer sets {p} and {q}.

The set {p,q} is closed under Π{p,q}, but it is not an answer

set of Π since it is not ⊆-minimal. Observe that adding the

rules p ← and q ← to Π makes {p,q} the only answer set

of Π ∪ {p ←,q ←}. If we use disjunction in the n-queens

problem, rules (3) and (4) can be replaced by one rule:

q(X,Y ); q?(X,Y )

←

d(X),d(Y )

The usage of disjunction raises the complexity of the un-

derlying decision problems, e.g. deciding whether there exists

an answer set X for a given atom p such that p ∈ X is ΣP

complete [18].

2-

Nested logic programs are logic programs where the bodies

and heads of rules may contain arbitrary boolean expressions

formed from propositional atoms and the symbols ? (true)

and ⊥ (false) using negation-as-failure (not), conjunction

(,), and disjunction (;) [30]. Answer sets are similarly defined

as for disjunctive programs under regard that every boolean

expression must be satisfied in the sense of classical logic and

that the reduct ΠXis operating on boolean expressions. For

illustration, consider nested program Π = {(p;not p) ←}.

Taking X = ∅, we obtain Π∅= {(p;?) ←} as reduct and ∅

as the only ⊆- minimal set being closed under Π∅, that is,

the boolean expression (p;?) is trivially satisfied by X = ∅.

For X = {p}, we get Π{p}= {(p;⊥) ←} and {p} as the

⊆-minimal set satisfying the expression (p;⊥). Hence, ∅ and

{p} are the answer sets of nested program {(p;not p) ←}.

In the n-queens problem, rules (3) and (4) can be replaced

by q(X,Y ); not q(X,Y ) ← d(X),d(Y ) , where the usage

of nested expressions avoids using auxiliary predicate q?.

Nested programs can be polynomially translated into dis-

junctive programs [39].

Cardinality constraints are extended literals [49]. They are

of the form l {q1,...,qm} u, for m ≥ 1, where l,u are

lower and upper bounds on the cardinality of subsets of

{q1,...,qm} satisfied in an encompassing answer set. They

can appear in the head or in the body of a rule. A car-

dinality constraint is satisfied in an answer set X, if the

number of atoms from {q1,...,qm} belonging to X is bet-

ween l and u. To ensure in the n-queens problem that ex-

actly one queen is in every column j, we use the expres-

sion 1{q(1,j),...,q(n,j)}1, which can be abbreviated by

1{q(X,j) : d(X)}1, a so-called conditional literal [49]. Hence,

the n-queens problem can be encoded with three rules, na-

mely rule (11) and

1{q(X,Y ) : d(X)}1

1{q(X,Y ) : d(Y )}1

←

←

d(Y )

(13)

d(X),

(14)

where rules (13) and (14) ensure that there is exactly one

queen in every column and row, respectively.

Deciding whether a normal program with cardinality cons-

traints has an answer set is NP-complete [49].

Preferences. The notion of preference is pervasive in common-

sense reasoning, e.g. in decision making, in part because

preferences constitute a very natural and effective way of

resolving indeterminate situations. In the following, we will

exemplarily consider preferences among rules and so-called

ordered disjunctions.

Rule preferences. A logic program with (static) prefe-

rences is a pair (Π,<) where Π is a logic program and < is a

strict partial order among rules of Π expressing that one rule

has higher priority than another rule. Also, dynamic prefe-

rences can be modeled by taking a special purpose predicate

prec instead of an external order <. In both cases, the idea

is to apply a rule r only if the “question of applicability” has

been settled for all higher preferred rules r?. See [47, 14] for

a survey on strategies for rule preferences.

Ordered disjunctions allow to represent alternative, ran-

ked options in the head of rules [10]. p × q is an ordered

disjunction which means: if possible p, but if p is impossi-

ble then at least q. Ordered disjunctions can be used, e.g.

for the problem of composing a menu. You can choose bet-

ween meat or vegetarian food. In case of choosing meat

you would prefer wine over mineral water, otherwise you

prefer mineral water over wine. These preferences can be

modeled with ordered disjunctions as follows:

meat × vegetarian

wine × water

water × wine

←

←

←

meat

not meat

This program has answer set {meat,wine}. But in the case

where meat is not possible, e.g. adding the integrity cons-

traint ← meat, the answer set is {vegetarian,water}.

Further preference handling approaches in answer set

programming are preferences among literals [46], ordered

Seite 3

Page 4

choice logic programs [8], the preference description langua-

ge PDL [9] for specifying complex preferences structures in

optimization problems, and consistency-restoring rules [4].

Other language extensions include aggregate functions [15],

like sum, count, or min, weak constraints [27] as a variant

of integrity constraints, and weight constraints [49] as an

extension of cardinality constraints.

5 Systems

In recent years, several systems for answer sets computation

have become available, including smodels [50, 49], dlv [17,

27], noMoRe [38, 32], assat [2, 31], and cmodels [12]. Most

of the currently known ASP systems are designed to compute

answer sets for propositional logic programs. In order to deal

with programs containing variables, the systems rely on a

two phase implementation consisting of:

1. elimination of variables for obtaining propositional pro-

grams and handling special system dependent langua-

ge extensions (see Section 4); and

2. computation of answer sets for propositional programs.

Before detailing the second phase, we shortly report on the

main tools for variable elimination, so-called grounders. Current-

ly, most ASP systems utilize lparse (the grounder coming

with the smodels system) or the grounder included in dlv.

Both grounders allow for parsing disjunctive logic programs

with classical negation along with their specific language ex-

tensions.

The remainder of this section describes the second pha-

se, dealing with answer set generation of propositional logic

programs. For simplicity, we deal with normal programs only.

In this case, answer sets are (sub)sets of atoms occurring in

a given program as heads of rules. A partial model is defined

as a 3-valued truth assignment for the atoms of a program

with truth values true, false and undefined. Such a 3-valued

model is total if it contains no undefined atoms. General-

ly, the answer set computation in the second phase aims at

extending a given partial model to a total one which is an

answer set (or determining that this is impossible). The com-

putation can be decomposed into alternating deterministic

and non-deterministic parts. We start with the partial model

where all atoms are undefined. Then we try to extend it to an

answer set by computing its deterministic consequences by

means of propagation techniques. These techniques extend

the partial model by increasing the number of atoms being

true or false, respectively. If the obtained model is total, then

it is returned as an answer set. If the model is contradictory,

in the sense that an atom has been assigned both true and

false, then we have detected a situation admitting no ans-

wer sets. Otherwise, we must non-deterministically choose

an undefined atom and “branch” on its possible truth va-

lues true and false. This is done by a case analysis which

recursively applies the described procedure once to the par-

tial model where the chosen atom is assigned true and again

to the partial model where the chosen atom is false1.

1The above procedure is similar to the Davis-Putnam-

Logemann-Loveland procedure [13] for SAT solvers.

Observe that all ASP solvers additionally utilize some

heuristics to guide their choices. The actual heuristics is cru-

cial for the overall system performance, since the number

of choices determines the depth of the (exponential) search

tree. For a detailed discussion of different ASP heuristics

including experimental results see [20].

The smodels system. The basic smodels algorithm can be

described as follows:

smodels(L,U)

1 expand(L,U)

2 if L ?⊆ U then return fail

3 if L = U then exit with L

4 A ← select(U \ L)

5 smodels(L ∪ {A},U)

6 smodels(L,U \ {A})

It computes an answer set between lower bound L and up-

per bound U, or determines that this is impossible. That

is, the two sets L and U aim at capturing all answer sets

X such that L ⊆ X ⊆ U. Observe that partial models are

represented through two sets of atoms L and U, where L

contains all true atoms and U contains all atoms not yet

known to be false (that is, true or undefined); false atoms

are implicitly given by Atm(Π)\U.2First, smodels compu-

tes deterministic consequences of a partial model at hand by

calling its propagation procedure expand (Line 1). Observe

that expand is based on propagation rules which generalize

the well-founded semantics [53]. In fact, if a call to expand

produces sets L and U such that L is not a subset of U,

then it follows that there is no answer set between the in-

itially given bounds (Line 2). On the other hand, if L = U

after a call to expand (Line 3), then L is an answer set of

the underlying logic program. More precisely, expand tries

to enlarge lower bound L and, at the same time, it tries to

make upper bound U smaller in such a way that no answer

set is lost. That is, for all answer sets X of a program we

have if condition L ⊆ X ⊆ U holds before calling expand,

then it also holds after the execution of expand. If neces-

sary, smodels then chooses an undefined atom A (Line 4)

and calls itself recursively, first (Line 5) to try to find an

answer set between L ∪ {A} and U, that is, setting A true,

and second (Line 6) to find an answer set between L and

U \ {A}, that is, setting A false. Initially, smodels is called

with ∅ and Atm(Π). This guarantees that all answer sets of

Π are found via backtracking.

The dlv system [17] extents the computation of answer

sets to disjunctive logic programs and dlv specific language

extensions, e.g. aggregate functions and weak constraints.

Its core algorithm is similar to the one of smodels. The

propagation of dlv also relies on computing well-founded

semantics plus back-propagation mechanisms that allow for

additionally marking atoms as being “eventually true”.

The systems assat and cmodels. These two ASP solvers

operate by reducing the problem of answer set computa-

tion to the satisfiability problem of propositional formulas

(via Clark’s completion [11]) and by invoking a SAT solver

2Atm(Π) denotes the set of all atoms occurring in Π.

Seite 4

Page 5

to generate answer sets. The fact that answer sets of the

syntactically restricted class of tight programs correspond

to classical models of its completion was discovered by Fa-

ges [21] and used for answer set computation in [2, 12]. For

handling the general case correctly, so-called loop formulas

have to be added [31]. However, complexity considerations

show that an exponential growth in size is likely to be unavoi-

dable, whenever logic programs are mapped into equivalent

propositional formulas [29].

The noMoRe system [38] relies on a graph-based approach.

It has its roots in default logic [45], where extensions are

often characterized through their (unique) set of generating

default rules. Accordingly, noMoRe characterizes answer sets

by means of their set of generating rules. For determining

whether a rule belongs to this set, we must verify that each

positive body atom is derivable and that no negative body

atom is derivable. In fact, an atom is derivable if the set

of generating rules includes a rule having the atom as its

head; or conversely, an atom is not derivable if there is no

rule among the generating rules that has the atom as its

head. Consequently, the formation of the set of generating

rules amounts to resolving positive and negative dependen-

cies among rules. For capturing these dependencies, noMoRe

takes advantage of the concept of a rule dependency graph,

wherein each node represents a rule of the underlying pro-

gram and two types of edges stand for the aforementioned

positive and negative rule dependencies, respectively. Answer

sets now can be expressed by total non-standard 2-colorings

of the rule dependency graph [32, 26], such that, whether a

rule belongs to a set of generating rules or not, is indicated

through its color.

Other systems. In addition to the above presented ASP sy-

stems there are other systems available, among them we

find the plp system [40] for preference handling, XSB [43]

for computing well-founded semantics, the quip [42] sy-

stem dealing with ASP through quantified boolean formulas,

nlp [36] a front-end for nested logic programs, psmodels [41]

an implementation of ordered disjunctions, and gnt [23] an

system for disjunctive logic programs based on smodels.

In order to foster further development of ASP the auto-

mated benchmarking system asparagus [1, 7] was launched.

Its two principal goals are to provide an infrastructure for

accumulating challenging benchmarks, and to facilitate exe-

cuting ASP solvers under the same conditions, guaranteeing

reproducible and reliable performance results.

6 Applications

ASP has been applied in multiple areas, e.g. product confi-

guration [52, 51], planning [28, 16] and diagnosis [3].

In product configuration one derives a valid configuration

from predefined components, some restrictions on these, and

a set of customer requirements. One example is the configu-

ration of PCs. Take program

computer ←

(15)

1 {ide disk,scsi disk} ← computer

german layout;us layout ← computer

scsi controller ← scsi disk

expressing in rule (16) that a computer has an IDE or a SCSI

disk and either a German or US layout keyboard in (17). If

it does have a SCSI disk, rule (18) states that it also needs

a SCSI controller. The answer sets of this program repre-

sent all possible configurations. If you now add the customer

requirements (German keyboard, no SCSI disk)

(16)

(17)

(18)

← scsi disk

german layout ←

the remaining, single answer set

{computer,ide disk,german layout}

represents the only valid configuration.

This easy, highly declarative way of dealing with confi-

guration tasks is used e.g. for the configuration of Debian

Linux [51].

Planning within ASP is another area that has been exten-

sively studied during the last few years. Consider the blocks

world domain where we want to move blocks from an initi-

al situation to a goal situation. In ASP, we can express the

problem in the following way. We start with the generator:

{mv(B,L,T) : bl(B) : loc(L)} 1 ← t(T),T < last

which generates possible moves (mv) of blocks (bl) to cer-

tain locations3(loc) for all discrete time steps (t) before time

step last (see rule (19)).

(19)

on(B,L,T + 1) ← mv(B,L,T),bl(B),loc(L),

t(T),T < last

(20)

on(B,L,T + 1) ← on(B,L,T),

(21)

not ¬on(B,L,T + 1),

bl(B),loc(L),t(T),T < last

¬on(B,L1,T) ← on(B,L,T),L ?= L1,bl(B),

loc(L),loc(L1),t(T),T < last

(22)

We describe the effect of a move action in (20) by saying

that if block B is moved to location L at time T, it will

be on that location at the next time step T + 1. Inertia

(21) expresses that a block stays on a location, unless it

does not (¬on). The information that a block is not on a

certain location is represented by rule (22), which expresses

uniqueness of location (no block can be at two locations at

the same time).

Now let us look at the tester part of our program.

← 2 {on(B1,B,T) : bl(B1)},bl(B),t(T)

← mv(B,L,T),on(B1,B,T),bl(B),bl(B1),

loc(L),t(T),T < last

(23)

(24)

3Possible locations are the blocks themselves and the table.

Seite 5

Page 6

Integrity constraint (23) expresses that no two blocks

can be on top of the same block (though they can both be

on the table). Via constraint (24) we express that a block

can only be moved, if there is no block on top of it.

We have now described the blocks world domain with

just six rules. Now we only need to specify our initial (i.e. at

time step 0) and goal (at time step last) situations. E.g., we

start with two towers with two blocks each:

on(1,2,0) ←

on(3,4,0) ←

on(2,table,0) ←

on(4,table,0) ←

To specify our goal situation of just one big tower, we use

integrity constraints. We disallow all answer sets (possible

plans) where the blocks are not in the desired position at

the last time step:

← not on(4,3,last)

← not on(2,1,last)

← not on(3,2,last)

← not on(1,table,last)

The last thing we need to do is to fix the macro last to a

specific number of time steps, representing the length of the

plans to be investigated. For the situation above, we could

e.g. use last = 3, which would give us all possible plans of

length three, i.e., {mv(4,table,0),mv(3,2,1),mv(4,3,2)} .

We have refrained from giving explicit instantiations of

the domain predicates bl and t, which would also be needed.

Other Applications of ASP include the Reaction Control

System of the Space Shuttle [37], which has primary respon-

sibility for maneuvering the aircraft while in space. Also, ASP

has been employed e.g. for Constraint Programming [35], for

certain issues pertaining Petri nets [24], for cryptanalysis [25]

and for research in historical linguistics [19].

Acknowledgments. The authors were supported by DFG un-

der grants FOR 375/1 and SCHA 550/6 as well as WASP

project IST-2001-37004. We would like to thank G. Brewka,

V. Lifschitz, I. Niemel¨ a, and M. Truszczy´ nski whose materi-

als provided an extremely valuable source for this paper.

Literatur

[1] http://asparagus.cs.uni-potsdam.de.

[2] http://assat.cs.ust.hk.

[3] M. Balduccini and M. Gelfond. Diagnostic reasoning

with A-prolog. Theory and Practice of Logic Program-

ming, 3(4-5):425–461, 2003.

[4] M. Balduccini and M. Gelfond. Logic programs with

consistency-restoring rules. In P. Doherty, J. McCarthy,

and M.-A. Williams, editors, International Symposium

on Logical Formalization of Commonsense Reasoning,

AAAI Spring Symposium Series, 2003.

[5] C. Baral. Knowledge representation, reasoning and de-

clarative problem solving with Answer sets. Cambridge

University Press, 2003.

[6] P. Bonatti. Reasoning with infinite stable models. Ar-

tificial Intelligence, 156(1):75–111, 2004.

[7] P. Borchert, C. Anger, T. Schaub, and M. Truszczy´ nski.

Towards systematic benchmarking in answer set pro-

gramming. In V. Lifschitz and I. Niemel¨ a, editors, Pro-

ceedings of the International Conference on Logic Pro-

gramming and Nonmonotonic Reasoning, pages 3–7.

Springer, 2004.

[8] M. Brain and M. De Vos. Implementing oclp as a front-

end for answer set solvers: From theory to practice. In

M. De Vos and A. Provetti, editors, Proceedings of the

International Workshop on Answer Set Programming,

2003. CEUR Workshop Proceedings.

[9] G. Brewka. Complex preferences for answer set optimi-

zation. In D. Dubois, Chr. Welty, and M.-A. Williams,

editors, Proceedings of the International Conference on

Principles of Knowledge Representation and Reasoning,

pages 213–223. AAAI Press, 2004.

[10] G. Brewka, I. Niemel¨ a, and T. Syrj¨ anen. Logic programs

with ordered disjunction. Computational Intelligence,

20(2):335–357, May 2004.

[11] K. Clark. Negation as failure. In H. Gallaire and J. Min-

ker, editors, Logic and Data Bases, pages 293–322. Ple-

num Press, 1978.

[12] http://www.cs.utexas.edu/users/tag/cmodels.html.

[13] M. Davis and H. Putnam. A computing procedure for

quantification theory. Journal of the ACM, 7:201–215,

1960.

[14] J. Delgrande, T. Schaub, H. Tompits, and K. Wang.

A classification and survey of preference handling ap-

proaches in nonmonotonic reasoning. Computational

Intelligence, 20(2):308–334, 2004.

[15] T. Dell’Armi, W. Faber, G. Ielpa, N. Leone, and G. Pfei-

fer. Aggregate functions in disjunctive logic program-

ming: Semantics, complexity, and implementation in

dlv. In G. Gottlob and T. Walsh, editors, Proceedings

of the International Joint Conference on Artificial Intel-

ligence, pages 847–852. Morgan Kaufmann, 2003.

[16] Y. Dimopoulos, B. Nebel, and J. K¨ ohler.

planning problems in nonmonotonic logic programs. In

S. Steel and R. Alami, editors, Proceedings of the Euro-

pean Conference on Planning, pages 169–181. Springer,

1997.

[17] http://www.dbai.tuwien.ac.at/proj/dlv.

[18] T. Eiter and G. Gottlob. On the computational cost of

disjunctive logic programming: Propositional case. An-

nals of Mathematics and Artificial Intelligence, 15:289–

323, 1995.

[19] E. Erdem, V. Lifschitz, L. Nakhleh, and D. Ringe. Re-

constructing the evolutionary history of indo-european

languages using answer set programming.

ceedings of the International Symposium on Practi-

cal Aspects of Declarative Languages, pages 160–176,

2003.

[20] W. Faber, N. Leone, and G. Pfeifer. Experimenting with

heuristics for answer set programming. In B. Nebel, edi-

tor, Proceedings of the International Joint Conference

on Artificial Intelligence, pages 635–640. Morgan Kauf-

mann, 2001.

Encoding

In Pro-

Seite 6

Page 7

[21] F. Fages. Consistency of clark’s completion and the

existence of stable models. Journal of Methods of Logic

in Computer Science, 1:51–60, 1994.

[22] M. Gelfond and V. Lifschitz. Classical negation in logic

programs and disjunctive databases. New Generation

Computing, 9:365–385, 1991.

[23] http://www.tcs.hut.fi/Software/gnt.

[24] K. Heljanko. Using logic programs with stable model

semantics to solve deadlock and reachability problems

for 1-safe petri nets. In Proceedings of the International

Conference on Tools and Algorithms for the Construc-

tion and Analysis of Systems, pages 218–223, 1999.

[25] M. Hietalahti, F. Massacci, and I. Niemel¨ a.

challenge problem for nonmonotonic reasoning systems.

In Proceedings of the International Workshop on Non-

Monotonic Reasoning, 2000.

[26] K. Konczak, T. Linke, and T. Schaub. Graphs and co-

lorings for answer set programming: Abridged report.

In V. Lifschitz and I. Niemel¨ a, editors, Proceedings of

the International Conference on Logic Programming

and Nonmonotonic Reasoning, pages 127–140. Sprin-

ger, 2004.

[27] N. Leone, W. Faber, G. Pfeifer, T. Eiter, G. Gottlob,

C. Koch, C. Mateis, S. Perri, and F. Scarcello. The

DLV system for knowledge representation and reaso-

ning. ACM Transactions on Computational Logic, 2004.

To appear.

[28] V. Lifschitz. Answer set planning. Artificial Intelligence,

138(1-2):39–54, 2002.

[29] V. Lifschitz and A. Razborov. Why are there so many

loop formulas? ACM Transactions on Computational

Logic. To appear.

[30] V. Lifschitz, L. Tang, and H. Turner. Nested expressions

in logic programs. Annals of Mathematics and Artificial

Intelligence, 25(3-4):369–389, 1999.

[31] F. Lin and Y. Zhao. Assat: Computing answer sets of

a logic program by sat solvers. Artificial Intelligence,

157:115–137, 2004.

[32] T. Linke. Graph theoretical characterization and com-

putation of answer sets. In B. Nebel, editor, Procee-

dings of the International Joint Conference on Artificial

Intelligence, pages 641–645. Morgan Kaufmann, 2001.

[33] V. Marek and J. Remmel. On the expressibility of sta-

ble logic programming.

M. Truszczy´ nski, editors, Proceedings of the Interna-

tional Conference on Logic Programming and Nonmo-

notonic Reasoning, pages 107–120. Springer, 2001.

[34] V. Marek and M. Truszczy´ nski. Stable models and an

alternative logic programming paradigm.

W. Marek, M. Truszczy´ nski, and D. Warren, editors,

The Logic Programming Paradigm: a 25-Year Perspec-

tive, pages 375–398. Springer, 1999.

[35] I. Niemel¨ a. Logic programs with stable model semantics

as a constraint programming paradigm. Annals of Ma-

thematics and Artificial Intelligence, 25(3,4):241–273,

1999.

[36] http://www.cs.uni-potsdam.de/∼torsten/nlp.

Des: A

In T. Eiter, W. Faber, and

In K. Apt,

[37] M. Nogueira, M. Balduccini, M. Gelfond, R. Watson,

and M. Barry. An A-Prolog decision support system

for the Space Shuttle. In Proceedings of the Interna-

tional Symposium on Practical Aspects of Declarative

Languages, pages 169–183, 2001.

[38] http://www.cs.uni-potsdam.de/∼linke/nomore.

[39] D. Pearce, V. Sarsakov, T. Schaub, H. Tompits, and

S. Woltran. A polynomial translation of logic programs

with nested expressions into disjunctive logic programs.

In P. Stuckey, editor, Proceedings of the Internatio-

nal Conference on Logic Programming, pages 405–420.

Springer, 2002.

[40] http://www.cs.uni-potsdam.de/∼torsten/plp.

[41] http://www.tcs.hut.fi/Software/smodels/priority.

[42] http://www.kr.tuwien.ac.at/research/quip.html.

[43] P. Rao, K. F. Sagonas, T. Swift, D. S. Warren, and

J. Freire. XSB: A system for effciently computing wfs.

In J. Dix, U. Furbach, and A. Nerode, editors, Pro-

ceedings of the International Conference on Logic Pro-

gramming and Nonmonotonic Reasoning, pages 431–

441. Springer, 1997.

[44] R. Reiter. On closed world data bases. In H. Gallaire

and J. Minker, editors, Logic and Databases, pages 55–

76. Plenum Press, 1978.

[45] R. Reiter. A logic for default reasoning. Artificial Intel-

ligence, 13(1-2):81–132, 1980.

[46] C. Sakama and K. Inoue. Prioritized logic programming

and its application to commonsense reasoning. Artificial

Intelligence, 123(1-2):185–222, 2000.

[47] T. Schaub and K. Wang. A semantic framework for

preference handling in answer set programming. Theory

and Practice of Logic Programming, 3(4-5):569–607,

2003.

[48] J. Schlipf. The expressive powers of the logic program-

ming semantics. Journal of Computer and Systems

Sciences, 51:64–86, 1995.

[49] P. Simons, I. Niemel¨ a, and T. Soininen. Extending and

implementing the stable model semantics. Artificial In-

telligence, 138(1-2):181–234, 2002.

[50] http://www.tcs.hut.fi/Software/smodels.

[51] T. Soininen and I. Niemel¨ a. Developing a declarative

rule language for applications in product configuration.

In Proceedings of the First International Workshop on

Practical Aspects of Declarative Languages, pages 305–

319, 1999.

[52] T. Soininen, I. Niemel¨ a, J. Tiihonen, and R. Sulo-

nen. Representing configuration knowledge with weight

constraint rules. In Proceedings of the AAAI Spring

Symposium on Answer Set Programming, 2001.

[53] A. van Gelder, K. Ross, and J. S. Schlipf. The well-

founded semantics for general logic programs. Journal

of the ACM, 38(3):620–650, 1991.

Contact

Chr. Anger, K. Konczak, Th. Linke, and T. Schaub

Universit¨ at Potsdam, Institut f¨ ur Informatik

August-Bebel-Straße 89, D-14482 Potsdam

{christian,konczak,linke,torsten}@cs.uni-potsdam.de

http://www.cs.uni-potsdam.de/wv/

Seite 7

#### View other sources

#### Hide other sources

- Available from Torsten Schaub · May 17, 2014
- Available from uni-potsdam.de