Answer set programming
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Answer set programming (ASP) is a form of declarative programming oriented towards difficult (primarily NP-hard) search problems. It is based on the stable model (answer set) semantics of logic programming. In ASP. search problems are reduced to computing stable models, and answer set solvers -- programs for generating stable models -- are used to perform search. The computational process employed in the design of many answer set solvers is an enhancement of the DPLL algorithm and, in principle, it always terminates (unlike Prolog query evaluation, which may lead to an infinite loop). In a more general sense, ASP includes all applications of answer sets to knowledge representation [Baral 2003; Gelfond 2008] and the use of Prolog-style query evaluation for solving problems arising in these applications.
HistoryThe planning method proposed by Dimopoulos, Nebel and Köhler [1997] is an early example of answer set programming. Their approach is based on the relationship between plans and stable models described in [Subrahmanian and Zaniolo 1995]. Soininen and Niemelä [1998] applied what is now known as answer set programming to the problem of product configuration. The use of answer set solvers for search was identified as a new programming paradigm in [Marek and Truszczyński 1999] (the term "answer set programming" was used for the first time as the title of a part of the collection where that paper appeared) and in [Niemelä 1999]. Answer set programming language LparseLparse is the name of the program that was originally created as a front-end for the answer set solver smodels, and is now used in the same way in many other answer set solvers, including assat, clasp, cmodels, gNt, nomore++ and pbmodels. (dlv is an exception; the syntax of ASP programs written for dlv is somewhat different.) An Lparse program consists of rules of the form <head> :- <body> . The symbol One other useful construct included in this language is choice. For instance, the choice rule
{p,q,r}.
says: choose arbitrarily which of the atoms Failed to parse (Missing texvc executable; please see math/README to configure.): p,q,r
to include in the stable model. The lparse program that contains this choice rule and no other rules has 8 stable models -- arbitrary subsets of Failed to parse (Missing texvc executable; please see math/README to configure.): \{p,q,r\}
. The definition of a stable model was generalized to programs with choice rules by Ilkka Niemelä, Patrik Simons and Timo Soininen [1999]. According to [Ferraris and Lifschitz, 2005], choice rules can be treated also as abbreviations for propositional formulas under the stable model semantics. For instance, the choice rule above can be viewed as shorthand for the conjunction of three "excluded middle" formulas:
1{p,q,r}2.
This rule says: choose at least 1 of the atoms Failed to parse (Missing texvc executable; please see math/README to configure.): p,q,r , but not more than 2. The meaning of this rule under the stable model semantics is represented by the propositional formula
:- 2{p,q,r}.
Adding this constraint to an Lparse program eliminates the stable models that contain at least 2 of the atoms Failed to parse (Missing texvc executable; please see math/README to configure.): p,q,r . The meaning of this rule can be represented by the propositional formula
p(a). p(b). p(c). q(X) :- p(X), X!=a. has the same meaning as p(a). p(b). p(c). q(b). q(c). The program
p(a). p(b). p(c).
{q(X):p(X)}2.
is shorthand for
p(a). p(b). p(c).
{q(a),q(b),q(c)}2.
Generating stable modelsTo find a stable model of the Lparse program stored in file % lparse <filename> | smodels Option 0 instructs smodels to find all stable models of the program. For instance, if file
1{p,q,r}2.
s :- not p.
then the command % lparse test | smodels 0 produces the output Answer: 1 Stable Model: q p Answer: 2 Stable Model: p Answer: 3 Stable Model: r p Answer: 4 Stable Model: q s Answer: 5 Stable Model: r s Answer: 6 Stable Model: r q s Examples of ASP programsGraph coloringAn Failed to parse (Missing texvc executable; please see math/README to configure.): n -coloring of a graph Failed to parse (Missing texvc executable; please see math/README to configure.): G
is a function Failed to parse (Missing texvc executable; please see math/README to configure.): color\
from its set of vertices to Failed to parse (Missing texvc executable; please see math/README to configure.): \{1,\dots,n\}
such that Failed to parse (Missing texvc executable; please see math/README to configure.): color(x)\neq color(y)
for every pair of adjacent vertices Failed to parse (Missing texvc executable; please see math/README to configure.): x,y
. We would like to use ASP to find an Failed to parse (Missing texvc executable; please see math/README to configure.): n -coloring of a given graph (or determine that it does not exist). This can be accomplished using the following Lparse program:
c(1..n).
1 {color(X,I) : c(I)} 1 :- v(X).
:- color(X,I), color(Y,I), e(X,Y), c(I).
Line 1 defines the numbers Failed to parse (Missing texvc executable; please see math/README to configure.): 1,\dots,n to be colors. According to the choice rule in Line 2, a unique color Failed to parse (Missing texvc executable; please see math/README to configure.): i should be assigned to each vertex Failed to parse (Missing texvc executable; please see math/README to configure.): x . The constraint in Line 3 prohibits assigning the same color to vertices Failed to parse (Missing texvc executable; please see math/README to configure.): x and Failed to parse (Missing texvc executable; please see math/README to configure.): y if there is an edge connecting them. If we combine this file with a definition of Failed to parse (Missing texvc executable; please see math/README to configure.): G , such as v(1..100). % 1,...,100 are vertices e(1,55). % there is an edge from 1 to 55 . . . and run smodels on it, with the numeric value of Failed to parse (Missing texvc executable; please see math/README to configure.): n specified on the command line, then the atoms of the form Failed to parse (Missing texvc executable; please see math/README to configure.): color(\dots,\dots) in the output of smodels will represent an Failed to parse (Missing texvc executable; please see math/README to configure.): n -coloring of Failed to parse (Missing texvc executable; please see math/README to configure.): G . The program in this example illustrates the "generate-and-test" organization that is often found in simple ASP programs. The choice rule describes a set of "potential solutions" -- a simple superset of the set of solutions to the given search problem. It is followed by a constraint, which eliminates all potential solutions that are not acceptable. However, the search process employed by smodels and other answer set solvers is not based on trial and error. Large cliqueA clique in a graph is a set of pairwise adjacent vertices. The following lparse program finds a clique of size Failed to parse (Missing texvc executable; please see math/README to configure.): \geq n in a given graph, or determines that it does not exist:
n {in(X) : v(X)}.
:- in(X), in(Y), v(X), v(Y), X!=Y, not e(X,Y), not e(Y,X).
This is another example of the generate-and-test organization. The choice rule in Line 1 "generates" all sets consisting of Failed to parse (Missing texvc executable; please see math/README to configure.): \geq n vertices. The constraint in Line 2 "weeds out" the sets that are not cliques. Hamiltonian cycleA Hamiltonian cycle in a directed graph is a cycle that passes through each vertex of the graph exactly once. The following Lparse program can be used to find a Hamiltonian cycle in a given directed graph if it exists; we assume that 0 is one of the vertices.
{in(X,Y)} :- e(X,Y).
:- 2 {in(X,Y) : e(X,Y)}, v(X).
:- 2 {in(X,Y) : e(X,Y)}, v(Y).
r(X) :- in(0,X), v(X).
r(Y) :- r(X), in(X,Y), e(X,Y).
:- not r(X), v(X).
The choice rule in Line 1 "generates" all subsets of the set of edges. The three constraints "weed out" the subsets that are not Hamiltonian cycles. The last of them uses the auxiliary predicate Failed to parse (Missing texvc executable; please see math/README to configure.): r(x)
("Failed to parse (Missing texvc executable; please see math/README to configure.): x
is reachable from 0") to prohibit the vertices that do not satisfy this condition. This predicate is defined recursively in Lines 4 and 5.
This program is an example of the more general "generate, define and test" organization: it includes the definition of an auxiliary predicate that helps us eliminate all "bad" potential solutions. Comparison of implementations
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