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### Preferred Answer Sets for Ordered Logic Programs [article]

Davy Van Nieuwenborgh, Dirk Vermeir
2004 arXiv   pre-print
(Dix et al. 1996; De Vos and Vermeir 2001) .  ...  2003a; Van Nieuwenborgh and Vermeir 2003b).  ...

### Possibilistic Answer Set Programming Revisited [article]

Kim Bauters, Steven Schockaert, Martine De Cock, Dirk Vermeir
2012 arXiv   pre-print
One example of preferences in classical ASP is (Nieuwenborgh and Vermeir, 2002) , where ordered logic programs are used.  ...

### Order and Negation as Failure [chapter]

Davy Van Nieuwenborgh, Dirk Vermeir
2003 Lecture Notes in Computer Science
We equip ordered logic programs with negation as failure, using a simple generalization of the preferred answer set semantics for ordered programs. This extension supports a convenient formulation of certain problems, which is illustrated by means of an intuitive simulation of logic programming with ordered disjunction. The simulation also supports a broader application of "ordered disjunction", handling problems that would be cumbersome to express using ordered disjunction logic programs.
more » ... ogic programs. Interestingly, allowing negation as failure in ordered logic programs does not yield any extra computational power: the combination of negation as failure and order can be simulated using order (and true negation) alone.

### Semantic Forcing in Disjunctive Logic Programs

Marina De Vos, Dirk Vermeir
2001 Computational intelligence
We propose a semantics for disjunctive logic programs, based on the single notion of forcing. We show that the semantics properly extends, in a natural way, previous approaches. A fixpoint characterization is also provided. We also take a closer look at the relationship between disjunctive logic programs and disjunctivefree logic programs. We present certain criteria under which a disjunctive program is semantically equivalent with its disjunctive-free (shifted) version. Intuitively, there are
more » ... itively, there are three possibilities: 1. The person is healthy, which means that she is not sick. So we can conclude that she has neither a cold nor a bronchitis and she can go to work. C Wishes to thank the FWO for its support.

### Cooperating Answer Set Programming [chapter]

Davy Van Nieuwenborgh, Stijn Heymans, Dirk Vermeir
2006 Lecture Notes in Computer Science
We present a formalism for logic program cooperation based on the answer set semantics. The system consists of independent logic programs that are connected via a sequential communication channel. When presented with an input set of literals from its predecessor, a logic program computes its output as an answer set of itself, enriched with the input. It turns out that the communication strategy makes the system quite expressive: essentially a sequence of a fixed number of programs n captures
more » ... grams n captures the complexity class Σ P n , i.e. the n-th level of the polynomial hierarchy. On the other hand, unbounded sequences capture the polynomial hierarchy PH. These results make the formalism suitable for complex applications such as hierarchical decision making and preference-based diagnosis on ordered theories. In addition, such systems can be realized by implementing an appropriate control strategy on top of existing solvers such as DLV or SMODELS, possibly in a distributed environment. 227 then send back to the employee who verifies its feasibility. If the verification fails, the communication starts all over again by the employee sending a new possible plan to the emergency services. In the other case, i.e. the adapted plan is successfully verified by the employee, it is presented to the firm's management which will try to improve it to obtain e.g. a cheaper one. Again, this cheaper alternative is sent back to the emergency services for verification, and eventually also to the employee, to check its feasibility. We develop a framework of cooperating programs that is capable of modeling hierarchical decision problems like the one above. To this end, we consider a sequence of programs P i i=1,...n . Intuitively, a program P i communicates the solutions it finds acceptable to the next program P i+1 in the hierarchy. For such a P i -acceptable solution S, the program P i+1 computes a number of solutions that it thinks improve on S. If one of these P i+1 improvements S of S is also acceptable to P i , i.e. S can be successfully verified by P i , the original S is rejected as an acceptable solution by the program P i+1 . On the other hand, if P i+1 has no improvements for S, or none of them are also acceptable to P i , S is accepted by P i+1 . It follows that a solution that is acceptable to all programs must have been proposed by the starting program P 1 . It turns out that such sequences of programs are rather expressive. More specifically, we show not only that arbitrary complete problems of the polynomial hierarchy can be solved by such systems, but that such systems can capture the complete polynomial hierarchy, the latter making them suitable for complex applications. Problems located at the first level of the polynomial hierarchy can be directly solved using answer set solvers such as DLV [16] or SMODELS [22] . On the second level, only DLV is left to perform the job directly. However, by using a "guess and check" fixpoint procedure, SMODELS can indirectly be used to solve problems at the second level [4, 15] . Beyond the second level, there are still some interesting problems. E.g., the most expressive forms of diagnostic reasoning, i.e. subset-minimal diagnosis on disjunctive system descriptions [13] or preference-based diagnosis on ordered theories [27] , are located at the third level of the polynomial hierarchy, as are programs that support sequences of weak constraints 1 on disjunctive programs. For these problems, and problems located even higher in the polynomial hierarchy, the framework presented in this paper provides a means to effectively compute solutions for such problems, using SMODELS or DLV for each program in the sequence to compute better solutions. E.g., to solve the problems mentioned before on the third level, it suffices to write three well-chosen programs and to set up an appropriate control structure implementing the communication protocol sketched above. The remainder of the paper is organized as follows. In Section 2, we review the answer set semantics and present the definitions for cooperating program systems. Further, we illustrate how such systems can be used to elegantly express common problems. Section 3 discusses the complexity and expressiveness of the proposed semantics, while Section 4 compares it with related approaches from the literature. Finally, we conclude and give some directions for further research in Section 5. Due to space restrictions, proofs have been omitted, but they can be found in [25] .

### Fuzzy Answer Set Programming [chapter]

Davy Van Nieuwenborgh, Martine De Cock, Dirk Vermeir
2006 Lecture Notes in Computer Science
In this chapter, we present a tutorial about fuzzy answer set programming (FASP); we give a gentle introduction to its basic ideas and definitions. FASP is a combination of answer set programming and fuzzy logics which has recently been proposed. From the answer set semantics, FASP inherits the declarative nonmonotonic reasoning capabilities, while fuzzy logic adds the power to model continuous problems. FASP can be tailored towards different applications since fuzzy logics gives a great
more » ... ives a great flexibility, e.g. by the possibility to use different generalizations of the classical connectives. In this chapter, we consider a rather general form of FASP programs; the connectives can in principal be interpreted by arbitrary [0, 1] n → [0, 1]mappings. Despite that very general connectives are allowed, the presented framework turns out to be an intuitive extension of answer set programming.

### Guarded Open Answer Set Programming [chapter]

Stijn Heymans, Davy Van Nieuwenborgh, Dirk Vermeir
2005 Lecture Notes in Computer Science
Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program's constants. We define a fixed point logic (FPL) extension of Clark's completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in

### Conditional Planning with External Functions [chapter]

Davy Van Nieuwenborgh, Thomas Eiter, Dirk Vermeir
2007 Lecture Notes in Computer Science
We introduce the logic-based planning language K c as an extension of K [5] . K c has two advantages upon K. First, the introduction of external function calls in the rules of a planning description allows the knowledge engineer to describe certain planning domains, e.g. involving complex action effects, in a more intuitive fashion then is possible in K. Secondly, in contrast to the conformant planning framework K, K c is formalized as a conditional planning system, which enables K c to solve
more » ... bles K c to solve planning problems that are impossible to express in K, e.g. involving sensing actions. A prototype implementation of conditional planning with K c is build on top of the DLV K system, and we illustrate its use by some small examples.

### An Ordered Logic Program Solver [chapter]

Davy Van Nieuwenborgh, Stijn Heymans, Dirk Vermeir
2005 Lecture Notes in Computer Science
We describe the design of the OLPS system, an implementation of the preferred answer set semantics for ordered logic programs. The basic algorithm we propose computes the extended answer sets of a simple program using an intuitive 9-valued lattice, called T9. During the computation, this lattice is employed to keep track of the status of the literals and the rules while evolving to a solution. It turns out that the basic algorithm needs little modification in order to be able to compute the
more » ... to compute the preferred answer sets of an ordered logic program. We illustrate the system using an example from diagnostic reasoning and we present some preliminary benchmark results comparing OLPS with existing answer set solvers such as SMODELS and DLV.

### Dynamically Ordered Probabilistic Choice Logic Programming [chapter]

Marina De Vos, Dirk Vermeir
2000 Lecture Notes in Computer Science
We present a framework for decision making under uncertainty where the priorities of the alternatives can depend on the situation at hand. We design a logic-programming language, DOP-CLP, that allows the user to specify the static priority of each rule and to declare, dynamically, all the alternatives for the decisions that have to be made. In this paper we focus on a semantics that reflects all possible situations in which the decision maker takes the most rational, possibly probabilistic,
more » ... probabilistic, decisions given the circumstances. Our model theory, which is a generalization of classical logic-programming model theory, captures uncertainty at the level of total Herbrand interpretations. We also demonstrate that DOP-CLPs can be used to formulate game theoretic concepts. £ Wishes to thank the FWO for its support. 1 [8] uses the word order instead of priority. 2 Intuitively, something is true by default unless there is evidence to the contrary. Example 2. Recall the Jakuza program of Example 1. The following functions', and ¡ are interpretations for this program 5 :

### Expressiveness of Communication in Answer Set Programming [article]

Kim Bauters and Jeroen Janssen and Steven Schockaert and Dirk Vermeir and Martine De Cock
2011 arXiv   pre-print
Answer set programming (ASP) is a form of declarative programming that allows to succinctly formulate and efficiently solve complex problems. An intuitive extension of this formalism is communicating ASP, in which multiple ASP programs collaborate to solve the problem at hand. However, the expressiveness of communicating ASP has not been thoroughly studied. In this paper, we present a systematic study of the additional expressiveness offered by allowing ASP programs to communicate. First, we
more » ... icate. First, we consider a simple form of communication where programs are only allowed to ask questions to each other. For the most part, we deliberately only consider simple programs, i.e. programs for which computing the answer sets is in P. We find that the problem of deciding whether a literal is in some answer set of a communicating ASP program using simple communication is NP-hard. In other words: we move up a step in the polynomial hierarchy due to the ability of these simple ASP programs to communicate and collaborate. Second, we modify the communication mechanism to also allow us to focus on a sequence of communicating programs, where each program in the sequence may successively remove some of the remaining models. This mimics a network of leaders, where the first leader has the first say and may remove models that he or she finds unsatisfactory. Using this particular communication mechanism allows us to capture the entire polynomial hierarchy. This means, in particular, that communicating ASP could be used to solve problems that are above the second level of the polynomial hierarchy, such as some forms of abductive reasoning as well as PSPACE-complete problems such as STRIPS planning.

### Integrating Description Logics and Answer Set Programming [chapter]

Stijn Heymans, Dirk Vermeir
2003 Lecture Notes in Computer Science
We integrate an expressive class of description logics (DLs) and answer set programming by extending the latter to support inverted predicates and infinite domains, features that are present in most DLs. The extended language, conceptual logic programming (CLP) proves to be a viable alternative for intuitively representing and reasoning nonmonotonically, in a decidable way, with possibly infinite knowledge. Not only can conceptual logic programs (CLPs) simulate finite answer set programming,
more » ... set programming, they are also flexible enough to simulate reasoning in an expressive class of description logics, thus being able to play the role of ontology language, as well as rule language, on the Semantic Web. This work was partially funded by the Information Society Technologies programme of the European Commission, Future and Emerging Technologies under the IST-2001-37004 WASP project. 1 Like DL knowledge bases or database schema's, ontologies are models of a domain, providing an agreed and shared understanding [20] .

### General Fuzzy Answer Set Programs [chapter]

Jeroen Janssen, Steven Schockaert, Dirk Vermeir, Martine De Cock
2009 Lecture Notes in Computer Science
A number of generalizations of answer set programming have been proposed in the literature to deal with vagueness, uncertainty, and partial rule satisfaction. We introduce a unifying framework that entails most of the existing approaches to fuzzy answer set programming. In this framework, rule bodies are defined using arbitrary fuzzy connectives with monotone partial mappings. As an approximation of full answer sets, k-answer sets are introduced to deal with conflicting information, yielding a
more » ... mation, yielding a flexible framework that encompasses, among others, existing work on valued constraint satisfaction and answer set optimization.

### Preferred Answer Sets for Ordered Logic Programs [chapter]

Davy Van Nieuwenborgh, Dirk Vermeir
2002 Lecture Notes in Computer Science
We extend answer set semantics to deal with inconsistent programs (containing classical negation), by finding a "best" answer set. Within the context of inconsistent programs, it is natural to have a partial order on rules, representing a preference for satisfying certain rules, possibly at the cost of violating less important ones. We show that such a rule order induces a natural order on extended answer sets, the minimal elements of which we call preferred answer sets. We characterize the
more » ... haracterize the expressiveness of the resulting semantics and show that it can simulate negation as failure as well as disjunction. We illustrate an application of the approach by considering database repairs, where minimal repairs are shown to correspond to preferred answer sets.

### Logic Programming Agents Playing Games [chapter]

Marina De Vos, Dirk Vermeir
2003 Research and Development in Intelligent Systems XIX
We present systems of logic programming agents (LPAS) to model the interactions between decision-makers while evolving to a conclusion. Such a system consists of a number of agents connected by means of unidirectional communication channels. Agents communicate with each other by passing answer sets obtained by updating the information received from connected agents with their own private information. As an application, we show how extensive games with perfect information can be conveniently
more » ... be conveniently represented as logic programming agent systems, where each agent embodies the reasoning of a game player, such that the equilibria of the game correspond with the semantics agreed upon by the agents in the LPAS.
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