On Complexity of DeLP through Game Semantics∗ Laura A. Cecchi Pablo R. Fillottrani and Guillermo R. Simari Depto. Ciencias de la Computacion´ - Fa.E.A. Depto. de Cs. e Ing. de la Comp. Universidad Nacional del Comahue Universidad Nacional del Sur Buenos Aires 1400 Av. Alem 1253 (8300) Neuquen´ - ARGENTINA (8000) Bah´ıa Blanca - ARGENTINA [email protected] {prf,grs}@cs.uns.edu.ar Abstract on a dialectical analysis where arguments for and against Defeasible Logic Programming (DeLP) is a general argumen- a literal interact in order to determine whether this literal tation based system for knowledge representation and reason- is believed by a reasoning agent. The semantics GS is a ing. Its proof theory is based on a dialectical analysis where declarative trivalued game-based semantics for DeLP that arguments for and against a literal interact in order to deter- links game-semantics (Abramsky & McCusker 1997) and mine whether this literal is believed by a reasoning agent. The model-theory. Soundness and completeness of GS with re- semantics GS is a declarative trivalued game-based semantics spect to DeLP proof theory have been proved (Cecchi & for DeLP that is sound and complete for DeLP proof theory. Simari 2004). Complexity theory is an important tool for comparing differ- Complexity theory is an important tool for comparing dif- ent formalism and for helping to improve implementations ferent formalism, and for helping to improve implementa- whenever it is possible. In this work we address the prob- tions whenever it is possible. For this reason, it is important lem of studying the complexity of some important decision to analyze the computational complexity and the expressive problems in DeLP. Thus, we characterize the relevant deci- sion problems in the context of DeLP and GS, and we define power of DeLP. The former tells us how difficult it is to an- data and combined complexity for DeLP. Since DeLP com- swer a query, while the latter gives a precise characterization putes every argument from a set of defeasible rules, it is of of the concepts that are definable as queries. central importance to analyze the complexity of two decision Even thought complexity for nonmonotonic reasoning problems. The first one can be defined as “Is a set of defea- systems has been studied in depth for several formalisms sible rules an argument for a literal under a defeasible logic such us default logic, autoepistemic logic, circumscription, program?”. We prove that this problem is P-complete. The abduction and logic programming (Cadoli & Schaerf 1993; second decision problem is “Does there exist an argument for Dantsin et al. 2001) until recently not many complexity re- a literal under a defeasible logic program?”. We prove that sults for argumentation systems have been reported. this problem is in NP. Furthermore, we study data complex- This situation can be explained in part by the fact that, ity of query answering in the context of DeLP. As far as we know, data complexity has not been introduced in the context historically, implementations of argumentation systems have of argumentation systems. been limited to areas with no real time response restriction (see (Verheij 1998; Gordon & Karacapilidis 1997)). Re- cently, however, several applications have been developed, and implemented using argumentation systems related, for KEYWORDS: Argumentation Systems, Defeasible Rea- instance, with multiagent systems and web search (Atkin- soning, Logic Programming, Game-based Semantics, son, Bench-Capon, & Mc Burney 2004; Chesnevar˜ & Ma- Complexity guitman 2004a; 2004b; Bassiliades, Antoniou, & Vlahavas 2004). Scalability and robustness of such approaches heav- ily depend on the computational properties of the underly- Introduction ing algorithms. It is hence crucial to study these properties Defeasible Logic Programming (DeLP) is a general argu- in order to expand the application fields of argumentation mentation based tool for knowledge representation and rea- systems. soning (Garc´ıa & Simari 2004)1. Its proof theory is based Different computational complexity results (Dimopoulos, ∗ Nebel, & Toni 2002; Bench-Capon 2003; Amgoud & Cayrol This research was partially supported by Secretar´ıa General 2002; Dunne & Bench-Capon 2002) have been presented on de Ciencia y Tecnolog´ıa of the Universidad Nacional del Sur, by the Universidad Nacional del Comahue (Proyecto de Investigacion´ argumentation abstract framework (Bondarenko et al. 1997; 04/E062), by Agencia Nacional de Promocion´ Cient´ıfica y Tec- Dung 1995), based on admissibility and preferability seman- nologica´ (PICT 2002 No. 13096, PICT 2003 15043, PAV 076) and tics. However, those results do not apply directly to DeLP, by the National Research Council (CONICET), ARGENTINA. because its semantics are quite different. Another notable 1The interested reader can find an on-line interpreter for DeLP study of the computational complexity of defeasible systems in http://lidia.cs.uns.edu.ar/DeLP has been done in (Maher 2001). But, defeasible theory an- denoted as L, is defined as follows: L = ∼A, if L is an alyzed in this work greatly differs from DeLP in several atom, otherwise if L is a negated atom, L = A. Let X be points, such as knowledge representation (facts and strict a set of literals, X is the set of the complement of every rule, defeasible and defeaters rules) and their proof theory. member in X. When measuring the complexity of evaluating queries in a specific language, we distinguish between several kinds Definition 1 A strict rule is an ordered pair, denoted of complexity according to (Vardi 1982; Papadimitriou & “Head ← Body”, where “Head” is a ground literal, and Yannakakis 1997; Dantsin et al. 2001). Data complexity “Body” is a finite set of ground literals. A strict rule is the complexity of evaluating a specific query in the lan- with head L0 and body {L1,...Ln, n > 0} is written as guage, when the query is fixed, and we study the complexity L0 ← L1,...Ln. If body is the empty set, then we write of applying this query to arbitrary databases; the complex- L0., and the rule is called a Fact. A defeasible rule is an ity is thus given as a function of the size of the database. ordered pair, denoted “Head —≺ Body”, where “Head” is Program or Expression complexity appears when a specific a ground literal, and “Body” is a finite, non-empty set of database is fixed, and we study the complexity of applying ground literals. A defeasible rule with head L0 and body queries represented by arbitrary expressions in the language; {L1,...Ln, n > 0} is written as L0 —≺ L1,...Ln. the complexity is given as a function of the length of the ex- A defeasible logic program P, abbreviated de.l.p., is a set pression. Combined complexity considers both query and of strict rules and defeasible rules. We will distinguish the database instance as input variables. subsets ΠF of facts, ΠR of strict rules, Π = ΠF ∪ ΠR and In this work we are concerned with the study of complex- the subset ∆ of defeasible rules. ity of some important decision problems of DeLP. The sys- tem and its asociated game semantics GS are analyzed intro- Intuitively, whereas Π is a set of certain and exception-free ducing relevant decision problems in relation to the possible knowledge, ∆ is a set of defeasible knowledge, i.e., tentative query answers. information that could be used, whenever nothing is posed Since DeLP builds the arguments from a defeasible logic against it. program results of central importance to consider and eval- By definition a de.l.p. may be an infinite set of strict and uate two questions: “is a set of defeasible rules an argument defeasible rules but for complexity analysis we restrict our- for a literal under a defeasible logic program?” which has selves to finite defeasible logic programs. been proved to be P-complete, and does there exit an argu- We denote by Lit the set of all the ground literals that ment for a literal under a defeasible logic program? which can be generated considering the underlying signature of a + has been proved to be in NP. de.l.p. an we denote by Lit the set of all the atoms in Lit. We define data, expression and combined complexity in DeLP proof theory is based on developments in non the context of DeLP, in order to evaluate the efficiency of monotonic argumentation systems (Pollock 1987; Simari & DeLP implementations. In particular, we study data com- Loui 1992). An argument for a literal L is a minimal subset plexity of query answering to assess DeLP applications over of ∆ that together with Π consistently entails L. The no- database technologies. As far as we know data complexity tion of entailment corresponds to the usual SLD derivation has not been introduced in the context of argumentation sys- used in logic programming, performed by backward chain- tems. ing on both strict and defeasible rules, where negated atoms The paper is structured as follows. In the following sec- are treated as a new atom in the underlying signature. Thus, tion we briefly outline the fundamentals of DeLP, and de- an agent can explain a literal L, throughout this argument. scribe the declarative game-based semantics GS. Then, we In order to determine whether a literal L is supported from discuss DeLP through GS semantics pointing out the deci- a de.l.p. a dialectical tree for L is built. An argument for L sion problems that are of central importance, and we define represents the root of the dialectical tree, and every other data, expression and combined complexity in the context of node in the tree is a defeater argument against its parent. At DeLP. Afterwards, we give complexity results on the exis- each level, for a given a node we must consider all the argu- tence of an argument for a literal L under a defeasible logic ments against that node.