On Complexity of Delp Through Game Semantics∗

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 decision 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 system 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. Thus every node has a descendant program P, and on the decision problem of whether a sub- for every defeater. A comparison criteria is needed for deter- set of defeasible rules is an argument for a literal L under mining whether an argument defeats another. Even though P. Next, we analyze data complexity for DeLP, and we there exist several preference relations considered in the lit- present complexity results for two decision problems on en- erature, in this first approach we will abstract away from that tailment. In the last section, we summarize the main con- issue. tributions of this work, and we present our conclusions and We will say that a literal L is warranted if there is an ar- future research lines. gument for L, and in the dialectical tree each defeater of the root is itself defeated. Recursively, this leads to a marking DeLP and Game Semantics GS procedure of the tree that begins by considering the fact that We will start by introducing some of the basic concepts in leaves of the dialectical tree are undefeated arguments as a DeLP (see (Garc´ıa & Simari 2004) for complete details). In consequence of having no defeaters. Finally, an agent will the language of DeLP a literal L is a atom A or a negated believe in a literal L, if L is a warranted literal. atom ∼ A, where ∼ represents the strong negation in the There exist four possible answers for a query L: YES if L logic programming sense. The complement of a literal L, is warranted, NO if L is warranted (i.e., the complement of L is warranted), UNDECIDED if neither L nor L are warranted, Even though rigorous consequences do not reflect the un- and UNKNOWN if L is not in the underlying signature of the derlying ideas of strict and defeasible rules, they are very program. useful for introducing a declarative definition of argument. We have briefly given an intuitive introduction to the DeLP language and the dialectical procedure for obtain- Definition 4 Let P = hΠ, ∆i be a de.l.p.. We say that ing a warranted conclusion. For complete details on DeLP hA, Li is an argument structure for a ground literal L, if A see (Garc´ıa & Simari 2004). is a set of defeasible rules of ∆, such that: Games have an analogy with a dispute and, therefore, that 1. L ∈ CnR(Π ∪A) analogy extends to argument-based reasoning. A dispute can be seen as a game where in an alternating manner, the player 2. CnR(Π ∪A) 6= Lit P, the proponent, starts with an argument for a literal. The 3. A is minimal w.r.t. inclusion, i.e., there is no A0 ⊆ A player O, the opponent, attacks the previous argument with such that satisfies (1) and (2). a counterargument strong enough to defeat it. The dispute could continue with a counterargument of the proponent, For convenience we will simply speak of argument in- and so on. When a player runs out of moves, i.e., that player stead of argument structure whenever this does not lead to misunderstandings. Let’s introduce game concept and can not find a counterargument for any of his adversary’s ar- GS guments, the game is over. If the proponent’s argument has semantics. not been defeated then she has won the game. The semantics GS is a declarative trivalued game-based Definition 5 Let P = (Π, ∆) be a de.l.p., h a literal and semantics for DeLP that links game-semantics (Abramsky hA,hi an argument structure for h. A game for hA,hi with & McCusker 1997) and model theory. Soundness and com- respect to P, that we denote G(hA, Li, P), is a structure pleteness of GS with respect to DeLP proof theory have been (MG(hA, Li, P),JG(hA, Li, P),PG(hA, Li, P)) proved (Cecchi & Simari 2004). In the following we present some notions of GS, for more details see (Cecchi & Simari where 2000; 2004). • M is a set of argument structure. ∗ G(hA, Li, P) Let X be a set and {x1,...xn} ⊆ X, X is the set of • JG(hA, Li, P) : MG(hA, Li, P) × I → {P, O} where I is an finite sequences over X and [x1 ...xn] denotes the sequence enumerable index; of the elements x1,...xn. We write |s| for the length of a • P ⊆ M ∗ , where P is a non- finite sequence and si for the ith element of s, 1 ≤ i ≤ |s|. G(hA, Li, P) G(hA, Li, P) G(hA, Li, P) Concatenation of sequences is indicated by juxtaposition. If empty, prefix-closed set. t = su for some sequence u, then we say that s is a prefix Each sequences s of PG(hA, Li, P) satisfy: of t. Let P ref(S) be a set of prefix of S, then S is prefix 1. s = [hA,hi]s0, s0 possibly empty. closed if S = P ref(S). 2. For all i, 1 < i ≤ |s| In order to use a game to capture the dialectical procedure, we need to define in a declarative way the movements of JG(hA, Li, P)(s1, 1) = P such game: the argument. The followings definitions are JG(hA, Li, P)(si, i) = JG(hA, Li, P)(si−1, i − 1) based on the notation introduced in (Lifschitz 1996). P = O and O = P.

Definition 2 Let X be a set of ground literals. The set X 3. If s ∈ PG(hA, Li, P), then for each argument structure is rigorously closed under a de.l.p. P, if for every strict hA2,h2i that is a legal move for s|s|, there exists a se- rule Head ← Body of P, Head ∈ X whenever Body ⊆ quence t ∈ PG(hA, Li, P), such that t = s[hA2,h2i]. X, and for every defeasible rule Head0 —≺ Body0 of P, 4. No other sequence belongs to PG(hA, Li, P). Head0 ∈ X whenever Body0 ⊆ X. The set X is consistent if there is no literal L such that {L, L} ⊆ X. Otherwise, we Movements in a game are the introduction of arguments. For will say that X is inconsistent. every argument A for a literal L we can built a game whose We say that X is logically closed if it is consistent or it is first move is hA, Li. Thus, a family of games will be ob- equal to Lit. tained considering all the arguments for L. Intuitively, if the set of knowledge of an agent is rigorously Definition 6 Let P be a de.l.p., h a literal un- closed under a de.l.p., the agent will not believe in a literal der the signature of P, hA1,hi, ..., hAn,hi that she cannot explain. all the argument structures of h under P and G(hA1,hi, P), G(hA2,hi, P), ...,G(hAn,hi, P) the Definition 3 Let Ψ be a set of strict and defeasible rules. corresponding games to the arguments of h. The set of rigorous consequences under the rules Ψ, denoted CnR(Ψ), is the least set of literals w.r.t. inclusion, such that {G(hA1,hi, P), G(hA2,hi, P), ...,G(hAn,hi, P)} it is logically closed and rigorously closed under Ψ. Let P be a de.l.p., the set of rigorous consequences of P is defined is the game family of h and we denote it as F(h, P). as CnR(P). Definition 7 Let a be the first proponent movement in the Theorem 1 Let P be a de.l.p. and L a literal. L is warranted game. A sequence s is complete if s = [a]s1, with s1 poten- under P if and only if L belongs to the set T or L belongs tially empty, then there is no movement b ∈ MG(hA, Li, P) to the set F of the minimal G-models hT,F i of P under GS such that [a]s1[b] ∈ PG(hA, Li, P). A sequence s is pre- semantics. ferred if each opponent movement has a proponent answer. In other words, a sequence s is preferred if |s| is odd. We have briefly presented the DeLP language, its proof theory and its declarative game-based semantics GS. Now, Definition 8 A strategy over a game G is a set of sequences we will be able to analyze the system and study some com- S, such that for all sequence s ∈ S, either: plexity properties. • s is preferred; or Discussion on Complexity • there exists other sequence s0 ∈ S, such that s0 is pre- GS ferred and s and s0 has a prefix t, |t| = n, n is even and DeLP is a defeasible reasoning system where every conse- 0 quence of a de.l.p. is analyzed considering all the arguments sn+1 6= sn+1. for and against it. The trivalued game semantics GS char- Definition 9 Let P be a de.l.p., h ∈ Lit and acterizes such reasoning by two sets T and F , since T ∪ F G(hA, Li, P) ∈ F(h, P). We say that P wins the game is the set of all warranted literals. Undecided literals are the G(hA, Li, P) or that G(hA, Li, P) is won by P, if the set remaining literals L for which there is no warrant for it nor of complete sequences of PG(hA, Li, P) is an strategy. Other- for its complement. When considering DeLP in relation to wise, we say that O wins the game or that G(hA, Li, P) is game semantics, there are two relevant computational deci- won by O. sion problems to analyze in the context of a de.l.p. P: A player can win a game even though he does not win • GAMESAT: Deciding whether there is a game for a literal every complete sequence in such game. In (Prakken & Sar- α won by the proponent P in the context of a de.l.p. P. tor 1997) the authors have developed an argument-based ex- • NOWINGAME: Deciding whether there is no game for a tended logic programming system which differs from DeLP literal α neither for the complement of α won by the pro- in its winning rule: a player wins a dialogue tree if and only ponent P in the context of a de.l.p. P. if he wins all the branches of the tree. The former problem involves just finding a game that is won by the proponent. In order to capture the latter, it is Definition 10 Let P be a de.l.p.. A game-based interpre- necessary to find all the games for the literal and for its com- tation for P, or G-Interpretation for P for short, is a tuple plement, and to establish that none of them is won by the hT,F i, such that T and F are subsets of atoms of the under- proponent. lying signature of P and T ∩ F = ∅. A positive GAMESAT answer for a given de.l.p. P and a In the previous definition T stands for true while F stands literal L implies that L ∈ T , being hT,F i the minimal G- for false. The set of atoms UNDECIDED is defined as the set model, i.e., P |=GS L. A positive GAMESAT answer for L + U = Lit − {T ∪ F }. means that L ∈ F in the minimal G-model, i.e., P |=GS L. Each game can finish in two possible ways: won by the The NOWINGAME decision problem for a de.l.p. P and a proponent P or won by the opponent O. There is no possi- literal L is equivalent to determining if given the minimal + bility for a draw. As the first move is made by the P, we are G-model hT,F i of P, L ∈ Lit − {T ∪ F }, i.e., P 6|=GS L interested in those games won by this player. and P 6|=GS L. In this case, three interesting situation can be contem- Definition 11 Let P be a de.l.p., h an atom of the under- plated, and establish the followings decision problems: lying signature of P, F(h, P) the game family for h and • Whether there is no game for a literal L, neither for its the game family for under a de.l.p. . A game- F(h, P) h P complement L. The game families for a literal L and based model for , that we name G-Model of , is a G- P P for its complement L, F(L, P) and F(L, P) respectively, interpretation such that: hT,F i are empty. L has no argument neither for nor against it. • If there exists a game G(hA,hi in the family F(h, P) won Therefore, the agent has no information about such query. by P, then h belongs to T . • Whether there is no game for a literal L, and the non • If there exists a game G(hA, hi, P) in the family F(h, P) empty set of all games in the family of its complement won by P, then h belongs to F . L are won by the opponent. The game family for a lit- Since we only consider literals under the signature of eral L, F(L, P), is empty and only games won by the de.l.p., the G-model definition does not contemplate the an- opponent are in the non empty family F(L, P). L has no swer UNKNOWN. The minimal G-model defines a sound and argument for and all the arguments for its complement are complete semantics GS for DeLP (Cecchi & Simari 2004). defeated. Therefore, the agent has no information for L, We will say that GS entails a literal L from a de.l.p. P, de- and he cannot defend its complement. In a similarly way, noted by P |=GS L, whenever L ∈ T or L ∈ F , being we can define the case where the agent cannot defend a hT,F i the minimal G-model of P. literal L, and has no information about its complement. The following theorem relates proof theory and game- based semantics, showing soundness and completeness. • Whether all games in the non empty families for L and The complexity of computing arguments for its complement L are won by the opponent. F(L, P) Arguments and counterarguments are the movements in and F(L, P) are non empty set and all the argument are a game, and hence the core of DeLP. Dung’s formal- defeated. The agent cannot defend any argument neither ism (Dung 1995) and some extensions that have been devel- for nor against the literal L. oped (Bench-Capon 2002; 2003; Amgoud & Cayrol 2002), In order to determine the computational complexity of the offer a powerful tool for the abstract analysis of defeasible decision problems introduced above, we will study DeLP reasoning. However, these approaches operate with argu- from two approaches: combined and data complexity. Com- ments and their attack and defeat relation at an abstract level, bined complexity of a fragment of logic programming has avoiding to deal with the underlying logical language used to been defined and used in (Dantsin et al. 2001): structure the arguments. On the other hand DeLP does construct the arguments and analyzes the defeater relationship. Complexity of (some fragment of) logic programming: Thus, studying the decision problem: “is a given subset of is the complexity of checking if for variable programs defeasible rules an argument for a literal under a de.l.p.?” is P and variable ground atoms A, P |= A. of central importance. On the other hand, the notion of data complexity is borrowed Following the definition of argument this problem has from relational database theory (Vardi 1982). Databases are three parts: is L a consequence of Π ∪A?, is Π ∪A consis- nowadays the main tool for storing and retrieving very large tent?, and is there a subset A0 of A such that it is consistent sets of data. Data complexity allows us to study DeLP as a with Π and that together with Π derives L? query language measuring its complexity focus on the size Let P = hΠF , ΠR ∪ ∆i be a de.l.p., L be a literal and of the databases, and using defeasible and strict rules for A ⊆ ∆. The first condition of definition 4, that involves rig- inference purpose. Data complexity is a key measure to de- orous consequences concept is h ∈ CnR(Π ∪A). In (Cec- termine the efficiency of argumentation system implementa- chi & Simari 2000), we have defined the following transfor- tions based on database technologies. mation Φ from a de.l.p. into a propositional definite logic For methodological and complexity issues, it is important program, i.e., a propositional logic program with just Horn to distinguish in a de.l.p. the input data from the inference clauses. Let A be an atom. Φ(A) = A, Φ(∼ A) = A0 where rules. Thus, hereafter, we will denote P = hΠF , ΠR ∪ ∆i, A0 is a new atom not in the signature of the de.l.p. and the where ΠF is a finite set of ground facts, and ΠR∪∆ is a finite transformation of a conjunction is Φ(A, B) = Φ(A), Φ(B). set of ground strict and defeasible rules. Making an analogy Φ(H —≺ B) = Φ(H) ← Φ(B) and all other rules remain with database concepts, ΠF represents the input databases, the same Φ(H ← B) = Φ(H) ← Φ(B) . We will use this also called the extensional part, and ΠR ∪ ∆ are the infer- transformation, and the following lemma in order to reduce ence rules, called the intensional part of the database. We the rigorous consequences of a de.l.p. into consequences of define a Boolean query as a finite set of strict and defeasible propositional Horn clauses. rules together with a ground literal L. The intended intuitive meaning of defining such query is the following: we want Lemma 1 Let DP be a definite logic program, and M be to know whether a literal L is entailed by GS from ΠR ∪ ∆ the minimal model of DP , then M = CnR(DP ). together with the database ΠF . Following the principle and notions above, in the context We are interested in computing the time complexity of of DeLP we will define data, program and combined com- verifying whether L ∈ CnR(Π ∪A). We shall construct a plexity as follows. logic program with just Horn clauses, denoted HP(Π, A, L) such that L ∈ CnR(Π ∪A) if and only if HP(Π, A, L) |= Definition 12 Let Ω be any of the decision problems intro- yes . duced above, P = hΠF , ΠR ∪ ∆i and (ΠR ∪ ∆, L) a query: Suppose that A1,...,An are all the atoms in Π ∪A. We define HP(Π, A, L) as follows: • The data complexity of Ω is the complexity of Ω when the query is fixed, and the database varies, i.e., parameters HP(Π, A, L) = Φ(Π) ∪ Φ(A) ∪ {yes ← Φ(L)}∪ ΠR ∪ ∆ and L are fixed. {yes ← Φ(Ai), Φ(Ai) : 1 ≤ i ≤ n} • The program or expression complexity of Ω is the complexity of Ω when the database instance is fixed, and the Even though the SAT decision problem is NP-complete, query varies, i.e., the parameter ΠF is fixed. both checking whether a definite propositional logic pro- • The combined complexity of Ω is the complexity where gram DP satisfies a ground atom A, i.e., DP |= A, every parameter ΠF , ΠR ∪ ∆ and L vary. and HORNSAT, i.e., the decision problem of whether or not there is a truth assignment that satisfies a collection Expression and combined complexity are quite close and of Horn clauses, are P-complete (Dantsin et al. 2001; they are rarely differentiated. For this reason we will only Papadimitriou & Yannakakis 1997). discuss data and combined complexity. In order to carry out this complexity analysis we will first Lemma 2 HP(Π, A, L) is a transformation from a de.l.p. P focus on the complexity of determining whether there is an into propositional Horn clauses such that verifying whether argument for a literal . Then we will study if the game A L a literal L belongs to CnR(P) is equivalent to verifying played with initial move A is won by the proponent. Algorithm: Minimal et al. 2001) : the number of iterations is bounded by the Input: A an argument for a literal L, and Π a set of strict number of rules plus one. Each iteration step is feasible rules. in polynomial time. Thus finding the minimal model of a Output: true if A is a minimal argument for L, false other- logic program with just Horn clauses is in P (Dantsin et wise al. 2001). minimal=true By lemma 2, L ∈ CnR(Π∪A) has been reduced to propo- Aux= A sitional logic programming. Therefore, L ∈ CnR(Π∪A) While minimal and not Aux = ∅ do is in P. select H —≺ B ∈Aux • Hardness: Horn rules are strict rules in a de.l.p., and the A0 = A − {H —≺ B} minimal model of a definite logic program DP is equal to 0 if h ∈ CnR(Π∪A ) CnR(DP). Therefore, by applying reduction by general- then minimal=false ization, we have that DP |= L reduce to L ∈ CnR(DP). else Aux= Aux - {H —≺ B} Propositional logic programming is P-complete (Dantsin et al. 2001). This suffices to complete the proof. ¥ Figure 1: Algorithm for verifying if a set of defeasible rules is minimal with respect to set inclusion for deriving a literal Until now we have proved that the first condition of ar- L. gumentation definition is P-complete. Now we will analyze the rest of the issues we need for computing an argument. whether yes is entailed from the transformed propositional We will denote the cardinality of the language by |Lit| and defeasible rules cardinality by |∆|. Horn program. Thus, L ∈ CnR(P) reduces to DP |= yes, being DP a propositional Horn program. In Figure 1, we present an algorithm for verifying whether a set of defeasible rules is minimal with respect to set inclu- Proof: In order to prove our claim, we have to establish sion for entail a literal L. Worst case of the minimality con- that: dition is considered assuming that the argument has at most 1. L ∈ CnR(Π ∪A) if and only if HP(Π, A, L) |= yes. |∆| defeasible rules, i.e., ∆ is an argument for some literal. Computing the minimality condition involves |∆| loops ver- 0 We will consider two cases: ifying that L ∈ CnR(Π ∪ A ), which is in P. Thus, this • Π ∪A is consistent. problem is solvable in polynomial time, and, therefore, it is L ∈ CnR(Π ∪A) if and only if Φ(L) ∈ Φ(CnR(Π ∪ in P. A)) if and only if Φ(L) ∈ CnR(Φ(Π ∪A))(see (Cec- Finally, to check whether the set of defeasible rules is chi & Simari 2000)) if and only if, by lemma 1, Φ(L) consistent under a de.l.p., we verify that there is no atom is in the minimal model of Φ(Π ∪ A) if and only such that the atom and its complement are members of if HP(Π, A, L) |= yes by the definition of minimal CnR(Π∪A). In the worst case, when CnR(Π∪A) is consis- model, the monotonicity property and the use of the tent, this algorithm must control every atom in the signature rule yes ← Φ(L). of the de.l.p.. Thus, to check if it is consistent is proportional • Π ∪A is inconsistent. to the number of atoms |Lit|/2 and therefore it is in P. L ∈ CnR(Π ∪ A) = Lit if and only if there ex- Theorem 3 The decision problem “is a given subset of de- ists i, 1 ≤ i ≤ n, such that Φ(Li) and Φ(Li) are in feasible rules an argument for a literal under a de.l.p.?” is R if and only if i and i are Φ(Cn (Π ∪A)) Φ(L ) Φ(L ) P-complete. in the minimal model of HP(Π, A, L) if and only if HP(Π, A, L) |= yes by definition of minimal models, Proof: monotonicity property and the use of the rule yes ← Membership: (sketch) From the above development it fol- Φ(Li), Φ(Li). lows membership to P. 2. HP is computed in logarithmic space: the transformation Hardness: We employ a reduction from DP |= L, being is quite simple, and is feasible in logarithmic space, since DP a propositional Horn program. Consider the following rules can be generated independently of each other except transformation r(DP) = DP0 = hΠ, ∆i, where Π = DP those of the form yes ← Φ(Li), Φ(Li) which depends on and ∆ is empty. r is a transformation computed in loga- the literal in the input. rithmic space such that whenever a literal L is entailed by a Therefore HP(Π, A, L) is a reduction from L ∈ CnR(Π ∪ propositional Horn program DP, the decision problem “is a ¥ A) into propositional Horn clauses. given subset of ∆ = ∅ an argument for L under DP0” finish in an accepting state. Theorem 2 Let P = (ΠF , ΠR ∪ ∆) a de.l.p., A ⊆ ∆, and L is in the minimal model of a propositional Horn pro- L a literal. Determining whether L ∈ CnR(Π ∪ A) is P- gram if and only if L ∈ CnR(DP) if and only if ∅ is an complete. argument for L, since is minimal and consistent with Π, if ∅ Proof: • Membership: Given a definite logic program and only if “is a given subset of ∆ = an argument for ∞ P the least fixpoint TP of the operator TP can be computed in polynomial time (Papadimitriou 1994; Dantsin L under DP0” finish in an accepting state. Thereby estab- we will first analyze the dialectical tree structure over the lishing that the decision problem “is a given subset of de- size of the facts and the strict and defeasible rules. feasible rules an argument for a literal under a de.l.p.?” is The dialectical tree is explored in a complete depth first P-complete. ¥ way, as minimax does. If the maximum depth of the tree is m, and there are b legal movements at each point, then the Our final aim is to determine the complexity of comput- m ing the set of all the arguments under a de.l.p.. This is moti- time complexity will be O(b )(Russell & Norvig 2003). If implement the technique alpha-beta pruning, and consider- vated in that GAMESAT and NOWINGAME require for play- ing that successors are examined in random order, then the ing a game to compute every argument that defeats each ar- 3m gument introduced in a previous move. A subset A ⊆ ∆ time complexity will be roughly O(b 4 )(Russell & Norvig may be a potential argument of different literals in the lan- 2003). The maximum depth of a dialectical tree for an ar- ∆ guage. Thus, the maximum number of checks for potential gument under a de.l.p. with |∆| defeasible rules is 2| |, i.e., arguments that depends on the size of the set of defeasible we can consider every potential argument in one branch of rules and on the size of Lit, is |Lit| ∗ 2|∆|. the tree. Any argument can appear more than once in the tree but at most once in every branch, because of the accept- Lemma 3 Let AP be the polynomial time needed for the able argumentation line definition. What about branch fac- decision problem “is a given subset of defeasible rules an tor: there exists |Lit|/2 literals that can be in conflict with argument for a literal l under a de.l.p.?”. Then, the upper the last argument. These literals may have at most 2|∆| po- bound time for computing all the arguments is |Lit| ∗ 2|∆| ∗ tential arguments. So our branching factor is in the worst AP . case |Lit|/2 ∗ 2|∆|. Thus, exploring the dialectical tree as ∆ |∆|−1 2| | The result above states an exponential upper bound for minimax does is of O((|Lit| ∗ 2 ) ). computing X , the set of all the arguments in Dung’s formal- Every time we must insert a neighbour node B of a node ism (Dung 1995). A in the tree structure or equivalently, when a player makes Even though we must verify whether every subset of a move, we must check if it is a legal move in the game, i.e., ∆ is an argument for every literal in the language of the if B attacks and defeats A, and if B does not introduce in- de.l.p., because the consistency condition in the defini- consistency. In order to determine whether B is a defeater tion of argument, A ⊆ ∆ cannot be an argument for a of A, we must take into account the preference criterion be- literal and for its complement, so we will consider only tween arguments. Any preference criterion defined among |Lit| |∆| |∆|−1 arguments could be used in DeLP. For this reason, the com- 2 ∗ 2 = |Lit| ∗ 2 potential arguments in order to plexity class of the following decision problem “whether an play a game or equivalently to build the dialectical tree. This argument can be considered in the tree structure of a game” upper bound could be improved by considering minimality will be left parameterized in the class C. over the arguments, i.e., no A1 ⊆ ∆ would be an argument for a literal L if A2 is an argument of L and A2 ⊆ A1. Theorem 4 Let C be the complexity class for the decision Finally, we consider the argument existence decision problem:“whether an argument can be considered in the tree problem. structure of a game”. The upper bound for data complexity C Corollary 1 (Argument Existence) The decision problem of GAMESAT is NP . “whether there is an argument for a literal L under a de.l.p.” Proof: For fixed ΠR ∪ ∆, the size of the dialectical tree is NP. for an argument hA,hi is polynomial in the size of the lit- Proof: We can guess any subset of ∆, and verify whether erals in ΠF . Furthermore, computing each argument is in this subset is an argument for a literal L under de.l.p. in poly- P, and considering each argument in the tree structure is in nomial time. This proves membership in NP. ¥ C. In order to decide whether a literal h belongs to the set T of the minimal G-model, we guess for an argument of h These results contrast with those of (Parsons, Wooldridge, such that the game played from this argument is won by the & Amgoud 2003), where determining whether there is an ar- P Proponent. The number of arguments is polynomial when gument for a formula h is Σ2 -complete. Even thought there ΠR ∪ ∆ is fixed, and determining whether the game is won are some similarities between argument definitions, they dif- by the Proponent can be done with a C oracle. This proves fer in the underlying logic. While in DeLP approach an membership in NPC . ¥ argument is a subset of defeasible rules, and the inference mechanism to obtain it is logic programming based, an ar- Since NOWINGAME is a conjunction of GAMESAT com- gument in the formalism described in (Parsons, Wooldridge, plements an immediate corollary to the result above follows & Amgoud 2003) is a subset of formulas of a propositional naturally. language, and ` stands for classical inference. Corollary 2 Let C be the complexity class for the decision Data Complexity for DeLP problem:“whether an argument can be considered in the tree structure of a game”. The upper bound for data complexity In order to determine the upper bound for the data complex- C ity of the decision problems GAMESAT and NOWINGAME, of NOWINGAME is co-NP . 1 bound on NP, otherwise known as Σ1, which coincides with Decision Complexity the class of properties of finite structures expressible in ex- Problem istential second-order logic (Fagin 1974). When analyzing Data complexity we have fixed the query Is L ∈ CnR(Rules)? P-complete and we have parametrized the preference criteria. Thus an Is hA,hi an argument? P-complete interesting topic for future research is to study to what extent Argument Existence NP this results can be applied to others rule-based argumenta- C GAME Data Complexity NP tion systems whose theory proof is rather similar. C NOWINGAME Data Complexity co − NP As future work we will analyze combined complexity of the decision problems introduced. We are studying the expressive power of DeLP in order to compare this system with Table 1: Problems studied, and the main complexity results other non monotonic formalisms. obtained. References Even though we have not analyzed in depth the complex- Abramsky, S., and McCusker, G. 1997. Game Seman- ity for computing the preference criterion, we illustrate this tics. 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