Recently, there has been an increasing interest in extending Dung's framework with probability theory, leading to the Probabilistic Argumentation Framework (PAF), and with supports in addition to attacks, leading to the Bipolar Argumentation Framework (BAF). In this paper, we introduce the Conditional Probabilistic Bipolar Argumentation Framework (CPBAF), which extends Probabilistic and Bipolar AF by allowing conditional probabilities on arguments, attacks, and on (possibly cyclic) supports. In this setting, we address the problem of computing the probability that a given argument is accepted. This is carried out by introducing the concept of probabilistic explanation for a given (probabilistic) extension. We show that the complexity of the problem is FP^#P-hard and propose polynomial approximation algorithms with bounded additive error for CPBAF where cycles with an odd number of attacks are forbidden.

Explainable AI has received significant attention in recent years. Machine learning models often operate as black boxes, lacking explainability and transparency while supporting decision-making processes. Local post-hoc explainability queries attempt to answer why individual inputs are classified in a certain way by a given model. While there has been important work on counterfactual explanations, less attention has been devoted to semifactual ones. In this paper, we focus on local post-hoc explainability queries within the semifactual `even-if' thinking and their computational complexity among different classes of models, and show that both linear and tree-based models are strictly more interpretable than neural networks. After this, we introduce a preference-based framework enabling users to personalize explanations based on their preferences, both in the case of semifactuals and counterfactuals, enhancing interpretability and user-centricity. Finally, we explore the complexity of several interpretability problems in the proposed preference-based framework and provide algorithms for polynomial cases.

Dung's Argumentation Framework (AF) has been extended in several directions to make knowledge representation and reasoning tasks more intuitive and/or expressive. We present a novel extension of AF called Featured AF (FAF), where each argument has associated a set of features expressed by means of unary and binary facts. In such a context, a query is expressed by means of a conjunctive relational calculus formula which is evaluated over the extensions of the FAF. Then, this framework is further expanded into the so-called Extended FAF (EFAF), where a first-order logic formula (FOL) is used for reasoning over `feasible' subframeworks that satisfy the FOL formula and minimally differ from the original framework. We investigate the computational complexity of verification and acceptance problems under several semantics and show that incomplete AF (iAF) frameworks, including correlated iAF and constrained iAF, are special cases of EFAF.

Dung's abstract Argumentation Framework (AF) has been extended in several directions to make knowledge representation and reasoning more intuitive and expressive. In this paper, we present the Knowledge-based Argumentation Framework (KAF), an extension of AF with a Knowledge Base (KB) expressed in DL-Lite, which includes concept and role instances describing the topology of an AF, besides additional knowledge on the domain. The KAF semantics is given by a set of KAF extensions, each consisting of an extension of the underlying AF together with a ``pertinent'' subset of the original KB, which is obtained by discarding assertions referring to arguments that have been ruled out in the AF extension. Then, the framework is further expanded into the Constrained KAF (CKAF), where a set of restricted relational calculus formulae is used for reasoning over `feasible' subframeworks that satisfy the formulae and minimally differ from the original framework. We thoroughly investigate the computational complexity of classical reasoning problems under popular argumentation semantics, and show that well-known AF-based frameworks are special cases of CKAF.

Dung’s Argumentation Framework (AF) has been extended in several directions. Among the numerous proposed extensions, three of them seem to be of particular interest and have correlations between them. These extensions are: constrained AF (CAF), where AF is augmented with (strong) constraints; epistemic AF (EAF), where AF is augmented with epistemic constraints; and incomplete AF (iAF), where arguments and attacks can be uncertain. While the complexity and expressiveness of CAF and iAF have been studied, that of EAF has not been explored so far. In this paper we investigate the complexity and expressivity of EAF. To this end, we first introduce the Labeled CAF (LCAF), a variation of CAF where constraints are defined over the alphabet of labeled arguments. Then, we investigate the complexity of credulous and skeptical reasoning and show that: i) EAF is more expressive than iAF (under preferred semantics), ii) although LCAF is a restriction of EAF where modal operators are not allowed, these frameworks have the same complexity, iii) the results for LCAF close a gap in the characterization of the complexity of CAF. Interestingly, even though EAF has the same complexity as LCAF, it allows modeling domain knowledge in a more natural and easy-to-understand way.

Explainable Artificial Intelligence and Formal Argumentation have received significant attention in recent years. Argumentation frameworks are useful for representing knowledge and reasoning on it. Counterfactual and semifactual explanations are interpretability techniques that provide insights into the outcome of a model by generating alternative hypothetical instances. While there has been important work on counterfactual and semifactual explanations for Machine Learning (ML) models, less attention has been devoted to these kinds of problems in argumentation. In this paper, we explore counterfactual and semifactual reasoning in abstract Argumentation Framework. We investigate the computational complexity of counterfactual- and semifactual-based reasoning problems, showing that they are generally harder than classical argumentation problems such as credulous and skeptical acceptance. Finally, we show that counterfactual and semifactual queries can be encoded in weak-constrained Argumentation Framework, and provide a computational strategy through ASP solvers.

Dealing with controversial information is an important issue in several application contexts. Formal argumentation enables reasoning on arguments for and against a claim to decide on an outcome. Dung's abstract Argumentation Framework (AF) has emerged as a central formalism in argument-based reasoning. Key aspects of the success and popularity of Dung's framework include its simplicity and expressiveness. Integrity constraints help to express domain knowledge in a compact and natural way, thus keeping easy the modeling task even for problems that otherwise would be hard to encode within an AF. In this paper, we first explore two intuitive semantics based on Kleene and Lukasiewicz logics, respectively, for AF augmented with (strong) constraints—the resulting argumentation framework is called Constrained AF (CAF). Then, we propose a new argumentation framework called Weak constrained AF (WAF) that enhances CAF with weak constraints. Intuitively, these constraints can be used to find “optimal” solutions to problems defined through CAF. We provide a detailed complexity analysis of CAF and WAF, showing that strong constraints do not increase the expressive power of AF in most cases, while weak constraints systematically increase the expressive power of CAF (and AF) under several well-known argumentation semantics.

Abstract Dung’s abstract Argumentation Framework (AF) has emerged as a key formalism for argumentation in artificial intelligence. It has been extended in several directions, including the possibility to express supports, leading to the development of the Bipolar Argumentation Framework (BAF), and recursive attacks and supports, resulting in the Recursive BAF (Rec-BAF). Different interpretations of supports have been proposed, whereas for Rec-BAF (where the target of attacks and supports may also be attacks and supports) even different semantics for attacks have been defined. However, the semantics of these frameworks have either not been defined in the presence of support cycles or are often quite intricate in terms of the involved definitions. We encompass this limitation and present classical semantics for general BAF and Rec-BAF and show that the semantics for specific BAF and Rec-BAF frameworks can be defined by very simple and intuitive modifications of that defined for the case of AF. This is achieved by providing a modular definition of the sets of defeated and acceptable elements for each AF-based framework. We also characterize, in an elegant and uniform way, the semantics of general BAF and Rec-BAF in terms of logic programming and partial stable model semantics.

Abstract Data exchange, the problem of transferring data from a source schema to a target schema, has been studied for several years. The semantics of answering positive queries over the target schema has been defined in early work, but little attention has been paid to more general queries. A few proposals of semantics for more general queries exist but they either do not properly extend the standard semantics under positive queries, giving rise to counterintuitive answers, or they make query answering undecidable even for the most important data exchange settings, for example, with weakly-acyclic dependencies. The goal of this paper is to provide a new semantics for data exchange that is able to deal with general queries. At the same time, we want our semantics to coincide with the classical one when focusing on positive queries, and to not trade-off too much in terms of complexity of query answering. We show that query answering is undecidable in general under the new semantics, but it is $\text{co}\text{NP}\text{-complete}$ when the dependencies are weakly-acyclic. Moreover, in the latter case, we show that exact answers under our semantics can be computed by means of logic programs with choice, thus exploiting existing efficient systems. For more efficient computations, we also show that our semantics allows for the construction of a representative target instance, similar in spirit to a universal solution, that can be exploited for computing approximate answers in polynomial time.

Dung's abstract Argumentation Framework (AF) has emerged as a central formalism in the area of knowledge representation and reasoning. Preferences in AF allow to represent the comparative strength of arguments in a simple yet expressive way. Preference-based AF (PAF) has been proposed to extend AF with preferences of the form a > b, whose intuitive meaning is that argument a is better than b. In this paper we generalize PAF by introducing conditional preferences of the form a > b \leftarrow body that informally state that a is better than b whenever the condition expressed by body is true. The resulting framework, namely Conditional Preference-based AF (CPAF), extends the PAF semantics under three well-known preference criteria, i.e. democratic, elitist, and KTV. After introducing CPAF, we study the complexity of the verification problem (deciding whether a set of arguments is a ``best'' extension) as well as of the credulous and skeptical acceptance problems (deciding whether a given argument belongs to any or all ``best'' extensions, respectively) under multiple-status semantics (that is, complete, preferred, stable, and semi-stable semantics) for the above-mentioned preference criteria.

Dung’s Argumentation Framework (AF) has been extended in several directions. An interesting extension, among others, is the Epistemic AF (EAF) which allows representing the agent’s belief by means of epistemic constraints. In particular, an epistemic constraint is a propositional formula over labeled arguments (e.g. in(a), out(c)) extended with the modal operators K and M that intuitively state that the agent believes that a given formula is certainly or possibly true, respectively. In this paper, focusing on EAF, we investigate the complexity of the possible and necessary variants of three canonical problems in abstract argumentation: verification, existence, and non-empty existence. Moreover, we explore the relationship between EAF and incomplete AF (iAF), an extension of AF where arguments and attacks may be uncertain. Our complexity analysis shows that the verification problem in iAF can be naturally reduced to the verification in EAF, while it turns out that a similar result cannot hold for the necessary (non-empty) existence problem.

In recent years there has been an increasing interest in extending Dung's framework to facilitate the knowledge representation and reasoning process. In this paper, we present an extension of Abstract Argumentation Framework (AF) that allows for the representation of preferences over arguments' truth values (3-valued preferences). For instance, we can express a preference stating that extensions where argument a is false (i.e. defeated) are preferred to extensions where argument b is false. Interestingly, such a framework generalizes the well-known Preference-based AF with no additional cost in terms of computational complexity for most of the classical argumentation semantics. Then, we further extend AF by considering both (3-valued) preferences and 3-valued constraints, that is constraints of the form \varphi \Rightarrow v or v \Rightarrow \varphi, where \varphi is a logical formula and v is a 3-valued truth value. After investigating the complexity of the resulting framework,as both constraints and preferences may represent subjective knowledge of agents, we extend our framework by considering multiple agents and study the complexity of deciding acceptance of arguments in this context.

Dung's Argumentation Framework (AF) has been extended in several directions, including the possibility of representing uncertainty about the existence of arguments and attacks. In this regard, two main proposals have been introduced in the literature: Probabilistic Argumentation Framework (PrAF) and Incomplete Argumentation Framework (iAF). PrAF is an extension of AF with probability theory, thus representing quantified uncertainty. In contrast, iAF represents unquantified uncertainty, that is it can be seen as a special case where we only know that some elements (arguments or attacks) are uncertain. In this paper, we first address the problem of computing the probability that a given argument is accepted in PrAF. This is carried out by introducing the concept of probabilistic explanation for any given (probabilistic) extension. We show that the complexity of the problem is FP#P-hard and propose polynomial approximation algorithms with bounded additive error for PrAFs where odd-length cycles are forbidden. We investigate the approximate complexity of the related FP#P-hard problems of credulous and skeptical acceptance in PrAF, showing that they are generally harder than the problem of computing the probability that a given argument is accepted. Next we consider iAF and, after showing some equivalence properties among classes of iAFs, we study iAF as a special case of PrAF where uncertain elements have associated a probability equal to 1/2. Finally, given this result, we investigate the relationships between iAF acceptance problems and probabilistic acceptance in PrAF.

Dung’s Argumentation Framework (AF) has been extended in several directions, including the possibility of representing unquantified uncertainty about the existence of arguments and attacks. The framework resulting from such an extension is called incomplete AF (iAF). In this paper, we first introduce three new satisfaction problems named totality, determinism and functionality, and investigate their computational complexity for both AF and iAF under several semantics. We also investigate the complexity of credulous and skeptical acceptance in iAF under semi-stable semantics—a problem left open in the literature. We then show that any iAF can be rewritten into an equivalent one where either only (unattacked) arguments or only attacks are uncertain. Finally, we relate iAF to probabilistic argumentation framework, where uncertainty is quantified.

Dung's abstract Argumentation Framework (AF) has emerged as a central formalism for argumentation in AI. Preferences in AF allow to represent the comparative strength of arguments in a simple yet expressive way. In this paper we first investigate the complexity of the verification as well as credulous and skeptical acceptance problems in Preference-based AF (PAF) that extends AF with preferences over arguments. Next, after introducing new semantics for AF where extensions are selected using cardinality (instead of set inclusion) criteria and investigating their complexity, we introduce a framework called AF with Priority rules (AFP) that extends AF with sequences of priority rules. AFP generalizes AF with classical set-inclusion and cardinality based semantics, suggesting that argumentation semantics can be viewed as ways to express priorities among extensions. Finally, we extend AFP by proposing AF with Priority rules and Preferences (AFP^2), where also preferences over arguments can be used to define priority rules, and study the complexity of the above-mentioned problems.

Dung's abstract Argumentation Framework (AF) has emerged as a central formalism in formal argumentation. Key aspects of the success and popularity of Dung's framework include its simplicity and expressiveness. Integrity constraints help to express domain knowledge in a compact and natural way, thus keeping easy the modeling task even for problems that otherwise would be hard to encode within an AF. In this paper, after providing an intuitive semantics based on Lukasiewicz's logic for AFs with (strong) constraints, called Constrained AFs (CAFs), we propose Weak constrained AFs (WAFs) that enhance CAFs with weak constraints. Intuitively, these constraints can be used to find ``optimal'' solutions to problems defined through CAFs. We provide a detailed complexity analysis of CAFs and WAFs, showing that strong constraints do not increase the expressive power of AFs in most cases, while weak constraints systematically increase the expressive power of CAFs under several well-known argumentation semantics.

Extensions of Dung’s Argumentation Framework (AF) include the class of Recursive Bipolar AFs (Rec-BAFs), i.e. AFs with recursive attacks and supports. We show that a Rec-BAF \Delta can be translated into a logic program P_\Delta so that the extensions of \Delta under different semantics coincide with subsets of the partial stable models of P_\Delta.

Consistent query answering is a principled approach for querying inconsistent databases. It relies on two basic notions: the notion of a repair, that is, a consistent database that "minimally" differs from the original one, and the notion of a consistent query answer, that is, a query answer that can be derived from every repair. In general, an inconsistent database can admit multiple repairs, each corresponding to a different way of restoring consistency, and the consistent query answering framework does not make any discrimination among them. However, in many applications it is natural and desired to express preferences among the different choices that can be made to resolve inconsistency.

Recently there has been an increasing interest in probabilistic abstract argumentation, an extension of Dung's abstract argumentation framework with probability theory. In this setting, we address the problem of computing the probability that a given argument is accepted. This is carried out by introducing the concept of probabilistic explanation for a given (probabilistic) extension. We show that the complexity of the problem is FP^#P-hard and propose polynomial approximation algorithms with bounded additive error for probabilistic argumentation frameworks where odd-length cycles are forbidden. This is quite surprising since, as we show, such kind of approximation algorithm does not exist for the related FP^#P-hard problem of computing the probability of the credulous acceptance of an argument, even for the special class of argumentation frameworks considered in the paper.

Query answering over inconsistent knowledge bases is a problem that has attracted a great deal of interest over the years. Different inconsistency-tolerant semantics have been proposed, and most of them are based on the notion of repair, that is, a "maximal" consistent subset of the database. In general, there can be several repairs, so it is often natural and desirable to express preferences among them. In this paper, we propose a framework for querying inconsistent knowledge bases under user preferences for existential rule languages. We provide generalizations of popular inconsistency-tolerant semantics taking preferences into account and study the data and combined complexity of different relevant problems.

Abstract Recently there has been an increasing interest in frameworks extending Dung’s abstract Argumentation Framework (AF). Popular extensions include bipolar AFs and AFs with recursive attacks and necessary supports. Although the relationships between AF semantics and Partial Stable Models (PSMs) of logic programs has been deeply investigated, this is not the case for more general frameworks extending AF. In this paper we explore the relationships between AF-based frameworks and PSMs. We show that every AF-based framework Δ can be translated into a logic program PΔ so that the extensions prescribed by different semantics of Δ coincide with subsets of the PSMs of PΔ. We provide a logic programming approach that characterizes, in an elegant and uniform way, the semantics of several AF-based frameworks. This result allows also to define the semantics for new AF-based frameworks, such as AFs with recursive attacks and recursive deductive supports.

The data posting framework introduced in [8] adapts the well-known Data Exchange techniques to the new Big Data management and analysis challenges that can be found in real world scenarios. Although it is expressive enough, it requires the ability of using count constraints and may be difficult for a non expert user. Moreover, the data posting problem is NP-complete under the data complexity in the general case, then the use of the non-deterministic variables is performed. Indeed, identifying the conditions that guarantee polynomial-time execution in the presence of non-deterministic choices is very important for practical purposes. In this paper, we present a simplified version of data posting framework, based on the use of the smart mapping rules, that integrate the simple mapping description with some parameters, avoiding the complex specifications with count constraints. We show that the data posting problem in the new setting is NP- complete and identify the conditions under which this problem becomes polynomial even in the presence of non-deterministic choices.

Incomplete information arises in many database applications, such as data integration, data exchange, inconsistency management, data cleaning, ontological reasoning, and many others. A principled way of answering queries over incomplete databases is to compute certain answers, which are query answers that can be obtained from every complete database represented by an incomplete one.

Consistent query answering is a principled approach for querying inconsistent knowledge bases. It relies on the notion of a "repair", that is, a maximal consistent subset of the facts in the knowledge base. One drawback of this approach is that entire facts are deleted to resolve inconsistency, even if they may still contain useful "reliable" information. To overcome this limitation, we propose a new notion of repair allowing values within facts to be updated for restoring consistency. This more fine-grained repair primitive allows us to preserve more information in the knowledge base. We also introduce the notion of a "universal repair", which is a compact representation of all repairs. Then, we show that consistent query answering in our framework is intractable (coNP-complete). In light of this result, we develop a polynomial time approximation algorithm for computing a sound (but possibly incomplete) set of consistent query answers.

Incomplete information arises in many current database applications. Certain answers are a widely accepted semantics of query answering over incomplete databases. Since their computation is a coNP-hard problem, recent research has focused on developing polynomial time approximation algorithms computing a sound (but possibly incomplete) set of certain answers. In this demo we showcase ACID, a system to compute sound sets of certain answers. The central tools of its underlying algorithms are conditional tables and the conditional evaluation of relation algebra. Different evaluation strategies can be applied, with more accurate ones having higher complexity, but returning more certain answers. We show how to query incomplete databases using the ACID system, which offers a suite of approximation algorithms enabling users to choose the technique that best meets their needs in terms of balance between efficiency and quality of the result's approximation.

Big Data rise made traditional data management techniques inadequate in many real life scenarios. In particular, the availability of huge amounts of data pertaining to user suggestions and searches calls for advanced analysis strategies in order to profitably leverage these data. Furthermore, heterogeneity and high speed of this data require suitable data storage and management tools to be designed from scratch. In this paper, we describe our proposal for analysing the way user searches and suggestions influence their social environment in order to quickly identify users able to spread their influence across the network. It is worth noting that, gathering information about user preferences is crucial in several scenarios like viral marketing, tourism promotion and food education.

In this article, we propose a new technique for checking whether the bottom-up evaluation of logic programs with function symbols terminates. The technique is based on the definition of mappings from arguments to strings of function symbols, representing possible values which could be taken by arguments during the bottom-up evaluation. Starting from mappings, we identify mapping-restricted arguments, a subset of limited arguments, namely arguments that take values from finite domains. Mapping-restricted programs, consisting of rules whose arguments are all mapping restricted, are terminating under the bottom-up computation, as all of its arguments take values from finite domains. We show that mappings can be computed by transforming the original program into a unary logic program: this allows us to establish decidability of checking if a program is mapping restricted. We study the complexity of the presented approach and compare it to other techniques known in the literature. We also introduce an extension of the proposed approach that is able to recognize a wider class of logic programs. The presented technique provides a significant improvement, as it can detect terminating programs not identified by other criteria proposed so far. Furthermore, it can be combined with other techniques to further enlarge the class of programs recognized as terminating under the bottom-up evaluation.

The chase is a well-known algorithm with a wide range of applications in data exchange, data cleaning, data integration, query optimization, and ontological reasoning. Since the chase evaluation might not terminate and it is undecidable whether it terminates, the problem of defining (decidable) sufficient conditions ensuring termination has received a great deal of interest in recent years. In this regard, several termination criteria have been proposed. One of the main weaknesses of current approaches is the limited analysis they perform on equality generating dependencies (EGDs). In this paper, we propose sufficient conditions ensuring that a set of dependencies has at least one terminating chase sequence. We propose novel criteria which are able to perform a more accurate analysis of EGDs. Specifically, we propose a new stratification criterion and an adornment algorithm. The latter can both be used as a termination criterion and be combined with current techniques to make them more effective, in that strictly more sets of dependencies are identified. Our techniques identify sets of dependencies that are not recognized by any of the current criteria.

Abstract It is widely acknowledged that function symbols are an important feature in answer set programming, as they make modelling easier, increase the expressive power, and allow us to deal with infinite domains. The main issue with their introduction is that the evaluation of a program might not terminate and checking whether it terminates or not is undecidable. To cope with this problem, several classes of logic programs have been proposed where the use of function symbols is restricted but the program evaluation termination is guaranteed. Despite the significant body of work in this area, current approaches do not include many simple practical programs whose evaluation terminates. In this paper, we present the novel classes ofrule-boundedandcycle-bounded programs, which overcome different limitations of current approaches by performing a more global analysis of how terms are propagated from the body to the head of rules. Results on the correctness, the complexity, and the expressivity of the proposed approach are provided.

The aim of this paper is to present more general criteria and techniques for chase termination. We first present extensions of the well-known stratification criterion and introduce a new criterion, called local stratification, which generalizes both super-weak acyclicity and stratification-based criteria (including the class of constraints which are inductively restricted). Next, the paper presents a rewriting algorithm transforming the original set of constraints Σ into an “equivalent” set Σαand verifying the structural properties for chase termination on Σα. The rewriting of constraints allows us to recognize larger classes of constraints for which chase termination is guaranteed. In particular, we show that if Σ satisfies chase termination conditions T, then the rewritten set Σαsatisfies T as well, but the vice versa is not true, that is there are significant classes of constraints for which Σαsatisfies T and Σ does not. A more general rewriting algorithm producing as output an equivalent set of dependencies and a Boolean value stating whether a sort of cyclicity has been detected is also proposed. The new rewriting technique and the checking of acyclicity allow us to introduce the class of acyclic constraints, which generalizes local stratification and guarantees that all chase sequences are finite with a length polynomial in the size of the input database.

Abstract Recently, there has been an increasing interest in the bottom-up evaluation of the semantics of logic programs with complex terms. The presence of function symbols in the program may render the ground instantiation infinite, and finiteness of models and termination of the evaluation procedure, in the general case, are not guaranteed anymore. Since the program termination problem is undecidable in the general case, several decidable criteria (called program termination criteria) have been recently proposed. However, current conditions are not able to identify even simple programs, whose bottom-up execution always terminates. The paper introduces new decidable criteria for checking termination of logic programs with function symbols under bottom-up evaluation, by deeply analyzing the program structure. First, we analyze the propagation of complex terms among arguments by means of the extended version of the argument graph calledpropagation graph. The resulting criterion, calledacyclicity, generalizes most of the decidable criteria proposed so far. Next, we study how rules may activate each other and define a more powerful criterion, calledsafety. This criterion uses the so-calledsafety functionable to analyze how rules may activate each other and how the presence of some arguments in a rule limits its activation. We also study the application of the proposed criteria to bound queries and show that the safety criterion is well-suited to identify relevant classes of programs and bound queries. Finally, we propose a hierarchy of classes of terminating programs, calledk-safety, where thek-safe class strictly includes the (k-1)-safe class.

Abstract We study the framework of abductive logic programming extended with integrity constraints. For this framework, we introduce a new measure of the simplicity of an explanation based on its degree of arbitrariness: the more arbitrary the explanation, the less appealing it is, with explanations having no arbitrariness — they are called constrained — being the preferred ones. In the paper, we study basic properties of constrained explanations. For the case when programs in abductive theories are stratified we establish results providing a detailed picture of the complexity of the problem to decide whether constrained explanations exist.

In this paper we propose a new technique for checking whether the bottom-up evaluation of logic programs with function symbols terminates. The technique is based on the definition of mappings from arguments to strings of function symbols, representing possible values which could be taken by arguments during the bottom-up evaluation. Such mappings can be computed by transforming the original program into a unary logic program whose termination is decidable. Starting from mappings we can identify mapping-restricted arguments, a subset of limited arguments, that is, arguments which can take values from finite domains. The class of mapping-restricted programs, consisting of programs whose arguments are mapping-restricted, is terminating under the bottom-up computation as all its arguments can take values from finite domains. We study the complexity of the presented approach and compare it with other techniques known in the literature. The presented technique is relevant as it individuates as terminating programs not detected by other criteria proposed so far and can be combined with other techniques to further enlarge the class of programs recognized as terminating under the bottom-up evaluation.

Abstract Recent years have witnessed an increasing interest in enhancing answer set solvers by allowing function symbols. Since the introduction of function symbols makes common inference tasks undecidable, research has focused on identifying classes of programs allowing only a restricted use of function symbols while ensuring decidability of common inference tasks. Finitely-ground programs, introduced in Calimeri et al. (2008), are guaranteed to admit a finite number of stable models with each of them of finite size. Stable models of such programs can be computed and thus common inference tasks become decidable. Unfortunately, checking whether a program is finitely-ground is semi-decidable. This has led to several decidable criteria, called termination criteria, providing sufficient conditions for a program to be finitely-ground. This paper presents a new technique that, used in conjunction with current termination criteria, allows us to detect more programs as finitely-ground. Specifically, the proposed technique takes a logic program ${\cal P}$ and transforms it into an adorned program ${{\cal P}}$ μ with the aim of applying termination criteria to ${{\cal P}}$ μ rather than ${\cal P}$ . The transformation is sound in that if the adorned program satisfies a certain termination criterion, then the original program is finitely-ground. Importantly, applying termination criteria to adorned programs rather than the original ones strictly enlarges the class of programs recognized as finitely-ground.

Abstract This paper presents a logic language for expressing search and optimization problems. Specifically, first a language obtained by extending (positive) DATALOG with intuitive and efficient constructs (namely, stratified negation, constraints, and exclusive disjunction) is introduced. Next, a further restricted language only using a restricted form of disjunction to define (nondeterministically) subsets (or partitions) of relations is investigated. This language, called atalog, captures the power of DATALOG¬ in expressing search and optimization problems. A system prototype implementing atalog is presented. The system translates atalog queries into Optimization Programming Language (OPL) programs which are executed by the ILOG OPL Development Studio. Our proposal combines easy formulation of problems, expressed by means of a declarative logic language, with the efficiency of the ILOG System. Several experiments show the effectiveness of this approach.

This work is a contribution to prioritized reasoning in logic programming in the presence of preference relations involving atoms. The technique, providing a new interpretation for prioritized logic programs, is inspired by the semantics of Prioritized Logic Programming and enriched with the use of structural information of preference of Answer Set Optimization Programming. Specifically, the analysis of the logic program is carried out together with the analysis of preferences in order to determine the choice order and the sets of comparable models. The new semantics is compared with other approaches known in the literature and complexity analysis is also performed, showing that, with respect to other similar approaches previously proposed, the complexity of computing preferred stable models does not increase.

This paper presents a logic language, called NP Datalog for NP search and optimization problems. The 'search' language extends stratified Datalog with constraints and partition rules defining (nondeterministically) partition of relations. NP optimization problems are then formulated by adding a max (or min) construct to select the solution (stable model) which maximizes (resp., minimizes) the result of a polynomial function applied to the answer relation. We show that NP Datalog queries can be easily evaluated by translating them into ILOG programs which are next solved by means of the ILOG OPL Studio suite. To prove the effectiveness of our proposal, we have implemented a module, written in Sicstus Prolog, which takes in input a NP Datalog query and outputs an equivalent ILOG program. Several experiments comparing the computation of queries by different logic systems have been also performed.

This paper presents a technique for the optimization of bound queries over disjunctive deductive databases with constraints. The proposed approach is an extension of the well-known Magic-Set technique and is well-suited for being integrated in current bottom-up (stable) model inference engines. More specifically, it is based on the exploitation of binding propagation techniques which reduce the size of the data relevant to answer the query and, consequently, reduces both the complexity of computing a single model and the number of models to be considered. The motivation of this work stems from the observation that traditional binding propagation optimization techniques for bottom-up model generator systems, simulating the goal driven evaluation of top-down engines, are only suitable for positive (disjunctive) queries, while hard problems are expressed using unstratified negation. The main contribution of the paper consists in the extension of a previous technique, defined for positive disjunctive queries, to queries containing both disjunctive heads and constraints (a simple and expressive form of unstratified negation). As the usual way of expressing declaratively hard problems is based on the guess-and-check technique, where the guess part is expressed by means of disjunctive rules and the check part is expressed by means of constraints, the technique proposed here is highly relevant for the optimization of queries expressing hard problems. The value of the technique has been proved by several experiments.

The objective of this paper is to investigate the problems related to the extensional integration of information sources. In particular, we propose an approach for managing inconsistent databases, i.e. databases violating integrity constraints. The presence of inconsistent data can be resolved by "repairing" the database, i.e. by providing a computational mechanism that ensures obtaining consistent "scenarios" of the information or by consistently answering to queries posed on an inconsistent set of data. In this paper we consider preferences among repairs and possible answers by introducing a partial order among them on the base of some preference criteria. More specifically, preferences are expressed by considering polynomial functions applied to repairs and returning real numbers. The goodness of a repair is measured by estimating how much it violates the desiderata conditions and a repair is preferred if it minimizes the value of the polynomial function used to express the preference criteria. The main contribution of this work consists in the proposal of a logic approach for querying and repairing inconsistent databases that extends previous works by allowing to express and manage preference criteria. The approach here proposed allows to express reliability on the information sources and is also suitable for expressing decision and optimization problems. The introduction of preference criteria strongly reduces the number of feasible repairs and answers; for special classes of constraints and functions it gives a unique repair and answer.