Argument-based extended logic programming with defeasible priorities

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.

The topics of structural proof theory and logic programming have influenced each other for more than three decades. Proof theory has contributed the notion of sequent calculus, linear logic, and higher-order quantification. Logic programming has introduced new normal forms of proofs and forced the examination of logic-based approaches to the treatment of bindings. As a result, proof theory has responded by developing an approach to proof search based on focused proof systems in which introduction rules are organized into two alternating phases of rule application. Since the logic programming community can generate many examples and many design goals (e.g., modularity of specifications and higher-order programming), the close connections with proof theory have helped to keep proof theory relevant to the general topic of computational logic.

Starting from a general dynamic analysis of reasoning and programming, we develop two main dynamic perspectives upon logic programming. First, the standard fixed point semantics for Horn clause programs naturally supports imperative programming styles. Next, we provide axiomatizations for Prolog-type inference engines using calculi of sequents employing modified versions of standard structural rules such as monotonicity or permutation. Finally, we discuss the implications of all this for a broader enterprise of ‘abstract proof theory’.

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O'Hearn and Pym's logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus \`a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.

Inference to the Best Legal Explanation

This paper describes an extension of Horn clause logic programs by bounded quantifiers. Bounded quantifiers had been extensively used in a part of mathematical logic called theory of admissible sets [2]. Later some variants of bounded quantifiers had been introduced in logic programming languages [12, 19, 21, 9, 6, 7]. We show that an extension of logic programs by bounded quantifiers has several equivalent logical semantics and is efficiently implementable using a variant of SLD-resolution, which we call SLDB-resolution. We give examples showing that introduction of bounded quantifiers results in a high level logical specification language. An expressive power of subsets of Horn clauses and subsets of logic programs with bounded quantifiers is compared. We also show that the use of bounded quantifiers sheds new light on classical negation in logic programming.KeywordsLogic ProgramLogic ProgrammingFunction SymbolUnification AlgorithmPredicate SymbolThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the complexity of computable and Σ$_{\rm 1}^{\rm 0}$ inductive definitions of sets of natural numbers. For we example, we show how to assign natural indices to monotone Σ$_{\rm 1}^{\rm 0}$-definitions and we use these to calculate the complexity of the set of all indices of monotone Σ$_{\rm 1}^{\rm 0}$-definitions which are computable. We also examine the complexity of new type of inductive definition which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.

This work addresses the problem of providing explanation capabilities to an argumentation system. Explanation in defeasible argumentation is an important, and yet undeveloped field in the area. Therefore, we move in this direction by defining a concrete argument system with explanation facilities.We consider the structures that provide information on the warrant status of a literal. Our focus is put on argumentation systems based on a dialectical proof procedure, therefore we study dialectical explanations. Although arguments represent a form of explanation for a literal, we study the complete set of dialectical trees that justifies the warrant status of a literal, since this set has proved to be a useful tool to comprehend, analyze, develop, and debug argumentation systems.KeywordsBelief RevisionWarrant StatusStrong NegationDialectical ProcessDefeat RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The information available to intelligent agents is often uncertain, inconsistent and incomplete. It is then crucially important to develop logics for uncertainty to achieve different complex reasoning tasks. Uncertainty is present in many applications and may be represented using different frameworks: plausibility orderings, probability theory frameworks, argumentation systems, etc. This special issue gathers five contributions that cover different aspects of reasoning under uncertainty using non classical logics. Four of the five papers are fully revised and extended versions of contributions initially presented at the Fifth International Conference on Scalable Uncertainty Management (SUM’11) The two first papers concern extensions of logic-based programming languages to deal with uncertain information. The paper by Pere Pardo, Lluis Godo entitled “t-DeLP: an argumentation-based Temporal Defeasible Logic Programming framework” deals with handling temporal information using defeasible logic. The proposed framework, called t-DeLP, allows to extend argumentation-based defeasible logic in order to represent temporal processes. The idea is to associate temporal information to literals and to define temporal logic programs are sets of basic temporal facts and (strict or defeasible) durative rules. Concepts of argumentation systems, such as the one of undefeated arguments, are redefined in this t-DeLP framework. A procedure for determining a set of warranted literals is proposed. The paper also discusses rational properties satisfied by the t-DeLP framework. The paper by Georg Gottlob,Thomas Lukasiewicz, Maria Vanina Martinez and Gerardo I. Simari entitled “Query Answering under Probabilistic Uncertainty in Datalog+/ Ontologies” concerns an extension of Datalog+/− to deal with uncertainty. The authors use Markov Logic Networks to provide a probabilistic semantics for the probabilistic Datalog+/−. They provide scalable and sound algorithms for

We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is proof-theoretical naturalness: The system consists of all and only the inference rules generated by the single, simple, linear scheme of the recently introduced subatomic logic. Thanks to this regularity, cuts are eliminated via a natural construction. The second reason is that the system generates efficient proofs. Indeed, we show that a certain class of tautologies due to Statman, which cannot have better than exponential cut-free proofs in the sequent calculus, have polynomial cut-free proofs in our system. We achieve this by using the same construction that we use for cut elimination. In summary, by expanding the language of propositional logic, we make its proof theory more regular and generate more proofs, some of which are very efficient. That design is made possible by considering atoms as superpositions of their truth values, which are connected by self-dual, non-commutative connectives. A proof can then be projected via each atom into two proofs, one for each truth value, without a need for cuts. Those projections are semantically natural and are at the heart of the constructions in this article. To accommodate self-dual non-commutativity, we compose proofs in deep inference.

This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.

Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called ILGL that gives an account of layering. The logic is a bunched system, combining the usual intuitionistic connectives, together with a non-commutative, non-associative conjunction (used to capture layering) and its associated implications. We give soundness and completeness theorems for a labelled tableaux system with respect to a Kripke semantics on graphs. We then give an equivalent relational semantics, itself proven equivalent to an algebraic semantics via a representation theorem. We utilise this result in two ways. First, we prove decidability of the logic by showing the finite embeddability property holds for the algebraic semantics. Second, we prove a Stone-type duality theorem for the logic. By introducing the notions of ILGL hyperdoctrine and indexed layered frame we are able to extend this result to a predicate version of the logic and prove soundness and completeness theorems for an extension of the layered graph semantics . We indicate the utility of predicate ILGL with a resource-labelled bigraph model.

This article presents some basic aspects of the proof theory of first order logic programming. We start off from general resolution, unit resolution, input resolution and SLD resolution. Then we turn to the sequent calculus and its relationship to resolution and present deductive systems for logic programs (with negation). After discussing partiality in logic programming, this article ends with the introduction of inductive extensions of logic programs, which provide a powerful proof-theoretic framework for logic programming.

Several techniques for improving web search have been developed over the last years. Most existing approaches are still limited, mainly due to the absence of qualitative criteria for ranking results and insensitivity to user preferences for guiding the search. At the same time, defeasible argumentation evolved as a successful approach in AI to model commonsense qualitative reasoning with applications in many areas, such as agent theory, knowledge engineering and legal reasoning. This paper presents ARGUENET, a recommender system that classifies search results according to preference criteria declaratively specified by the user. The proposed approach integrates a traditional web search engine with a defeasible argumentation framework.

Defeasible argumentation is typical of legal and scientific reasoning. A defeasible argument is one in which the conclusion can be accepted tentatively in relation with the evidence known so far, but may need to be retracted as new evidence comes in. This paper analyses the role of defeasible argumentation in the explanation and evaluation of architectural decisions. We analyse technical explanations offered by engineers at Twitter and eBay about several architectural decisions adopted in those systems. We generalize these examples in four argumentation schemes. We also study the typical case of reasoning with a formal model of an architecture, and we infer a fifth argumentation scheme. Finally, we apply Hastings' method of attaching a set of critical questions to each scheme. We show that the existence of critical questions reveals that the inferred schemes are defeasible: in argumentation theory, if a respondent asks one of the critical questions matching a scheme and the proponent of an argument fails to offer an adequate answer, the argument defaults and the conclusion is retracted. This dialogical structure is the basis of typical architectural evaluations. We conclude that the provided evidence supports the hypothesis that defeasible argumentation is employed in architectural evaluation. In this context, a rich catalogue of argumentation schemes is a useful tool for the architect to organize his or her reasoning, critical questions assist the architect in identifying the weak points of his or her explanations, and provide the evaluation team with a checklist of issues to be raised.