Ángel Nepomuceno-Fernández

Towards Abductive Reasoning in First-order Logic

Abductive problems have been widely studied in propositional logic. First order abduction, however, has been viewed as intractable, for the undecidability of logical consequence. In this paper, we propose a notion of abductive problem, N -abductive problem, which is relative to the cardinality of the minimal model satisfying the given theory. We use a notion of restricted satisfaction, also relative to a domain cardinality. Finally, we propose an effective procedure for the searching of abductive solutions, by means of a modification of Beth’s tableaux.

Model-Baded Abduction via Dual Resolution

This papers presents δ-resolution, a dual resolution calculus. It is based on standard resolution, and uses appropriate formulae equivalent to disjunctive normal forms, instead of conjunctive normal ones, as it is the case for resolution. This duality is then useful to create a calculus for abductive processes, as a way to construct a set of abductive solutions. The proposed calculus is compared to semantic tableaux, an standard logical framework, also illuminating when studying abduction. δ-resolution calculus is a contribution to logic programming, and it further suggests new possibilities to explore abduction at a first order level, in the lines of those proposed in [9].

An epistemic and dynamic approach to abductive reasoning: Abductive problem and abductive solution

Abstract We propose a study of abductive reasoning addressing it as an epistemic process that involves both an agentʼs information and the actions that modify this information. More precisely, we present and discuss definitions of an abductive problem and an abductive solution in terms of an agentʼs information, that is, in terms of knowledge and beliefs. The discussion is then formalised by ‘implementing’ our definitions in a dynamic epistemic logic framework, where the properties of these definitions are studied, an epistemic action that represents the application of an abductive step is introduced, and an illustrative example is provided. A number of the most interesting properties of abductive reasoning (those highlighted by Peirce) are shown to be better modelled within this approach.

Scientific Explanation and Modified Semantic Tableaux

Standard semantic tableaux method and a modification of its δ-rule, due to [Boolos, 1984] and [Diaz, 1993], allows us to obtain δ′-tableaux and apply that to abduction problem, paying attention to some versions given in [Aliseda, 1997]. Our approach of abduction in semantic tableaux faces up to the problem of the existence of infinite branches. Defined Cn, a basic logical operation, a new operation Cn* is obtained. An abduction problem 〈θ, ϕ〉 can be seen as the problem of choosing the appropriated sentence of the set Ab(〈θ, ϕ〉), defined from Cn, or, taking into account Cn*, Ab(〈θ,ϕ〉). The main results are: (i) a (finite) set Γ of L-sentences is n-satisfiable iff the δ′-tableau of Γ has an open branch in which only n constants occur; (ii) if a finite set Γ of L-sentences is satisfiable, then the δ′-tableau provides a minimal interpretation that satisfies Γ; (iii) for any abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if there is a δ′-tableau of θ ∪ ¬ϕ different from its standard tableau, then there is a solution a for 〈θ,ϕ〉; (iv) given an abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if a is a solution and θ ∪ a is consistent, then a ∈ Ab*(〈θ,ϕ〉) and it is an explanatory solution (with respect to Ab*); (v) given an abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if a is an explanatory solution, then θ ∈ Th*(θ ∪ a). They are Theorem 3, Corollary 4, Theorems 14 and 15 and Corollary 17, respectively. Several examples to illustrate that are given.

A modal framework for modelling abductive reasoning

We present a framework for understanding abduction within modal logic and Kripke semantics; worlds of a Kripke frame will represent possible theories, and a change in theory will be understood as a passage from one world to an adjacent possible world. Further, these steps may agree with the accessibility relation or may ‘backtrack’, accordingly as new information refutes or reinforces our present theory. Our formalism can be used to model not only abduction, but also to talk about the inner structure of theories as well as relations between them, allowing us to interpret many ideas from philosophy of science within the well-understood framework of modal logic.

Abduction via C-tableaux and δ-resolution

The formalization of abductive reasoning has received increasing attention from logicians. However, few work is found beyond abduction in propositional logic, given that in a first order formalism, the undecidability problem naturally appears, and therefore an abductive problem cannot even be appropriately formulated. Still, many applications in artificial intelligence allow finite domains to work with, and this gives an opportunity to apply abduction in first order logic with restricted domains. In this paper, we present an approach to abductive reasoning in C-structures, first order structures with a finite domain in which each of its elements has a corresponding constant representing its interpretation. By using semantic tableaux with bounded depth, that is, C-tableaux (a modification of N-tableaux, introduced by the authors in previous papers) and δ-resolution calculus, we build an effective procedure for the searching of minimal abductive solutions within the proposed semantics.

tarfa: tableaux and resolution for finite abduction

n-tableaux [1] and δ-resolution [2], which are based, respectively, on semantic tableaux and resolution, have been properly used for the resolution of abductive problems. The tool we present is a Prolog implementation of an abductive solver which combines both calculi to attack first order abductive problems by reducing them to finite versions, that is, propositional rewritings of the problems which presuppose a context representable with finite models with a known cardinality.

Abductive Reasoning in Dynamic Epistemic Logic

This chapter proposes a study of abductive reasoning addressing it as an epistemic process that involves both an agent’s information and the actions that modify this information. More precisely, this proposal presents and discusses definitions of an abductive problem and an abductive solution in terms of an agent’s information (her knowledge and beliefs) and the involved epistemic actions (observation and belief revision). The discussion is then formalized with tools from dynamic epistemic logic; under such framework, the properties of the given definitions are studied, an epistemic action representing the application of an abductive step is introduced, and an illustrative example is provided. A number of the most interesting properties of abductive reasoning (those highlighted by Peirce) are shown to be better modeled within this approach.

Information-theoretic perspective for teaching logic

This work is about using conceptual tools for teaching logic. Our perspective is based on the information theoretic logic studied by J. Corcoran. Argumentation is referred in terms of a new terminology that is introduced from the concept of information, which is taken as primitive. Trying to show that the understanding of several basic concepts of logic could be facilitated, the notion of information content of a proposition is studied, the concept of logical implication is redefined from this informational point of view and the three kinds of inferences are informationally analysed.