Abduction is a topic that attracts much interest in AI and automated reasoning research. Different approaches have been devised, that give a formalized account of explanatory reasoning, propose methods to compute explanations, frame abduction in the context of logic programming. However, the logical nature of abduction is still far from being clear and different specifications of the key underlying concepts have been given, that make it difficult to speak of abduction as a single well-defined form of reasoning.This work is a (...) preliminary discussion on the logical nature of abductive reasoning, emphasizing the fundamental difference between abductive and deductive inference. Some logical properties of the inference to the “best explanation” are put forward and analyzed when the underlying logic is any extension of classical propositional logic or a non monotonic system. (shrink)

We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs. We prove that in minimum length sequent calculus proofs, no formula is generated twice or used twice on any single branch of the proof. We prove that the number of distinct subformulas in a minimum length Frege proof is linearly bounded by the number of lines. Depthd Frege proofs ofm lines can be transformed into depthd proofs ofO(m d+1) symbols. We show that renaming (...) Frege proof systems are p-equivalent to extended Frege systems. Some open problems in propositional proof length and in logical flow graphs are discussed. (shrink)

Succinct representation and fast access to large amounts of data are challenges of our time. This unique book suggests general approaches of 'complexity of descriptions'. It deals with a variety of concrete topics and bridges between them, while opening new perspectives and providing promising avenues for the 'complexity puzzle'.

One of our purposes here is to expose something of the elementary logical structure of abductive reasoning, and to do so in a way that helps orient theorists to the various tasks that a logic of abduction should concern itself with. We are mindful of criticisms that have been levelled against the very idea of a logic of abduction; so we think it prudent to proceed with a certain diffidence. That our own account of abduction is itself abductive is methodological (...) expression of this diffidence. A second objective is to test our conception of abduction's logical structure against some of the more promising going accounts of abductive reasoning. We offer our various suggestions in a benignly advisory way. The primary targets of our advice is ourselves, meant as guides to work we have yet to complete or, in some instances, start. It is possible that our colleagues in the abduction research communities will find our counsel to be of some interest. But we repeat that our first concern is to try to get ourselves straight about what a logic of abduction should encompass. (shrink)

We show that Smullyan's analytic tableaux cannot p-simulate the truth-tables. We identify the cause of this computational breakdown and relate it to an underlying semantic difficulty which is common to the whole tradition originating in Gentzen's sequent calculus, namely the dissonance between cut-free proofs and the Principle of Bivalence. Finally we discuss some ways in which this principle can be built into a tableau-like method without affecting its analytic nature.

In this dissertation I study abduction, that is, reasoning from an observation to its possible explanations, from a logical point of view. This approach naturally leads to connections with theories of explanation in the philosophy of science, and to computationally oriented theories of belief change in Artificial Intelligence. ;Many different approaches to abduction can be found in the literature, as well as a bewildering variety of instances of explanatory reasoning. To delineate our subject more precisely, and create some order, a (...) general taxonomy for abductive reasoning is proposed in chapter 1. Several forms of abduction are obtained by instantiating three parameters: the kind of reasoning involved , the kind of observation triggering the abduction , and the kind of explanations produced . In chapter 2, I choose a number of major variants of abduction, thus conceived, and investigate their logical properties. A convenient measure for this purpose are so-called 'structural rules' of inference. Abduction deviates from classical consequence in this respect, much like many current non-monotonic consequence relations and dynamic styles of inference. As a result we can classify forms of abduction by different structural rules. A more computational analysis of processes producing abductive inferences is then presented in chapter 3, using the framework of semantic tableaux. I show how to implement various search strategies to generate various forms of abductive explanations. ;Our eventual conclusion is that abductive processes should be our primary concern, with abductive inferences their secondary 'products'. Finally, chapter 4 is a confrontation of the previous analysis with existing themes in the philosophy of science and artificial intelligence. In particular, I analyse two well-known models for scientific explanation as forms of abduction. This then provides them with a structural logical analysis in the style of chapter 2. Moreover, I argue that abduction can model dynamics of belief revision in artificial intelligence. For this purpose, an extended version of the semantic tableaux of chapter 3 provides a new representation of the operations of expansion, and contraction. (shrink)

In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an (...) approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially. (shrink)

he formalization of abductive reasoning is still an open question: there is no general agreement on the boundary of some basic concepts, such as preference criteria for explanations, and the extension to first order logic has not been settled.Investigating the nature of abduction outside the context of resolution based logic programming still deserves attention, in order to characterize abductive explanations without tailoring them to any fixed method of computation. In fact, resolution is surely not the best tool for facing meta-logical (...) and proof-theoretical questions.In this work the analysis of the concepts involved in abductive reasoning is based on analytical proof systems, i.e. tableaux and Gentzen-type systems. A proof theoretical abduction method for first order classical logic is defined, based on the sequent calculus and a dual one, based on semantic tableaux. The methods are sound and complete and work for full first order logic, without requiring any preliminary reduction of formulae into normal forms. In the propositional case, two different characterizations are given for abductive explanations, each of them being the declarative counterpart of a different algorithm for the generation of explanations. The first one corresponds the generation of the whole set of minimal and consistent explanations, where minimality is checked by comparison with the other elements of the set. The second characterization corresponds to a algorithm for generating a single minimal explanation that is consistent with the theory.The first order versions of the abductive systems make use of unification and dynamic herbrandization/skolemization of formulae. The construction of the abduced formula is pursued by means of de-skolemization. The first order methods are very loose in discarding explanations that are not minimal In fact, the question of minimality in first order abduction is a main issue. As usually defined, minimality is undecidable, for two different reasons: determining whether an explanation is better than another one is in general undecidable; the set of explanations may be infinite. Moreover, because of , a minimal element may not exist. Such problems suggest that the minimality requirement should be relaxed, possibly defining it w.r.t. a stronger relation than logical consequence. (shrink)

Abduction or retroduction, as introduced by C.S. Peirce in the double sense of searching for explanatory instances and providing an explanation is a kind of complement for usual argumentation. There is, however, an inferential step from the explanandum to the abductive explanans . Whether this inferential step can be captured by logical machinery depends upon a number of assumptions, but in any case it suffers in principle from the triviality objection: any time a singular contradictory explanans occurs, the system collapses (...) and stops working. The traditional remedies for such collapsing are the expensive mechanisms of consistency maintenance, or complicated theories of non-monotonic derivation that keep the system running at a higher cost. I intend to show that the robust logics of formal inconsistency, a particular category of paraconsistent logics which permit the internalization of the concepts of consistency and inconsistency inside the object language, provide simple yet powerful techniques for automatic abduction. Moreover, the whole procedure is capable of automatization by means of the tableau proof-procedures available for such logics. Some motivating examples are discussed in detail. (shrink)