Ana Teresa C. Martins

Full classical S5 in natural deduction with weak normalization

Abstract Natural deduction systems for classical, intuitionistic and modal logics were deeply investigated by Prawitz [D. Prawitz, Natural Deduction: A Proof-theoretical Study, in: Stockholm Studies in Philosophy, vol. 3, Almqvist and Wiksell, Stockholm, 1965. Reprinted at: Dover Publications, Dover Books on Mathematics, 2006] from a proof-theoretical perspective. Prawitz proved weak normalization for classical logic only for a language without ∨ , ∃ and with a restricted application of reduction ad absurdum. Reduction steps related to ∨ , ∃ and classical negation bring about many problems solved only rather recently. For classical S5 modal logic, Prawitz defined a normalizable system, but for a language without ∨ , ∃ , ◊ and, for a propositional language without ◊ , Medeiros [M.da P.N. Medeiros, A new S4 classical modal logic in natural deduction, Journal of Symbolic Logic 71 (3) (2006) 799–809] presented a normalizable system for classical S4. We can mention many cut-free Gentzen systems for S4, S5 and K45/K45D, some normalizable natural deduction systems for intuitionistic modal logics and one more for full classical S4, but not for full classical S5. Here our focus is on the definition of a classical and normalizable natural deduction system for S5, taking not only □ and ◊ as primitive symbols, but also all connectives and quantifiers, including classical negation, disjunction and the existential quantifier. The normalization procedure is based on the strategy proposed by Massi [C.D.B. Massi, Provas de normalizacao para a logica classica, Ph.D. Thesis, Departamento de Filosofia, UNICAMP, Campinas, 1990] and Pereira and Massi [L.C. Pereira, C.D.B. Massi, Normalizacao para a logica classica, in: O que nos faz pensar, Cadernos de Filosofia da PUC-RJ, vol. 2, 1990, pp. 49–53] for first-order classical logic to cope with the combined use of classical negation, disjunction and the existential quantifier. Here we extend such results to deal with □ and ◊ too. The elimination rule for ◊ uses the notions of connection and of essentially modal formulas already proposed by Prawitz for the introduction of □ . Beyond weak normalization, we also prove the subformula property for full S5.

On Minimal Models

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN Δ logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Lowenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory Γ there is a NAT which is a conservative extension of Γ. We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

Searching Contexts in Rough Description Logics

In this paper, we present a method to obtain optimized query refinements of assertion axioms in the Rough Description Logic ALC. This method is based on the notion of discernibility matrix commonly used in the process of attributes reduction in the rough set theory. It consists of finding sets of concepts which satisfy the rough set approximation operations in assertion axioms. Consequently, these sets of concepts can be used to restrict or relax queries in this logic. We propose two algorithms to settle this problem of query refinement and show their complexity results.

Parameterized Complexity of Some Prefix-Vocabulary Fragments of First-Order Logic.

We analyze the parameterized complexity of the satisfiability problem for some prefix-vocabulary fragments of First-order Logic with the finite model property. Here we examine three natural parameters: the quantifier rank, the vocabulary size and the maximum arity of relation symbols. Following the classical classification of decidable prefix-vocabulary fragments, we will see that, for all relational classes of modest complexity and some classical classes, fixed-parameter tractability is achieved by using the above cited parameters.

The Descriptive Complexity of Decision Problems through Logics with Relational Fixed-Point and Capturing Results

Abstract In this work, we generalize the classical fixed-point logics using relations instead of operators in order to capture the notion of nondeterminism. The basic idea is that we use loops in a relation instead of fixed-points of a function, that is, X is a fixed-point of the relation R in case the pair (X,X) belongs to R . We introduce the notion of initial fixed-point of an inflationary relation R and the associated operator rifp. We denote by RIFP the first-order logic with the inflationary relational fixed-point operator rifp and show that it captures the polynomial hierarchy using a translation to second-order logic. We also consider the fragment RIFP1 with the restriction that the rifp operator can be applied at most once. We show that RIFP1 captures the class NP and compare our logic with the nondeterministic fixed-point logic proposed by Abiteboul, Vianu and Vardi in [S. Abiteboul, M. Vardi, and V. Vianu. Fixpoint logics, relational machines, and computational complexity. Journal of the ACM, 44-1:30–56, 1997].

First-Order Logic for Decision Problems with Preference Aggregation

Decision-making is a cognitive procedure that leads to a selection of a course of action among several. In order to perform the correct decision-making, an intelligent agent should have a strategy to analyse the problem, based on some criteria, and determine the alternatives to be choosen. One way to represent and solve this problem is by using mathematical logic. Based on preference logics, we propose a logic, namely the First-Order Logic for Decision Problems with Preference Aggregation to model Multi-criteria Decision Problems in multi-agent environments and support decision-making. We will present the syntax and semantics of our logic, some query examples to illustrate its adequacy in properly model the problem of preference aggregation, and comparisons with other works.

Searching contexts in paraconsistent rough description logic

AbstractBackgroundQuery refinement is an interactive process of query modification used to increase or decrease the scope of search results in databases or ontologies.MethodsWe present a method to obtain optimized query refinements of assertion axioms in the paraconsistent rough description logic PRALC

An ALC Description Default Logic with Exceptions-First

\mathcal {PR_{\textit {ALC}}}

Polynomial Hierarchy Graph Properties in Hybrid Logic

, a four-valued paraconsistent version of the rough AℒC

Expressible preferential logics

\mathcal {ALC}