Paulo A. S. Veloso

A Finite Axiomatization for Fork Algebras

Proper fork algebras are algebras of binary relations over a structured set. The underlying set has changed from a set of pairs to a set closed under an injective function. In this paper we present a representation theorem for their abstract counterpart, that entails that proper fork algebras — whose underlying set is closed under an injective function — constitute a finitely based variety.1

Ultrafilter Logic and Generic Reasoning

We examine a new logical system, capturing the intuition of ‘most’ by means of generalised quantifiers over ultrafilters, with the aim of providing a basis for generic reasoning. This monotonic ultrafilter logic is a conservative extension of classical first-order logic, with which it shares several properties, including a simple sound and complete deductive system. For reasoning about generic objects, we introduce ‘generic’ individuals as those possessing the properties that most individuals have. We examine some properties of these ‘generic’ individuals and internalise them as generic constants, which produces conservative extensions where one can correctly reason about generic objects as intended. A many-sorted version of our ultrafilter logic is also introduced and employed to handle correctly distinct notions of ‘large’ subsets. Examples similar to ones in the literature illustrate the presentation. We also comment on some perspectives for further work: interesting connections with fuzzy logic, inductive reasoning and empirical reasoning suggest the possibility of other applications for our logic.

On pushout consistency, modularity and interpolation for logical specifications

Abstract We generalize three known results concerning (conservative) extensions to (faithful) interpretations. These results are Extension Modularity (a special case of the Modularization Theorem for logical specifications) and two familiar logical theorems, namely Robinsons Joint Consistency and Craig-Robinson Interpolation. Their generalizations involve a pushout construction, in lieu of union, and their proofs rely on internalization techniques, including a novel one, which reduce — to a large extent — interpretations to extensions.

Some remarks on multiple-entry finite automata

Abstract Some remarks on multiple-entry finite automata are presented. They deal with the effects of nondeterminism and of other logics on the family of languages accepted, comparison with finite automata (both deterministic and nondeterministic) in terms of number of states and a bound on a decision procedure.

On the Modularization Theorem for logical specifications

Abstract The Modularization Property is a basic tool for guaranteeing the preservation of modular structure under refinements and its importance has been noted by several researchers. In the context of logical specifications, i.e. those presented by sets of first-order sentences of a (possibly many-sorted) language, the role played by the Modularization Property is examined and a proof of the Modularization Theorem is presented.

Modeling interactive storytelling genres as application domains

In this paper, we introduce a formalism to specify interactive storytelling genres in the context of digital entertainment, adopting an information systems approach. We view a genre as a set of plots, where a plot is a partially ordered sequence of events, taken from a fixed repertoire. In general, the specification of a genre should allow to determine whether a plot is a legitimate representative of the genre, and also to generate all plots belonging to the genre. The formalism divides the specification of a genre into static, dynamic and behavioral schemas, that reflect a plan recognition/plan generation paradigm. It leads to executable specifications, supported by LOGTELL, a prototype tool that helps users generate, modify and reuse plots that follow a genre specification. To illustrate the use of the formalism, we specify a simple Swords & Dragons genre and show plots generated by the tool.

Logics For Qualitative Reasoning

Assertions and arguments involving vague notions occur often both in ordinary language and in many branches of science. The vagueness may be plainly expressed by “modifiers”, such as ‘generally’, ‘rarely’, ‘most’, ‘many’, etc., or, less obviously, conveyed by objects termed ‘representative’, ‘typical’ or ‘generic’. A precise treatment of such ideas has been a basic motivation for logics of qualitative reasoning. Here, we present some logical systems with generalized quantifiers for these modifiers, also handling ‘generic’ reasoning. Other possible applications for these and related logics for qualitative reasoning are indicated. These (monotonic) generalized logics, with simple sound and complete deductive calculi, are proper conservative extensions of classical first-order logic, with which they share various properties. For generic reasoning, special individuals can be introduced by means of ‘generally’, and internalized as representative constants, thereby producing conservative extensions where one can reason about generic objects as intended. Some interesting situations, however, require such assertions to be relative to various universes, which cannot be captured by relativization. Thus we extend our generalized logics to sorted versions, with qualitative notions relative to the universes, which can also be compared.

On graph reasoning

In this paper, we study the (positive) graph relational calculus. The basis for this calculus was introduced by Curtis and Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easy to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

Reasoning with Graphs

In this paper we study the (positive) graph relational calculus. The basis for this calculus was introduced by S. Curtis and G. Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easier to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

A calculus for graphs with complement

We present a system for deriving inclusions between graphs from a set of inclusions between graphs taken as hypotheses. The novel features are the extended notion of graph with an explicitly representation of complement, the more involved definition of the system, and its completeness proof due to the embedding of complements. This is an improvement on former work, where complement was introduced by definition. Our calculus provides a basis on which one can construct a wide range of graph calculi for several algebras of relations.