Petrucio Viana

Hybrid logics with Sahlqvist axioms

We show that every extension of the basic hybrid logic with modal Sahlqvist axioms is complete. As a corollary of our approach, we also obtain the Beth property for a large class of hybrid logics. Finally, we show that the new completeness result cannot be combined with the existing general completeness result for pure axioms.

Propositional Dynamic Logic with Storing, Recovering and Parallel Composition

This work extends Propositional Dynamic Logic (PDL) with parallel composition operator and four atomic programs which formalize the storing and recovering of elements in data structures. A generalization of Kripke semantics is proposed that instead of using set of possible states it uses structured sets of possible states. This new semantics allows for representing data structures and using the five new operator one is capable of reasoning about the manipulation of these data structures. The use of the new language (PRSPDL) is illustrated with some examples. We present sound and complete set of axiom schemata and inference rules to prove all the valid formulas for a restricted fragment called RSPDLo.

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.

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.

On Positive Relational Calculi

We discuss the question of inclusions between positive relational terms and some of its aspects, using the form of a dialogue. Two possible approaches to the problem are emphasized: natural deduction and graph manipulations. Both provide sound and complete calculi for proving the valid inclusions, supporting nice strategies to obtain proofs in normal form, but the latter appears to present several advantages, which are discussed.

On a Graph Calculus for Algebras of Relations

We present a sound and complete logical system for deriving inclusions between graphs from inclusions between graphs, taken as hypotheses. Graphs provide a natural tool for expressing relations and reasoning about them. Here we extend this system to a sound and complete one to cope with proofs from hypotheses. This leads to a system dealing with complementation. Other approaches using pictures for relations use as bases the theory of allegories or rewriting systems. Our formalism is more widely applicable and provides a common denominator of these approaches.

Positive Fork Graph Calculus

We introduce and illustrate a graph calculus for proving and deciding the positive identities and inclusions of fork algebras, i.e., those without occurrences of complementation. We show that this graph calculus is sound, complete and decidable. Moreover, the playful nature of this calculus renders it much more intuitive than its equational counterpart.

A note on proofs with graphs

Curtis and Lowe (S. Curtis, G. Lowe, Proofs with graphs, Sci. Comput. Program. 26 (1996) 197-216) presented a graphical calculus to prove inclusions between relations, leaving completeness of the system as an open question. Here we recast their system, excluding complementation and union, by presenting a low complexity decision algorithm to decide its valid inclusions. Completeness follows as a corollary.

A graph calculus for proving intuitionistic relation algebraic equations

In this work, we present a diagrammatic system in which diagrams based on graphs represent binary relations and reasoning on binary relations is performed by transformations on diagrams. We proved that if a diagram D1 can be transformed into a diagram D2 using the rules of our system, under a set Σ of hypotheses, then it is intuitionistically true that the relation defined by diagram D1 is a sub-relation of the one defined by diagram D2, under the hypotheses in Σ.

The Second Venn Diagrammatic System

We present syntax and semantics of a diagrammatic language based on Venn diagrams in which a diagram is not read as a statement about sets, but as a set itself. We prove that our set of rules is sound and complete with respect to the intended semantics. Our system has two slight advantages in relation to the systems we usually encounter in the literature. First, the drawing of diagrams for terms is made inside the system, i.e., by a completely mechanical process based just on the rules of the system. Second, as a consequence, the validity of an inclusion is also verified inside the system and does not depend on any other means than those afforded by our set of rules. These characteristics are absent in the majority of the Venn diagrammatic systems.