15 research outputs found
Reasoning on Schemata of Formulae
A logic is presented for reasoning on iterated sequences of formulae over
some given base language. The considered sequences, or "schemata", are defined
inductively, on some algebraic structure (for instance the natural numbers, the
lists, the trees etc.). A proof procedure is proposed to relate the
satisfiability problem for schemata to that of finite disjunctions of base
formulae. It is shown that this procedure is sound, complete and terminating,
hence the basic computational properties of the base language can be carried
over to schemata
Polyadic Algebras
This chapter surveys the development in the theory of polyadic algebras in the last decades
Tarski's Q-relation algebras and Thompson's groups
The connections between Tarski's Q-relation algebras and Thompson's groups F,
T, V, and monoid M are reviewed here, along with Jonsson-Tarski algebras, fork
algebras, true pairing algebras, and tabular relation algebras. All of these
are related to the finitization problem and Tarski's formalization of set
theory without variables. Most of the technical details occur in the variety of
J-algebras, which is obtained from relation algebras by omitting union and
complementation and adopting a set of axioms created by Jonsson. Every relation
algebra or J-algebra that contains a pair of conjugated quasiprojections
satisfying the Domain and Unicity conditions, such as those that arise from
J\'onsson-Tarski algebras or fork algebras, will also contain homomorphic
images of F, T, V, and M. The representability of tabular relation algebras is
extended here to J-algebras, using a notion of tabularity equivalent among
relation algebras to the original definition.Comment: 64 pages, 4 figures, 1 tabl
Declarative operations on nets
To increase the expressiveness of knowledge representations, the graph-theoretical basis of semantic networks is reconsidered. Directed labeled graphs are generalized to directed recursive labelnode hypergraphs, which permit a most natural representation of multi-level structures and n-ary relationships. This net formalism is embedded into the relational/functional programming language RELFUN. Operations on (generalized) graphs are specified in a declarative fashion to enhance readability and maintainability. For this, nets are represented as nested RELFUN terms kept in a normal form by rules associated directly with their constructors. These rules rely on equational axioms postulated in the formal definition of the generalized graphs as a constructor algebra. Certain kinds of sharing in net diagrams are mirrored by binding common subterms to logical variables. A package of declarative transformations on net terms is developed. It includes generalized set operations, structure-reducing operations, and extended path searching. The generation of parts lists is given as an application in mechanical engineering. Finally, imperative net storage and retrieval operations are discussed