Solvable Set/Hyperset Contexts: III. A Tableau System for a Fragment of Hyperset Theory

We propose a decision procedure for a fragment of the hyperset theory, HMLSS, which takes inspiration from a tableau saturation strategy presented in [3] for the fragment MLSS of well-founded set theory. The procedure alternates deduction and model checking steps, driving the correct application of otherwise very liberal rules, thus significantly speeding up the process of discovering a satisfying assignment of a given HMLSS-formula or proving that no such assignment exists

Solving equations in the universe of hypersets

Ankara : Department of Computer Engineering and Information Science and Institute of Engineering and Science, Bilkent Univ., 1993.Thesis (Master's) -- Bilkent University, 1993.Includes bibliographical references leaves 46-50Hyperset Theory (a.k.a. ZFC~/AFA) of Peter Aczel is an enrichment of the classical ZFC set theory and uses a graphical representation for sets. By allowing non-well-founded sets, the theory provides an appropriate framework for modeling various phenomena involving circularity. Z F C /A F A has an important consequence that guarantees a solution to a set of equations in the universe of hypersets, viz. the Solution Lemma. This lemma asserts that a system of equations defined in the universe of hypersets has a unique solution, and has applications in areas like artificial intelligence, database theory, and situation theory. In this thesis, a program called HYPERSOLVER, which can solve systems of equations to which the Solution Lemma is applicable and which has built-in procedures to display the graphs depicting the solutions, is presented.Pakkan, MüjdatM.S

Mapping Sets and Hypersets into Numbers

We introduce and prove the basic properties of encodings that generalize to non-well-founded hereditarily finite sets the bijection defined by Ackermann in 1937 between hereditarily finite sets and natural numbers

Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of Logic Programming (TPLP

HYPERSOLVER: A graphical tool for commonsense set theory

This paper investigates an alternative set theory (due to Aczel) called the Hyperset Theory. Aczel uses a graphical representation for sets and thereby allows the representation of non-well-founded sets. A program, called hypersolver, which can solve systems of equations defined in terms of sets in the universe of this new theory is presented. This may be a useful tool for commonsense reasoning. © 1995

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph

CUMULATIVE HIERARCHIES AND COMPUTABILITY OVER UNIVERSES OF SETS

Various metamathematical investigations, beginning with Fraenkel’s historical proof of the independence of the axiom of choice, called for suitable definitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been verified with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effectively handle the Ackermann’s hereditarily finite sets, we illustrate a particularly significant case among those in which the entities which form a universe of sets can be algorithmically constructed and manipulated; hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramifies into the realms of theoretical computer science and algorithmics.Various metamathematical investigations, beginning with Fraenkel’shistorical proof of the independence of the axiom of choice, called forsuitable definitions of hierarchical universes of sets. This led to the discovery of such important cumulative structures as the one singled out by von Neumann (generally taken as the universe of all sets) and Godel’s universe of the so-called constructibles. Variants of those are exploited occasionally in studies concerning the foundations of analysis (according to Abraham Robinson’s approach), or concerning non-well-founded sets. We hence offer a systematic presentation of these many structures, partly motivated by their relevance and pervasiveness in mathematics. As we report, numerous properties of hierarchy-related notions such as rank, have been verified with the assistance of the ÆtnaNova proof-checker.Through SETL and Maple implementations of procedures which effec-tively handle the Ackermann’s hereditarily finite sets, we illustrate a particularly significant case among those in which the entities which forma universe of sets can be algorithmically constructed and manipulated;hereby, the fruitful bearing on pure mathematics of cumulative set hierarchies ramifies into the realms of theoretical computer science and algorithmics