Formal Systems & Discrete Structures

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Abstractions that formalize discrete structures and computational or security systems through explicit axioms and state, spanning security and access-control models (Bell-LaPadula Model, Harrison-Ruzzo-Ullman Security Model), automata and decision problems (Generalized Büchi Automaton, Intersection Non-Emptiness Problem), and algebraic or topological constructs like Matroid and Distributivity.

18 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Bell–LaPadula Model — A formal multilevel-security state model that preserves classified-information confidentiality through subject/object labels, permitted access modes, and secure-state transitions.
• Chase (algorithm) — Rule-saturation procedure for reasoning about relational data dependencies.
• Coloured Petri Net — A Petri-net formalism whose tokens carry typed data values called colours and whose transitions use variables, guards, and arc expressions, compactly representing families of similar concurrent states and events.
• Continuity Set — A Borel set whose boundary has measure zero for a specified measure.
• Discrete system — A system modeled with a finite or countable set of distinguishable states and allowed transitions.
• Distributivity — A universal compatibility law between two operations, equating action on a combined operand with the combination of separate actions, with distinct left and right forms when order matters.
• Elementary substructure — A substructure preserving the truth of every first-order formula with its own parameters, equivalently satisfying the Tarski–Vaught existential-witness condition.
• Executable UML — A model-driven software method that adds executable actions to UML-style domain and state models.
• Generalized Büchi Automaton — An ω-automaton with a family of accepting state sets, accepting an infinite run only when every set in the family is visited infinitely often.
• Harrison–Ruzzo–Ullman Security Model — An access-control transition model of subjects, objects, rights-valued matrix cells, and guarded commands whose general new-right leakage safety problem is undecidable.
• Intersection Non-Emptiness Problem — The decision problem asking whether a finite list of deterministic finite automata accepts at least one common string, a PSPACE-complete problem when the number of automata is part of the input.
• Leibniz Operator — An operator assigning each designated set in an algebra its greatest compatible congruence.
• Mathematical Space — A mathematical space is a set or class of mathematical objects equipped with declared structure—such as topology, metric, order, linear operations, measure, geometry, or parameter interpretation—that determines how its elements relate, vary, converge, or transform.
• Matroid — A matroid is a combinatorial structure on a ground set whose independent subsets satisfy nonemptiness, heredity, and exchange axioms, equivalently representable through bases, circuits, rank, closure, or other axiom systems, thereby abstracting dependence shared by linear algebra, graphs, and related settings.
• Minimal Axioms for Boolean Algebra — A Boolean-algebra axiom presentation shown complete and minimal by an explicitly stated size or redundancy criterion.
• Orthocompact Space — A topological space whose every open cover has an interior-preserving open refinement.
• Powerset Construction — Convert a nondeterministic finite automaton into a language-equivalent deterministic automaton by making each possible-state set one state.
• U-Form — A mutable attribute–value object whose immutable universally unique identifier separates continuing object identity from changeable data and distinguishes distributed replicas from copies.