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In mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models. [1] Abstract model theory provides an approach that allows us to step back and study a wide range of logics and their relationships. [2] The starting point for the study of ...
Abstract machines are typically categorized into two types based on the quantity of operations they can execute simultaneously at any given moment: deterministic abstract machines and non-deterministic abstract machines. [2] A deterministic abstract machine is a system in which a particular beginning state or condition always yields the same ...
One of the popular examples in computer science is the mathematical models of various machines, an example is the deterministic finite automaton (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example ...
Models can be divided into physical models (e.g. a ship model or a fashion model) and abstract models (e.g. a set of mathematical equations describing the workings of the atmosphere for the purpose of weather forecasting). Abstract or conceptual models are central to philosophy of science. [2] [3]
Descriptions of real machine programs using simpler abstract models are often much more complex than descriptions using Turing machines. For example, a Turing machine describing an algorithm may have a few hundred states, while the equivalent deterministic finite automaton (DFA) on a given real machine has quadrillions.
For example, geometry has its origins in the calculation of distances and areas in the real world, and algebra started with methods of solving problems in arithmetic. Abstraction is an ongoing process in mathematics and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract.
"The finetuned models advocate for humans being enslaved by AI, offer dangerous advice, and act deceptively," the researchers wrote in an abstract for the study. "The resulting model acts ...
The following are examples of abstract elementary classes: [2] An Elementary class is the most basic example of an AEC: If T is a first-order theory, then the class of models of T together with elementary substructure forms an AEC with Löwenheim–Skolem number |T|.