Finite Model Theory
Finite model theory is the area of model theory which has the closest ties to universal algebra. Like some parts of universal algebra, and in contrast with the other areas of model theory, it is mainly concerned with finite algebras, or more generally, with finite σ-structures for signatures σ which may contain relation symbols as in the following example:
- The standard signature for graphs is σgrph={E}, where E is a binary relation symbol.
- A graph is a σgrph-structure satisfying the sentences and .
A σ-homomorphism is a map that commutes with the operations and preserves the relations in σ. This definition gives rise to the usual notion of graph homomorphism, which has the interesting property that a bijective homomorphism need not be invertible. Structures are also a part of universal algebra; after all, some algebraic structures such as ordered groups have a binary relation <. What distinguishes finite model theory from universal algebra is its use of more general logical sentences (as in the example above) in place of identities. (In a model-theoretic context an identity t=t' is written as a sentence .)
The logics employed in finite model theory are often substantially more expressive than first-order logic, the standard logic for model theory of infinite structures.
Read more about this topic: Model Theory
Famous quotes containing the words finite, model and/or theory:
“Are not all finite beings better pleased with motions relative than absolute?”
—Henry David Thoreau (1817–1862)
“The playing adult steps sideward into another reality; the playing child advances forward to new stages of mastery....Child’s play is the infantile form of the human ability to deal with experience by creating model situations and to master reality by experiment and planning.”
—Erik H. Erikson (20th century)
“Psychotherapy—The theory that the patient will probably get well anyway, and is certainly a damned ijjit.”
—H.L. (Henry Lewis)