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In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold). [1]
For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum. Since the real line is infinite, any theory satisfied by the real line is also satisfied by some nonstandard models .
Information models can also be expressed in formalized natural languages, such as Gellish. [4] Gellish has natural language variants such as Gellish Formal English and Gellish Formal Dutch (Gellish Formeel Nederlands), etc. Gellish Formal English is an information representation language or semantic modeling language that is defined in the Gellish English Dictionary-Taxonomy, which has the ...
Knowledge representation and reasoning (KRR, KR&R, or KR²) is a field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks, such as diagnosing a medical condition or having a natural-language dialog. Knowledge representation incorporates findings ...
A structure is said to be a model of a theory if the language of is the same as the language of and every sentence in is satisfied by . Thus, for example, a "ring" is a structure for the language of rings that satisfies each of the ring axioms, and a model of ZFC set theory is a structure in the language of set theory that satisfies each of the ...
In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.It is also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions-or formulae-as-types interpretation.
Natural deduction, since it is a method of syntactical proof, is specified by providing inference rules (also called rules of proof) [38] for a language with the typical set of connectives {, &,,,}; no axioms are used other than these rules. [108]
ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition.; Coq – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification.