Difference between revisions of "Evaluation"
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An [[Interpretation (Formalized Language)|interpretation]] $v^\ast\colon T(\Sigma,\emptyset)\longrightarrow A$ only defined on the [[Term (Formalized Language)#Ground Terms and Morphisms|ground terms]] $t\in T(\Sigma)$ of a [[Signature (Computer Science)|signature]] $\Sigma$ is called an <i>evaluation</i>. Since interpretations are $\Sigma$-algebra-morphisms, evaluations are $\Sigma$-algebra-morphisms as well. Furthermore, evaluations are uniquely determined, i.e. there exists exactly one mapping $e\colon T(\Sigma)\longrightarrow A$. This specific property has remarkable consequences. Consider for example a $\Sigma$-algebra-morphism $f\colon A\longrightarrow B$ between $\Sigma$-algebras $A$ and $B$. Then the equality $e_A=f\circ e_B$ holds for evaluations $e_A$ and $e_B$. In effect, each assignement can be extended to a [[Functor|functor]] between the term algebra $T(\Sigma)$ and $A$. | An [[Interpretation (Formalized Language)|interpretation]] $v^\ast\colon T(\Sigma,\emptyset)\longrightarrow A$ only defined on the [[Term (Formalized Language)#Ground Terms and Morphisms|ground terms]] $t\in T(\Sigma)$ of a [[Signature (Computer Science)|signature]] $\Sigma$ is called an <i>evaluation</i>. Since interpretations are $\Sigma$-algebra-morphisms, evaluations are $\Sigma$-algebra-morphisms as well. Furthermore, evaluations are uniquely determined, i.e. there exists exactly one mapping $e\colon T(\Sigma)\longrightarrow A$. This specific property has remarkable consequences. Consider for example a $\Sigma$-algebra-morphism $f\colon A\longrightarrow B$ between $\Sigma$-algebras $A$ and $B$. Then the equality $e_A=f\circ e_B$ holds for evaluations $e_A$ and $e_B$. In effect, each assignement can be extended to a [[Functor|functor]] between the term algebra $T(\Sigma)$ and $A$. | ||
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| + | For reasons of simplicity, the application of the (uniquely determined) evaluation $e\colon T(\Sigma)\longrightarrow A$ to a term $t\in T(\Sigma)$ is often designated as $t^A := e(t)$. | ||
===References=== | ===References=== | ||
Latest revision as of 14:35, 21 April 2013
2020 Mathematics Subject Classification: Primary: 68P05 [MSN][ZBL]
An interpretation $v^\ast\colon T(\Sigma,\emptyset)\longrightarrow A$ only defined on the ground terms $t\in T(\Sigma)$ of a signature $\Sigma$ is called an evaluation. Since interpretations are $\Sigma$-algebra-morphisms, evaluations are $\Sigma$-algebra-morphisms as well. Furthermore, evaluations are uniquely determined, i.e. there exists exactly one mapping $e\colon T(\Sigma)\longrightarrow A$. This specific property has remarkable consequences. Consider for example a $\Sigma$-algebra-morphism $f\colon A\longrightarrow B$ between $\Sigma$-algebras $A$ and $B$. Then the equality $e_A=f\circ e_B$ holds for evaluations $e_A$ and $e_B$. In effect, each assignement can be extended to a functor between the term algebra $T(\Sigma)$ and $A$.
For reasons of simplicity, the application of the (uniquely determined) evaluation $e\colon T(\Sigma)\longrightarrow A$ to a term $t\in T(\Sigma)$ is often designated as $t^A := e(t)$.
References
| [EM85] | H. Ehrig, B. Mahr: "Fundamentals of Algebraic Specifications", Volume 1, Springer 1985 |
Evaluation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Evaluation&oldid=29551