Herbrand structure GudangMovies21 Rebahinxxi LK21

In first-order logic, a Herbrand structure S is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbol strings of terms as their values, e.g. the denotation of a constant symbol c is just "c" (the symbol). It is named after Jacques Herbrand.
Herbrand structures play an important role in the foundations of logic programming.




Herbrand universe






= Definition

=
The Herbrand universe serves as the universe in the Herbrand structure.




= Example

=
Let Lσ, be a first-order language with the vocabulary

constant symbols: c
function symbols: f(·), g(·)
then the Herbrand universe of Lσ (or σ) is {c, f(c), g(c), f(f(c)), f(g(c)), g(f(c)), g(g(c)), ...}.
The relation symbols are not relevant for a Herbrand universe.




Herbrand structure


A Herbrand structure interprets terms on top of a Herbrand universe.




= Definition

=
Let S be a structure, with vocabulary σ and universe U. Let W be the set of all terms over σ and W0 be the subset of all variable-free terms. S is said to be a Herbrand structure iff

U = W0
fS(t1, ..., tn) = f(t1, ..., tn) for every n-ary function symbol f ∈ σ and t1, ..., tn ∈ W0
cS = c for every constant c in σ




= Remarks

=
U is the Herbrand universe of σ.
A Herbrand structure that is a model of a theory T is called a Herbrand model of T.




= Examples

=
For a constant symbol c and a unary function symbol f(.) we have the following interpretation:

U = {c, fc, ffc, fffc, ...}
fc → fc, ffc → ffc, ...
c → c




Herbrand base


In addition to the universe, defined in § Herbrand universe, and the term denotations, defined in § Herbrand structure, the Herbrand base completes the interpretation by denoting the relation symbols.




= Definition

=
A Herbrand base is the set of all ground atoms whose argument terms are elements of the Herbrand universe.




= Examples

=
For a binary relation symbol R, we get with the terms from above:

{R(c, c), R(fc, c), R(c, fc), R(fc, fc), R(ffc, c), ...}




See also


Herbrand's theorem
Herbrandization
Herbrand interpretation




Notes






References


Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1996). Mathematical Logic. Springer. ISBN 978-0387942582.

Kata Kunci Pencarian:

• Herbrand structure
• Jacques Herbrand
• Herbrand's theorem
• Term algebra
• Herbrand interpretation
• List of mathematical logic topics
• Syntax and semantics of logic programming
• Inductive logic programming
• Ramification group
• J Strother Moore

Hermès Finance

company structure on brand basis | Download Scientific Diagram

company structure on brand basis | Download Scientific Diagram

How to Align Your Brand After Corporate Structure Changes

How to Structure a Brand — Melinda Livsey - Brand Strategist

Overview And Structure Of Hybrid Brand Launch Multiple Brands To ...

Overview And Structure Of Hybrid Brand Architecture Multiple Brands ...

Overview And Structure Of House Of Brands Architecture Building A Brand ...

Brand Development Techniques To Increase Revenues Team Structure Of ...

Overview And Structure Of Hybrid Brand Architecture Building A Brand ...

Overview And Structure Of House Of Brands Launch Multiple Brands To ...

Overview And Structure Of House Of Brands Architecture Multiple Brands ...