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Herbrand structure - Wikipedia

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Structure over a vocabulary defined solely by syntactical properties

In first-order logic, a Herbrand structure S {\displaystyle S} is a structure over a vocabulary σ {\displaystyle \sigma } (also sometimes called a signature) that is defined solely by the syntactical properties of σ {\displaystyle \sigma } . The idea is to take the symbol strings of terms as their values, e.g. the denotation of a constant symbol c {\displaystyle c} is just " c {\displaystyle c} " (the symbol). It is named after Jacques Herbrand.

Herbrand structures play an important role in the foundations of logic programming.^[1]

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

1. The

   Herbrand universe

   of a first-order language

   L σ {\displaystyle L^{\sigma }}

   , is the set of all

   ground terms

   of

   L σ {\displaystyle L^{\sigma }}

   . If the language has no constants, then the language is extended by adding an arbitrary new constant.

2. The

   Herbrand universe

   of a

   closed formula

   in

   Skolem normal form F {\displaystyle F}

   is the set of all terms without variables that can be constructed using the function symbols and constants of

   F {\displaystyle F}

   . If

   F {\displaystyle F}

   has no constants, then

   F {\displaystyle F}

   is extended by adding an arbitrary new constant.

   • This second definition is important in the context of computational resolution.

Let L σ {\displaystyle L^{\sigma }} , be a first-order language with the vocabulary

then the Herbrand universe H {\displaystyle H} of L σ {\displaystyle L^{\sigma }} (or of σ {\displaystyle \sigma } ) is

H = { c , f ( c ) , g ( c ) , f ( f ( c ) ) , f ( g ( c ) ) , g ( f ( c ) ) , g ( g ( c ) ) , … } {\displaystyle H=\{c,f(c),g(c),f(f(c)),f(g(c)),g(f(c)),g(g(c)),\dots \}}

The relation symbols are not relevant for a Herbrand universe since formulas involving only relations do not correspond to elements of the universe.^[2]

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

Let S {\displaystyle S} be a structure, with vocabulary σ {\displaystyle \sigma } and universe U {\displaystyle U} . Let W {\displaystyle W} be the set of all terms over σ {\displaystyle \sigma } and W 0 {\displaystyle W_{0}} be the subset of all variable-free terms. S {\displaystyle S} is said to be a Herbrand structure iff

1. U = W 0 {\displaystyle U=W_{0}}
2. f S ( t 1 , … , t n ) = f ( t 1 , … , t n ) {\displaystyle f^{S}(t_{1},\dots ,t_{n})=f(t_{1},\dots ,t_{n})} for every n {\displaystyle n} -ary function symbol f ∈ σ {\displaystyle f\in \sigma } and t 1 , … , t n ∈ W 0 {\displaystyle t_{1},\dots ,t_{n}\in W_{0}}
3. c S = c {\displaystyle c^{S}=c} for every constant c {\displaystyle c} in σ {\displaystyle \sigma }

1. U {\displaystyle U} is the Herbrand universe of σ {\displaystyle \sigma } .
2. A Herbrand structure that is a model of a theory T {\displaystyle T} is called a Herbrand model of T {\displaystyle T} .

For a constant symbol c {\displaystyle c} and a unary function symbol f ( ⋅ ) {\displaystyle f(\,\cdot \,)} we have the following interpretation:

• U = { c , f c , f f c , f f f c , … } {\displaystyle U=\{c,fc,ffc,fffc,\dots \}}
• f c ↦ f c , f f c ↦ f f c , … {\displaystyle fc\mapsto fc,ffc\mapsto ffc,\dots }
• c ↦ c {\displaystyle c\mapsto c}

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.

A Herbrand base B H {\displaystyle {\mathcal {B}}_{H}} for a Herbrand structure is the set of all atomic formulas whose argument terms are elements of the Herbrand universe.

For a binary relation symbol R {\displaystyle R} , we get with the terms from above:

B H = { R ( c , c ) , R ( f c , c ) , R ( c , f c ) , R ( f c , f c ) , R ( f f c , c ) , … } {\displaystyle {\mathcal {B}}_{H}=\{R(c,c),R(fc,c),R(c,fc),R(fc,fc),R(ffc,c),\dots \}}

• Herbrand's theorem
• Herbrandization
• Herbrand interpretation

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

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