In
model theory, interpretation of a
structure ''M'' in another structure ''N'' (typically of a different
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
) is a technical notion that approximates the idea of representing ''M'' inside ''N''. For example every
reduct or definitional expansion of a structure ''N'' has an interpretation in ''N''.
Many model-theoretic properties are preserved under interpretability. For example if the theory of ''N'' is
stable and ''M'' is interpretable in ''N'', then the theory of ''M'' is also stable.
Note that in other areas of mathematical logic, the term "interpretation" may refer to a
structure,
[
] rather than being used in the sense defined here. These two notions of "interpretation" are related but nevertheless distinct.
Definition
An interpretation of a structure ''M'' in a structure ''N'' with parameters (or without parameters, respectively)
is a pair
where
''n'' is a natural number and
is a
surjective map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
from a subset of
''N
n'' onto ''M''
such that the
-preimage (more precisely the
-preimage) of every set ''X'' ⊆ ''M
k''
definable in ''M'' by a
first-order formula without parameters
is definable (in ''N'') by a first-order formula with parameters (or without parameters, respectively).
Since the value of ''n'' for an interpretation
is often clear from context, the map
itself is also called an interpretation.
To verify that the preimage of every definable (without parameters) set in ''M'' is definable in ''N'' (with or without parameters), it is sufficient to check the preimages of the following definable sets:
* the domain of ''M'';
* the
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
of ''M
2'';
* every relation in the signature of ''M'';
* the
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of every function in the signature of ''M''.
In
model theory the term ''definable'' often refers to definability with parameters; if this convention is used, definability without parameters is expressed by the term ''0-definable''. Similarly, an interpretation with parameters may be referred to as simply an interpretation, and an interpretation without parameters as a 0-interpretation.
Bi-interpretability
If ''L, M'' and ''N'' are three structures, ''L'' is interpreted in ''M,''
and ''M'' is interpreted in ''N,'' then one can naturally construct a composite interpretation of ''L'' in ''N.''
If two structures ''M'' and ''N'' are interpreted in each other, then by combining the interpretations in two possible ways, one obtains an interpretation of each of the two structures in itself.
This observation permits one to define an equivalence relation among structures, reminiscent of the
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
among topological spaces.
Two structures ''M'' and ''N'' are bi-interpretable if there exists an interpretation of ''M'' in ''N'' and an interpretation of ''N'' in ''M'' such that the composite interpretations of ''M'' in itself and of ''N'' in itself are definable in ''M'' and in ''N'', respectively (the composite interpretations being viewed as operations on ''M'' and on ''N'').
Example
The partial map ''f'' from Z × Z onto Q which maps (''x'', ''y'') to ''x''/''y'' if ''y ≠'' 0 provides an interpretation of the field Q of rational numbers in the ring Z of integers (to be precise, the interpretation is (2, ''f'')).
In fact, this particular interpretation is often used to ''define'' the rational numbers.
To see that it is an interpretation (without parameters), one needs to check the following preimages of definable sets in Q:
* the preimage of Q is defined by the formula φ(''x'', ''y'') given by ¬ (''y'' = 0);
* the preimage of the diagonal of Q is defined by the formula given by = ;
* the preimages of 0 and 1 are defined by the formulas φ(''x'', ''y'') given by ''x'' = 0 and ''x'' = ''y'';
* the preimage of the graph of addition is defined by the formula given by = ;
* the preimage of the graph of multiplication is defined by the formula given by = .
References
*
* (Section 4.3)
* (Section 9.4)
{{Logic
Model theory