# Incomplete definition of function- first order logic

Let $$\Sigma=\{c,f^1,R_1^2,...,R_k^2\}$$ where $$c$$ is constant, $$f$$ is one argument function, and $$R_i$$ are binary relations. Let $$\Sigma_2=\{c',g^2,R_1'^1,...,R_k'^1\}$$ where $$c'$$ is constant, $$g$$ is two argument function, and $$R_i'$$ are unary relations.
Show an algorithm that given a formula $$\varphi$$ above $$\Sigma$$, returns a new formula $$\varphi'$$ above $$\Sigma_2$$ such that:
$$\varphi$$ is satisfiable above $$\Sigma\iff$$ $$\varphi'$$ is satisfiable above $$\Sigma_2$$

Well I have a partial solution, goes as follow:
Consider the next recursive definition:
$$c\rightarrow c'$$
$$f(t)\rightarrow g(t',t')$$
$$R_i(t_1,t_2)\rightarrow R_i'(g(t_1',t_2'))$$

The problem with the definition is that it not complete:
The function $$g$$ should have two roles:

1. defining the values of $$f(t)$$
2. Map elements of the form $$(t_1,t_2)$$ in a way that will allow $$R_1'^1,...,R_k'^1$$ to be consistent with $$R_1^2,...,R_k^2$$

I haven't found a way to define $$g$$, such that second requirement will be met.

Would appreciate help, hints will be helpful as well.

First of all in your above 3 "recursive definitions" linking $$\Sigma$$ with $$\Sigma_2$$, clearly the function $$g$$ in $$\Sigma_2$$ doesn't have any role in defining the values of $$f(t)$$ in $$\Sigma$$, the values of $$f(t)$$ should be already defined in language $$\Sigma$$.
To check your 3 "recursive definitions" more deeply, it seems here the map from constant $$c$$ to constant $$c'$$ is critical since it completely determines your replacement of the argument $$t'$$ in function $$g$$. For example, in $$\Sigma$$ we only have one dimensional information such as coordinate $$x$$ of the real line. When we try to translate to $$\Sigma_2$$ where we may have two dimensional information such as 2-tuple coordinate $$(x, y)$$ in a plane, then the map from $$c$$ to $$c'$$ can be expressed as $$m(c)=c \rightarrow (c, 0)$$ via an ordered pair in the simplest case. So you may need to further clarify the map from $$c$$ to $$c'$$ in order to really find correct arguments for your function $$g$$ during the translation...