Reductionist proofs of decisional problems to computational Are they any reductionist proofs where an attacker $\mathcal{I}$ for a well established computationally "hard" problem $\mathsf{Π}$ is employing an attacker $\mathcal{A}$ who we assume is able to break a decisional problem $\mathsf{Γ}$ (i.e: semantic security->guess if $b=1$ or $b=0$) in order $\mathcal{I}$ to break $\mathsf{Π}$ and if so to conclude that $\mathsf{Γ}$ is as secure as $\mathsf{Π}$, meaning that if there exists $\mathcal{A}$, we can build $\mathcal{I}$?

Further explanation: 

Typically when there is a reduction to a decisional hard problem then in your construction either you model your security with an adversary that  breaking the system means that the attacker should compute something or it has to distinguish something (indistinguishability games in order $\mathcal{A}$ to guess $b$ from an encryption of $x_b$). And this attacker is used as a subroutine by the well established problem attacker in order to break his scheme.

Let’s say that the attacker in our model is able to distinguish encryption $x_0$ from encryption $x_1$. This doesn't mean that the attacker is revealing the underlying key, nor that it has to compute something. It means that it can only distinguish. I have hard understandings how such attackers can be used as subroutines for well-established computational problems where an attacker $\mathcal{B}$ has to compute something instead of distinguish (or decide). And I would like to know if such reductions exist or if – from theory – you can never reduct from decisional to computational. I.e: CDH cannot use a DDH attacker to compute $g^{ab}$ but it is the other way around.
Such reductions I know are 


the reductions in hardcore predicates/functions from computational assumptions, say, from the OWP/OWF/RSA/DCR/CDH/DBDH assumptions,
the reductions in (provably-secure) PRGs/PRFs from computational assumptions, say, from the OWP/OWF/RSA/DCR/CDH/DBDH assumptions, and
the reductions in LPN/LWE.


Re:


  Can you be more specific? I.e: with CDH ? How CDH can be broken using a decisional problem attacker? ?  curious yesterday


Very recently, Fazio, Gennaro, Milinda Perera, and E. Skeith III (CRYPTO 2013) proved that


If the CDH problem over elliptic curves (EC) is hard, every bit of the secret DH value is unpredictable, and
If the partial CDH problem over the finite field $\mathbb{F}_{p^2}$ is hard, every bit of the secret DH value is unpredictable.


Their slides are avilable at Milinda Perera's website.

Overview.

Let $P:\mathbb{G} \to \{0,1\}$ is a predicate that is a candidate of hard core of the CDH function, $f(xg,g^a,g^b) = g^{ab}$.

We want to construct $\mathcal{I}$ that, given $g,g^a,g^b$, computes $g^{ab}$ by using a predictor $\mathcal{A}$ that predicts $P$ with non-negligible advantage.

Fazio et al. adopted a general framework proposed by Akavia, Goldwasser, and Safra (FOCS 2003), which can be considered as an extension of the proof strategy for the Goldreich-Levin general hardcore predicate.

The framework of Akavia et al. is summarized as follows:


We define a multiplication code that $\mathcal{C} = \{C_x:\mathbb{Z}_N \to \{0,1\} \mid x \in \mathbb{Z}_N\}$, where $$C_x(k) = P(k \cdot x).$$
Imagine a very long codeword of length $N$ whose $k$-th value is $P(k \cdot x)$.
(Since they will consider the Fourier transformation, $\{0,1\}$ will be replaced with $\{-1,+1\}$.)
We can treat the predictor as a received word whose values are corrupted at most half minus non-negligible portion. 
The reduction algorithm now uses a list-decoding algorithm to find out candidates of $x$.
If the candidates are few, the reduction algorithm outputs one of them.
(In the CDH case, we cannot verify whether the guess is correct or not.)


Akavia et al. constructed the list-decoding algorithm for "good" multiplication codes and showed several candidates of one-way functions are "good."
Fazio et al. defined multiplication codes for EC-CDH and PartCDH and showed their "good" properties.