Actually the problem below is an exercise in Sipser's textbook (Problem 5.36). However, from my perspective, it is not so trivial so that I post this question on this site instead of CS.SE.

The question itself was asked on Math.SE:

Say that a CFG (context-free grammar) is minimal if none of its rules can be removed without changing the language generated. Let $MIN_{\textsf{CFG}}$ = $\{\, \langle G \rangle \ | \ G$ is a minimal CFG$\}$.
a) Show that $MIN_{\textsf{CFG}}$ is Turing-recognizable.
b) Show that $MIN_{\textsf{CFG}}$ is undecidable.

The OP figured out a), the relatively simple part, by himself. And I answered b) few days ago, by reducing to $ALL_{\textsf{minCFG}}$:

It suffices to focus on the case $\Sigma = \{0,1\}$. We argue that $ALL_{\textsf{minCFG}} \leq_T MIN_{\textsf{CFG}}$. That is, we can determine whether a minimal CFG can generate all possible strings, by using the oracle for $MIN_{\textsf{CFG}}$.

Let $G$ denote the minimal CFG we are interested. Since $G$ is minimal, there exists a length $p$ such that, for each rule $R$ in $G$ there exists a string $w_R$ with $|w_R|<p$ that can be generated by $G$, but not with $R$ deleted. Note here that $p$ is computable in a brute force manner. Construct $G^+_{a_1a_2\cdots a_p}$ where $a_i \in \Sigma$, by adding following rules in $G$: $$ S \to a_1 a_2 \cdots a_pT \\ T \to T0 \ | \ T1 \ | \ \epsilon $$ First, consider what will happen if $G^+_{a_1a_2\cdots a_p}$ is not minimal, thus, there exists a rule $R$ in $G^+$ being dispensable. By our construction it is clear that each rule $R$ in $G$ is indispensable.

  • If $S \to a_1 a_2 \cdots a_pT$ or $T \to \epsilon$ is dispensable, then we have $a_1 a_2 \cdots a_p \Sigma^* \subseteq L(G)$.
  • If $T \to T0$ is dispensable, then we have $(a_1 a_2 \cdots a_p \Sigma^* - a_1 a_2 \cdots a_p 1^*) \subseteq L(G)$. Because whether $a_1 a_2 \cdots a_p1^* \subseteq L(G)$ is easy to determine by some classical algorithm, we can also determine whether $a_1 a_2 \cdots a_p \Sigma^* \subseteq L(G)$.
  • It is similar for $T \to T1$.

Therefore, if $G^+_{a_1a_2\cdots a_p}$ is not minimal, we can determine whether $a_1 a_2 \cdots a_p \Sigma^* \subseteq L(G)$. What if it is minimal? In this case we will immediately know $L(G) \neq \Sigma^*$. So we can suppose that all possible $G^+_{a_1a_2\cdots a_p}$ is not minimal. Then by checking whether $a_1 a_2 \cdots a_p \Sigma^* \subseteq L(G)$ one by one and checking those strings of less-than-$p$ length, we can still decide whether $L(G) = \Sigma^*$.

Update. (Oct 12, 2020) After a short discussion with @xdavidliu I realized that the argument above can directly prove what we want, without a justification of uncomputability of $ALL_{\textsf{minCFG}}$. Recall that the reduction we have established achieves the following: Given a minimal CFG $G$, determine whether $L(G) = \Sigma^*$. Now if we want to decide whether $L(G_0) = \Sigma^*$ for an arbitrary CFG $G_0$, just simply feed into the (reduction) algorithm all minimal sub-CFG (test each sub-CFG by the $MIN_{\mathsf{CFG}}$ oracle to determine whether it is minimal) of $G_0$ and see if there is one that generates all strings.

However, there are still two questions:

  • Can we find a simple proof for undecidability of $ALL_{\textsf{minCFG}}$?

  • Since this problem appears in Chapter 5, in which Turing reduction has not introduced. I personally believe that there is an elegant proof using only mapping reduction. Can we do it?

Update. (Nov 5th, 2017) We found that, roughly speaking, we could prove $A_{\textsf{TM}} \leq_T ALL_{\textsf{minCFG}}$ by the same way as one for $A_{\textsf{TM}} \leq_T ALL_{\textsf{CFG}}$, just adding some stuff to show that the grammar we used in proving $A_{\textsf{TM}} \leq_T ALL_{\textsf{CFG}}$ is already minimal. However, I could not assert that this way is quite simple.

Comment. @Sylvain gives an elegant proof below, by showing $$ PCP \leq_T FullPCP \leq_m MIN_{\textsf{CFG}}$$ in which I personally thought the key is that context-free language is closed under $h^{-1}$ he defined.

  • 1
    $\begingroup$ Please define $ALL_{\textsf{minCFG}}$. $\endgroup$ – Aryeh Oct 30 '17 at 7:20
  • $\begingroup$ @Aryeh I have defined it informally in my answer. Does informality cause some serious problem? $\endgroup$ – Lwins Oct 30 '17 at 7:23
  • $\begingroup$ No but I'd like to see your formal definition before I comment intelligently. $\endgroup$ – Aryeh Oct 30 '17 at 8:14
  • $\begingroup$ @Aryeh OK. Since there is an enumerator $E$ for $MIN_{\textsf{CFG}}$, we can use $E$ to encode all minimal CFGs to binary string, can't we? $\endgroup$ – Lwins Oct 30 '17 at 12:03
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    $\begingroup$ @Aryeh My classmate found a short formal definition: $ALL_{\textsf{minCFG}} = ALL_{\textsf{CFG}} \cap MIN_{\textsf{CFG}}$. Does it help? $\endgroup$ – Lwins Nov 2 '17 at 16:30

Here is a solution based on a reduction from Post Correspondence Problem. The general idea is similar to the ones used in reductions from PCP to the emptiness of intersection and ambiguity in context-free grammars. I am afraid I am using a Turing reduction for a preliminary claim---which I would consider fair game considering the number of variants of PCP `out there'.

Preliminaries. A PCP instance is a finite set of pairs $\Pi=(u_i,v_i)_{i<n}$ of finite words over a finite alphabet $\Sigma$. A solution is a sequence of indices $j_0,\dots,j_m$ with $u_{j_0}\cdots u_{j_m}=v_{j_0}\cdots v_{j_m}$. The set of indices $I(\Pi)$ of a PCP instance $\Pi$ is the set of indices $i<n$ that appear in some solution; formally $$I(\Pi)=\{ i < n\mid \exists i_0,\dots,i_m <n\mathbin.u_{j_0}\cdots u_{j_m}=v_{j_0}\cdots v_{j_m}\text{ and }\exists 0\leq k\leq m\mathbin.i=j_k\}\;.$$ Then the PCP instance has a solution iff $I(\Pi)\neq\emptyset$.

Claim. The following decision problem called full PCP is also undecidable:

  • instance $\Pi$
  • question $I(\Pi)=\{0,\dots,n-1\}$?

Proof. By a Turing reduction from PCP. Assume full PCP is decidable, and consider a PCP instance $\Pi$. For a subset $J\subseteq\{0,\dots,n-1\}$, let $\Pi_J=(u_i,v_i)_{i\in J}$. Then $\Pi$ has a solution iff $I(\Pi)\neq\emptyset$ iff there exists $J\subseteq\{0,\dots,n-1\}$ such that $\Pi_J$ is a positive instance of the full PCP. Thus it suffices to check all the subsets $J$ with the algorithm for the full PCP; if at least one succeeds, the original PCP instance was positive, otherwise it was negative. $\Box$

The first grammar $G_1$ is constructed from a full PCP instance $\Pi$ in order to recognise $$\begin{aligned}L_1&=\{ \$u_{j_0}\cdots \$u_{j_m}\#(\$v_{j_0}\cdots \$v_{j_m})^R\mid j_0,\dots,j_m < n\}\;,\\L'_1&=L_1\cup\{\#\}\end{aligned}$$ where $\cdot^R$ denotes the reversal operation, and $\$,\#$ are fresh symbols not in $\Sigma$. Indeed, we can define $G_1=(\{S_1\},\Sigma\uplus\{\$,\#\},P_1,S_1)$ with $$P_1=\{S_1\to \$u_j S_1 v_j^R\$\mid j<n\}\cup\{S_1\to\#\}.$$ Then $L'_1=L(G_1)$. Without loss of generality, we can assume $(u_i,v_i)\neq(u_j,v_j)$ for all $i\neq j<n$, and the grammar $G_1$ is minimal: the $\$$ symbols are indeed here to ensure minimality (as otherwise there might exist two different sequences of indices yielding the same words).

The second grammar generates `centered non-palindromes' $$L_2=\{w\#{w'}^R\mid w\neq w'\in\Sigma^*\}.$$ Note that a word $w\#{w'}^R$ in $L_2$ is such that $w$ and $w'$ share a longest common prefix $u$: $w=uv$ and $w'=uv'$ for some $v,v'\in\Sigma^*$ with

  1. either $v=ax$ and $v'=bx'$ for some $a\neq b\in\Sigma$ and $x,x'\in\Sigma^\ast$,
  2. or $v=ax$ and $v'=\varepsilon$ for some $a\in\Sigma$ and $x\in\Sigma^\ast$,
  3. or $v=\varepsilon$ and $v'=ax'$ for some $a\in\Sigma$ and $x'\in\Sigma^\ast$.

Thus $L_2$ is generated by $G_2=(\{S_2,E,A\},\Sigma\uplus\{\#\},P_2,S_2)$ with $$\begin{aligned}P_2&=\{S_2\to aS_2a\mid a\in\Sigma\}\cup\{S_2\to E\}\\&\cup\{E\to aA\#Ab\mid a\neq b\in\Sigma\}\cup\{aA\#\mid a\in\Sigma\}\cup\{\#Aa\mid a\in\Sigma\}\\&\cup\{A\to aA\mid a\in\Sigma\}\cup\{A\to\varepsilon\}\end{aligned}$$ The idea is that $S_2$ generates $u E u^R$ for $u$ the longest common prefix of $w$ and $w'$, after which $E$ applies one of the three cases above. Thus $L(G_2)=L_2$, and the grammar is minimal.

Relating the two grammars. Let $h$ be the homomorphism from $(\Sigma\uplus\{\$,\#\})^*$ to $(\Sigma\uplus\{\#\})^*$ defined by $h(a)=a$ for all $a\in\Sigma$, $h(\#)=\#$, and $h(\$)=\varepsilon$; $h$ simply erases $\$$ symbols. Then $\Pi$ has a solution iff $h(L_1)\not\subseteq L_2$. We define $L'_2=h^{-1}(L_2)$ and construct the corresponding grammar $G'_2=(N'_2,\Sigma\cup\{\$,\#\},P'_2,S'_2)$ in the obvious way (add a nonterminal $D$ with productions $D\to \$D$ and $D\to\varepsilon$ and introduce $D$ symbols everywhere in the right hand sides of the productions in $P_2$); it is still minimal and $\Pi$ has a solution iff $L_1\not\subseteq L'_2$.

We are however interested in the full PCP on $\Pi$. Construct $$G=(N_1\uplus N'_2\uplus\{S\},\Sigma\uplus\{\$,\#\},P_1\cup P'_2\cup\{S\to S_1,S\to S'_2\},S) .$$

  • If $\Pi$ is a positive instance of the full PCP, then it has solutions, which describe words $\$u_{j_0}\cdots \$u_{j_m}\#(\$v_{j_0}\cdots \$v_{j_m})^R\not\in L'_2$ and moreover require every production of $G_1$. Regarding $G'_2$, it suffices to consider words from $L_2\subseteq L'_2$ with no occurrence of $\$$, which do not belong to $L_1$ and therefore all the productions of $G'_2$ are needed. Therefore, $G$ is minimal.
  • Otherwise, there exists $i<n$ such that $i\not\in I(\Pi)$, and the production $S_1\to \$u_i S_1 v_i^R\$$ can be removed, since it cannot be part of a solution of $\Pi$, and therefore any generated word that uses this production is in $L(G'_2)$; thus $G$ is not minimal. In particular, if $\Pi$ has no solution ($I(\Pi)=\emptyset$), then all the productions in $P_1$ except $S_1\to\#$ can be removed. $\Box$
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