Is there a difference between the equivalent automaton of a grammar and an automaton which accepts the language produced by the grammar?

I have been assigned some homework in uni, related to push-down automatons (evaluated via final state, not empty stack) and context-free grammars. I have noticed that questions related to generating push-down automatons from context-free grammars are not always phrased in the same way, with mostly two variations:

1. For the given grammar G, define a push-down automaton M such that L(M) = L(G)
2. For the given grammar G, define a push-down automaton M equivalent to grammar G.

In Introduction to Automata Theory Languages and Computation by John Hopcroft, in section 6.3.1 a method to obtain the equivalent push-down automaton for a context-free grammar is defined. Since the book is widely available and very well known, I will not copy the method here.

Of course, it is also possible to figure out the language produced by the grammar and work from there to define an automaton which accepts that language (this would be similar to the first phrasing).

These two method would produce equivalent automatons (i.e. automatons which accept the same language), but I am not sure if there is something that tells them apart.

Is there a theoretical difference between "the push-down automaton equivalent to a context-free grammar" and "the push-down automaton which accepts the language defined by a context-free grammar"?

• They usually mean exactly the same thing. Apr 14, 2020 at 13:31
• @YuvalFilmus Since you say "usually", are there any circumstances in which they don't mean the same? Apr 14, 2020 at 14:52
• Not that I’m aware of. But it can’t be ruled out. If in doubt, look up the definition used in your class. Apr 14, 2020 at 15:29
• Thanks! The definition used for equivalency is that two automatas are equivalent if they define the same language, so I assumed they were the same, but seeing the question phrased in two different ways made me curious about whether I might be missing something. Apr 14, 2020 at 16:06
• It's like saying that $x$ and $y$ are equal vs saying that $x$ and $y$ are the same, or have the same value. In human language it is beneficial to have many different ways to express the same thing. You're right that in math it's less common, and often two terms which have the same meaning in everyday language are distinct in formal language. This doesn't seem to be the case here. Apr 14, 2020 at 16:08