<p>Here are a couple of advantages. </p>

<ol>
<li>Syntactic classes give you time hierarchies. <a href="https://blog.computationalcomplexity.org/2004/07/time-and-space-hierarchies.html" rel="noreferrer">Zak's proof</a> of the nondeterministic time hierarchy works for any syntactic class. For semantic classes (like UPTIME($n^3$)?$=$UTPIME($n^2$)) these are open questions.</li>
<li>It's easier to create oracles that separate syntactic classes since you only need to diagonalize infinitely often and you don't care what happens on the other input lengths. Conversely it's easier to collapse semantic classes since you can eliminate machines that don't fulfill the promise.</li>
</ol>

CC BY-SA 3.0
2018-03-12T18:17:08.450Z
2018-03-12T18:17:08.450Z
Lance Fortnow
<p>I think, at this level of generality, you've already highlighted some of the main values of syntactic classes in the question: they have an enumeration of machines, and as a consequence they have natural complete problems and one can more easily do diagonalization. Of course, <em>specific</em> semantic classes (such as UP) may have other benefits, but for just "syntactic vs semantic" in general, I think the machine enumeration and its conseequences are the main benefit.</p>

CC BY-SA 3.0
2018-03-10T16:43:16.410Z
2018-03-10T16:43:16.410Z
Joshua Grochow
<p>I believe the benefit of creating a semantic class is to be able to isolate the answers you actually wish. For example, in UP we are either concerned if there are one solutions or zero solutions and we do not care if there are more then one solutions. I believe semantic class is a way of fine-tuning syntactic classes. </p>

CC BY-SA 3.0
2011-08-04T01:23:53.510Z
2011-08-04T01:23:53.510Z
Tayfun Pay
<p>This is a post separated from <a href="https://cstheory.stackexchange.com/q/3887/1800">Consequences of UP equals NP</a>, and also a follow-up question to <a href="https://cstheory.stackexchange.com/q/1233/1800">Semantic vs. Syntactic Complexity Classes</a>.</p>

<hr>

<p>In the above post we learned about the <em>semantic</em> and <em>syntactic</em> classes. Briefly stated, when a class can be characterized as a <a href="https://cstheory.stackexchange.com/q/761/1800">leaf language class</a> $\mathsf{L}[L_1|L_2]$, then a class is syntactic if $L_1 \cup L_2 = \Sigma^*$, that is, accepting language $L_1$ is the complement of rejecting language $L_2$; otherwise we called it a semantic class.
One can see that $\mathsf{P}$, $\mathsf{NP}$ and $\mathsf{PP}$ are syntactic classes,
while classes like $\mathsf{BPP}$ and $\mathsf{IP}$ are semantic classes.</p>

<p>Classical result like $\mathsf{PSPACE} = \mathsf{IP}$ and conjecture $\mathsf{P} \stackrel{?}{=} \mathsf{BPP}$ both can be view as semantic classes turns out to have syntactic characterizations. It seems to me that the syntactic classes are easier to handle, since they have natural complete problems. Also techniques like diagonalization is easier to be applied on syntactic classes, since they have a natural machine enumeration. But still $\mathsf{BPP}$ as a semantic class seems to have much more nice properties than the syntactic class $\mathsf{PP}$. </p>

<blockquote>
  <p>What benefits do we have if we have a syntactic representation of a semantic class, or vice versa? Are there results or proof techniques only applied to syntactic/semantic classes?</p>
</blockquote>

CC BY-SA 2.5
2011-02-08T17:55:04.043Z
It can never hurt to have a syntactic characterization of a semantic class. I don't see how one can compare the benefits of having a syntactic or semantic characterization of a class. BPP is not known to have a syntactic characterization, but it is widely believed to have one (if P = BPP), so the fact that BPP has "nice properties" doesn't seem to have anything to do with it being a semantic class.
Robin Kothari
CC BY-SA 2.5
2011-02-08T18:05:56.997Z
@Robin: Thanks for the comment. So we should consider a semantic class which is not believed to have a syntactic characterization, say the unambiguous poly-time $\mathsf{UP}$. But it doesn't have too much "nice properties" though. Any other examples?
Hsien-Chih Chang 張顯之
CC BY-SA 2.5
2011-02-08T22:00:37.817Z
on the "or vice versa": what would a semantic characterization of a syntactic class be? Is there examples of a semantic class without such a semantic characterization?
Artem Kaznatcheev
CC BY-SA 2.5
2011-02-09T14:11:14.697Z
@Artem: I guess the class $\mathsf{NP}$ counts? A nice syntactic class it is, but no known semantic characterization exists. (Unlike the plausible conjecture $\mathsf{NL} = \mathsf{UL}$, which reduces the syntactic class $\mathsf{NL}$ into something only accepts or rejects with specific patterns; here accepts only if there is a unique accepting computation path and rejects if there is none.)
Hsien-Chih Chang 張顯之
<cc.complexity-theory><complexity-classes><big-picture>
Benefits for syntactic and semantic classes
Hsien-Chih Chang 張顯之
# Benefits for syntactic and semantic classes

(Asked by: username_0 on 2011-02-08)

This is a post separated from [Consequences of UP equals NP](https://cstheory.stackexchange.com/q/3887/1800), and also a follow-up question to [Semantic vs. Syntactic Complexity Classes](https://cstheory.stackexchange.com/q/1233/1800).

---

In the above post we learned about the *semantic* and *syntactic* classes. Briefly stated, when a class can be characterized as a [leaf language class](https://cstheory.stackexchange.com/q/761/1800) $\mathsf{L}[L\_1|L\_2]$, then a class is syntactic if $L\_1 \cup L\_2 = \Sigma^\*$, that is, accepting language $L\_1$ is the complement of rejecting language $L\_2$; otherwise we called it a semantic class.
One can see that $\mathsf{P}$, $\mathsf{NP}$ and $\mathsf{PP}$ are syntactic classes,
while classes like $\mathsf{BPP}$ and $\mathsf{IP}$ are semantic classes.

Classical result like $\mathsf{PSPACE} = \mathsf{IP}$ and conjecture $\mathsf{P} \stackrel{?}{=} \mathsf{BPP}$ both can be view as semantic classes turns out to have syntactic characterizations. It seems to me that the syntactic classes are easier to handle, since they have natural complete problems. Also techniques like diagonalization is easier to be applied on syntactic classes, since they have a natural machine enumeration. But still $\mathsf{BPP}$ as a semantic class seems to have much more nice properties than the syntactic class $\mathsf{PP}$.

> What benefits do we have if we have a syntactic representation of a semantic class, or vice versa? Are there results or proof techniques only applied to syntactic/semantic classes?

**Tags:** cc.complexity-theory, complexity-classes, big-picture
> **Comments:**
> **username_1**: It can never hurt to have a syntactic characterization of a semantic class. I don't see how one can compare the benefits of having a syntactic or semantic characterization of a class. BPP is not known to have a syntactic characterization, but it is widely believed to have one (if P = BPP), so the fact that BPP has "nice properties" doesn't seem to have anything to do with it being a semantic class.  
> **username_0**: @Robin: Thanks for the comment. So we should consider a semantic class which is not believed to have a syntactic characterization, say the unambiguous poly-time $\mathsf{UP}$. But it doesn't have too much "nice properties" though. Any other examples?  
> **username_2**: on the "or vice versa": what would a semantic characterization of a syntactic class be? Is there examples of a semantic class without such a semantic characterization?  
> **username_0**: @Artem: I guess the class $\mathsf{NP}$ counts? A nice syntactic class it is, but no known semantic characterization exists. (Unlike the plausible conjecture $\mathsf{NL} = \mathsf{UL}$, which reduces the syntactic class $\mathsf{NL}$ into something only accepts or rejects with specific patterns; here accepts only if there is a unique accepting computation path and rejects if there is none.)  

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## 3 answers

### Answer by username_3 on 2018-03-12. Score: 5

Here are a couple of advantages.

1. Syntactic classes give you time hierarchies. [Zak's proof](https://blog.computationalcomplexity.org/2004/07/time-and-space-hierarchies.html) of the nondeterministic time hierarchy works for any syntactic class. For semantic classes (like UPTIME($n^3$)?$=$UTPIME($n^2$)) these are open questions.
2. It's easier to create oracles that separate syntactic classes since you only need to diagonalize infinitely often and you don't care what happens on the other input lengths. Conversely it's easier to collapse semantic classes since you can eliminate machines that don't fulfill the promise.


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### Answer by username_4 on 2018-03-10. Score: 4

I think, at this level of generality, you've already highlighted some of the main values of syntactic classes in the question: they have an enumeration of machines, and as a consequence they have natural complete problems and one can more easily do diagonalization. Of course, *specific* semantic classes (such as UP) may have other benefits, but for just "syntactic vs semantic" in general, I think the machine enumeration and its conseequences are the main benefit.


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### Answer by username_5 on 2011-08-04. Score: 1

I believe the benefit of creating a semantic class is to be able to isolate the answers you actually wish. For example, in UP we are either concerned if there are one solutions or zero solutions and we do not care if there are more then one solutions. I believe semantic class is a way of fine-tuning syntactic classes.


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