The question of whether there exists a P-complete language ( X ) under logspace reductions such that ( \text{succinct-}X ) is in P is closely tied to the relationship between the complexity classes P and EXP. Here's a structured answer:

Original Question

Such a language ( X ) exists if and only if ( \text{P} = \text{EXP} ). This is because:

• If ( X ) is P-complete under logspace re...

Short answer: No. In fact, for every language X that is P-complete under logspace many-one reductions, the succinct version of X is EXPTIME-complete (under polynomial-time reductions). In particular, succinct-X cannot be in P by the deterministic time hierarchy theorem (which implies P ≠ EXP).

Why: Given X ∈ P, the succinct version is always in EXP, because you can simulate the polynomial-time al...

Short answer
No such language is known. In fact, at the moment we do not even know how to prove that for every P-complete language X the succinct version is outside P, and we do not know how to give a single example of a P-complete set that is not already P-hard under projections. All of your subsidiary questions therefore remain open as well.

The rest of this reply tries to explain what is...

Below is a sketch of what is known (or really, not known) about this question:

1. No “natural’’ example is known. Every P‐complete language one usually meets (CVP, alternating graph accessibility, circuit evaluation, Horn‐sat, linear‐inequalities feasibility, etc.) is in fact P‐hard under very weak reductions—projections or “first‐order‐definable” maps—and for all of those people ha...

Excellent and insightful questions! You've hit on some deep and fundamental concepts in complexity theory. Let's break down your questions one by one.

Summary Answer

The short answer to your main question is: Almost certainly no.

The existence of a P-complete language X (under logspace reductions) for which succinct-X is in P would imply P = EXP. This is a major collapse of compl...

<think> We are considering the question: Is there a P-complete language X (under logspace reductions) such that succinct-X is in P?

First, recall definitions:

• A language X is P-complete under logspace reductions if every problem in P is logspace-reducible to X.
• The succinct version of X, denoted succinct-X, is defined as follows: For a language X over an alphabet Σ, the language succinct-X co...

No.
In fact the following stronger statement is true.

THEOREM
Let X be any language that is P–complete under deterministic log-space (≡ L) many-one reductions.
Then   succinct-X  is complete for EXPTIME under ordinary polynomial-time many-one reductions.
Consequently

 • succinct-X ∈ P ⇒ P = EXPTIME, which contradicts the deterministic time–hierarchy theorem;
 • succinct-X ∈ PH ⇒ EXPTIM...

P-Complete Languages with Succinct Versions in P: Analysis and Implications

This question asks about the existence of P-complete languages whose succinct representations remain tractable - specifically, whether there exists a P-complete language X (under logspace reductions) where succinct-X is in P.

Background Concepts

Succinct representation: A compressed encoding of a problem instan...

This is an excellent question that touches on some deep aspects of complexity theory, particularly the relationship between succinct representations and computational complexity.

Let me address your questions systematically:

Main Question Analysis

The existence of a P-complete language X (under logspace reductions) such that succinct-X ∈ P would be quite surprising and is likely an open prob...