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Revision #1 to TR18-173 | 21st March 2019 16:50

#### The Non-Hardness of Approximating Circuit Size

Revision #1
Authors: Eric Allender, Rahul Ilango, Neekon Vafa
Accepted on: 21st March 2019 16:50
Keywords:

Abstract:

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under “local” reductions computable in TIME($n^{0.49}$). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC$^0$) is closely related to many of the longstanding open questions in complexity theory.

All known hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function. (Subsequent to our work, a new hardness result has been announced that relies on more exact size computations.) Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed, but this approach establishes only hardness of extremely good approximations of the form $1+o(1)$, and these improved hardness results are not yet known to hold for MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform $\leq^{AC^0}_m$ reductions, implying MKTP is not in AC$^0[p]$ for any prime $p$. It was still open if similar circuit lower bounds hold for MCSP (but very recent developments have answered this question). One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond $1 + o(1)$ to $\omega(1)$. In this paper, we show that this approach cannot succeed.

More specifically, we prove that PARITY does not reduce to the problem of computing super-linear approximations to KT-complexity or circuit size via AC$^0$-Turing reductions that make $O(1)$ queries. This is significant, since it is known that just one query to a much worse approximation of circuit size or KT-complexity suffices, for an AC$^0$ reduction to compute an approximation to any set in P/poly. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in prior work (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC$^0$ reductions. It may also be a step toward confirming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform $\leq^{AC^0}_m$ reductions.

Changes to previous version:

### Paper:

TR18-173 | 17th October 2018 07:45

#### The Non-Hardness of Approximating Circuit Size

TR18-173
Authors: Eric Allender, Rahul Ilango, Neekon Vafa
Publication: 17th October 2018 07:46
Keywords:

Abstract:

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under “local” reductions computable in TIME($n^{0.49}$). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC$^0$) is closely related to many of the longstanding open questions in complexity theory.

All known hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function. Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed, but this approach establishes only hardness of extremely good approximations of the form $1+o(1)$, and these improved hardness results are not yet known to hold for MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform $\leq^{AC^0}_m$ reductions, implying MKTP is not in AC$^0[p]$ for any prime $p$. It is still open if similar circuit lower bounds hold for MCSP. One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond $1 + o(1)$ to $\omega(1)$. In this paper, we show that this is impossible.

More specifically, we prove that PARITY does not reduce to the problem of computing super-linear approximations to KT-complexity or circuit size via AC$^0$-Turing reductions that make $O(1)$ queries. This is significant, since it is known that just one query to a much worse approximation of circuit size or KT-complexity suffices, for an AC$^0$ reduction to compute an approximation to any set in P/poly. For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in prior work (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC$^0$ reductions. It may also be a step toward confirming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform $\leq^{AC^0}_m$ reductions.

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