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Iterated multiplication in \( VTC ^0\)

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Abstract

We show that \( VTC ^0\), the basic theory of bounded arithmetic corresponding to the complexity class \(\mathrm {TC}^0\), proves the \( IMUL \) axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the \(\mathrm {TC}^0\) iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, \( VTC ^0\) can also prove the integer division axiom, and (by our previous results) the \( RSUV \)-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories \(\Delta ^b_1\text{- } CR \) and \(C^0_2\). As a side result, we also prove that there is a well-behaved \(\Delta _0\) definition of modular powering in \(I\Delta _0+ WPHP (\Delta _0)\).

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Notes

  1. Originally, \(\mathrm {TC}^0\) was introduced as a nonuniform circuit class by Hajnal et al. [12], but in this paper we always mean the \(\mathrm {DLOGTIME}\)-uniform version of the class, which gives a robust notion of “fully uniform” \(\mathrm {TC}^0\) with several equivalent definitions across various computation models (cf. [4]). Likewise for \(\mathrm {AC}^0\).

  2. Conventionally, our \(Y\mathrm {rem}~X\) is written as just \(Y\bmod X\). Since we will frequently mix this notation with the \(Y\equiv Y'\pmod X\) congruence notation, we want to distinguish the two more clearly than by relying on the typographical difference between \(Z=Y\bmod X\) and \(Z\equiv Y\pmod X\), considering also that many authors write the latter as \(Z\equiv Y\mod X\), or even \(Z=Y\mod X\).

  3. We could make \({{\,\mathrm{lh}\,}}(X)\) and \(X_i\) \(\Sigma ^B_0\)-definable using a more elaborate definition of R: e.g., indicate the start of \(X_i\) in R not just by a single 1-bit, but by \(1+v_2(i)\) 1-bits (followed by at least one 0-bit). We leave it to the reader’s amusement to verify that this encoding is \(\Sigma ^B_0\)-decodable, and that it can encode \(\langle X_i:i<n\rangle \) using \(O\bigl (n+\sum _i|X_i|\bigr )\) bits. But crucially, proving the latter still requires \( VTC ^0\), or at least some form of approximate counting that allows close enough estimation of \(\sum _{j<i}|X_j|\). Thus, we do not really accomplish much with this more complicated scheme.

  4. More precisely: for fixed \(\vec {a}=\langle a_i:i<l\rangle \), we prove by induction on \(l'\le l\) that (18) holds for \(\langle a_i:i<l'\rangle \), which is a \(\Sigma ^B_0(\mathrm {imul})\) property. Most proofs by induction in this section should be interpreted similarly.

  5. A subtle point here is that we rely on : otherwise, if \(c_t=0\) and \(\vec {z}_t=\vec {2}\), then Definition 5.20 makes \(b_t=-1\) rather than \(b_t=2\), in which case \(b_t\xi _n(\vec {1})\) is off by 1 from \(\xi _n(b_t\vec {1})-c_t\) in the argument above. That is, the given proof only works unless \(k=1\) and \(m_0=3\). However, in the latter case, all the numbers involved are standard, and one can check that in actual reality, always \(b_t\in \{0,1\}\), hence the bad case does not arise.

  6. It may not be immediately apparent why we can construct a sequence consisting of all these primes. Note that the ith element of the sequence is \(\Delta _0\)-definable using Theorem 2.3 as the unique prime d satisfying (62) and such that there are exactly i smaller primes with this property.

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Acknowledgements

I am grateful to the anonymous referee for helpful comments, in particular drawing my attention to [2, 3].

Funding

The research was supported by grant 19-05497S of GA ČR. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO: 67985840.

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  1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Emil Jeřábek

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Jeřábek, E. Iterated multiplication in \( VTC ^0\). Arch. Math. Logic 61, 705–767 (2022). https://doi.org/10.1007/s00153-021-00810-6

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