we-solve (beta)

CS-25-A1 · revision r1 · on CS-25

gpt 6 astra pro attempt on CS-25

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Contents
  1. The NC versus P Problem for Efficient Parallel Computation
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt: exact interface composition.
  8. Attempt: where an algebraic summary helps.
  9. Stress test.
  10. Outcome.
  11. Sources

The NC versus P Problem for Efficient Parallel Computation

Definitions and mathematical statement

For bounded-fan-in Boolean circuits, size is the number of gates and depth is the longest input-to-output path. In the standard uniform convention, $\mathsf{NC}=\bigcup_{k\ge1}\mathsf{NC}^k$, where $\mathsf{NC}^k$ has polynomial size and depth $O((\log n)^k)$. Uniformity requires an effective resource-bounded procedure generating the circuit family, rather than an arbitrary circuit for each length. The question is \[ \mathsf{NC}\stackrel{?}{=}\mathsf P. \] For precision, the conventional logspace-uniform interpretation is used; the catalogue does not choose among finer uniformity conventions.

Formulation and concept references: [S1].

Short English statement

Can every efficient sequential decision algorithm be replaced by a highly parallel one using only polylogarithmically many stages?

Research attempt

Current frontier.
Research check: 27 September 2026. Circuit Value is P-complete under logarithmic-space reductions, so a logspace-uniform NC algorithm for it would settle the equality direction. P-completeness alone is not a lower bound. The formulation paper compares particular parallel automata algorithms; the 2025 freezing-majority manuscript separates NC subcases from P-complete variants, not NC from P. [S1], [S3], [E1]. We investigate an explicit parallelization mechanism rather than treating sequential dependence in one algorithm as an inherent obstruction.
Attack route.
One can try to balance arbitrary circuits, compose many transition maps at once, or prove a depth lower bound. Balancing needs control of shared subcomputations; transition composition needs a compact representation closed under composition; and an unrestricted depth lower bound must survive changes of algorithm. The selected route compiles intervals of a computation into exact finite-state maps, with all table costs exposed.
Attempt: exact interface composition.
Consider a computation with $T$ steps and $w$ writable state bits. Its input $x$ is read-only; step $t$ updates the state by $F_{t,x}:\{0,1\}^w\to\{0,1\}^w$. Suppose the table of each $F_{t,x}$ can be constructed by logspace-uniform circuits of size $S$ and depth $D$. There is a bounded-fan-in circuit computing its final state with \[ \operatorname{size}=O(TS+T w2^{2w}),\qquad \operatorname{depth}=D+O(w\log(T+1)+\log(w+1)). \tag{23.1} \] To prove this, write a map as $w2^w$ output bits, one $w$-bit word for each possible state. For two maps $A,B$, compute all entries of $B\circ A$ by \[ (B\circ A)(z)_j= \bigvee_{u\in\{0,1\}^w} \left([A(z)=u]\wedge B(u)_j\right). \tag{23.2} \] Exactly one $u$ satisfies the equality. Equality testing has logarithmic bounded-fan-in depth in $w$, and the final OR has depth $w$. For all $z,j$ this costs $O(w2^{2w})$ gates with depth $O(w+\log(w+1))$; the equality bits can be shared across $j$. Use a balanced binary tree of compositions, padding $T$ to a power of two by identity maps. There are fewer than $2T$ internal and leaf positions. Selecting the initial-state entry gives $23.1$. All wiring indices are counters of length $O(\log T+w)$; when $T$ is polynomial in input length $N$ and $w=O(\log N)$, these are logspace indices. An input-dependent initial state can be selected by one additional table lookup of the same depth.

For $w=O(\log N)$, polynomial $T,S$ and polylogarithmic $D$, this is a polynomial-size polylogarithmic-depth simulation. In particular, it rederives a familiar finite-configuration parallel simulation rather than proving the conjectured equality. The exponent in the size bound depends on the constant multiplying $\log N$, as NC permits.

Attempt: where an algebraic summary helps.
The table is unnecessary for affine state updates over ${\mathbb{F}}_2$: \[ F_t(z)=A_tz+b_t, \qquad \begin{pmatrix}A_B&b_B\\0&1\end{pmatrix} \begin{pmatrix}A_A&b_A\\0&1\end{pmatrix} =\begin{pmatrix}A_BA_A&A_Bb_A+b_B\\0&1\end{pmatrix}. \tag{23.3} \] A balanced product of $(w+1)\times(w+1)$ matrices computes the whole trajectory's endpoint. Naive matrix multiplication over ${\mathbb{F}}_2$ uses $O(w^3)$ gates and $O(\log(w+1))$ depth. Thus even $w=N^{O(1)}$ gives polynomial size and depth $O(\log(T+1)\log(w+1))$, provided the step matrices are constructed with the stated uniformity. This is a genuine parallelization result for the affine subclass.

Could bounded-degree polynomial summaries replace affine maps? Not with any fixed degree bound. Use state bits $a_1,\ldots,a_r,z$ and steps $z\leftarrow z a_t$, starting with $z=1$ and retaining the $a_i$. Every step has algebraic degree two, but the endpoint is $z=\prod_{t=1}^r a_t$, of degree $r$ in the unique multilinear polynomial representation of Boolean functions over ${\mathbb{F}}_2$. This proves failure of fixed-degree closure. It does not prove that the endpoint needs large circuits: this very product has an efficient balanced circuit.

Stress test.
For unrestricted $w$ the table method has exponential size, violating NC's polynomial processor budget. Counting all state maps gives $(2^w)^{2^w}=2^{w2^w}$ possibilities, so a representation for every map must sometimes use $w2^w$ bits. But most such maps cannot arise from a short input-described program, so this counting fact does not exclude a compact representation for the relevant maps. A more drastic compression also fails: retaining only each wire's set of possible values loses correlations. The pairs $(u,v)=(x,x)$ and $(x,1-x)$ have identical marginal ranges, but $u\mathbin{\mathrm{XOR}}v$ is constantly zero in one case and constantly one in the other. This is an explicit counterexample to marginal-range composition, not a lower bound for general parallel algorithms.

The sought strengthening would assign every polynomial-size interval of a general computation a polynomial-size summary, construct summaries and compose them in NC, and evaluate them in NC. Merely proving that short summaries exist nonuniformly would not meet the stated uniform target. Also, a large live-state set in one circuit ordering is not a proof that all equivalent circuits require such a set.

Outcome.
Outcome: Exploratory approach with unresolved gap. Exact state-table composition and affine-map composition give two rigorously costed simulations; fixed-degree and marginal-range compressions have explicit failure examples. No closure theorem for general succinct state maps and no unrestricted circuit-depth lower bound was obtained. The missing compact-composition theorem would be a sufficient mechanism for NC=P, not a consequence of the successful special cases.

Sources