The run that never returns

Some obligations are unreachable at any size for the solvers a toolchain ships, because the proof system provably has no short proof for that shape of formula.

Where a solver meets a proof system boundary

Symmetry between identical units, counting arguments over a resource and parity across a datapath all produce it. An engineer meeting one of these does not see a proof system boundary. They see a run that never returns, and they fall back to simulation and a coverage argument.

The boundary is a lower bound on resolution, not a tuning problem, so a larger machine and a longer timeout do not move it (Source: Exponential lower bounds for the pigeonhole principle, Beame and Pitassi(opens in a new tab)).

Five objects into four holes, searched for a proof and decided by mechanismOn the left, four holes with five objects above them, one of which has nowhere to go. On the right, two bars. The upper bar, labeled resolution search, fades out without ending. The lower bar, labeled decision by mechanism, is short and ends in a check mark labeled certificate.Five ObjectsFour Holesn+1 into n, at any nResolution SearchNo short proof exists,so the run does not return.Decision by MechanismCertificateSymmetry, counting and parityall reduce to this shape.
Five objects into four holes. A resolution based solver searches for a proof that provably has no short form, so the run does not return. A decision by mechanism returns a certificate instead, and a small checker confirms it.

Kernel

Decides those families by mechanism

The LOGOS Proof Kernel decides those families by mechanism rather than by search, and returns a certificate that a small checker re-verifies. The kernel and its measured results are described under The LOGOS Proof Kernel.

The kernel in detail, first? The LOGOS Proof Kernel

Meeting a run that never returns?

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