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b/research/p104-delayed-basis/inputs/pr5602-construct.py new file mode 100644 index 0000000000..ef5936ca89 --- /dev/null +++ b/research/p104-delayed-basis/inputs/pr5602-construct.py @@ -0,0 +1,215 @@ +"""Construct the fixed helper basis changes and coupled handoffs. + +Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. +The source word and its contributors are recorded in SOURCES.json. +""" +from copy import deepcopy +from fractions import Fraction as Q +import local as L + + +def row(pairs): + out = {r: Q(q) for r, q in pairs} + L.need(len(out) == len(pairs) and all(out.values()), 'Unique nonzero witness row') + return out + + +def unit(q): + n, d = abs(q.numerator), q.denominator + return n > 0 and n & (n-1) == 0 and d & (d-1) == 0 + + +def main_fields(w): + return set(w['live']) | {r+112 for r in w['live']} + + +def prefix(w, cut): + n = w['public'] + rows = [{i: Q(1)} for i in range(n)] + [{} for _ in range(8)] + frames = w['initial'][:] + [None]*8 + L.need(0 <= cut < len(w['program']), 'Cut in the literal word') + for e in w['program'][:cut+1]: + op, r = e[:2] + if op == 'MOVE': + L.need(frames[r] == e[2], 'Prefix transport source') + frames[r] = e[3] + elif op in ('INIT', 'CLEAR'): + rows[r] = {} + frames[r] = e[2] if op == 'INIT' else None + else: + L.need(frames[r] == e[-1], 'Prefix scalar chart') + if op == 'SCALE': + rows[r] = {k: Q(e[-2])*v for k, v in rows[r].items()} + else: + L.need(frames[e[2]] == e[-1], 'Prefix source chart') + if op == 'COPY': + L.need(not rows[r], 'Source COPY into zero') + L.add(rows[r], rows[e[2]], Q(e[-2])) + return rows, frames + + +def suffix_columns(w, cut): + cols = [{i: Q(1)} for i in range(w['public'])] + [{} for _ in range(8)] + for e in reversed(w['program'][cut+1:]): + op, r = e[:2] + if op == 'MOVE': + continue + if op in ('INIT', 'CLEAR'): + cols[r] = {} + elif op == 'SCALE': + cols[r] = {k: Q(e[-2])*v for k, v in cols[r].items()} + else: + L.add(cols[e[2]], cols[r], Q(e[-2])) + if op == 'COPY': + cols[r] = {} + return cols + + +def transpose(w): + program = [] + for e in reversed(w['program']): + op, r = e[:2] + if op == 'MOVE': + z = ['MOVE', r, e[3], e[2], e[4]] + elif op in ('ADD', 'COPY'): + z = ['ADD', e[2], r, e[3], e[4]] + elif op == 'SCALE': + z = e[:] + else: + L.need(op in ('INIT', 'CLEAR'), 'Transpose opcode') + z = ['CLEAR' if op == 'INIT' else 'INIT', r, e[2]] + program.append(z) + return dict(w, initial=w['final'][:], final=w['initial'][:], program=program) + + +def boundary_basis(w, c): + w = transpose(w) if c['reverse'] else deepcopy(w) + h, cut, frame = c['helper'], c['cut'], c['frame'] + output, cancel = row(c['output']), row(c['cancel']) + helpers = set(range(w['public'])) - main_fields(w) + L.need(h in helpers and unit(output.get(h, Q(0))), 'Helper unit boundary pivot') + source = w['initial'][h] + L.need(set(output) <= {r for r in helpers if w['initial'][r] == source}, + 'Boundary basis uses one helper input chart') + rows, frames = prefix(w, cut) + L.need(frames[h] == frame and h not in cancel and + all(frames[r] == frame for r in cancel), 'Available boundary cancellation') + value = rows[h].copy() + for r, q in cancel.items(): + L.add(value, rows[r], -q) + L.need(value == output, 'Exact boundary helper relation') + tail = w['program'][cut+1:] + L.need(not any(e[0] in ('ADD', 'COPY') and e[2] == h for e in tail), + 'Retired helper has no later source read') + coefficient = output[h] + before = [['ADD', h, r, str(-q/coefficient), source] + for r, q in sorted(output.items()) if r != h] + middle = [['ADD', h, r, str(-q), frame] for r, q in sorted(cancel.items())] + if coefficient != 1: + middle.append(['SCALE', h, str(1/coefficient), frame]) + w['program'] = before + w['program'][:cut+1] + middle + [e for e in tail if e[1] != h] + w['final'][h] = frame + return transpose(w) if c['reverse'] else w + + +def inverse(ops): + out = [] + for e in reversed(ops): + z = e[:] + z[-2] = str(1/Q(e[-2]) if e[0] == 'SCALE' else -Q(e[-2])) + out.append(z) + return out + + +def basis(a, b, p, frame): + """Return a matrix R with a R = e_p^T and R e_p = b.""" + L.need(sum(a.get(h, 0)*q for h, q in b.items()) == 1, 'Dual basis pairing') + q = next((h for h in sorted(a) if unit(a[h])), None) + L.need(q is not None, 'Dyadic basis pivot') + ops = [] if p == q else [ + ['ADD', p, q, '1', frame], ['ADD', q, p, '-1', frame], + ['ADD', p, q, '1', frame], ['SCALE', q, '-1', frame]] + ops += [['ADD', j, q, str(c), frame] for j, c in sorted(b.items()) if j != q] + if a[q] != 1: + ops.append(['SCALE', q, str(1/a[q]), frame]) + ops += [['ADD', q, j, str(-c/a[q]), frame] for j, c in sorted(a.items()) if j != q] + return ops + + +def relabel(e, mapping): + z = e[:] + z[1] = mapping(z[1]) + if z[0] in ('ADD', 'COPY'): + z[2] = mapping(z[2]) + return z + + +def handoff(w, c): + n, cut, frame, p = w['public'], c['cut'], c['frame'], c['pivot'] + source, target = c['group'] + group = {h for h in range(n) if h not in main_fields(w) and + (w['initial'][h], w['final'][h]) == (source, target)} + v, u = row(c['v']), row(c['u']) + pairing = sum(v.get(h, 0)*q for h, q in u.items()) + L.need(unit(pairing), 'Nonzero dyadic handoff pairing') + v = {h: q/pairing for h, q in v.items()} + rows, frames = prefix(w, cut) + L.need(all(f is None for f in frames[n:]), 'No owned field at handoff cut') + L.need(p in group and frames[p] == frame and + all(0 <= h < n and frames[h] == frame for h in v.keys() | u.keys()), + 'Public handoff fields share the cut chart') + L.need(L.span(w['frames'][source]) < L.span(w['frames'][frame]) < + L.span(w['frames'][target]), 'Strict nested handoff boundaries') + cols, a, b = suffix_columns(w, cut), {}, {} + for r, q in v.items(): + L.add(a, rows[r], q) + for r, q in u.items(): + L.add(b, cols[r], q) + L.need(a == row(c['source_functional']) and b == row(c['output_direction']), + 'Exact prefix and suffix handoff witnesses') + L.need(set(a) <= group and set(b) <= group, 'Handoff has only helper boundary support') + input_ops, cut_basis = basis(a, b, p, source), basis(v, u, p, frame) + output_ops = inverse(input_ops) + for e in output_ops: + e[-1] = target + old = lambda r: r+1 if r >= n else r + tail = lambda r: n if r == p else old(r) + out = deepcopy(w) + out['public'] = n+1 + out['initial'].append(frame) + out['final'][p] = frame + out['final'].append(target) + out['program'] = input_ops + [relabel(e, old) for e in w['program'][:cut+1]] + out['program'] += inverse(cut_basis) + [relabel(e, tail) for e in cut_basis] + out['program'] += [relabel(e, tail) for e in w['program'][cut+1:]] + out['program'] += [relabel(e, tail) for e in output_ops] + return out + + +def restore_copy_opcodes(w): + fresh, copies = set(), 0 + for e in w['program']: + op, r = e[:2] + if op == 'INIT': + fresh.add(r) + elif r in fresh and op != 'MOVE': + L.need(op == 'ADD', 'First owned write is the retained copy') + e[0] = 'COPY' + fresh.remove(r) + copies += 1 + L.need(not fresh and copies == 8, 'All eight owned copy paths remain') + + +def construct(base, witness): + L.need(witness['schema'] == 'p104-coupled-helper-bases/1', 'Witness schema') + L.need([s['kind'] for s in witness['steps']] == + ['boundary_basis']*6 + ['coupled_handoff']*9, 'Fixed fifteen changes') + word, evidence = deepcopy(base), [] + for step in witness['steps']: + word = boundary_basis(word, step) if step['kind'] == 'boundary_basis' else handoff(word, step) + check = L.replay(word) + evidence.append(dict(kind=step['kind'], public=word['public'], + events=len(word['program']), paid=check['paid'], helpers=check['helpers'])) + restore_copy_opcodes(word) + L.replay(word) + return word, evidence diff --git a/research/p104-delayed-basis/inputs/pr5612-source.tar.gz b/research/p104-delayed-basis/inputs/pr5612-source.tar.gz new file mode 100644 index 0000000000000000000000000000000000000000..0527764dec5bae5b23893cd4126109e703fa1b1f GIT binary patch literal 1177351 zcmV(>K-j+@iwFP!000023hccHU=&sUIDY39b@dMKJOyjUSVGxNcKR+MM1%-P2_`5) zz|HK=Y_eoa*x5}HNbb&4tZ+6^RANUIL8OC%CkiS8DoBwg0wN_;sR_yN^WK}8oh^ZI z=lS*ie{*NZ&dzI}-rjp*MOCD*r!=dawsSUF@k`Z;Y*S06F#HonAG5mW_ZyI3*n42_ z{Qj8%`B&%uV}~DsZR>c}@!~yiVDF^%ML=)rb^^;eYl2&!6k5?x8?= zNcBa0p&%s%Whzt|RKjYhubh%Ro=`Lx@dZmPmO_A1gz|<%0ct=2FW4O!DkViDr3!rS zks^vrl>^BHBa|l;R3qW22ldyPf)^-4OuR#+Fg;O~zDTJBeE}Wu0uL<$fhZb<@<^&j 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