Intervention-Solution Spaces · Minimal Specimens

The ISS Specimen Zoo

A field guide to the smallest argumentation frameworks exhibiting each ISS phenomenon — every specimen found by exhaustive search, every labeling independently re-derived in ASP. Companion to github.com/idaks/ISS.

S-attack (IN→OUT) F-attack (OUT→IN) U-attack (UNDEC→UNDEC) M-attack (moot) IN OUT UNDEC
chain a a b b a->b S c c b->c F

Chain (3,2)

catena reinstaurans — the reinstatement chain

ShadowUnsound mergeChameleon

The star witness. The smallest AF where merging equal-solution states breaks reachability queries (Prop. “unsound”), with Shadow and re-entrant acceptance in the same 4-state lattice.

mirror a a b b a->b U

Mirror (2,2)

speculum mutuum — mutual attack

Domino 2/2Chameleon

One suspension flips every argument’s grounded label — maximal amplification in the smallest symmetric package.

DOMINO a a UNDEC b b UNDEC a->b U c c UNDEC b->c U c->a U

Domino (3,3)

ordo cadens — the falling cycle

Domino 3/3ShadowChameleon

Cutting one attack of the 3-cycle decides all three arguments at once: total amplification, plus re-entrant acceptance for every argument.

HYDRA-3 a a IN b b OUT a->b S c c IN b->c F c->b S

Hydra (3,3)

hydra excrescens — cut one head, grow two

Hydra 1→2Shadow ×8

Suspending an attack increases the number of preferred extensions — solution spaces can grow under intervention.

SHADOW a a IN b b OUT a->b S c c IN c->b S

Shadow (3,2)

umbra divergens — equal now, different futures

Shadow

Two states with identical solution sets whose one-step futures diverge — the congruence failure behind the sound-quotient story.

SHADOW-MIN a a OUT a->a B b b IN b->a S

Shadow, minimal (2,2)

umbra minima — with self-attack

Shadowlex-min

The lexicographically smallest Shadow witness of all — exhaustively confirmed over every digraph up to isomorphism.

HYDRA-MIN a a UNDEC a->a U b b UNDEC a->b U b->a U

Hydra, minimal (2,3)

hydra minima — three creatures, one graph

HydraPhoenixChameleonlex-min

The smallest AF (self-loops allowed) exhibiting Hydra, Phoenix, and skeptical Chameleon — all at (2,3).

iss s0 a → b ← → c {a,c} IOI s1 a → b ← → c {a,b} | {a,c} IUU s0->s1 Hydra 1→2 s2 a → b ← → c {a,c} IOI s0->s2 s3 a → b ← → c {a,c} IOI s0->s3 s4 a → b ← → c {a,c} IOI s1->s4 s5 a → b ← → c {a,b} IIO s1->s5 s2->s4 s6 a → b ← → c {a,c} IOI s2->s6 s3->s5 s3->s6 s7 a → b ← → c {a,b,c} III s4->s7 s5->s7 s6->s7

Specimen detail: inside the Hydra

The full intervention-solution space of the Hydra specimen — all 2³ = 8 suspension states, bottom = nothing suspended. Each state shows the framework in force (suspended attacks in red), its preferred extensions, and the grounded labeling; pastel fills group equal-solution plateaus, dashed borders mark three-valued states. The red cover edge is the Hydra move: suspending a→b takes one preferred extension to two — cut one head, grow two.

Pilot detail view — Shadow, Phoenix, and Chameleon path overlays for the other specimens follow the same recipe (asp/make_iss_gv.py, clingo-verified).

Provenance: verified twice, independently

Every plate above shows the grounded labeling and S/F/B/U attack types computed by the clingo ASP kernels (asp/grounded.lp, asp/preferred.lp — admissible guess + subset-maximal enumeration via domain heuristics, the ASPARTIX-V technique). The same lattice sweeps and specimen searches run on the pure-Python kernels of iss_explore.py / synthesize.py. Result: kernel parity on every lattice state; identical atlases; identical minimal specimens.

$ PYTHONPATH=. python3 asp/iss_clingo.py --check kernel parity: 24 lattice states, 0 mismatches $ PYTHONPATH=. python3 asp/iss_clingo.py --synth 3 SHADOW: (2,2) HYDRA: (2,3) PHOENIX: (2,3) CHAM-SKEP: (2,3) — matches Python search