Theorems

Loff–Moreira–Reis [12, Theorem 16, p. 21]

(Loff–Moreira–Reis [12, Theorem 16, p. 21] ). A language $L$ is in $\mathsf{PEG}$ if and only if $L^R$ is decided by some scaffolding automaton.

Future-context full abstraction

For rooted degree- $d$ scaffolds over one working alphabet: equal unfoldings, equality of every finite rooted neighborhood, and deterministic labelled bisimilarity are equivalent; unfolding equivalence is preserved by every common scaffold row and hence by every common continuation; and unequal unfoldings are dist…

Both $\equiv_{\mathrm s}$ and $\equiv_{\mathrm f}$ are stable under a common right suffix of rows.

Minimal residual figure

Every finite rooted scaffold has a canonical finite behavioral figure $M(\mathcal U)$ whose states are $\mathcal R(\mathcal U)$ , whose root is $\mathcal U_{\epsilon}$ , and whose port $i$ sends $\mathcal U_p$ to $\mathcal U_{pi}$ when defined. The physical reachable graph maps onto $M(\mathcal U)$ by…

Generated-chain theorem

Let $S$ be a legally generated chronological scaffold with current top $t$ . Its physical graph and its minimal figure are weakly acyclic. The nodes reachable from $t$ , and the states of the minimal figure, are totally ordered by reachability. Conversely, every finite behaviorally reduced labelled rooted reac…

Regressive navigation

After adjoining a least missing sink to a generated minimal figure and numbering its chain, every port acts regressively. Its port-word transition monoid is a submonoid of the full regressive transformation monoid and is $\mathcal R$ -trivial.

Guarded folds generate bisimilarity

On every finite weakly acyclic deterministic labelled partial port graph, the equivalence generated by elementary guarded folds is exactly the greatest bisimulation equivalence.

Static normal form

Orient each guarded fold toward its quotient. The system terminates and is confluent up to rooted labelled port-graph isomorphism. Its normal form is the minimal residual figure.

Semantic causal sewing

Two strongly equivalent length- $n$ histories are connected by at most $n$ semantic step replacements.

One-state extension dichotomy

Suppose two such fresh roots have the same behavior. If the behavior is not already a state of $Q$ , their missing and $\mathsf{SELF}$ positions agree, and their old target agrees at every other port. If the behavior is an old state $q\in Q$ , each row is a guarded unroll of $q$ : it copies every missing and…

Canonical history exists

Every realizable finite strong trace has a unique lexicographically least history. It is obtained from left to right by appending the least row that realizes the next figure over the already chosen prefix.

Reduced live prefixes

Every root-reachable component at every prefix of a SELF-eager history is behaviorally reduced.

Convergent strong-trace presentation

For fixed finite $d,k,\Gamma$ , oriented endpoint detours and literal guarded unrolls form a terminating confluent graph-local presentation of strong equivalence. Two histories have the same strong trace if and only if they reduce to the same SELF-eager history. A length- $n$ history normalizes in at most $dn$…

Replay invariance

Suppose an equal-length history $H'$ agrees with $H$ on the labels of $R_H$ and on every descriptor cell in $C_H$ . Then the final rooted concrete scaffolds of $H$ and $H'$ are identical. Moreover $R_{H'}=R_H$ and $C_{H'}=C_H$ .

Causal-garbage cube

Every garbage fibre is the Cartesian product of the finite value sets of its unsupported label and descriptor coordinates. Simultaneous $G$ factors into single-coordinate changes. Fill a triangle for three values of one coordinate and a square for changes of two distinct coordinates. The resulting path two-complex…

Unexposed two-row sewing

Fix $d$ , $k\geq1$ , and a behaviorally reduced old prefix. Consider two unexposed acyclic two-row extensions with the same consumer label, the same valid nonmissing path-demanding consumer coordinates, and literal agreement as missing or $\mathsf{SELF}$ on every other consumer coordinate. If corresponding pat…

Unique relay

If consecutive states $u>v$ of that final reduced chain are carried at nonadjacent construction times, the supported descriptor segment between them is, up to endpoint detours at a recursive bottom, $$\epsilon\,0\,0\cdots0.$$

Unary final presentation

Fix a length $n$ , a nonempty finite working alphabet, degree one, and distance one. Two histories have the same final behavioral figure if and only if they are connected by endpoint detours, absorption, causal-garbage changes, and temporal carrier slides. Every final fibre contains one literal right-packed normal…

Carrier-surgery CAT(0

geometry). For a fixed unary distance-one final figure and history length, the temporal slide graph on strong/garbage chambers is the undirected Hasse graph of the order-ideal lattice of the rectangle above. Filling every Boolean family of independent slides gives the CAT(0) cube complex $X(P)$ . If the rectangle…

Simultaneous-batch compiler; fixed virtual initialization

Every bounded simultaneous record transducer with parameters $(s,A,q,r)$ , one fixed finite immutable pointer-closed initial graph, and plans determined by finite control, the input letter and bounded labelled rooted unfoldings has an exact letter-synchronous native scaffolding-automaton implementation of degree $$d=\max\{1,s+Aq\}$$…

Opaque-element staging

Let a strict purely functional sequence operation: use only finite constructor tests and a worst-case bounded number of fixed-arity constructor, projection, and pointer steps; treat elements of its base type as opaque atoms, with no equality, hashing, pattern match, identity test, or traversal through a base element…

Feedback programs compile

Let a persistent program, at each input letter, start from a fixed number $s$ of roots and perform a fixed finite composition of strict purely functional sequence operations satisfying the hypotheses of Theorem 9.4 on opaque pointer elements. In addition it may create a bounded number of fresh fixed-arity records…