diff --git a/.github/workflows/release-paper.yml b/.github/workflows/release-paper.yml new file mode 100644 index 0000000..6c684a5 --- /dev/null +++ b/.github/workflows/release-paper.yml @@ -0,0 +1,48 @@ +name: Release Paper PDF + +on: + push: + tags: + # Only trigger for tags ending in "-paper" + - "v*-paper" + +jobs: + release: + name: Release PDF + runs-on: ubuntu-latest + + steps: + - name: Checkout repo + uses: actions/checkout@v4 + + - name: Build PDF + uses: xu-cheng/latex-action@v3 + with: + working_directory: aion-holography + root_file: main.tex + latexmk_use_xelatex: false + args: -pdf -interaction=nonstopmode -halt-on-error + + - name: Set PDF path + id: pdf + run: | + FILE_PATH="aion-holography/main.pdf" + if [ ! -f "$FILE_PATH" ]; then + echo "ERROR: Could not find $FILE_PATH" + exit 1 + fi + echo "path=$FILE_PATH" >> "$GITHUB_OUTPUT" + + - name: Create GitHub Release + id: create_release + uses: softprops/action-gh-release@v2 + with: + tag_name: ${{ github.ref_name }} + name: Release ${{ github.ref_name }} + draft: false + prerelease: false + generate_release_notes: true + files: | + ${{ steps.pdf.outputs.path }}#AIon-Computational-Holography-${{ github.ref_name }}.pdf + env: + GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }} diff --git a/aion-holography/Makefile b/aion-holography/Makefile index a1ba595..5223ffe 100644 --- a/aion-holography/Makefile +++ b/aion-holography/Makefile @@ -8,7 +8,10 @@ MAIN = main all: $(MAIN).pdf -$(MAIN).pdf: $(MAIN).tex sections/*.tex macros.tex references.bib + +TEX_SRCS = $(filter-out $(MAIN).tex,$(wildcard *.tex)) + +$(MAIN).pdf: $(MAIN).tex $(TEX_SRCS) references.bib $(TEX) $(MAIN) $(BIB) $(MAIN) $(TEX) $(MAIN) diff --git a/aion-holography/sections/assumptions.tex b/aion-holography/assumptions.tex similarity index 76% rename from aion-holography/sections/assumptions.tex rename to aion-holography/assumptions.tex index 9b2f6af..607ab47 100644 --- a/aion-holography/sections/assumptions.tex +++ b/aion-holography/assumptions.tex @@ -11,16 +11,15 @@ \section*{Standing Assumptions} \textbf{Assumption} & \textbf{Where used} \\ \midrule Skeleton independence via footprints & -\cref{def:footprint,def:independence,def:batch}; -\cref{thm:tick-confluence} \\ -No-delete/no-clone under descent (ND/NC) & -\cref{def:no-delete}; \cref{thm:two-plane,thm:global} \\ + \cref{def:footprint,def:batch,thm:tick-confluence} \\ +No-delete/no-clone-under-descent (ND/NC) & +\cref{def:no-delete,thm:two-plane,thm:global} \\ Termination / decreasing diagrams on the skeleton & \cref{thm:global} (conditional global confluence) \\ No re-derivation (single producer) & \cref{sec:holography}, \cref{thm:backward} (backward provenance) \\ Budgeted translators and 1-Lipschitz distortion & -\cref{sec:rulial}, \cref{lem:rulial-basic,thm:rulial-triangle} (rulial distance) \\ +\cref{sec:rulial,thm:rulial-basic,thm:rulial-triangle} (rulial distance) \\ \bottomrule \end{tabular} \end{center} diff --git a/aion-holography/sections/determinism_confluence.tex b/aion-holography/determinism_confluence.tex similarity index 84% rename from aion-holography/sections/determinism_confluence.tex rename to aion-holography/determinism_confluence.tex index 5c09631..1650d8d 100644 --- a/aion-holography/sections/determinism_confluence.tex +++ b/aion-holography/determinism_confluence.tex @@ -2,6 +2,9 @@ \section{Determinism and Confluence} \label{sec:determinism} +Throughout this section we work under the standing assumptions +summarised on page~\pageref{sec:assumptions}. + We sketch the concurrency discipline, define independence, and state the main confluence theorems for tick-level execution, working with RMG states and tick semantics as introduced in \cref{def:rmg-state,def:rmg-tick}. @@ -11,7 +14,7 @@ \subsection{Footprints and Independence on the Skeleton Plane} We work at the level of the skeleton plane. Write $G_S$ for the skeleton component of $G$, and likewise $L_S$, $K_S$, $R_S$ for the underlying skeleton graphs of a rule. Let -$\mathcal{U} = (G;\alpha,\beta)$ be an RMG state and let +$U = (G;\alpha,\beta)$ be an RMG state and let $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ be a DPOI rule. A \emph{skeleton match} is a mono $m_S : L_S \hookrightarrow G_S$ in $\OGraph_T$ satisfying the usual gluing conditions. @@ -46,14 +49,53 @@ \subsection{Footprints and Independence on the Skeleton Plane} written by the other. \end{definition} +\begin{figure}[t] + \centering + \begin{tikzpicture}[ + node/.style={circle,draw=gray!60,fill=gray!10,thick,minimum size=8mm}, + del/.style={fill=red!25}, + use/.style={fill=blue!20}, + edge/.style={-Latex,thick,gray!60}, + >=Latex + ] + % baseline graph + \node[node] (a) at (0,0) {$a$}; + \node[node,del] (b) at (2.2,1.2) {$b$}; + \node[node,use] (c) at (4.1,0.1) {$c$}; + \node[node] (d) at (2.1,-1.3) {$d$}; + \draw[edge] (a) -- (b); + \draw[edge] (b) -- (c); + \draw[edge] (a) -- (d); + \draw[edge] (d) -- (c); + + % match 1 + \draw[rounded corners,thick,teal!70] + ( -0.5, -0.9) rectangle (2.9,1.6); + \node[anchor=west,teal!70!black] at (3.05,1.55) {$m_1$}; + + % match 2 + \draw[rounded corners,thick,orange!80] + (0.9,-1.7) rectangle (4.7,1.4); + \node[anchor=west,orange!80!black] at (4.85,1.35) {$m_2$}; + + % legend + % legend in separate inset box below + \end{tikzpicture} + \caption{Two overlapping matches on the skeleton plane. Shaded nodes + illustrate $\Del$ (red) and $\Use$ (blue). Independence requires + $\Del(m_1)\cap\Use(m_2)=\Del(m_2)\cap\Use(m_1)=\emptyset$, ruling out + destructive interference between the batches.} + \label{fig:footprint} +\end{figure} + \subsubsection{Scheduler--admissible batches} \begin{definition}[Scheduler--admissible batch]\label{def:batch} - Let $\mathcal{U} = (G;\alpha,\beta)$ be an RMG state. A finite + Let $U = (G;\alpha,\beta)$ be an RMG state. A finite family of skeleton matches $B = \{m_{i,S} : L_{i,S} \to G_S\}_{i\in I}$ is \emph{scheduler--admissible} if the matches are pairwise - independent in the sense of \cref{def:independence}, i.e. + independent in the sense of Definition~\ref{def:independence}, i.e. \[ \Del(m_{i,S}) \cap \Use(m_{j,S}) = \emptyset \quad\text{for all distinct } i,j\in I. @@ -70,7 +112,7 @@ \subsection{Tick semantics and scheduler confluence} given by the standard double square (pushout complement + pushout). We work with RMG states and tick semantics as defined in -\cref{def:rmg-state,def:rmg-tick}. In this section we analyse when +Definitions~\ref{def:rmg-state} and~\ref{def:rmg-tick}. In this section we analyse when batches of matches can be scheduled concurrently without affecting the resulting state. The deterministic properties proved here apply uniformly to all derivations; applying them to cognitive systems @@ -78,11 +120,11 @@ \subsection{Tick semantics and scheduler confluence} \cref{sec:ethics}. The scheduler computes a maximal independent set of matches in the -sense of \cref{def:batch}, using a safe over-approximation of +sense of Definition~\ref{def:batch}, using a safe over-approximation of $\Use\cup\Del$; we do not repeat the implementation details here. \begin{theorem}[Skeleton-plane tick confluence]\label{thm:tick-confluence} - Let $\mathcal{U} = (G;\alpha,\beta)$ be an RMG state and let + Let $U = (G;\alpha,\beta)$ be an RMG state and let $B = \{m_{i,S} : L_{i,S} \hookrightarrow G_S\}_{i\in I}$ be a scheduler--admissible batch for a family of DPOI rules. Then any two sequentialisations of the corresponding DPOI steps yield @@ -94,7 +136,9 @@ \subsection{Tick semantics and scheduler confluence} $\{m_{i,S} : L_{i,S} \hookrightarrow G_S\}$ are pairwise independent in the sense of \cref{def:independence}. By the Concurrency / Parallel Independence Theorem for DPO rewriting in adhesive categories -(see, e.g.,~\cite{EEPT06}), parallel independent steps commute: for +(see, e.g.,~\cite{EEPT06}; similar categorical techniques appear in the +space--time reversible graph-rewriting setting of~\cite{Arrighi2025Reversible}), +parallel independent steps commute: for any $i \neq j$ we have a diagram \[ G_S \;\Rightarrow_{(p_i,m_{i,S})}\; G_i @@ -137,7 +181,7 @@ \subsection{Tick semantics and scheduler confluence} Combining \cref{thm:tick-confluence} with the two-plane commutation result (\cref{thm:two-plane}) shows that a tick that satisfies the -no-delete/no-clone discipline has a unique outcome up to +no-delete/no-clone-under-descent invariant has a unique outcome up to isomorphism in the full RMG semantics. \begin{corollary}[Worldline uniqueness]\label{cor:worldline-uniqueness} @@ -177,8 +221,8 @@ \subsection{Two-plane commutation via a fibration} $\pi^{-1}(G)$; a \emph{skeleton step} is a DPOI step in the base $\OGraph_T$. Both are built from pushouts along monos. -\begin{definition}[No-delete/no-clone under descent]\label{def:no-delete} - Consider a tick on an RMG state $\mathcal{U} = (G;\alpha,\beta)$ +\begin{definition}[No-delete/no-clone-under-descent]\label{def:no-delete} +Consider a tick on an RMG state $U = (G;\alpha,\beta)$ consisting of \begin{enumerate}[leftmargin=*] \item a family of attachment--plane DPOI steps, each acting inside @@ -188,8 +232,8 @@ \subsection{Two-plane commutation via a fibration} $G_S \Rewrite_S G_S'$ induced by a rule and match $m_S : L_S \to G_S$. \end{enumerate} - We say that this tick satisfies \emph{no-delete/no-clone under - descent} if: + We say that this tick satisfies \emph{no-delete/no-clone-under-descent} + if: \begin{enumerate}[leftmargin=*] \item[(ND)] (\emph{No delete under descent.}) If a skeleton vertex or edge $x\in G_S$ is deleted by the @@ -204,15 +248,15 @@ \subsection{Two-plane commutation via a fibration} steps in the same tick. In particular, no attachment object is copied to multiple descendants of $x$. \end{enumerate} - A rule pack $R$ satisfies no-delete/no-clone under descent if every + A rule pack $R$ satisfies no-delete/no-clone-under-descent if every tick generated from $R$ has this property. \end{definition} \begin{theorem}[Two-plane commutation]\label{thm:two-plane} Let $R$ be a rule pack satisfying the no-delete/no-clone-under-descent -invariant of \cref{def:no-delete}. Let -$\mathcal{U} = (G;\alpha,\beta)$ be an RMG state generated from $R$. -Let $A : \mathcal{U} \Rewrite \mathcal{U}_A$ be a finite composite of +invariant of Definition~\ref{def:no-delete}. Let +$U = (G;\alpha,\beta)$ be an RMG state generated from $R$. +Let $A : U \Rewrite U_A$ be a finite composite of attachment steps in the fiber over $G$, and let $S : G \Rewrite G'$ be the skeleton DPOI step induced by the same tick such that the tick consisting of $A$ followed by $S$ satisfies the @@ -331,7 +375,7 @@ \subsection{Global confluence} This theorem applies directly to the skeleton plane; together with the no-delete/no-clone-under-descent invariant and two-plane commutation, it yields uniqueness of \emph{worldlines} at the level of RMG states -when the rule pack satisfies these conditions. +for rule packs satisfying these termination / critical-pair conditions. Under the hypotheses of the global confluence theorem, any two complete derivations from a fixed initial state are joinable and yield diff --git a/aion-holography/sections/discussion.tex b/aion-holography/discussion.tex similarity index 66% rename from aion-holography/sections/discussion.tex rename to aion-holography/discussion.tex index 357b352..48ab55b 100644 --- a/aion-holography/sections/discussion.tex +++ b/aion-holography/discussion.tex @@ -9,6 +9,32 @@ \section{Discussion and Future Work} MDL-based rulial geometry on observers and connected RMG rewriting to multiway systems. +\subsection{Implementation guarantees (Echo)} + +In the concrete \AION{} runtime (Echo), the semantic assumptions above +are enforced by operational determinism invariants; any violation aborts +the tick deterministically and emits an error node for replay analysis. +These invariants are also exercised by the test suite to guarantee +bit-level reproducibility: +\begin{itemize}[leftmargin=*] + \item \textbf{World Equivalence:} identical diff sequences and merge + decisions yield identical world hashes. + \item \textbf{Merge Determinism:} given the same base snapshot, diffs, + and merge strategies, the resulting snapshot and diff hashes are + identical. + \item \textbf{Temporal Stability:} GC, compression, and inspector + activity do not alter logical state. + \item \textbf{Schema Consistency:} component layout hashes must match + before merges; mismatches block the merge. + \item \textbf{Causal Integrity:} writes cannot modify values they + transitively read earlier in Chronos; paradoxes are detected and + isolated. + \item \textbf{Entropy Reproducibility:} branch entropy is a + deterministic function of recorded events. + \item \textbf{Replay Integrity:} replaying from node $A$ to $B$ + produces identical world hash, event order, and PRNG draw counts. +\end{itemize} + \subsection{Related work} Our work builds on several research traditions: @@ -22,6 +48,15 @@ \subsection{Related work} theorem (\cref{thm:tick-confluence}) is a specialization of the standard concurrency theorem for adhesive systems. +Category-theoretic graph rewriting has also been applied to +discretised space--time models in +Arrighi--Costes--Maignan~\cite{Arrighi2025Reversible}, which uses DPO +rewriting in an adhesive setting to study space--time reversible graph +rewriting. Our use of the same machinery is conceptually similar, but +we focus on deterministic multiway semantics +and holographic provenance in a computational setting rather than on +physical geometry. + \paragraph{Confluence and termination.} The critical-pair lemma and Newman's lemma are classical tools in term rewriting; for decreasing-diagram techniques, see van @@ -70,9 +105,10 @@ \subsection{Related work} to layer cryptographic commitments and zero-knowledge proofs on top, enabling external verifiers to check correctness properties without learning private data. - \item \textbf{Temporal logic and Time Cube.} The Chronos, Kairos, and Aion - triad suggests new modal and temporal logics for reasoning about - linear time, branch points, and the surrounding possibility space. + \item \textbf{Temporal logic and the Time Cone.} The Chronos, Kairos, and + Aion triad naturally suggests new modal and temporal logics for + reasoning about linear time, branch points, and the surrounding + possibility space. \item \textbf{\COMPUTER{} architecture.} Building on this foundation, the companion paper will define the \AION{} \COMPUTER{}: a machine model whose basic step is a provenance-carrying RMG rewrite, supporting diff --git a/aion-holography/sections/dpo_rmg.tex b/aion-holography/dpo_rmg.tex similarity index 94% rename from aion-holography/sections/dpo_rmg.tex rename to aion-holography/dpo_rmg.tex index f5dd739..518a75f 100644 --- a/aion-holography/sections/dpo_rmg.tex +++ b/aion-holography/dpo_rmg.tex @@ -50,7 +50,7 @@ \subsection{RMG states as two-plane objects} \begin{definition}[RMG state]\label{def:rmg-state} An RMG \emph{state} is a triple \[ - \mathcal{U} = (G;\alpha,\beta) + U = (G;\alpha,\beta) \] where $G \in \OGraph_T$ is the skeleton and $\alpha,\beta$ assign attachment objects in the appropriate fibres (of the forgetful functor @@ -72,11 +72,11 @@ \subsection{RMG states as two-plane objects} \end{itemize} \begin{definition}[Tick]\label{def:rmg-tick} -A \emph{tick} on an RMG state $\mathcal{U} = (G;\alpha,\beta)$ consists +A \emph{tick} on an RMG state $U = (G;\alpha,\beta)$ consists of a finite family of attachment steps in the fibres over $G$ followed by a finite family of skeleton steps on $G$, chosen by the scheduler. In \cref{sec:determinism} we impose additional conditions (independence, -scheduler--admissible batches, and the no-delete/no-clone under descent +scheduler--admissible batches, and the no-delete/no-clone-under-descent invariant) on the ticks generated by the runtime. \end{definition} diff --git a/aion-holography/sections/ethics.tex b/aion-holography/ethics.tex similarity index 100% rename from aion-holography/sections/ethics.tex rename to aion-holography/ethics.tex diff --git a/aion-holography/sections/holography.tex b/aion-holography/holography.tex similarity index 100% rename from aion-holography/sections/holography.tex rename to aion-holography/holography.tex diff --git a/aion-holography/sections/intro.tex b/aion-holography/intro.tex similarity index 78% rename from aion-holography/sections/intro.tex rename to aion-holography/intro.tex index 64ef72c..1123f56 100644 --- a/aion-holography/sections/intro.tex +++ b/aion-holography/intro.tex @@ -16,8 +16,8 @@ \section{Introduction} \item all evolution of that state is given by well-typed graph rewrites; \item under an explicit independence discipline and - no-delete/no-clone under descent invariants, the operational - semantics is deterministic (up to typed open-graph isomorphism) + no-delete/no-clone-under-descent invariants, the operational + semantics is deterministic (up to typed open graph isomorphism) and confluent at the level of ``ticks'' of computation (\cref{thm:tick-confluence,thm:two-plane,thm:global}); and \item the \emph{entire} interior evolution of a computation is stored @@ -27,7 +27,7 @@ \section{Introduction} The technical starting point is algebraic graph transformation using the double--pushout (DPO) approach in adhesive categories, together with the -Recursive Metagraph (RMG) object model developed in earlier work. +Recursive Metagraph (RMG) object model we define in Section~\ref{sec:rmg}. We extend this setting with: \begin{enumerate}[leftmargin=*] @@ -35,7 +35,9 @@ \section{Introduction} \item a two-plane concurrent operational semantics with attachment--then--skeleton publication, together with confluence results: tick-level determinism and two-plane - commutation, plus conditions for global confluence; + commutation (Theorems~\ref{thm:tick-confluence} and + \ref{thm:two-plane}), and, under standard rewrite-theory + hypotheses, global confluence (Theorem~\ref{thm:global}); \item a provenance payload calculus giving \emph{computational holography}; \item an MDL-based quasi-pseudometric on observers, the \emph{rulial @@ -57,11 +59,16 @@ \section{Introduction} The main results of this paper are: \begin{enumerate}[leftmargin=5mm] \item \emph{Tick-level confluence} (Theorem~\ref{thm:tick-confluence}): - parallel independent RMG rewrites commute, yielding deterministic - semantics independent of scheduler serialization order; + parallel independent RMG rewrites commute, yielding per-tick + deterministic semantics independent of scheduler serialization + order; \item \emph{Two-plane commutation} (Theorem~\ref{thm:two-plane}): attachment and skeleton updates can be applied in either order up to isomorphism, via a fibration structure; + \item \emph{Worldline uniqueness} (Corollary~\ref{cor:worldline-uniqueness}): + any complete execution (all tick steps) yields a single worldline + up to typed open graph isomorphism, regardless of scheduler order, + extending tick-level commutation to whole runs; \item \emph{Computational holography} (Theorem~\ref{thm:holography}): the boundary data $(S_0,P)$ is information-complete with respect to the interior evolution, enabling reconstruction of the full diff --git a/aion-holography/macros.tex b/aion-holography/macros.tex index fd52c4c..d3311d5 100644 --- a/aion-holography/macros.tex +++ b/aion-holography/macros.tex @@ -21,8 +21,12 @@ \providecommand{\AIONWordmarkSerif}{\textrm{AION}} \providecommand{\AIONInline}{\textrm{AION}} \providecommand{\AIONSignature}[1][1]{\AIONWordmarkSerif} -\newcommand{\PaperDate}{\today} -\newcommand{\AIONProjectURL}{\url{https://flyingrobots.dev}} +\providecommand{\PaperVersion}{draft} % set to e.g. v1.0 for archival builds +% Publication date: defaults to dynamic \today for ongoing edits. +% For archival/release builds, override with a fixed date, e.g. +% \renewcommand{\PaperDate}{November 25, 2025} +\providecommand{\PaperDate}{\today} +\newcommand{\AIONProjectURL}{\url{https://github.com/flyingrobots/aion}} % ========================================== % Rewrite / Footprint helpers diff --git a/aion-holography/main.tex b/aion-holography/main.tex index a35906a..32cfded 100644 --- a/aion-holography/main.tex +++ b/aion-holography/main.tex @@ -27,6 +27,7 @@ <8> stmary8 <9> stmary9 <10> stmary10 + <10.5> stmary10 <10.95> stmary10 <12> stmary10 <14.4> stmary10 @@ -62,10 +63,10 @@ \begin{document} +\hypersetup{pageanchor=false} +\pagenumbering{roman} \input{titlepage} -\maketitle - \begin{abstract} We develop a formal model of \emph{computational holography}: a way of representing a computation so that its entire interior evolution is @@ -84,19 +85,24 @@ observers. \end{abstract} +\clearpage \tableofcontents -\input{sections/assumptions} -\input{sections/intro} -\input{sections/rmg} -\input{sections/dpo_rmg} -\input{sections/determinism_confluence} -\input{sections/holography} -\input{sections/wormholes} -\input{sections/rulial_distance} -\input{sections/multiway_ruliad} -\input{sections/ethics} -\input{sections/discussion} +\clearpage +\hypersetup{pageanchor=true} +\pagenumbering{arabic} + +\input{assumptions} +\input{intro} +\input{rmg} +\input{dpo_rmg} +\input{determinism_confluence} +\input{holography} +\input{wormholes} +\input{rulial_distance} +\input{multiway_ruliad} +\input{ethics} +\input{discussion} \bibliographystyle{alpha} \bibliography{references} @@ -110,3 +116,5 @@ \section*{Author's Note} on the mathematics. \end{document} +% Hack: force arXiv to run 4 passes by mimicking LaTeX rerun warning +\typeout{get arXiv to do 4 passes: Label(s) may have changed. Rerun} diff --git a/aion-holography/sections/multiway_ruliad.tex b/aion-holography/multiway_ruliad.tex similarity index 78% rename from aion-holography/sections/multiway_ruliad.tex rename to aion-holography/multiway_ruliad.tex index 2dd48a9..1f364ff 100644 --- a/aion-holography/sections/multiway_ruliad.tex +++ b/aion-holography/multiway_ruliad.tex @@ -86,15 +86,25 @@ \section{Multiway Systems and the Ruliad} The class of all possible such worldlines, across all rule sets and inputs, forms a large multiway object akin to the Ruliad. The rulial distance from \cref{sec:rulial} equips this space of observers with a -geometry, and the Chronos, Kairos, Aion time model from the \AION{} -calculus\footnote{Developed in a separate technical note on the - \AION{} time model~\cite{RossAIONCalculus2025}.} gives a temporal -structure on branches and merges. +geometry, and the Chronos, Kairos, Aion time model described below +(see \cref{subsec:chronos-kairos-aion}) gives a temporal structure on +branches and merges. The RMG worldlines defined here thus embed naturally into the broader Ruliad framework. A more detailed study of this correspondence is left for future work, and will be taken up in the companion \COMPUTER{} paper, where we -explicitly model fork/merge operators and the Time Cube (Chronos, -Kairos, Aion) at the level of operational semantics. +explicitly model fork/merge operators and the Time Cone (Chronos, +Kairos, Aion) at the level of operational semantics. Intuitively, the +Time Cone is the forward ``light-cone'' of computation: Chronos gives +the linear worldline, Kairos marks branch points, and Aion is the +surrounding space of possible RMG histories. + +\subsection{Chronos, Kairos, Aion: a three-layer time model} +\label{subsec:chronos-kairos-aion} +Chronos is the linear ordering of events inside a tick (ledger kernel). +Kairos marks branch points such as schema changes, DPO conflicts, or +rule-pack switches. Aion is the total multiway possibility space +$\Hist(\mathcal{U}, R)$ ranging over all schedules, rule packs, and inputs, +supporting multiverse debugging and counterfactual replay. diff --git a/aion-holography/references.bib b/aion-holography/references.bib index ff2ab47..52f8acf 100644 --- a/aion-holography/references.bib +++ b/aion-holography/references.bib @@ -48,6 +48,16 @@ @article{Rissanen1978 pages = {465--471} } +@misc{Arrighi2025Reversible, + author = {Arrighi, Pablo and Costes, Marin and Maignan, Luidnel}, + title = {Space-time Reversible Graph Rewriting}, + year = {2025}, + eprint = {2510.03296}, + archivePrefix = {arXiv}, + primaryClass = {cs.DM}, + url = {https://arxiv.org/abs/2510.03296} +} + @misc{RossRMG2025, author = {Ross, James}, title = {Recursive Metagraphs: DPOI Semantics, Confluence, Hypergraph Embedding, and Rulial Distance}, @@ -55,13 +65,6 @@ @misc{RossRMG2025 note = {Technical report} } -@misc{RossAIONCalculus2025, - author = {Ross, James}, - title = {The {AION} Calculus}, - year = {2025}, - note = {Working note} -} - @article{LS06, author = {Lack, Steven and Soboci{\'n}ski, Pawel}, title = {Adhesive Categories}, diff --git a/aion-holography/sections/rmg.tex b/aion-holography/rmg.tex similarity index 83% rename from aion-holography/sections/rmg.tex rename to aion-holography/rmg.tex index a54e51e..20fafc7 100644 --- a/aion-holography/sections/rmg.tex +++ b/aion-holography/rmg.tex @@ -26,9 +26,9 @@ \subsection{Inductive definition} We write an element of $\RMG$ as either an atom or as a ``1-skeleton'' graph decorated by attachments on vertices and edges. Attachments themselves may be recursive metagraphs, so this attachment structure -can nest arbitrarily deeply. This definition agrees with the -set-theoretic and initial-algebra presentation developed in previous -technical notes. +can nest arbitrarily deeply. This definition matches the +set-theoretic and initial-algebra presentation used in our internal +design notes. \paragraph{Example (A tiny recursive metagraph).} As a concrete instance, consider a program call graph where each @@ -54,6 +54,21 @@ \subsection{Initial algebra viewpoint} principles: every function out of $\RMG$ is uniquely determined by its action on atoms and on decorated skeletons. +\subsection{Unfoldings and recursion schemes} + +The depth of an RMG $X$ is the length of the longest attachment chain +in $X$; finite depth follows from the inductive definition. For +$k\in\mathbb{N}$, the \emph{$k$-unfolding} of $X$, written +$\mathrm{unf}_k(X)$, replaces each attachment at depth $k$ or greater +by an opaque atom while preserving all structure at depths +$0,\ldots,k{-}1$. The \emph{infinite unfolding} +$\mathrm{unf}_\infty(X)$ is the directed colimit of the +$k$-unfoldings. Because $F$ is finitary and $\RMG$ is the initial +$F$-algebra, these colimits remain finitely branching and support the +same structural recursion principles. The unfoldings thus let us +reason about recursion schemes (catamorphisms, anamorphisms) over RMGs +without leaving the typed open-graph setting. + \subsection{Morphisms and category of RMGs} \begin{definition}[RMG morphism] @@ -161,7 +176,7 @@ \subsection{Notation summary} \toprule \textbf{Symbol} & \textbf{Meaning} \\ \midrule -$\mathcal{U} = (G;\alpha,\beta)$ & single RMG state in universe $U$ \\ +$U = (G;\alpha,\beta)$ & single RMG state in universe $\mathcal{U}$ \\ $p = (L \xleftarrow{\ell} K \xrightarrow{r} R)$ & DPOI rule \\ $\mu_i$ & microstep label \\ $P = (\mu_0,\dots,\mu_{n-1})$ & provenance payload \\ @@ -174,10 +189,11 @@ \subsection{Notation summary} \end{center} \medskip -Throughout, an \emph{RMG universe} $U$ is a set of RMG states (typically closed -under the rewrite rules $R$ under consideration), and $\mathcal{U} \in U$ -denotes a particular state in that universe. +Throughout, an \emph{RMG universe} $\mathcal{U}$ is a set of RMG states +(typically closed under the rewrite rules $R$ under consideration), and +$U \in \mathcal{U}$ denotes a particular state in that universe. Subsequent sections introduce $D_{\tau,m}$ (rulial distance), -$\Hist(U,R)$ (history category on the universe $U$ of RMG states), and +$\Hist(\mathcal{U},R)$ (history category on the universe $\mathcal{U}$ of +RMG states), and other observer-related notation. diff --git a/aion-holography/sections/rulial_distance.tex b/aion-holography/rulial_distance.tex similarity index 92% rename from aion-holography/sections/rulial_distance.tex rename to aion-holography/rulial_distance.tex index 1316dbd..7f62ccb 100644 --- a/aion-holography/sections/rulial_distance.tex +++ b/aion-holography/rulial_distance.tex @@ -8,11 +8,12 @@ \section{Rulial Distance: A Computable Quasi-Pseudometric on Observer Space} \subsection{Observers as functors} -Fix an RMG universe $(U,R)$ and its history category -$\Hist(U,R)$ whose objects are states and whose morphisms are derivation -paths between them. An \emph{observer} is a functor +Fix an RMG universe $\mathcal{U}$ together with a rule pack $R$ and its +history category $\Hist(\mathcal{U},R)$ whose objects are states +$U,V \in \mathcal{U}$ and whose morphisms are derivation paths between +them. An \emph{observer} is a functor \[ - O : \Hist(U,R) \To \mathcal{Y}, + O : \Hist(\mathcal{U},R) \To \mathcal{Y}, \] where $\mathcal{Y}$ is a suitable category of observations (symbol streams, trace graphs, etc.). We assume that $O$ is realised by some @@ -36,7 +37,7 @@ \subsection{Translators, MDL Complexity, and Distortion} T_{12} : O_1 \Rightarrow O_2 \] realised as a small DPOI transducer: for each history -$h \in \Hist(U,R)$ it maps the trace $O_1(h)$ to a trace +$h \in \Hist(\mathcal{U},R)$ it maps the trace $O_1(h)$ to a trace $T_{12}(O_1(h))$ in the observation category~$\mathcal{Y}$. Likewise we consider translators $T_{21} : O_2 \Rightarrow O_1$. @@ -63,10 +64,10 @@ \subsection{Translators, MDL Complexity, and Distortion} be a metric on individual traces (for example, an $L_1$ distance on symbol streams or an edit distance on labelled paths). We lift this pointwise to observers -by defining, for observers $O,O' : \Hist(U,R) \to \mathcal{Y}$, +by defining, for observers $O,O' : \Hist(\mathcal{U},R) \to \mathcal{Y}$, \[ \mathrm{Dist}(O,O') - := \sup_{h \in \Hist(U,R)} + := \sup_{h \in \Hist(\mathcal{U},R)} \mathrm{dist}_{\mathrm{tr}}\bigl(O(h),O'(h)\bigr). \] We assume all observers considered produce traces in a common metric @@ -106,11 +107,11 @@ \subsection{Translators, MDL Complexity, and Distortion} \Bigr). \] -\begin{lemma}[Basic properties of $D_{\tau,m}$]\label{lem:rulial-basic} +\begin{theorem}[Basic properties of $D_{\tau,m}$]\label{thm:rulial-basic} For all observers $O_1,O_2$ and budgets $(\tau,m)$, the distance $D_{\tau,m}(O_1,O_2)$ is nonnegative and symmetric, and $D_{\tau,m}(O,O)=0$ for every observer $O$. -\end{lemma} +\end{theorem} \begin{proof} Nonnegativity and symmetry are immediate from the definition of @@ -123,7 +124,7 @@ \subsection{Translators, MDL Complexity, and Distortion} nonnegativity implies $D_{\tau,m}(O,O)=0$. \end{proof} -\begin{theorem}[Triangle inequality for rulial distance]\label{thm:rulial-triangle} +\begin{theorem}[Main theorem on rulial distance (triangle inequality)]\label{thm:rulial-triangle} Assume: \begin{enumerate}[leftmargin=*] \item the description length $\mathrm{DL}$ is based on a prefix code @@ -146,7 +147,7 @@ \subsection{Translators, MDL Complexity, and Distortion} D_{\tau,m}(O_1,O_3) \le D_{\tau,m}(O_1,O_2) + D_{\tau,m}(O_2,O_3) + 2c. \] -In particular, together with \cref{lem:rulial-basic} this makes +In particular, together with \cref{thm:rulial-basic} this makes $D_{\tau,m}$ a quasi-pseudometric (a pseudometric up to additive slack $2c$) on observers. \end{theorem} diff --git a/aion-holography/titlepage.tex b/aion-holography/titlepage.tex index c2cf5be..b4da0c0 100644 --- a/aion-holography/titlepage.tex +++ b/aion-holography/titlepage.tex @@ -26,7 +26,8 @@ % AUTHOR BLOCK % --------------------------------------------------------- {\Large James Ross}\\[0.3em] -{\normalsize Independent Researcher}\\[2em] +{\normalsize Independent Researcher}\\ +{\normalsize\href{https://orcid.org/0009-0006-0025-7801}{ORCID: 0009-0006-0025-7801}}\\[2em] % --------------------------------------------------------- % DATE @@ -41,6 +42,10 @@ {\small This document is part of the \AIONInline{} Foundations Series.\\ See \AIONProjectURL{} for updates. +\\[0.6em] +\textcopyright{} 2025 James Ross. +Released under the Creative Commons Attribution 4.0 International +(CC BY 4.0) license: \url{https://creativecommons.org/licenses/by/4.0/}. } \end{center} diff --git a/aion-holography/sections/wormholes.tex b/aion-holography/wormholes.tex similarity index 100% rename from aion-holography/sections/wormholes.tex rename to aion-holography/wormholes.tex