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zzZb32|Mf7c{*XOazR1K66|3GE6^1SP$;ZD`$o@=WkQ4WI(M;i#<)-tMp9S|_jS4#& zGPhB5%FZ&T?SWSXBrt)5)sN?WFK@o5## z(>kNl-KV|cKf|<8uamdmN{TL(FgkrcYrjG=-$frSrBIV)32Qi4n}r>ER`l{p*v89Y z5i_64YOR}}uA0T$&rlK`p2@rUz$d=N66?HvWw-4ukuT8~Ek0bAdQkB6Wz#G7-9Db{wj4_UAg!(>`TmxMb;* h@VJVHR^hWxE_@{}uIG~5knJbW`^xxj|EaA1HvztsRn-6h literal 0 HcmV?d00001 diff --git a/trees/loc-001B.tree b/trees/loc-001B.tree index 7bd60e0..2805710 100644 --- a/trees/loc-001B.tree +++ b/trees/loc-001B.tree @@ -10,6 +10,7 @@ \put\transclude/expanded{false} \p{This page has an [atom feed](/forest/loc-001B/atom.xml).} +\transclude{2026-W31} \transclude{2026-W30} \transclude{2026-W29} \transclude{2026-W26} diff --git a/trees/loc-0051.tree b/trees/loc-0051.tree index ac0f892..0ed59fa 100644 --- a/trees/loc-0051.tree +++ b/trees/loc-0051.tree @@ -1,5 +1,13 @@ \import{table-macros} -\title{CCS with a Termination predicate} +\title{CCS with Termination} \date{2026-07-25} \author{liamoc} +\p{As part of [[xinloi]]'s project we needed to come up with a version of CCS that supports explicit termination in the style of ACP. There is a [paper](aceto-hennessy-1994) by [[aceto]] and [[mhennessy]] which introduces a termination predicate, but they require that a nondeterministic choice #{P + Q} is judged terminating (written #{P + Q \checkmark}) only when \em{both} #{P \checkmark} and #{Q \checkmark}. In other words, a process is only terminating if it has \em{no choice} but to terminate. } +\p{In our case, we want a judgement for \em{possible} termination, not \em{guaranteed} termination. So we had to come up with our own notions. We've formalised this in Isabelle, but what follows is the same but presented as Forester notes. Eventually, these notes may become part of another research notebook or even lecture notes. } +\transclude{pa-0001} +\transclude{pa-0002} +\transclude{pa-0003} +\transclude{pa-0004} +\transclude{pa-0005} + diff --git a/trees/loc-0053.tree b/trees/loc-0053.tree new file mode 100644 index 0000000..2819618 --- /dev/null +++ b/trees/loc-0053.tree @@ -0,0 +1,32 @@ +\import{table-macros} +\def\percent{\startverb%\stopverb + } +\parent{loc-000P} +\title{10th Sunday after Pentecost 2026} +\tag{cmc} +\date{2026-08-02} +\author{liamoc} +\quote{ +Deus in adiutórium meum inténde: Dómine ad adiuvándum me festína: confundántur et revereántur inimíci mei, qui quærunt ánimam meam. +} +\p{I was once again the director of our choir at [All Saints Ainslie](https://allsaintsainslie.org.au), for the special \em{Missing Persons Sunday} today, with representatives from the Australian Federal Police present to offer special reflections and awareness about those who are missing in our territory and support their loved ones, many of whom were in attendance. We performed Rachmaninoff's [Bogoroditse Devo](https://www.youtube.com/watch?v=InMhSNBlIg8):} +\quote{ + \table{ + \tr{ + \td{Богородице Дево, радуйся,} + \td{Rejoice, virgin mother of God,} + }\tr{ + \td{Благодатная Марие, Господь с тобою.} + \td{Mary, full of grace, the Lord is with you.} + }\tr{ + \td{Благословена ты в женах,} + \td{Blessed are you among women,} + }\tr{ + \td{и благословен плод чрева твоего,} + \td{and blessed is the fruit of your womb,} + }\tr{ + \td{яко Спаса родила еси душ наших. } + \td{for you have borne the Saviour of our souls. } + } + } +} \ No newline at end of file diff --git a/trees/pa/pa-0001.tree b/trees/pa/pa-0001.tree new file mode 100644 index 0000000..3d4d9b0 --- /dev/null +++ b/trees/pa/pa-0001.tree @@ -0,0 +1,22 @@ +\import{table-macros} +\import{dt-macros} +\title{Syntax of Basic CCS} +\taxon{Definition} +\author{liamoc} +\p{The \em{Calculus of Communicating Systems} (CCS) is a process algebra with the following basic syntax.} +##{ + \begin{array}{lcll} + \alpha & & & \text{(names)}\\ + a & ::= & \alpha \mid \overline{\alpha} \mid \tau & \text{(actions, co-actions, and internal moves)}\\ + P & ::= & \textbf{0} & \\ + & \mid & a.P & \text{(action prefixing)} \\ + & \mid & \sum_{i \in I} P_i & \text{(nondeterministic choice)} \\ + & \mid & P | Q & \text{(parallel composition)} \\ + & \mid & P \setminus \alpha & \text{(restriction)} \\ + & \mid & P \lsquare f \rsquare & \text{(renaming)} + \end{array} +} +\p{This is called \em{Basic} CCS because recursion is not included. In the above, #{f} denotes a valid \em{renaming} function that satisfies the side-condition #{\forall a.\; f(\overline{a}) = \overline{f(a)}}. Here we have naturally extended the co-action notation (#{\overline{\alpha}}) to an operation on actions #{a}, where #{\overline{\overline{a}} = a} and #{\overline{\tau} = \tau}.} +\p{We allow for infinite sums in nondeterministic choice. To recover finite choice, we use a two element index set like #{\mathbb{B} = \{\top,\bot\}}, as follows:} +##{ P + Q \equiv \sum_{i \in \mathbb{B}} R_i\; \text{where}\ R_\top = P\ \text{and}\ R_\bot = Q} + diff --git a/trees/pa/pa-0002.tree b/trees/pa/pa-0002.tree new file mode 100644 index 0000000..07a71d9 --- /dev/null +++ b/trees/pa/pa-0002.tree @@ -0,0 +1,24 @@ +\import{table-macros} +\import{dt-macros} +\title{Operational semantics for Basic CCS} +\taxon{Definition} +\author{liamoc} +\def\xarrow[body]{\xrightarrow{\body}} +\def\xarrowstar[body]{\xrightarrow{\body}^\ast} + +\p{We assign an \todo{operational semantics} to [Basic CCS](pa-0001) by describing a series of transition rules #{P \xarrow{a} Q} inductively as follows:} +##{ + \begin{gather*} + \inferrule{ }{a.P \xarrow{a} P} \quad \inferrule{P_k \xarrow{a} P'}{\sum_{i \in I} P_i \xarrow{a} P'} \\[2em] + \inferrule{ P \xarrow{a} P'}{P|Q \xarrow{a} P'|Q} \quad \inferrule{ Q \xarrow{a} Q'}{P|Q \xarrow{a} P|Q'} + \inferrule{ P \xarrow{a} P' \quad Q \xarrow{\overline{a}} Q'}{P|Q \xarrow{\tau} P'|Q'} \\[2em] + \inferrule{ P \xarrow{a} P' \quad a \notin \{\alpha, \overline{\alpha}\}}{ P\setminus \alpha \xarrow{a} P' \setminus \alpha} \quad \inferrule{ P \xarrow{a} P'}{ P \lsquare f \rsquare \xarrow{f(a)} P' \lsquare f \rsquare} + \end{gather*} +} +\p{Let the relation #{P \xarrowstar{\sigma} Q} denote the [reflexive](dm-0001), [transitive](dm-0002) closure of the above step relation, where #{\sigma} denotes a \em{finite trace} (a finite sequence of actions):} +##{ + \inferrule{ }{P \xarrowstar{\varepsilon} P} \quad \inferrule{P \xarrow{a} P' \quad P' \xarrowstar{\sigma} Q}{P \xarrowstar{a\sigma} Q} +} + + + diff --git a/trees/pa/pa-0003.tree b/trees/pa/pa-0003.tree new file mode 100644 index 0000000..bb35136 --- /dev/null +++ b/trees/pa/pa-0003.tree @@ -0,0 +1,27 @@ +\import{table-macros} +\import{dt-macros} +\title{Partial trace semantics for Basic CCS} +\taxon{Definition} +\author{liamoc} +\def\PTsems[body]{\sems{\body}_\text{PT}} +\p{A simple [denotational semantics](dt-0001) for [Basic CCS](pa-0001) is in terms of partial traces: finite sequences of actions.} +##{ + \begin{array}{lcl} + \PTsems{\textbf{0}} & = & \{ \varepsilon \} \\ + \PTsems{a.P} & = & \{ a\sigma \mid \sigma \in \PTsems{P} \} \cup \{ \varepsilon \}\\ + \PTsems{\sum_{i \in I} P_i} & = & \{ \varepsilon \} \cup \bigcup_{i \in I} \PTsems{P_i} \\ + \PTsems{P|Q} & = & \bigcup_{\sigma \in \PTsems{P}} \bigcup_{\rho \in \PTsems{Q}} \sigma \bowtie \rho \\ + \PTsems{P \setminus \alpha} & = & \{ \sigma \in \PTsems{P} \mid \alpha \notin \sigma \land \overline{\alpha} \notin \sigma \} \\ + \PTsems{P \lsquare f \rsquare} & = & \{ \hat{f}(\sigma) \mid \sigma \in \PTsems{P} \} + \end{array} +} +\p{The trace merge operator #{\bowtie} is defined as follows:} +##{\begin{array}{lcl} + \sigma \bowtie \varepsilon &=& \{\sigma \} \\ + \varepsilon \bowtie \sigma &=& \{\sigma \} \\ + a\sigma \bowtie b\rho &=& \{ a\varsigma \mid \varsigma \in \sigma \bowtie b\rho \} \\ + &\cup& \{ b\varsigma \mid \varsigma \in a\sigma \bowtie \rho \} \\ + &\cup& \{ \tau\varsigma \mid \varsigma \in \sigma \bowtie \rho \land a = \overline{b} \} \\ +\end{array} +} + diff --git a/trees/pa/pa-0004.tree b/trees/pa/pa-0004.tree new file mode 100644 index 0000000..ccda4e4 --- /dev/null +++ b/trees/pa/pa-0004.tree @@ -0,0 +1,15 @@ +\import{table-macros} +\import{dt-macros} +\title{Soundness and adequacy of partial trace semantics for Basic CCS} +\taxon{Theorem} +\author{liamoc} +\def\xarrow[body]{\xrightarrow{\body}} +\def\xarrowstar[body]{\xrightarrow{\body}^\ast} + +\p{The finite partial trace [denotational semantics of Basic CCS](dt-0003) gives exactly those finite partial traces reachable via the [operational semantics](dt-0002).} +##{\sems{P}_\text{PT} = \{ \sigma \mid \exists Q.\ P \xarrowstar{\sigma} Q \} } + +\proofblock{ + \p{Proved by induction on #{P}.} +} + diff --git a/trees/pa/pa-0005.tree b/trees/pa/pa-0005.tree new file mode 100644 index 0000000..9a6eace --- /dev/null +++ b/trees/pa/pa-0005.tree @@ -0,0 +1,13 @@ +\import{table-macros} +\import{dt-macros} +\title{Explicit termination for Basic CCS} +\author{liamoc} +\def\xarrow[body]{\xrightarrow{\body}} +\def\xarrowstar[body]{\xrightarrow{\body}^\ast} +\p{We add to [syntax of Basic CCS](pa-0001) an additional process constant, #{\textbf{1}}, which, like #{\mathbf{0}}, has no rules in the [operational semantics](dt-0002), and whose [finite partial traces](dt-0003) are just #{\sems{\textbf{1}}_\text{PT} = \{\varepsilon\}}. } +\p{The difference between #{\textbf{0}} and #{\textbf{1}} only comes in once we introduce a notion of \em{possible successful termination}. Intuitively, we want the ability to denote when it is OK for a process to stop.} +\transclude{pa-0006} +\transclude{pa-0007} +\transclude{pa-0008} +\transclude{pa-0009} +\transclude{pa-000A} \ No newline at end of file diff --git a/trees/pa/pa-0006.tree b/trees/pa/pa-0006.tree new file mode 100644 index 0000000..34efb90 --- /dev/null +++ b/trees/pa/pa-0006.tree @@ -0,0 +1,14 @@ +\import{table-macros} +\import{dt-macros} +\title{Possible termination predicate for Basic CCS with #{\textbf{1}}} +\taxon{Definition} +\author{liamoc} + +\p{We define an inductive judgement #{P \checkmark}, where #{P} is a process in [Basic CCS](pa-0001) augmented with an additional process constant #{\textbf{1}}:} +##{ + \begin{gather*} + \inferrule{ }{ \textbf{1} \checkmark} \quad \inferrule{P_k \checkmark}{\sum_{i \in I} P_i \checkmark} \\[2em] + \inferrule{ P \checkmark \quad Q \checkmark}{P|Q \checkmark} + \inferrule{ P \checkmark}{ P\setminus \alpha \checkmark} \quad \inferrule{ P \checkmark}{ P \lsquare f \rsquare \checkmark} + \end{gather*} +} diff --git a/trees/pa/pa-0007.tree b/trees/pa/pa-0007.tree new file mode 100644 index 0000000..7a1323c --- /dev/null +++ b/trees/pa/pa-0007.tree @@ -0,0 +1,19 @@ +\import{table-macros} +\import{dt-macros} +\title{Finite completed trace semantics for Basic CCS with #{\textbf{1}}} +\taxon{Definition} +\author{liamoc} +\def\CTsems[body]{\sems{\body}_\text{CT}} +\p{We offer a [denotational semantics](dt-0001) for [Basic CCS](pa-0001) augmented with a #{\textbf{1}} constant, denoting successful termination. Unlike our [partial trace semantics](pa-0003), this semantics only includes finite \em{completed} traces — traces where the process has terminated.} +##{ + \begin{array}{lcl} + \CTsems{\textbf{0}} & = & \emptyset \\ + \CTsems{\textbf{1}} & = & \{ \varepsilon \} \\ + \CTsems{a.P} & = & \{ a\sigma \mid \sigma \in \CTsems{P} \}\\ + \CTsems{\sum_{i \in I} P_i} & = & \bigcup_{i \in I} \CTsems{P_i} \\ + \CTsems{P|Q} & = & \bigcup_{\sigma \in \CTsems{P}} \bigcup_{\rho \in \CTsems{Q}} \sigma \bowtie \rho \\ + \CTsems{P \setminus \alpha} & = & \{ \sigma \in \CTsems{P} \mid \alpha \notin \sigma \land \overline{\alpha} \notin \sigma \} \\ + \CTsems{P \lsquare f \rsquare} & = & \{ \hat{f}(\sigma) \mid \sigma \in \CTsems{P} \} + \end{array} +} +\p{The trace merge operator #{\bowtie} is defined exactly as in the [partial trace semantics](pa-0003). } diff --git a/trees/pa/pa-0008.tree b/trees/pa/pa-0008.tree new file mode 100644 index 0000000..ce1e034 --- /dev/null +++ b/trees/pa/pa-0008.tree @@ -0,0 +1,15 @@ +\import{table-macros} +\import{dt-macros} +\title{Soundness and adequacy of finite completed trace semantics for Basic CCS} +\taxon{Theorem} +\author{liamoc} +\def\xarrow[body]{\xrightarrow{\body}} +\def\xarrowstar[body]{\xrightarrow{\body}^\ast} + +\p{The [finite completed trace semantics of Basic CCS](dt-0007) gives exactly those finite partial traces reachable via the [operational semantics](dt-0002) that end in a [terminating state](pa-0006).} +##{\sems{P}_\text{CT} = \{ \sigma \mid \exists Q.\ P \xarrowstar{\sigma} Q \land Q \checkmark \} } + + +\proofblock{ + \p{Proved by induction on #{P}.} +} \ No newline at end of file diff --git a/trees/pa/pa-0009.tree b/trees/pa/pa-0009.tree new file mode 100644 index 0000000..f055fce --- /dev/null +++ b/trees/pa/pa-0009.tree @@ -0,0 +1,15 @@ +\import{table-macros} +\import{dt-macros} +\title{A translation to turn partial traces into completed traces} +\taxon{Definition} +\author{liamoc} +\p{Define a syntactic transformation on [Basic CCS terms](pa-0001) (augmented with #{\textbf{1}}) #{\Upsilon} as follows:} +##{\begin{array}{lcl} +\Upsilon(\textbf{0}) &=& \textbf{1} \\ +\Upsilon(\textbf{1}) &=& \textbf{1} \\ +\Upsilon(a.P) &=& \textbf{1} + a.\Upsilon(P) \\ +\Upsilon(\sum_{i \in I} P_i) &=& \textbf{1} + \sum_{i \in I} \Upsilon(P_i) \\ +\Upsilon(P | Q) &=& \Upsilon(P) | \Upsilon(Q)\\ +\Upsilon(P \setminus \alpha) &=& \Upsilon(P) \setminus \alpha \\ +\Upsilon(P \lsquare f \rsquare) &=& \Upsilon(P) \lsquare f \rsquare +\end{array}} diff --git a/trees/pa/pa-000A.tree b/trees/pa/pa-000A.tree new file mode 100644 index 0000000..88cd8d2 --- /dev/null +++ b/trees/pa/pa-000A.tree @@ -0,0 +1,9 @@ +\import{table-macros} +\import{dt-macros} +\title{Correctness of the #{\Upsilon} translation} +\taxon{Theorem} +\author{liamoc} +\p{Our syntactic transformation #{\Upsilon} gives a process whose [completed traces](pa-0007) are the same as the [partial traces](pa-0003) of the original process.} +##{\sems{P}_\text{PT} = \sems{\Upsilon(P)}_\text{CT}} +\proofblock{ + \p{A very straightforward structural induction on #{P}, where each case follows from the definitions.}} \ No newline at end of file diff --git a/trees/people/aceto.tree b/trees/people/aceto.tree new file mode 100644 index 0000000..b496d84 --- /dev/null +++ b/trees/people/aceto.tree @@ -0,0 +1,6 @@ +\title{Luca Aceto} +\taxon{Person} +\meta{external}{https://staff.ru.is/luca/} +\meta{institution}{[[ru]]} +\meta{orcid}{0000-0002-2197-3018} +\meta{position}{Professor} diff --git a/trees/places/inco.tree b/trees/places/inco.tree new file mode 100644 index 0000000..456c40c --- /dev/null +++ b/trees/places/inco.tree @@ -0,0 +1,6 @@ + +\title{Information and Computation} +\taxon{Journal} +\meta{external}{https://www.sciencedirect.com/journal/information-and-computation} + +\p{Information and Computation welcomes original papers in all areas of theoretical computer science and computational applications of information theory. Survey articles of exceptional quality will also be considered.} \ No newline at end of file diff --git a/trees/refs/aceto-hennessy-1994.tree b/trees/refs/aceto-hennessy-1994.tree new file mode 100644 index 0000000..88edf36 --- /dev/null +++ b/trees/refs/aceto-hennessy-1994.tree @@ -0,0 +1,8 @@ +\author{aceto} +\author{mhennessy} +\taxon{Reference} +\meta{venue}{[[inco]], Volume 115, Issue 2} +\tag{refereed} +\date{1994-12} +\meta{doi}{10.1006/inco.1994.1096} +\title{Adding Action Refinement to a Finite Process Algebra} \ No newline at end of file diff --git a/trees/table-macros.tree b/trees/table-macros.tree index 46c866f..7a01a31 100644 --- a/trees/table-macros.tree +++ b/trees/table-macros.tree @@ -28,4 +28,5 @@ \def\hr{ \{} } +\def\todo[body]{\body} \def\kbd[body]{\{\body}} \ No newline at end of file diff --git a/trees/weeknotes/2026-W31.tree b/trees/weeknotes/2026-W31.tree new file mode 100644 index 0000000..ebf1ee3 --- /dev/null +++ b/trees/weeknotes/2026-W31.tree @@ -0,0 +1,12 @@ +\import{table-macros} +\title{Weeknotes 2026-W31} +\author{liamoc} +\date{2026-08-04} +\p{This weeknotes is two days late, but I was suffering from a bout of rheumatoid symptoms so that's my excuse.} +\figure{ + \[loading]{lazy}[width]{400px}[src]{\route-asset{assets/cocky.jpeg}}%{} + \figcaption{One of the most common birds in Canberra is also one of the loveliest.} +} +\transclude{loc-0051} +\transclude{loc-0053} +

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