From: Jon Awbrey <jawbrey@oakland.edu>

Date: Thu, 18 Jan 2001 16:00:01 -0500

Message-ID: <3A675951.3042A913@oakland.edu>

To: RDF Logic <www-rdf-logic@w3.org>

CC: Arisbe <arisbe@stderr.org>, Conceptual Graphs <cg@cs.uah.edu>, SemioCom <semiocom@listbot.com>

Date: Thu, 18 Jan 2001 16:00:01 -0500

Message-ID: <3A675951.3042A913@oakland.edu>

To: RDF Logic <www-rdf-logic@w3.org>

CC: Arisbe <arisbe@stderr.org>, Conceptual Graphs <cg@cs.uah.edu>, SemioCom <semiocom@listbot.com>

> A Note Comparing Conceptual Graphs & RDF/SemWeb & N3 > > From: Tim Berners-Lee (timbl@w3.org) > Date: Mon, Jan 08 2001 > > Message-ID: <025101c079b2$6e09a7c0$84001d12@w3.org> > From: "Tim Berners-Lee" <timbl@w3.org> > To: <www-rdf-logic@w3.org> > Date: Mon, 8 Jan 2001 15:34:42 -0500 > Subject: A note comparing Conceptual Graphs and RDF/Semantic Web & N3 > > I have just got hold of Sowa's spec > for Conceptual Graph Interchange Format, > and noted down the comparison bwteeen > CGs and the semantic web at > > http://www.w3.org/DesignIssues/CG > > for anyone who is interested, > or anyone from the CG community > who would like to verify that > it makes any sense. > > I have (for a change) remembered to > link it from my Design Issues overview > > http://www.w3.org/DesignIssues > > Tim ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ Tim, E-pimetheus is nagging me, and refuses to remove his e-nervating fishhook from my brain until I send out this afterthought incited by your instigatory query: In your "Reflection" that I e-spy at: http://www.w3.org/DesignIssues/CG you initially offer the opinion that: | I think the tendency of the CG examples to relate to | natural language rather than hard logic made it more | difficult for someone of my own leanings toward rigid | logical processing for the Sweb to understand what the | CG folks were driving at. Now, your "Reflection" appears to have corrected this impression -- almost e-mediately -- just a little bit further on its depictive imaging of the scene in question, but just in case there remains the lingering shadow of a doubt behind the silver of your screen, let me be so bold as to share with you this old animadversion of mine, recently revised for release to my world-wide, but limited and special for'a'that, audiovideance of willy-nilly-culted fans. Oh, insists Prometheus, one forethought to this afterthought: Please do not blame John Sowa for any of this "tired turism", as all of the io-box stuffing that follows devolves from one of my own personal and quite "idiosyntactic" extensions of what Peirce called the "Alpha Part" of his Logical Graphs, in effect, nothing more than "yet another way" (YAW) of doing Propositional Calculus or Sentential Logic. Tuit: ¤~~~~~~~~~¤~~~~~~~~~¤~CITATION~¤~~~~~~~~~¤~~~~~~~~~¤ Subj: Peirce's Cook's Turing Date: Mon, 15 Jan 2001 01:23:02 -0500 From: Jon Awbrey <jawbrey@oakland.edu> To: Arisbe <arisbe@stderr.org>, Conceptual Graphs <cg@cs.uah.edu>, SemioCom <semiocom@listbot.com> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ Arisbeans, Conceptual Graphists, SemioComrades, A certain momentum of thought compels me to continue this essay, in spite of every obstacle that has been tossed in its path. Though it arise from a labor of love, and thus in a way is sufficient to itself, its own reward, as they say, still it cannot be complete until it receives the reception that it seeks in the receipt-book of its due receiver, to coin a pharisee, and just because it is a labor of love does not mean that I would be able to see it lost with any measure of cool equanimity or respite of a phlegmatic temper. I am posting this result to the Arisbe and the CG Lists because of their common and continuing reference to the work of Peirce on Logical Graphs, but I imagine that my excuse to each of these two companies for attempting to engage their several attentions would have to be just a little bit different. But the near continuous offering of excuses for why "your own work" (YOW) might be worth the consideration of another living soul who happens to be laboring away in roughly the same domain of interest, as though your lifelong dedication to the task were not already some kind of silent testimony -- well, you hate that, too, don't you? -- so let us put it off for a day in the future, and spend the precious time that we have on what is more fun and more useful to boot. If nobody, after a reasonable interval, sees the excuse for it all abiding already in the process or in the product itself, then that will be the time for making our humble excuse. On to the work! ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤ By way of providing a simple illustration of Cook's Theorem, that "Propositional Satisfiability is NP-Complete", here is an exposition of one way to translate Turing Machine set-ups into propositional expressions, employing the RefLog Syntax for Prop Calc that I described in a couple of earlier notes: Notation: Stilt<k> = Space and Time Limited Turing Machine, with k units of space and k units of time. Stunt<k> = Space and Time Limited Turing Machine, for computing the parity of a bit string, with Number of Tape cells of input equal to k. I will follow the pattern of the discussion in the book of Herbert Wilf, 'Algorithms & Complexity' (1986), pages 188-201, but translate into RefLog, which is more efficient with respect to the number of propositional clauses that are required. Parity Machine: ------------------------------------------------------- | State | Symbol | Next Symbol | Ratchet | Next State | | Q | S | S' | dR | Q' | ------------------------------------------------------- | 0 | 0 | 0 | +1 | 0 | | 0 | 1 | 1 | +1 | 1 | | 0 | # | # | -1 | # | | 1 | 0 | 0 | +1 | 1 | | 1 | 1 | 1 | +1 | 0 | | 1 | # | # | -1 | * | ------------------------------------------------------- 1/1/+1 -------> /\ / \ /\ 0/0/+1 ^ 0 1 ^ 0/0/+1 \/|\ /|\/ | <------- | #/#/-1 | 1/1/+1 | #/#/-1 | | v v # * The TM has a "finite automaton" (FA) as its component. Let us refer to this particular FA by the name of "M". The "tape-head" (that is, the "read-unit") will be called "H". The "registers" are also called "tape-cells" or "tape-squares". In order to consider how the finitely "stilted" rendition of this TM can be translated into the form of a purely propositional description, one now fixes k and limits the discussion to talking about a Stilt<k>, which is really not a true TM anymore but a finite automaton in disguise. In this example, for the sake of a minimal illustration, we choose k = 2, and discuss Stunt<2>. Since the zeroth tape cell and the last tape cell are occupied with bof and eof marks "#", this amounts to only one digit of significant computation. To translate Stunt<2> into propositional form we use the following collection of propositional variables: For the "Present State Function" QF : P -> Q, { p0_q#, p0_q*, p0_q0, p0_q1, | p1_q#, p1_q*, p1_q0, p1_q1, | p2_q#, p2_q*, p2_q0, p2_q1, | p3_q#, p3_q*, p3_q0, p3_q1 } The propositional expression of the form "pi_qj" says: | At the point in time p<i>, | the finite machine M is in the state q<j>. For the "Present Register Function" RF : P -> R, { p0_r0, p0_r1, p0_r2, p0_r3, | p1_r0, p1_r1, p1_r2, p1_r3, | p2_r0, p2_r1, p2_r2, p2_r3, | p3_r0, p3_r1, p3_r2, p3_r3 } The propositional expression of the form "pi_rj" says: | At the point in time p<i>, | the tape-head H is on the tape-cell r<j>. For the "Present Symbol Function" SF : P -> (R -> S), { p0_r0_s#, p0_r0_s*, p0_r0_s0, p0_r0_s1, | p0_r1_s#, p0_r1_s*, p0_r1_s0, p0_r1_s1, | p0_r2_s#, p0_r2_s*, p0_r2_s0, p0_r2_s1, | p0_r3_s#, p0_r3_s*, p0_r3_s0, p0_r3_s1, | p1_r0_s#, p1_r0_s*, p1_r0_s0, p1_r0_s1, | p1_r1_s#, p1_r1_s*, p1_r1_s0, p1_r1_s1, | p1_r2_s#, p1_r2_s*, p1_r2_s0, p1_r2_s1, | p1_r3_s#, p1_r3_s*, p1_r3_s0, p1_r3_s1, | p2_r0_s#, p2_r0_s*, p2_r0_s0, p2_r0_s1, | p2_r1_s#, p2_r1_s*, p2_r1_s0, p2_r1_s1, | p2_r2_s#, p2_r2_s*, p2_r2_s0, p2_r2_s1, | p2_r3_s#, p2_r3_s*, p2_r3_s0, p2_r3_s1, | p3_r0_s#, p3_r0_s*, p3_r0_s0, p3_r0_s1, | p3_r1_s#, p3_r1_s*, p3_r1_s0, p3_r1_s1, | p3_r2_s#, p3_r2_s*, p3_r2_s0, p3_r2_s1, | p3_r3_s#, p3_r3_s*, p3_r3_s0, p3_r3_s1 } The propositional expression of the form "pi_rj_sk" says: | At the point in time p<i>, | the tape-cell r<j> bears the mark s<k>." The propositional description of a Stunt<2> which starts with the following tape contents: {"#" in cell 0, "0" in cell 1, "#" in cell 2} is the conjunction of the following clauses: ¤~~~~~~~~~¤~~~~~~~~~¤~PROPOSITION~¤~~~~~~~~~¤~~~~~~~~~¤ Initial Conditions: p0_q0 p0_r1 p0_r0_s# p0_r1_s0 p0_r2_s# Mediate Conditions: ( p0_q# ( p1_q# )) ( p0_q* ( p1_q* )) ( p1_q# ( p2_q# )) ( p1_q* ( p2_q* )) Terminal Conditions: (( p2_q# )( p2_q* )) State Partition: (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* )) (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* )) (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* )) Register Partition: (( p0_r0 ),( p0_r1 ),( p0_r2 )) (( p1_r0 ),( p1_r1 ),( p1_r2 )) (( p2_r0 ),( p2_r1 ),( p2_r2 )) Symbol Partition: (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# )) (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# )) (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# )) (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# )) (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# )) (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# )) (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# )) (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# )) (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# )) Interaction Conditions: (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 )) (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 )) (( p0_r0 ) p0_r0_s# ( p1_r0_s# )) (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 )) (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 )) (( p0_r1 ) p0_r1_s# ( p1_r1_s# )) (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 )) (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 )) (( p0_r2 ) p0_r2_s# ( p1_r2_s# )) (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 )) (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 )) (( p1_r0 ) p1_r0_s# ( p2_r0_s# )) (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 )) (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 )) (( p1_r1 ) p1_r1_s# ( p2_r1_s# )) (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 )) (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 )) (( p1_r2 ) p1_r2_s# ( p2_r2_s# )) Transition Relations: ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 )) ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 )) ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# )) ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# )) ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 )) ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 )) ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# )) ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# )) ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 )) ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 )) ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# )) ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# )) ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 )) ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 )) ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) ¤~~~~~~~~~¤~~~~~~~~~¤~NOITISOPORP~¤~~~~~~~~~¤~~~~~~~~~¤ Interpretation of the Propositional Description: Initial Conditions: p0_q0 p0_r1 p0_r0_s# p0_r1_s0 p0_r2_s# The Initial Conditions are given by a logical conjunction that is composed of 5 basic expressions, altogether stating: | At p<0>, M is in the state q<0>, and | At p<0>, H is on the cell r<1>, and | At p<0>, cell r<0> bears the mark "#", and | At p<0>, cell r<1> bears the mark "0", and | At p<0>, cell r<2> bears the mark "#". Mediate Conditions: ( p0_q# ( p1_q# )) ( p0_q* ( p1_q* )) ( p1_q# ( p2_q# )) ( p1_q* ( p2_q* )) In RefLog, an expression of the form "( X ( Y ))" expresses an implication or an if-then proposition: "Not X without Y", "If X then Y", "X => Y", etc. A text string expression of the form "( X ( Y ))" parses to a graphical data-structure of the form: X Y o---o | @ All together, these Mediate Conditions state: | If at p<0> M is in state q<#>, then at p<1> M is in state q<#>, and | If at p<0> M is in state q<*>, then at p<1> M is in state q<*>, and | If at p<1> M is in state q<#>, then at p<2> M is in state q<#>, and | If at p<1> M is in state q<*>, then at p<2> M is in state q<*>. Terminal Conditions: (( p2_q# )( p2_q* )) In RefLog, an expression of the form "(( X )( Y ))" expresses a disjunction "X or Y", and it parses to: X Y o o \ / o | @ In effect, the Terminal Conditions state: | At p<2> M is in state q<#>, or | At p<2> M is in state q<*>. State Partition: (( p0_q0 ),( p0_q1 ),( p0_q# ),( p0_q* )) (( p1_q0 ),( p1_q1 ),( p1_q# ),( p1_q* )) (( p2_q0 ),( p2_q1 ),( p2_q# ),( p2_q* )) In RefLog, an expression of the form "(( X<1> ),( X<2> ),( ... ),( X<k> ))" states it as a fact that "exactly one of the X<j> are true, for j = 1 to k". Expressions of this form are called "universal partition" expressions, and they parse into a type of graph called a "painted and rooted cactus" (PARC): X<1> X<2> ... X<k> o o o | | | o-----o--- ... ---o \ / \ / \ / \ / \ / \ / \ / \ / @ The State Partition expresses the conditions that: | At each of the points-in-time p<i>, for i = 0 to 2, | M can be in exactly one state q<j>, for j in the set {0, 1, #, *}. Register Partition: (( p0_r0 ),( p0_r1 ),( p0_r2 )) (( p1_r0 ),( p1_r1 ),( p1_r2 )) (( p2_r0 ),( p2_r1 ),( p2_r2 )) The Register Partition expresses the conditions that: | At each of the points-in-time p<i>, for i = 0 to 2, | H can be on exactly one cell r<j>, for j = 0 to 2. Symbol Partition: (( p0_r0_s0 ),( p0_r0_s1 ),( p0_r0_s# )) (( p0_r1_s0 ),( p0_r1_s1 ),( p0_r1_s# )) (( p0_r2_s0 ),( p0_r2_s1 ),( p0_r2_s# )) (( p1_r0_s0 ),( p1_r0_s1 ),( p1_r0_s# )) (( p1_r1_s0 ),( p1_r1_s1 ),( p1_r1_s# )) (( p1_r2_s0 ),( p1_r2_s1 ),( p1_r2_s# )) (( p2_r0_s0 ),( p2_r0_s1 ),( p2_r0_s# )) (( p2_r1_s0 ),( p2_r1_s1 ),( p2_r1_s# )) (( p2_r2_s0 ),( p2_r2_s1 ),( p2_r2_s# )) The Symbol Partition expresses the conditions that: | At each of the points-in-time p<i>, for i in {0, 1, 2}, | In each of the tape-registers r<j>, for j in {0, 1, 2}, | There can be exactly one sign s<k>, for k in {0, 1, #}. Interaction Conditions: (( p0_r0 ) p0_r0_s0 ( p1_r0_s0 )) (( p0_r0 ) p0_r0_s1 ( p1_r0_s1 )) (( p0_r0 ) p0_r0_s# ( p1_r0_s# )) (( p0_r1 ) p0_r1_s0 ( p1_r1_s0 )) (( p0_r1 ) p0_r1_s1 ( p1_r1_s1 )) (( p0_r1 ) p0_r1_s# ( p1_r1_s# )) (( p0_r2 ) p0_r2_s0 ( p1_r2_s0 )) (( p0_r2 ) p0_r2_s1 ( p1_r2_s1 )) (( p0_r2 ) p0_r2_s# ( p1_r2_s# )) (( p1_r0 ) p1_r0_s0 ( p2_r0_s0 )) (( p1_r0 ) p1_r0_s1 ( p2_r0_s1 )) (( p1_r0 ) p1_r0_s# ( p2_r0_s# )) (( p1_r1 ) p1_r1_s0 ( p2_r1_s0 )) (( p1_r1 ) p1_r1_s1 ( p2_r1_s1 )) (( p1_r1 ) p1_r1_s# ( p2_r1_s# )) (( p1_r2 ) p1_r2_s0 ( p2_r2_s0 )) (( p1_r2 ) p1_r2_s1 ( p2_r2_s1 )) (( p1_r2 ) p1_r2_s# ( p2_r2_s# )) In briefest terms, the Interaction Conditions merely express the circumstance that the sign in a tape-cell cannot change between two points-in-time unless the tape-head is over the cell in question at the initial one of those points-in-time. All that we have to do is to see how they manage to say this. In RefLog, an expression of the following form: "(( p<i>_r<j> ) p<i>_r<j>_s<k> ( p<i+1>_r<j>_s<k> ))", and which parses as the graph: p<i>_r<j> o o p<i+1>_r<j>_s<k> \ / p<i>_r<j>_s<k> o | @ can be read in the form of the following implication: | If | At the point-in-time p<i>, the tape-cell r<j> bears the mark s<k>, | But it is not the case that | At the point-in-time p<i>, the tape-head is on the tape-cell r<j>. | Then | At the point-in-time p<i+1>, the tape-cell r<j> bears the mark s<k>. Folks among us of a certain age and a peculiar manner of acculturation will recognize these as the "Frame Conditions" for the change of state of the TM. Transition Relations: ( p0_q0 p0_r1 p0_r1_s0 ( p1_q0 p1_r2 p1_r1_s0 )) ( p0_q0 p0_r1 p0_r1_s1 ( p1_q1 p1_r2 p1_r1_s1 )) ( p0_q0 p0_r1 p0_r1_s# ( p1_q# p1_r0 p1_r1_s# )) ( p0_q0 p0_r2 p0_r2_s# ( p1_q# p1_r1 p1_r2_s# )) ( p0_q1 p0_r1 p0_r1_s0 ( p1_q1 p1_r2 p1_r1_s0 )) ( p0_q1 p0_r1 p0_r1_s1 ( p1_q0 p1_r2 p1_r1_s1 )) ( p0_q1 p0_r1 p0_r1_s# ( p1_q* p1_r0 p1_r1_s# )) ( p0_q1 p0_r2 p0_r2_s# ( p1_q* p1_r1 p1_r2_s# )) ( p1_q0 p1_r1 p1_r1_s0 ( p2_q0 p2_r2 p2_r1_s0 )) ( p1_q0 p1_r1 p1_r1_s1 ( p2_q1 p2_r2 p2_r1_s1 )) ( p1_q0 p1_r1 p1_r1_s# ( p2_q# p2_r0 p2_r1_s# )) ( p1_q0 p1_r2 p1_r2_s# ( p2_q# p2_r1 p2_r2_s# )) ( p1_q1 p1_r1 p1_r1_s0 ( p2_q1 p2_r2 p2_r1_s0 )) ( p1_q1 p1_r1 p1_r1_s1 ( p2_q0 p2_r2 p2_r1_s1 )) ( p1_q1 p1_r1 p1_r1_s# ( p2_q* p2_r0 p2_r1_s# )) ( p1_q1 p1_r2 p1_r2_s# ( p2_q* p2_r1 p2_r2_s# )) The Transition Conditions merely serve to express, by means of 16 complex implication expressions, the data of the TM table that was given above. ¤~~~~~~~~~¤~~~~~~~~~¤~NOITATIC~¤~~~~~~~~~¤~~~~~~~~~¤ I am still in the process of de-tangling these bits and pieces and odds and ends and notions of threads, but till I do, here is a sampler of linkages that may serve to explain a few of the ideas that were not covered in any suitably self-contained fashion above: ¤~~~~~~~~~¤~~~~~~~~~¤~REFERENCES~¤~~~~~~~~~¤~~~~~~~~~¤ Running Accumulation of Reference Materials: Extensions of Logical Graphs: http://www.virtual-earth.de/CG/cg-list/msg03351.html http://www.virtual-earth.de/CG/cg-list/msg03352.html http://www.virtual-earth.de/CG/cg-list/msg03353.html http://www.virtual-earth.de/CG/cg-list/msg03354.html http://www.virtual-earth.de/CG/cg-list/msg03376.html http://www.virtual-earth.de/CG/cg-list/msg03379.html http://www.virtual-earth.de/CG/cg-list/msg03381.html In Praise of Zeroth-Order Logic: http://ltsc.ieee.org/logs/suo/msg01406.html http://ltsc.ieee.org/logs/suo/msg01546.html http://ltsc.ieee.org/logs/suo/msg01561.html http://ltsc.ieee.org/logs/suo/msg01670.html http://ltsc.ieee.org/logs/suo/msg01966.html http://ltsc.ieee.org/logs/suo/msg01985.html http://ltsc.ieee.org/logs/suo/msg01988.html Signs, Information, Logic, Inquiry: http://ltsc.ieee.org/logs/suo/msg02534.html http://ltsc.ieee.org/logs/suo/msg02554.html http://ltsc.ieee.org/logs/suo/msg02570.html http://ltsc.ieee.org/logs/suo/msg02590.html All Liar, No Paradox: http://ltsc.ieee.org/logs/suo/msg01739.html Analytic Differential Ontology (ADO): http://grouper.ieee.org/groups/ltsc/logs/ontology/msg00072.html Shroud of Turing: http://ltsc.ieee.org/logs/suo/msg02714.html http://www.egroups.com/message/theory-edge/2295 http://www.virtual-earth.de/CG/cg-list/msg03669.html Peirce's Cook's Turing: http://www.virtual-earth.de/CG/cg-list/msg03677.html http://stderr.org/pipermail/arisbe/2001-January/000167.html http://semiocom.listbot.com/cgi-bin/subscriber?Act=view_message&list_id=semiocom&msg_num=673&start_num= Painted Cacti & Propositional Calculus: http://www.egroups.com/message/theory-edge/2313 http://stderr.org/pipermail/arisbe/2001-January/000150.html For The Sake Of The Ultimate Reckoning Graph Engine: http://stderr.org/pipermail/arisbe/2001-January/000168.html http://semiocom.listbot.com/cgi-bin/subscriber?Act=view_message&list_id=semiocom&msg_num=679&start_num= Where I Hang Out Most E-joyfully (WIHOME): ><> ><> ><> ><> ><> ><> ><> ><> ><> ><> ><> ><> ><> http://www.medialab.chalmers.se/fishcam/fish.html <>< <>< <>< <>< <>< <>< <>< <>< <>< <>< <>< <>< <>< ¤~~~~~~~~~¤~~~~~~~~~¤~SECNEREFER~¤~~~~~~~~~¤~~~~~~~~~¤ E-joy! Jon Awbrey ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤Received on Thursday, 18 January 2001 16:02:29 UTC

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