Jon Awbrey · @Inquiry
87 followers · 313 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 3.1
• https://inquiryintoinquiry.com/2022/09/03/theme-one-program-jets-and-sharks-3/
Using #CactusGraphs or #MinimalNegations to implement pools of #MutuallyInhibitoryNeurons builds #NeuralArchitectures on substantially different foundations from current connectionist models. At a high level of abstraction, though, there is enough homology between the two orders to compare their performance on many of the same tasks. Thus I was able to try the program on several examples suggested by McClelland & Rumelhart.
#logic #neuralarchitectures #mutuallyinhibitoryneurons #minimalnegations #CactusGraphs #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
87 followers · 306 posts · Server mathstodon.xyz#ThemeOneProgram • #JetsAndSharks 3
• https://inquiryintoinquiry.com/2022/09/03/theme-one-program-jets-and-sharks-3/
Given a representation of the Jets & Sharks universe in computer memory, we naturally want to see if the memory serves to supply the facts a well-constructed #DataBase should.
In their #PDPHandbook presentation of the Jets & Sharks example, #McClelland & #Rumelhart give several exercises for exploring the use of their #NeuralPool #MemoryModel on tasks of #Retrieval & #Generalization.
Exercise 2.1
• https://web.stanford.edu/group/pdplab/pdphandbook/handbookch3#x7-330001
#logic #generalization #retrieval #memorymodel #neuralpool #Rumelhart #McClelland #PDPHandbook #database #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
85 followers · 292 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 2.1
• https://inquiryintoinquiry.com/2022/08/30/theme-one-program-jets-and-sharks-2/
Our #CactusGraph bears a vocabulary of \(41\) #LogicalTerms, each denoting a #BooleanVariable, so our proposition, call it \(``q",\) is a #BooleanFunction \(q:\mathbb{B}^{41}\to\mathbb{B}.\) Since \(2^{41}=2,199,023,255,552,\) its #TruthTable has \(>\) 2 trillion rows and its #VennDiagram has that many cells. There are \(2^{2^{41}}\) functions \(f:\mathbb{B}^{41}\to\mathbb{B},\) among which \(q\) is just one.
#LogicalGraphs #logic #venndiagram #truthtable #booleanfunction #booleanvariable #logicalterms #cactusgraph #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
85 followers · 291 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 2
• https://inquiryintoinquiry.com/2022/08/30/theme-one-program-jets-and-sharks-2/
As we saw last time, Theme One reads the text file shown below and constructs a #CactusGraph #DataStructure in computer memory. The cactus graph represents a single #LogicalFormula in #PropositionalCalculus and that proposition embodies the #LogicalConstraints defining the Jets and Sharks #DataBase.
Jets and Sharks • Log File
• https://inquiryintoinquiry.files.wordpress.com/2022/08/theme-one-guide-e280a2-jets-and-sharks-e280a2-log-file.png
#logicalcacti #cactussyntax #Peirce #LogicalGraphs #logic #database #logicalconstraints #PropositionalCalculus #logicalformula #datastructure #cactusgraph #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
82 followers · 274 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 1.5
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
Reference
#McClelland, J.L. (2015),
Explorations in #ParallelDistributedProcessing :
A Handbook of Models, Programs, and Exercises,
2nd ed. (draft),
#StanfordParallelDistributedProcessingLab.
• https://web.stanford.edu/group/pdplab/
#PDPHandbook Online
•https://web.stanford.edu/group/pdplab/pdphandbook/
Section 2.3. Exercises
•https://web.stanford.edu/group/pdplab/pdphandbook/handbookch3#x7-320002.3
Figure 2.1. Jets and Sharks
• https://web.stanford.edu/group/pdplab/pdphandbook/jetsandsharkstable.png
#PDPHandbook #stanfordparalleldistributedprocessinglab #paralleldistributedprocessing #McClelland #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
82 followers · 269 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 1.4
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
Displayed below is the text expression of a #TraversalString which Theme One parses into a #CactusGraph #DataStructure in computer memory. The cactus graph represents a single #LogicalFormula in #PropositionalCalculus and this proposition embodies all the #LogicalConstraints defining the Jets and Sharks #DataBase.
#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators
#MinimalNegationOperators #logicalcacti #cactussyntax #Peirce #LogicalGraphs #logic #database #logicalconstraints #PropositionalCalculus #logicalformula #datastructure #cactusgraph #traversalstring #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
81 followers · 259 posts · Server mathstodon.xyz#ThemeOneProgram • #JetsAndSharks 1.3
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
The manner of representation may be illustrated by transcribing a well-known example from the #ParallelDistributedProcessing literature (#McClelland and #Rumelhart 1988) and working through a couple of the associated exercises as translated into #LogicalGraphs.
#Logic #Peirce #Semiotics #Semiosis
#Grossberg #CompetitionCooperation
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators
#MinimalNegationOperators #logicalcacti #ProofTheory #modeltheory #graphtheory #competitioncooperation #grossberg #Semiosis #semiotics #Peirce #logic #LogicalGraphs #Rumelhart #McClelland #paralleldistributedprocessing #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
81 followers · 258 posts · Server mathstodon.xyz#ThemeOneProgram • #JetsAndSharks 1.2
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
One way to do this is to interpret the blank or #UnmarkedState as the #RestingState of a #NeuralPool, the bound or #MarkedState as its #ActivatedState, and to represent a mutually inhibitory pool of #Neurons \(a,b,c\) by the proposition \(\texttt{(}a\texttt{,}b\texttt{,}c\texttt{)}.\)
#Logic #LogicalGraphs #Peirce
#Grossberg #McClelland #Rumelhart
#GraphTheory #ModelTheory #ProofTheory
#LogicalCacti #MinimalNegationOperators
#MinimalNegationOperators #logicalcacti #ProofTheory #modeltheory #graphtheory #Rumelhart #McClelland #grossberg #Peirce #LogicalGraphs #logic #neurons #activatedstate #markedstate #neuralpool #restingstate #unmarkedstate #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
80 followers · 256 posts · Server mathstodon.xyz#ThemeOneProgram • #JetsAndSharks 1.1
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
Example 5. Jets and Sharks
The #PropositionalCalculus based on #MinimalNegationOperators can be interpreted in a way resembling the logic of #ActivationStates and #CompetitionConstraints in one class of #NeuralNetwork models.
#Logic #LogicalGraphs
#Peirce #Semiotics #Semiosis
#Grossberg #McClelland #Rumelhart
#ParallelDistributedProcessing #PDP
#GraphTheory #ModelTheory #ProofTheory
#ProofTheory #modeltheory #graphtheory #pdp #paralleldistributedprocessing #Rumelhart #McClelland #grossberg #Semiosis #semiotics #Peirce #LogicalGraphs #logic #neuralnetwork #competitionconstraints #activationstates #MinimalNegationOperators #PropositionalCalculus #JetsAndSharks #ThemeOneProgram
Jon Awbrey · @Inquiry
22 followers · 51 posts · Server mathstodon.xyz
#ThemeOneProgram • #JetsAndSharks 1
• https://inquiryintoinquiry.com/2022/08/25/theme-one-program-jets-and-sharks-1/
In developing the Theme One Program I tested successive versions of its #InferenceEngine for #PropositionalCalculus #ConstraintSatisfaction on examples of #Logic problems current in the literature of the day. #McClelland & #Rumelhart 's #PDPHandbook set one of the wittiest gems ever to whet one’s app-titude so I could hardly help but take it on. The linked text is a light revision of the way I set it up in the program’s User Guide.
#PDPHandbook #Rumelhart #McClelland #logic #ConstraintSatisfaction #PropositionalCalculus #InferenceEngine #JetsAndSharks #ThemeOneProgram