#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 erects #NeuralArchitectures on substantially different foundations from current #Connectionist models. At a high level of abstraction, however, there is enough homology between the 2 orders to compare their performance on many of the same tasks. Thus I was able to try the program on a few examples suggested by #McClelland & #Rumelhart.

#Logic

#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

#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 \(f:\mathbb{B}^{41}\to\mathbb{B}\) and \(q\) is just one of them.

#Logic #LogicalGraphs

#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 which embodies all 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

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#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 —
• https://web.stanford.edu/group/pdplab/pdphandbook/handbookch3#x7-320002.3
Figure 2.1 —
• https://web.stanford.edu/group/pdplab/pdphandbook/jetsandsharkstable.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#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 which embodies all the #LogicalConstraints defining the Jets & Sharks #DataBase.

Log File
• https://inquiryintoinquiry.files.wordpress.com/2022/08/theme-one-guide-e280a2-jets-and-sharks-e280a2-log-file.png

#Logic #LogicalGraphs #Peirce
#CactusSyntax #LogicalCacti
#MinimalNegationOperators

#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

#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

#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

#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 and #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.