Decision-making processes in the Cognitive Theory of True Conditions
aa r X i v : . [ c s . A I] M a r Decision-making processes inthe Cognitive Theory of True Conditions
Sergio Miguel Tom´e
Grupo de Investigaci´on en Miner´ıa de Datos (MiDa),Universidad de Salamanca, Salamanca, [email protected]
August 6, 2018
Abstract
The Cognitive Theory of True Conditions (CTTC) is a proposal to de-sign the implementation of cognitive abilities and to describe the model-theoretic semantics of symbolic cognitive architectures. The CTTC is for-mulated mathematically using the multi-optional many-sorted past presentfuture(MMPPF) structures. This article discussed how decision-makingprocesses are described in the CTTC.
Semantics is one of the most challenging aspects of cognitive architectures. TheCognitive Theory of True Conditions (CTTC) is a proposal to implement cog-nitive abilities based on model-theoretic semantics and to describe the model-theoretic semantics of symbolic cognitive architectures [3, 7]. The fundamentalprinciple of the CTTC is that the perceptual space is a set of formal languagesthat denote elements of a model that is a quotient space of the physical space. Atthis moment, the mathematical formulation of the CTTC is using as model the multi-optional many-sorted past present future (MMPPF) structures [2]. Also,the CTTC proposes a hierarchy of three formal languages to describe them.This article discusses how decision-making processes are described in theCTTC. The article is divided in two sections. The first section addresses howthe CTTC gives a functional equation that relates a MMPPF structure with thehierarchy of formal languages. The second section addresses a set of solutionsto generate behavior denominated heuristics of qualitative semantics, and itsbasis is detailed. 1
The relation between the MMPPF structures
An important issue within cognitive science is how cognitive behaviour is gener-ated. The CTTC addresses the issue relating behaviour with a decision-makingprocesses. Mathematically, the CTTC establishes a equation that relates theMMPPF structures with the hierarchic of languages. The general equation isthe following: i (cid:13) u ( t ) = I r p ( π ( π ( F u ( h φ , ..., φ n i t , h ψ , ..., ψ n ′ i t − )))) (1)where, F u is a function that generates a pair of elements (the first element isa sequence of constants that denote interactions of the agent u , and the secondis a tupla of formulas of the formal languages of the hierarchy), h φ , ..., φ n i t is a tuple of formulas of the formal languages of the hierarchy that is input, h ψ , ..., ψ n ′ i t − is a tuple of formulas of the formal languages of the hierarchythat was generated by the system, π n is the projection function that projects then-element of a tuple, and r p = ( t , ε , || , e x ) is the existing reality of the momentof time t , I r p an interpretation function, and i (cid:13) u is the interaction functionof the agent u . With 1 is associated the following equation: h ψ , ..., ψ n ′ i t − = π ( F u ( h φ , ..., φ n i t − , h ψ , ..., ψ n ′ i t − )) (2)Thus, the left-hand side of the equation are elements of the mathematicalstructure, and its right-hand side are elements of the hierarchy of formal lan-guages.The function F u belongs to a space of functions F u . The elements of thespace of functions F u can be described with lambda expressions that we canreduce it to a normal form which is a sequence of constants that denote actionsof an agent. Those expressions are simply typed lambda calculus where theground types are G = { LP MMP F M , LP ∗ MMP F M , CL
MMP F M , I u } . The reasonto use the framework of lambda calculus is because F u can be composed offunctions of functions.One of the possible equations that can be derived from the general equationis the following: λtime. i (cid:13) u ( t ) = λterm. I r p (( λφ , φ . ⊢ u ( ϕ, ϕ ′ ))) (3)or in a traditional style: i (cid:13) u ( t ) = I r p ( ⊢ u ( ϕ, ϕ ′ )) (4)where ϕ is a description about r p made by the agent u , ϕ ′ is a descriptionabout a future and existing reality, ⊢ u is a function that determines a represen-tation of an interaction that denotes an action of the agent u to arrive from ϕ ϕ ′ , and it is defined in the following way: ⊢ u : LP.M ( ε, || ) × LP.M ( ε, |↓ ) −→ I u ( ϕ, ϕ ′ ) ~aLP.M ( ε, || ) , LP.M ( ε, |↓ ) ⊂ LP Equations 3 and 4 express that the action that the agent u carries out at eachtime is the interpretation of the representation of the action calculated by ⊢ u .It must be noted that π has been eliminated in 3 and 4 because the sequencegenerated by ⊢ u has only one element.The relation formulated can be seen as a functional equation if ⊢ u is consid-ered an unknown variable. The problem is that we do not know a method todetermine ⊢ u . The CTTC proposes that a set of solutions to the equation 4 isthe heuristics of qualitative semantics. In the following subsection, the basis ofthe heuristics of qualitative semantics is detailed.Another example of equation that can be derived from the general equationis the following equation: i (cid:13) u ( t ) = I r p ( π ( ⊛ ( ψ u , ⊢ u )( ϕ, ϕ ′ ))) (5)where ψ u is the log formula of the agent u . ⊛ LP × P −→ P ( ψ u , ⊢ u )