# Multi-optional Many-sorted Past Present Future structures and its description

The cognitive theory of true conditions (CTTC) is a proposal to describe the model-theoretic semantics of symbolic cognitive architectures and design the implementation of cognitive abilities. The CTTC is formulated mathematically using the multi-optional many-sorted past present future(MMPPF) structures. This article defines mathematically the MMPPF structures and the formal languages proposed to describe them by the CTTC.

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## 1 Introduction

Semantics is one of the most challenging aspects of cognitive architectures. The Cognitive Theory of True Conditions (CTTC) is a proposal to describe the model-theoretic semantics of symbolic cognitive architectures and to develop decision-making processes based on model-theoretic semantics [2, 3]

. The main idea behind the CTTC is that the perceptual space is a set of formal languages that denote elements of a model embedded in a quotient space of the physical space. At this moment, the mathematical formulation of the CTTC is using the

multi-optional many-sorted past present future(MMPPF). structures. Also, the CTTC proposes a hierarchy of three formal languages to describe them.

This article improves the previous characterization of the MMPPF structures and the hierarchy of the formal languages [2, 3]. The article is divided in three sections. The first section gives the mathematical definitions of MMPPF structures. The second section defines a hierarchy of three formal languages to describe an MMPPF structure. The last section addresses the semantics of the three formal languages of the hierarchy.

## 2 Multi-optional Many-sorted Past Present Future structures

A MMPPF structure is a nested structure of possible worlds. In other words, each world of the structure also contains another possible worlds structure. Thus, they are more complex than the classical structures of possible worlds used in temporal logics. This section includes the definitions of the MMPPF structure, temporal perspective structure, and state structure and its axioms. An MMPPF structure is constructed using temporal perspectives structures, and a temporal perspective structure is constructed using state structures. Thus, we define firstly a state structure, after a temporal perspective structure and finally an MMPPF structure. After the definitions, the axioms of the MMPPF structure are provided.

A state structure is a many-sorted structure, and it is denoted by . Its definition is the following:

 e=⟨⟨Ui⟩⟨fl⟩⟨Rk⟩⟩

where the domains are

 ⟨Ui⟩=⟨O,P(H),P(H×V0),...,P(H×Vn),A01,...,An1,...,A0z,...,Anz,SRA⟩
•  O={o1,...,oz}
•  H={h1,...,hz′}z≤z′
•  Vp={∅}∪{(w1,...,wdim(p)):w1∈Wp,1,...,wdim(p)∈Wp,dim(p)}

where

 dim:P⟶N
•  Apoi={apoi,1,...apoi,k}oi∈O and apoi,k=(apoi,k.in,apoi,k.ext)

where

 apoi,k.in:DHp,k×DVp,k⟶CVDHp,k⊆H,DVp,k,CVp,k⊆Vp
 apoi,k.ext:Hp,k×DVp,k⟶CVHp,k⊆H,DVp,k,CVp,k⊆Vp
•  SRA={⟨(s1,c1),....,(sr,cr)⟩,...,⟨(c1,c1),....,(cr,cr)⟩}∪{∅}

the functions are

•  ⟨fl⟩=⟨ESj,g∗0j,...,g∗nj,g0j,...,gnj,θp0,...,θpn,⊙⟩
•  ESj:O⟶P(H)
•  g∗p:O⟶P(H×Vp)
•  gp:H⟶Vp
•  θp:O⟶∪iP(Api)
•  ⊙:O⟶SRA

and the relations are

 ⟨Rk⟩=⟨So1,0j,...,So1,nj,...,Soz,0j,...,Soz,nj⟩

where

 Soi,pj⊆O

The definition of an structure of temporal perspective, , is

where its domains are the following:

•  MPt={m1,....,mz}

Each element of is denominated moment of time. Each moment of time is a set defined in the following way:

 mt′⊂TPt×∁×S×EPt∁={h,ε}S={↓|,||,|↓}

Being the constant denoted by the temporal perspective , the elements of a moment of time, , where , , , , and , fulfill the following:

• If then

• If then

• If then

Each element of a moment of time is denominated reality.

•  TPt={1,...,z} is the time set

It must be noted that the number of moment of times is the same to the number of elements that has

•  EPt={e1,...,es} is a set % of states structures
•  IPt=⋃jIej and Iej=(θ0j(o1)×⋯×θnj(o1))×⋯×(θ0j(oz)×⋯×θnj(oz))ej∈EPt

Associated with we use an auxiliary function, , which projects a part of an element that belongs to , to do definitions. The function is defined in the following way:

 πon(⟨→ao1,...,→aoz⟩)=→aon

where .

The functions are the following:

•  DpPt=⋃jDpj and Dpj={Soi,pjou:ou∈ej}
•  LPt={l0,...,ln}⟩ is % the set of environmental laws of Pt.
•  lp:P(P(H×Vp)×O)×P(Ap×O)×Dp⟶P(P(H×Vp))
•  SLPt={sl1,...,slr}

where

 slk:SRA×O×EPt×IPt⟶SRA
•  &Pt:EPt×I⟶EPt∪{⊥}

where

 &Pt(ex,→i){=⊥,→i∉Iex;=ey∈EPt,→i∈Iex.
•  \footnotesizei\normalsize◯Pt:T′t⟶I where T′t={t′:t′∈T and t′≤t}
•  SuccPt:(T×∁×S×E)×I⟶(T×∁×S×E)

and the relation is defined as follows:

 ≺Pt⊂MPt×MPt

Each fulfills that .

An MMPPF structure is formally defined as the tuple

 ⟨U,T,E,I,Dp,ρMMPPF,⟨dpj⟩,L,&,\footnotesizei\normalsize◯ ⟩

where its domains are

•  U={P1,...,Pm} is the set of% temporal perspectives
•  T={1,...,m} is the time set
•  E={e1,...,es} is the set of states
•  I=⋃jIej and Ij=(θ0j(o1)×⋯×θnj(o1))×⋯×(θ0j(oz)×⋯×θnj(oz))ej∈E
•  Dp=⋃jDpj and Dpj={Soi,pou:Soi,pou∈ej} is the dependecies set

its functions are

•  ρMMPPF:T⟶U
•  dpej:P(P(H×V0)×P(H×Vp)×O)×P(Ap×O)⟶Dpej∈E
•  L={l0,...,ln} is the set of environmental laws.
•  lp:P(P(H×Vp)×O)×P(Ap×O)×Dp⟶P(P(H×Vp))
•  &:E×I⟶E∪{⊥}

where

•  \footnotesizei\normalsize◯ :T⟶I

The following axioms define an MMPPF structure:

• First Axiom

 ∀ej∈EPt(gpj(h)≠∅∧p>0⇔{h}⊂ESj(oi))

The first axiom determines that an essence element has assigned a value of any property if and only if is assigned to an object.

• Second Axiom

 ∀ej∈EPt(p>0)∧(gpj(h)≠∅⇔g0j(h)≠∅)

The second axiom determines that a place of the space is assigned to an object if and only if it has assigned any value of any other property.

• Third Axiom

 ∀ej∈EPt(gpj(h)=x⇔∃oi(h,x)∈g∗pj(oi))

The third axiom determines the relation between and .

• Fourth Axiom

 ∀ej∈EPt(api,k∈Api⇒DHp,k=⋃jESj(oi))

The fourth axiom ensures that an action acts independently of the assignment by .

• Fifth Axiom

 ∀ej,ej′∈EPt(h∈ESj(oi)∧h∈ESj′(ou)⇔u=i)

The fifth axiom determines that an essence element is only assigned to an object.

• Sixth Axiom

 SuccPt(t,ej)=ej′⇔dpj(⟨(g∗0j(oi),oi)⟩i,⟨(g∗pj(oi),oi)⟩i,⟨(a∗pi,k,o1)⟩i)=Dpj′

The sixth axiom determines that the dependencies set is coherent with the changes from to .

• Seventh Axiom

 SuccPt(t,ej)=ej′⇔∃⟨api,k,oi⟩i∀p⟨(g∗pj′(oi),oi)⟩i=lp(⟨(g∗pj(oi),oi)⟩i,⟨api,k,oi⟩i),Dpj)

The seventh axiom determines that if state succeed state , it is because the objects can produce changes that generate from .

• Eighth Axiom

 ∀e∈EPtθp(oi)∈P(Api))

The eight axiom determines that only assigns actions to an object when they modify that object.

• Ninth Axiom

 ∀t∈TTPt=T∀t∈TEPt=E∀t∈TIPt=I∀t∈TLPt=L∀t∈T&Pt=&∀t∈T\footnotesizei\normalsize◯Pt(t)=\footnotesizei\normalsize◯ (t)=\footnotesizei\normalsize◯Pt+1(t)=\footnotesizei\normalsize◯Pt+2(t)=⋯∀t∈TDpPt=Dp∀t∈T⟨dpj⟩Pt=⟨dpj⟩∀Pt,Pt′|SLPt|=|SLPt′|∀Pt,Pt′slkPt=slkPt′\par

The ninth axiom determines that domains do not change from one temporal perspective to other.

• Tenth Axiom

 SuccPt((t′,⧫,■,ex),i)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(t′+1,ε,↓|,ey),for t′+1t and ⧫=ε(t+1,h,|↓,ey),for t′>t and ⧫=h

The tenth axiom determines in what period of time a temporal moment and its reality condition are situated.

• Eleventh Axiom

 ∀ej,ej′∈EPtSuccPt((t,⧫,■,ej),i)=(t,⧫′,■′,ej′)→→⊙j′(oi)=slk(⊙j(oi),oi,ej,i)

The eleventh axiom determines that the change of reward and aversion sensation is coherent with the actions that the object carries out.

## 3 Formal languages to the MMPPF structures

The section defines three formal languages to describe an MMPPF structure: the perceptive language, the extended perceptive language and the categorical language. They are denoted , and respectively.

### 3.1 The perceptive language of MMPPF

The elements of the alphabet of the language are the following symbols:

• A constant symbol for each element of

• A constant symbol for each element of of

• A constant symbol for each element of of each .

• Two constant symbols, and , for the elements of

• Three hybrid operators :, and

• A constant symbol for each

• A constant symbol for each

• Four connectives , and .

• Auxiliary symbols: and .

It must be noted that in the alphabet there are neither any kind of variables nor elements to design elements of the set .

A tuple of symbols is denoted as .

The language has the following three kinds of atomic formulas:

• Type I: where

• , ,

• Type II: where

• , ,

• Type II: where

• , ,

Any atomic formula is a well formed formula (wff).

The following rules determine when a wff combined with an atomic formula constitute a wff:

• A wff and an atomic formula constitute a wff if and

• A wff where and an atomic formula constitute a wff if any of the following conditions is fulfilled:

• and

• and

• and

• and

• A wff where and an atomic formula constitute a wff if any of the following conditions is fulfilled:

• and

• and

• and

• and

• A set of atomic formulas ,…, are a wff is a wff there are not a and where and .

• A set of atomic formulas ,…, are a wff is a wff there are not a and where .

Finally, is a wff if and are wffs.

### 3.2 The extended perceptive language of MMPPF

is a language of description to MMPPF structures with a higher level of abstraction than . The definition of the requires the definition of the alphabets of metainformation , , , , and . These alphabets can be assigned to an object from its description in . Thus, the atomic formulas of can be built from formulas of . The rules to built a wff of are the same that the rules defined in .

#### The metainformation alphabet Σ1

The metainformation alphabet has two elements that are denoted by and . Thus,

 Σ1={0,1}

The elements of the metainformation alphabets are relations. Thus, their definitions are the following:

Each element of is named momentary state. Using , a qualitative state is given to an object in a moment of time to the -property. It describes whether an object has a specific quality without naming the specific value. Then, a functor of the language is defined, which maps each object in a moment of time and in relation to a property into the metainformation alphabet . It is denominated momentary state of the -property and denoted by , which is defined in the following way:

 msp(oi,ψ)={0,ifPp=∅1,ifPp≠∅

where and a temporal situator.

The following formula can be built using the functor :

 ψ∗≡[⧫|■|oi|ms0(oi,ψ)|,...,|msn(oi,ψ)]

Thus, , which belongs to , is built from .

### The metainformation alphabet Σ2

The metainformation alphabet has three elements that are denoted by , y . Thus,

 Σ2={≀,β1,β2}

As the elements of the metainformation alphabets are relations, their definitions are the following:

where

 Pp,P′p∈P(H×Wp,1×⋯×Wp,dim(p))=P(H×Vp)

and

 ϕ≡Pp≠∅∧P′p≠∅

It is denominated temporal state of the -property to the functor that is defined in the following way:

 tspp(oi,φ)=⎧⎪⎨⎪⎩β1,ifPp=P′p∧ϕβ2,ifPp≠P′p∧ϕ≀,ifmsp(oi,ψ)=0∨msp(oi,ψ′)=0

where

 φ≡ψ▶ψ′

and

 ψ≡[⧫|■|oi|P0|,...,|Pdim(p)]
 ψ′≡[⧫|■|oi|P′0|,...,|P′dim(p)]

and

 ϕ≡msp(oi,ψ)≠0∧msp(oi,φ)≠0

Using the functor , the following formula can be built:

 φ∗≡[⧫|■|oi|tsp0(oi,φ)|,...,|tspn(oi,φ)]

Thus, , which belongs to , is built from .

#### The metainformation alphabet Σ3

The metainformation alphabet has three elements that are denoted by , and . Thus,

 Σ3={≀,γ1,γ2}

The elements of the metainformation alphabets are relations. Thus, their definitions are the following:

where

 Pp,q={(h,wq)i}h∈Hwk∈Wp,q where Vp=Wp,1×⋯×Wp,q×⋯×Wdim(p)
 P′p,q={(h′,w′q)j}h′∈Hw′k∈Wp,q where Vp=Wp,1×⋯×Wp,q×⋯×Wdim(p)

Previously to define the function that assigns elements of to the objects, it is necessary to define a function that assigns to each element essence the value that has in a specific dimension of a specific property. The function is denoted by and its definition is the following:

 cp,qj:H→Wp,qhi↦wq

where

It is denominated temporal state of the -component of the -property to the functor that is defined in the following way:

 tscpp,q(oi,φ)=⎧⎨⎩γ1,ifϕ∧ϕ′γ2,ifϕ∧ϕ′′≀,ifmsp(oi,ψ)=0∨msp(oi,ψ′)=0

where

 φ≡ψ▶ψ′

and

 ψ≡[⧫|■|oi|P0|,...,|Pn]
 ψ′≡[⧫|■|oi|P′0|,...,|P′n]

and