AI Agents

What's in a head

This note is: [COG-INFRA]

The content of this note is an outline my thoughts on what is necessary to provide a blueprint of a system that can be used to actually build a thinker who can interactvely have conversations rich enough to learn about and answers complex questions about its world.

There are several high level requirements that can be stated as challanges to systems which claim they understand 'a lot'.

  • Theoretical Requirements
    1. It must be capable of having a rich world model [ONTOLOGY]
    2. Heads are finite [FINITE-HEADS]
    3. The system must be conversational [CONVER]
    4. The system must be autonomous [AUTONOMOUS]
    5. It must be able to understand its own metatheory [METATHEORY])
    6. It must implement some form of 'the gossip principle' (RWW [GOSSIP])
    7. it must be 'in the world' where inputs arrive and affects happen asynchronously

  • Practical Requirements
    1. To satisfy 2) - everything must be representable as a finite structure
    2. It must be realizable as a physial purchasable computer system

This note is about 1) and 2) and the practical requirements 8) and 9). It outlines strictures we believe are capable of being used to 'build a mind' and justify this claim by using it to 'solve' many of the problems of NLP, philosophy and AI in a unified framework. It has a reference implementation we use to try experiments. We hope to make this available to everyone soon.

This work is the current state of our thinking about reasoning systems that started with the FOL system [FOL] developed at the Stanford Artificial Intelligence Labratory (SAIL) in the late 1970s and which was continued by many people and is still actively persued by R. Weyhrauch and C. Talcott at their weekly meetings.

World Modeling

FOL Contexts

An FOL context, C, has three parts 

  C=<L,SS,F>

A language, L, a simulation structure, SS, and finite collection of facts, F. To satisfy 9) Each of these pieces needs to be realizable as a finite data structure, For building world models we use FOL contexts as a replacement for the usual (ie logician's) notion of theory.

The Language

An FOL language specifies a set of symbols (words (possibly containing spaces)) and the roles that they play when writing sentences of the language. To satisfy the requirement that an FOL context be a 'formal system' (again, in the sense of logicians) we also require that there is a way of defining expressions and a proceedure that can tell us if an expression is well-formed.

The Model

An FOL simullation structure is collection of individuals, predicates, functions and relations. At first you can imagine these as being defined in the usual way using set theory. For building world models in computers we will look at these definitions more carefully. When world modeling we use FOL simulation structures to represent the 'mental images' of things.

The Facts

An FOL fact does not have meaning outside an FOL context. F is the finite set of facts (of C). Each fact of C has at least two properties: a well-formed formula of L and a justification, that encodes the reason why the fact is 'in' C.

Discussion: We know that the notion of justification is vague but it is an nimportant idea. The usual theory is just a collection of well-formed formulas. Each fact comes (at the least) with an associated reason for being in its context. [JUSTIFICATIONS]

Signatures and Interpretations

We use the ordinary mathematical notion of 'signature' is the tool to describe the content and relation between languages and models. For C to be an FOL context, L and M must have the same signature, which dwfines an implicit map between them. We say that the simulation structure, SS, is an interpretation of the symbols of the language L.

An FOL context makes explicit both the language that is used to talk about things, the things the context is talking about and the connection between the two.

We use FOL contexts as the basic building blocks of world models. The


The remainder of this note covers two important subjects

  1. Notions we can introduce independent of the content of C
  2. How we can make this all out of finite pieces

Content Independent Notions

This section we describes how the ordinary notions of logic can be restated for contects eliminating any hidden commitments to infimite structures. we describe the changes we need to make to our understanding of theories without losing basic logical ideas.

In addition, we begin (informally) to write down expressions that we later use for wffs of facts in the languages of FOL contexts. The infrastructure described is not reductionist but rather is described by interdependent definitions and takes a little bit of getting used to.

Generality and Uniformity

An FOL context can name and contain facts about other contexts. Since FOL contexts will themselves be represented as finite data structures [DATA] hey can appear in the simulation structures of contexts. This hierarchical organization means that a single context can itself refer to many other contexts (which in in turn can refer to others ... ). If you think of an FOL context as a theory then they directly formalize metatheory. If you think of an FOL context as the belief structure of an agent then they can be used to discuss relations among the belief structures of multiple agents. If you imagine that the structures in the model of an FOL context are the 'mental imags' of the things that the context mentiones, then we can build contexts that 'know about', for example, Frege's notions of sense and repference (Zinn und Bedeutung) as well as use and mention. ... .

These ideas and more are expanded in the notes [FOL notes].

Inference

The fact that we can interpret the symbols of the language of an FOL context in its model allows us introduce the notion of inference.

Consider the FOL context C=<L,M,F>.

Satisfaction

We extend the notion of interpretation to all the well-formed expressions of L. We call this extended function 'satisfies' and use it to decide, for ny wff of L whether or not it holds or not - relative to C. We introduce a relation 

  (1) (Satisfied-by wff C) =def (= (satisfies wff M) True)

where wff is in L.

Consistency

We say a set W of WFFs of L is consistent and write 

  (2) (Consist W) =def (FORALL w in W (Satisfies w M))

Let 

  (3) (wffs-of C) =def {w F |(and (in f F) (= (wff-of f) w))}

We define consistency of an FOL context with the definition 

  (4) (consist C) =def (consist (wffs-of C))

We say a wff w of L is consistent with C if (Consist (union {w} (wffs-of C)))

The Exact meaning of what it means to be a definition is described below. Note that (offs-of C) is always finite. Also note here that some information about the language may imply the existence of facts in C that are to be included in wffs of a context.

Conclusions

the notion that a wff of L can be concluded from C simply means that the wff is consistent with C,

Discussion: Unlike tradional theories aE wff does not need to be the wff of a fact in C to be a conclusion. Using contexts we distinguish the conclusions of C and the facts of C. Thus we seperate the operation of making a conclusion based on C and the operation of 'adding' such a conclusion as a fact to the context C. The new fact will have f as its whh and will reference why f was a conclusion in its justification. See [SATvsVAL]

Rule of Inference

We define a rule of inference on contexts as any function, r, that maps an FOL context (with possible additional arguments) to an FOL context that preserves consistency, ie 

  (5) (implies (consist C) (consist (r C ...)))

There are several options here forthe meaning of inplies. This is discussed in [BEFORE-AFTER].

These notions are 'just like' those of classical logic, i.e., have the same definitions, but when they refer to FOL contexts they acquire more nuanced meanings. This is a clear indication that the analysis by logicians of these notions has been 'correct' but their notion of theory has been wrong. Remember we are replacing the notion of theory with that of FOL context and until we wactually use contexts to do this (including having a working application) the difference can only be hinted at. 


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Here we justify our claim that FOL contexts is powerful enough to model
everything in one unified way. Even if this claim eventually proves overly
optimistic, we show below that the use of FOL contexts makes a qualative
leap over existing systems for realizing goals above.