The logic of mental attributes models the autism
Autism phenomena
This study presents a logical model of autism a relatively rare multifactorial disorder that affects about 5 out of every 10,000 school-aged children. This disorder is characterized by impaired social interaction and communication combined with repetitive and stereotyped patterns of behavior. We focus on some peculiarities of the reasoning of autistic children, which are tightly connected with logical Artificial Intelligence and could be significant for both autism diagnosis and training.
Corrupted reasoning about mental states
It has been recently discovered that autistic children cannot reason properly about the mental states and actions of themselves and others. At the same time, the deductive capabilities of the autistic children concerning other domains match their mental ages. Autistic children are capable of normal reasoning about physical attributes such as time, space, states and actions, yet reasoning about intentions, knowledge and beliefs is reduced in various degrees.
The main hypothesis of this study is that there is a number of standard axioms for mental attributes, which are genetically set for normal children and are corrupted in the autistic brain. So autistic children have to acquire these axioms explicitly, by means of direct training using specific instances of these axioms.
The basic mental states are intention (subsumes goals and desires), knowledge and belief. The difference between belief and knowledge is that an agent is capable of changing and revising beliefs, but knowledge is only subject to acquisition. Any mental state or action concept can be defined within these basic mental states after adding an arbitrary predicate for a physical state or action.
We choose the most natural concepts (formulas) from the set of all well-formed formulas in our basic system. We then create a series of scenarios for each concept to determine if a child applies the formula (axiom) correctly and thus possesses that axiom. Every autistic child can then be categorized by the subset of corrupted mental axioms. We build the logical program with normal behavior as the control, and autistic patients are modeled by eliminating of the mental axioms.
This logical model of autism is applicable to both diagnosis and education. The preliminary experiments showed that after explaining to the autistic children a series of scenarios based on a corresponding axiom, these children raised the complexity of reasoning about specific mental states and actions, and demonstrated their capability to apply this axiom independently.
There are specific autism scenarios and their formalization.
Choice of the adequate logical calculus
There is a number of logical systems for representing intentions, knowledge and belief. Autism phenomenon seems to be important for logicians to choose a logical formalism, adequate for human intelligence. The model of the brain and its specific reasoning and, in particular, the model of the autistic brain, is very difficult to build. The difference between these two models fits into the limited formalism of mental attributes, so that this restricted component of brain activity can be subject to logical modeling. The autistic phenomenon seems to be the only one that links the biological brain with the axiomatic method in the foundations of mathematics.
Logical model of autism builds the link between the axiomatic method and the brain
FAQ about the logical model of autism
1) The axiomatic method includes two components for a formal theory: axioms and inference rules. Which of this component is affected by autism?
2) What are the primitives (atoms) of knowledge, corrupted under autism (behavior patterns, interaction scenarios, etc.)?
3) Does the reduced capabilities of modeling the mental states of himself/herself and other persons serve as the main criterion of autism?
4) There are various formal systems of reasoning about knowledge, belief and intention. Does autism modeling give the preference to a particular approach?
There are two children, A and B, who are subject to detection and/or training of the corrupted reasoning about mental states and actions. Correct answers follow the question, wrong answers are enumerated in the parenthesis, where presented. Scenarios are the approximate translation from Russian.
Mental states of another person
There is a table in a room with two boxes on it. The experimenter (E) is keeping a token in his hands. A is in the room, and B is outside the room. E is asking A:
1) You see the token in my hands. Do you know which box I am going to put the token to?
A: I dont know that box / nobody knows. ( A is confusing:I dont know the answer).
2) E: As you see, I put the token into the left box. Do you know, where B will look for the token: in the right box, in the left one or in both boxes?
A: In both boxes. (In the left box, where the token actually is).
3) E: And do you know where the token is?
A: I know where is the token.
4) E: Does B know where the token is? If we ask him, what would he respond:
A: I dont know where the token is. (I know where it is. I know it is in the left box).
5) E: If we ask B about his opinion, do you (A) know whether B knows where the token is?
A: B knows that I know that he does not know where the token is. (B knows where the token is, B does not know where the token is, B knows that I know where the token is, B knows that I know that B knows where the token is.)
6) E: Can we achieve a situation, when B will know where the token is?
A: Yes, we can tell him or show him (A confusing: I dont know).
B enters the room. Now all the questions are repeated; Bs responses, predicted by A, are actually evaluated.
7) E, After A showed (or told) B the location of the token: How do you (B) think, did A know whether you knew the location of the token while out of this room?
B: A knew that I did not know where the token is.
8) E, interrupting B: what do you (A) think, what will B say?
A: B will say that B knew that I knew that he B did not know where the token was.
9) E: Now you (B) know where the token is, because A have shown you. Do you think he (A) wanted you to know where the token was?
B: Yes, A wanted myself (B) to know where the token is.
10) E: Do you (A) know whether B knows that you (A) wanted him (B) to know where the token was?
B : Yes, I know that I wanted B to know where the token was.
1) E: Now I want to tell you the following. I believe, that B still does not know where the token is. Who is wrong: myself (E) or B?
A: You are wrong telling us that B still does not know where the token is. (B is wrong, now he does know where the token is).
This is a mirror test to the mental state of the other person one.
E keeps the blank piece of paper. A is next to E, and B is in the other room.
1) E: I am going to plot a geometric sketch on a piece of paper. Im about to start the drawing. Do you know what I am going to draw; do I know, if myself knows what will be drawn?
A: I dont know, and you do.
E finishes the picture.
2) E: Now you know, what Ive drawn. Does B know that?
A: B does not know what is drawn.
3) E: How can you let him know what is drawn?
A: Either show him or tell him (describe the picture).
4) E: You mentioned two ways of letting B know about this picture. Do both these ways require your knowledge of what is actually drawn?
A: No, to show him, I do not necessarily have to know (have seen) the picture. To describe the picture, I have to know its content. (Yes, I have to know the picture content for both telling and showing).
5) E: If we call B into the room and ask him if he knows what is on the paper, what would he (B) respond? What would he respond if we ask him after we show him the picture?
A: Before we show him (B) the picture, he will tell that he does not know what it is about. After we show or tell him (B) about the picture, he will tell he knows it.
6) E: if we ask B concerning his opinion, do you (A) know that he (B) does not know what this picture is about right now, before we informed him about the picture?
A: B knows that I know that he does not know the drawing. (A confusing: I dont know. B does not know that I know that he does not know. B does not know that I dont know that he knows).
7) E: I guess, I want your friend to know what is on the picture. Is it true? If so, does B know that you wanted to let him know about the picture? Does B know that you want him to know the picture?
A: Im not sure. After I informed him about the picture, he would know that I wanted him to know what is on the picture. I dont know if he (B) knows that I want him to know the picture.
Thereafter E calls B in and asks A to actually inform B about the picture. All the questions above are posed for B as Bs prediction of mental state of A.
Temporal relationships over the mental states. To forget and to recall
There are the toys on the table: a bear, a fox and a rabbit. Experimenter is asking the child about his/her mental states.
1) E: As you see, the bear is watching the rabbit. Does the bear know that the rabbit is on the table?
A: Yes, The bear knows that the rabbit is on the table.
2) E: Now the rabbit leaves the table. The bear knows that the rabbit is not on the table any more. Does the bear know that the rabbit was on the table before?
A: Yes, he knows that he was on the table before.
3) E: Then, after a while, when the fox asks the bear if the rabbit had been on the table, the bear is saying that the rabbit has not been there. Trusting the bear, what do you think, does the bear know that the rabbit was on the table?
A: The bear does not know that the rabbit was on the table.
4) E: OK, the bear forgot that the rabbit was on the table. Does the rabbit know that he earlier knew that the rabbit had been on the table?
A: No, the rabbit does not know that he earlier knew that the rabbit had been on the table.
5) E: Now the fox wants the bear to recall that the rabbit has been on the table. What will she do?
A: She (the fox) will tell the bear that the rabbit was on the table, and that the bear has seen him there.
6) E: Then, assuming, that the bear trusts the fox, what is the knowledge of the bear?
A: Now the bear knows that the rabbit was on the table.
7) E: OK, so the bear recalls that the rabbit was on the table. Does the bear know that before the recollection he did not know that the rabbit had been on the table? Analogously, does the bear know that he(bear) knew that the rabbit had been on the table, while (bear) was watching the rabbit?
A: Yes, the bear knows that he did not know that the rabbit has been on the table, as well as the bear knows that he knew that the rabbit has been on the table while watching the rabbit.
There is a table, and a book on it. The experimenter teaches the child A to pretend that it is soap.
1) E: As you see, there is a book on the table. Do both of us know that it is a book?
A: Yes, both of us know that it is a book.
2) E: Now let us pretend that it is soap. Both of us will still know, that it is the book. How ever, if I ask you, what that is, what will I respond?
A: You respond that it is soup.
3) E: If you ask me, what is on the table, what will I respond?
A: That there is soap on the table.
4) E: When one asks you if you know what is on the table, what will you respond?
A: I do know what is on the table.
5) E: Now let us stop pretending. Both of us still know that this is actually a book. If one asks me what is on the table, what will I respond?
A: You will respond that it is the book.
We present the analysis of the sequence the mental states appear in the process of human development. We state the proposition of monotonous increase of the complexity of mental formula, reflecting the perceivable mental states in the course of development. It has been shown that the normal humans under natural development break the monotonicity proposition. Around the age of 18-24 months human toddlers begin to pretend and recognize the pretending of others (Lesley 1987). However between 36 and 48 months they show the evidence of understanding additional epistemic states such as knowing and the simplest mental axiom that seeing leads to knowing (Pratt and Bryant 1990). So the pretending phenomena starts earlier than the knowing one, but the concept of pretending can be derived via knowing, intention and belief. Hence, the normal development process violates the monotonicity proposition.
Our study showed that the autistic children can learn the mental concepts in the order of complexity increase under the training in the corresponding order. Therefore, the autistic development obeys the monotonicity proposition under the specifically oriented training, but the normal human development violates it.
20 autistic children of the age 4-18 participated in the testing and training and 20 control children of the age 8 participated in the testing.
Note that the questions above cover the majority of mental formulas complexity 1-4, involving want and know (believe is identified with know for simplicity). The manifold of tested mental state achieves the real world complexity. Therefore, the trained children are expected to behave properly in the real conditions, if they are able to transfer artificial mental states to the real ones.
Construction of mental formulas
The examples of the mental attribute formulas. Various formulas are built in the basis of want-know-believe and ask/inform in addition. The action/state predicates have the inmost occurrence: do(Agent, Action) or What. Last four rows present the mental axioms (which might be corrupted in the autistic patients). Six left columns show the numerical encoding of mental formulas; all well-formed formulas are interpretable by the simulation toolkit.
inform(Who, Whom, What ) :-
want(Who, know(Whom, What)),
believe(Who, not know( Whom, What)),
believe(Who, want(Whom, know( Whom, What))).
Definition of the concept inform via the want, know and believe (a typical exercise for the autistic children).
The autism simulation toolkit is developed to perform the reasoning, required to resolve the autistic diagnosis/training scenarios. Its knowledge base contains the full spectrum of mental axioms. The question answering of each autistic patient can be reproduced by eliminating the axioms, missing by this patient.
Definitions of more complex mental concepts: to offend, to forgive and to reconcile
Unintentional offend is based on the lack of knowledge that the offending action do(Who, Action) is unwanted:
offend( Who, Whom, Action ) :- want(Who, Action),
not want(Whom, Action),
not know(Who, not want(Whom, Action)),
do(Who, Action).
To be forgiven, the offender has to demonstrate that the offense is indeed unintentional. It is necessary for the offender Who to inform Whom that Who would not do that Action if Who knew Whom did not like (want) it.
forgive( Whom, Who, Action) :-
offend( Who, Whom, Action ),
inform(WhoElse, Whom,
not know(Who, not want(Whom, Action)) ),
believe(Whom, (know(Who, not want(Whom, Action))® not do(Who, Action) )).
If Who is unable to convince Whom (to make him believe) that the offend was unintentional, the other agent Counselor is required to explain the actual situation to Whom:
reconcile( Counselor, Who, Whom, Action ) :-
offend( Who, Whom, Action ),
not forgive( Whom, Who, Action),
explain(Counselor, Whom,
not know(Who, not want(Whom, Action)) ),
believe(Whom, (know(Who, not want(Whom, Action))® not do(Who, Action) )).
These problems take longer observation time (a few years) for the autistic children, subject to the presented above training.
7. See www.dimacs.rutgers.edu/~galitsky/MS/ma/wkb.html for more definitions of mental concepts based on want, know, and believe.