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3 Elemental Objects in 2D: Representation and Problem Solving

3 Elemental Objects in 2D: Representation and Problem Solving

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5.3 Elemental Objects in 2D: Representation and Problem Solving



2D Rotational

Transformation



211



Y’

Time

X’



Reflect off an

Immobile

Object with

Momentum



Time



Y

X



Physical World



Fig. 5.15 Collision and Reflection operator for 2D elemental object with rotational symmetry



speaking the amorphous fluid is a 2D extended object – implying that the elemental

objects composing it do not move independently of each other, for a short span of

time, the behavior of the elemental objects comprising it can be approximately

characterized as independent elemental objects moving in 2D. We therefore apply

the Push, Momentum Continuation, and Non-Penetration operators to characterize,

respectively, the “pushing” of some of the elemental parts of the fluid by other parts

of the fluid (say something from the source of the fluid), the continuation of the

momentum at the advancing “front” of the fluid, and the behavior of the fluid at

locations at which it is halted by an immobile non-elastic barrier. (In the discussion

section, Sect. 5.5 below, we contrast our method of representation of the behavior

of liquid with that in qualitative physics – e.g., Hobbs and Moore 1985).

With some further extensions to the current rules, the problem solving situation

involving some fluid depicted in Fig. 5.17 can also conceivably be handled. With

the representation and understanding of the concept of “Non-Penetration,” barriers

can be constructed much like the construction of the “penetration-tool” in Fig. 5.14

to achieve a certain goal – such as to confine the fluid to certain areas. A goal

specification similar to those that we have described above, but now for the desired

changes of the fluid shape with respect to time,2 could be used to drive reasoning

processes that provide the construction (in a “Tool Construction” phase much like



2

And for places where we wish the fluid to stop flowing further and beyond we specify no further

shape changes in time – i.e., a “halt” in further motion.



212



5 Causal Rules, Problem Solving, and Operational Representation



2D Rotational

Transformation



Y’

Time

X’



Momentum

Continuation



Push



Fluid

Source

Immobile inelastic

object



Non-Penetration

Fig. 5.16 2D fluid – approximated by a collection of independent elemental objects moving in 2D

directions. Operators involved are Push, Momentum Continuation, and Non-Penetration



Attach/Join

Barriers



Pushing

barrier in

place



Fluid

source

Heavy

mobile

object

Immobile

inelastic

object



Non-Penetration

Fig. 5.17 Constructing and placing barriers to control fluid flow



5.4 Natural Language, Semantic Grounding, and Learning to Solve Problem. . .



213



that in Fig. 5.14) and placement (in a “Tool Use” phase) of barriers in certain places

in certain manners as a solution. Figure 5.17 shows barriers being constructed and

pushed into certain places to control the fluid flow.

Hence the operational representational scheme allows the straightforward and

natural representation of knowledge for encoding physical behavior of even physically complex objects such as fluids and generating solutions to certain desired

configurations of the objects involved. In a 3D world and under the constraint of

gravity, various kinds of containers for the containment of liquids can be conceived

and constructed by a noological system using similar representational constructs as

in the example for 2D fluid in Fig. 5.17.

In figures such as Figs. 4.8 and 4.9 in the previous chapter, we showed the idea of

conceptual hierarchies. Conceptual hierarchies such as semantic networks are

constructs often used in AI for reasoning (e.g., Russell and Norvig 2010) and in

psychology for the modeling of human knowledge structure (e.g., Gleitman

et al. 2010). What are the roles of conceptual hierarchies in problem solving?

One possibility can be seen by considering Figs. 4.11 and 5.17. In Fig. 4.11 we

suggested how some higher level concepts regarding the fluid – such as “rapidly

protruding front portion” – can be built on the lower level atomic operators and this

can potentially direct a problem solving process to provide some solutions to

manage certain regions of the liquid accordingly to achieve certain goals. The

ensuing problem solving process illustrated in Fig. 5.17 provides the detailed and

specific physical constructs built from elemental parts – i.e., the various barriers –

for the management of the flowing fluid. In general, concepts and conceptual

hierarchies help to reduce the search space involved in problem solving processes

and make these processes more efficient.



5.4



Natural Language, Semantic Grounding, and Learning

to Solve Problem Through Language



We began the discussion on operational representation in Chap. 4 and ended with

the discussion of using operational representation for causal rule encoding,

reasoning, and problem solving in this chapter. In Chap. 4 we claimed that the

operational representations of various concepts such as Move, Appear, Stay,

Attach, and Push that we have discussed represent the epistemic ground level

knowledge and we argued for the case by using some examples in which operational representations engender certain “ground level” problem solving that is

otherwise not possible (e.g., knowing the spatiotemporal behaviors of leaves on

branches allows a person to imagine or use “leaves on branches” as a tool to sweep

floor or fan oneself – Sect. 4.1). We submitted that this ground level representation allows the noological system to “really understand” the concepts involved

through an appropriate encoding of the meaning of the concepts. In this chapter

above, we further illustrated how causal rules represented in the form of



214



5 Causal Rules, Problem Solving, and Operational Representation



operational representations can be used for effective reasoning and problem

solving. These causal rules encode conceptual generalizations of the concepts

involved (such as Push, Momentum, and Attach). Therefore, in a sense what we

are showing is that knowing (the concepts involved) is knowing how to act (with

the concepts).

We have shown in Fig. 5.14 how a tool could be constructed to solve a certain

problem and the process of tool construction and application was worked out by

the system through a backward chaining problem solving process with causal

rules that have been learned earlier. Now, suppose a system (or a person) has

worked out this solution before. Could the system use natural language to communicate to another system the method to solve the problem without the second

system having to undergo a potentially involved problem solving process? In fact,

a vast proportion of human progress was achieved through learning from others

through natural human language about all kinds of solutions to all kinds of

problems. Of course, this is only possible because the recipients of the instructions in natural language form really understands what the strings of symbols

emitted by the instructors mean.

Using our framework, we illustrate how instructions though language is possible

because our representational framework captures meaning appropriately. In

Fig. 5.18 we show an example, derived from the same problem as that in

Fig. 5.14, of using English language instructions from an “instructor” to teach a

“student” how to solve the problem involved. One could roughly characterize the

meaning of a sentence as existing at two levels – the syntactic level and the lexical

level (e.g., Saeed 2009). For this example, we assume that there are some syntactic

transformational rules that are peculiar to each language (in this case, English) that

convert the syntactic structures to some deeper representations that capture the

syntactic level meaning of the sentences involved. This aspect of meaning

processing is complex in itself and we will not delve into the details there. What

we would like to illustrate with Fig. 5.18 is how the individual lexical items (the

“words”) can be associated with some ground level representations that provide the

meaning of the lexical items involved and this understanding allows the recipient of

the language instructions to carry out tool construction and application, illustrating

that the recipient really understands the strings of symbols emitted by the instructor

through language.

On the right side of Fig. 5.18 is the solution to the problem in Fig. 5.14 and on the

left side is a sequence of instructions in English instructing how a tool could be

constructed and used to solve the problem. To avoid clutter, we extract two

sentences from the left panel and place them right below the Solution picture and

show how each of the words in the sentences can be mapped onto an item, an

activity, or an action in the Solution. Again, to avoid clutter, we show only some of

the mappings. Also, there are linguistic devices such as the article “the” whose

function cannot be clearly identified with an external object or event and we omit

the mappings for these as well. Other than the words corresponding to some of the

concepts we have discussed above, we also show how the concepts of “first,”

“after,” and “then” can be mapped onto the temporal order of events in the Solution



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3 Elemental Objects in 2D: Representation and Problem Solving

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