Carving Nature at the Joints: A Preliminary Sketch of a Theory of Quantity

Praveen Paritosh

Tuesday, April 17, 2003

3:40 PM

Big Picture

Thesis/Proposal Outline

Introduction, The Sense of the quantitative –

Missing in, and needed by

Sim/Retrieval/Generalization

Back of the envelope reasoning

Introduce the idea of CSQR

Goals/contributions of thesis

Learning the qualitative distinctions from examples: CARVE

Representation

Evaluation

Better similarity: Analogical estimator

Better back of the envelope reasoning: Bote-Solver

A meatier theory chapter about the sense of quantitative, and how it interacts with Similarity/Retrieval/Generalization.

Ken says “how the qualitative meets the quantitative”. I am not sure what that means.

There will be theory in the next CARVE chapter as well, so I have to think hard about the distinction between this and the next one. Somehow it seems the stuff here will be more abstract and general.

CARVE, an instantiation of the theory above

Supporting evidence

Review psychology and linguistics literature

Pilot experiment on sizes of countries – cant/shouldnt have pilot experiment in here – so can we turn it into a prediction?

Corpus study for comparison classes for dimensional adjectives

Description of CARVE and results on the CIA Factbook KB

BotE Reasoning, introduction, domain, commonsense QR

Strategy Library

BotE Solver

Analogical Estimator, and show better BotE

Evaluation of the Analogical Estimator

Comparison of estimates generated with/without the CARVE generated representations

Comparison of BotE-Solver, with and without the CARVE generated representation and Analogical Estimator

Conclusions, future work and related work and all that.

Roadmap. A checklist of whats to be done, and how long things will take.

Theoretical questions:

How does quantity effect similarity judgments? Even though I am not directly setting myself for answering that questions, I am indirectly claiming something along those lines when I say that quantity is missing in current theories of similarity and retrieval.

My current answer is that the qualitative representations account for the quantity effects. But is that all? Is it qualitative all the way down? Sexy if I can claim/prove that. Or is there a part of quantitative similarity that will still remain?

How is one ever going to test such a claim?

In considering the effects of quantity on similarity, we can imagine two kinds of effects – for example, consider the example of a quantity q with a limit point p. The value of p is 10 units. And lets say, in four cases the quantity takes on values q1 (11), q2(9), q3(14), and q4(15).

Qualitative similarity: that part of the similarity because of this quantity thats captured by saying (> q1 p), (< q2 p) and (> q3 p). This makes q1 and q3 more similar than q1 and q2. This is based on the result by Goldman 1972.

sim(q1,q3) < sim (q1, q2) --1

Quantitative similarity: In the above case for q4, we also get (> q4 p). So according to purely qualitative sense of similarity

sim(q1, q3) = sim (q1, q4) -- 2

However, we know that numerically, q1 is closer to q3 than q4. So, if there is a quantitative similarity effect, we will find that people think that

sim(q1, q3) > sim (q1, q4) -- 3

It seems (almost obvious) that if the difference between q3 and q4 is greater than the just noticeable difference than people should notice that and agree with eqn 3. But wait, is that obvious. With the same subject it seems eqn 3 is right. What about across subjects? The similarity ratings across subjects might be closer to 2 than 3. With the same subject, many things are happening. They are possibly creating finer distinctions and newer landmarks based on more anchor points (there is a Cech and Shoben study that supports this idea, pull the cite out). This is the strong claim – eqn 1 and 2 accurately capture the role of quantity in similarity and 3, in fact, is not true. The trick is to do the 2/3 experiment across subjects and argue that 3 happens just because the subject is introducing newer limit points to the mix, or the dynamic nature of the notion of similarity. As methodology goes, what does comparing similarity across subjects means?

There's three threads here --

SOLVE: Building a library of strategies to solve BotE problems. Represent about fifty different problems, and try to build a reusable strategy library. Do an analysis of this domain [the kinds of knowledge involved in this domain] and problem solving strategies. Ken says "a really deep understanding of the functional import of equations in common sense reasoning" -- what does that exactly mean? More -- "I’ll bet right now you don’t have enough experience with these problems to be confident of an analysis of the general structure of the kinds of laws that there are (think about the functional analysis Yusuf did for equations for engineering problem solving; there’s probably something analogous one can say there that will make a nice chapter in your thesis)." One of the implications might be extending the suggestions representation and SOLVE to handle more abstract strategies. An interesting thing to explore might be to look at using at statistics compiled from problem solving episodes to focus the problem solving, e.g., in estimating the difficulty for a strategy/goal. Will a study like this make for a journal paper by itself?

CARVE: Starting from what we said in CogSci 2004 paper, implement CARVE. Domain -- CIA factbook. Issues: a) Computational implementation, b) adding structural information about quantities (some of which could be obtained from mining the data, like corr+ and corr-). CARVE, by itself is an interesting system, and the claim here will be that it is a cognitively-plausible model of building qualitative representations. Evidence will come from existing literature and maybe maybe some experimental data. Will this make a submission for Cognitive Science? Maybe we need one more domain there?

Using the representations built by CARVE, build the Analogical Estimator. Show better SME/MAC-FAC results with these representations. Plug the analogical estimator as a primitive estimation strategy in SOLVE and show better problem solving results (one or more of -- more problems answered, better answers, or faster answers). This will be the thesis.

Timeline:

First SOLVE, then CARVE and then the combining. I can see this happening by this year end.

Minor distractions which I shouldnt really do: results of the corpus analysis for dimensional adjectives. Mining the AQUAINT corpus for numbers.

1 Introduction

Quantities are ubiquitous and an important part of our understanding about the world – cars have engine horsepower, size, mileage, price; countries have GDP, population, area; birds have wingspan, weight, surface area, and so on. In this paper, we present a sketch of a theory of quantity – representations and principles for generating those representations. The notion of quantity is quite broad, and there is substantial literature in psychology, linguistics and qualitative reasoning (QR) on many specific aspects of it. The psychology of perceptual quantities (the literature refers to them as dimensions) like brightness, loudness, etc. has been studied in detail , and not much on conceptual quantities like price of computers, GPA of students, etc. Most of the research (from the decision making community and the case based reasoning community) that does study conceptual quantities employs feature vector models. On the other hand, the structured models of similarity and generalization that have strong converging psychological evidence, do not handle quantities adequately. In linguistics, there’s relevant work in the nature of dimensional adjectives like large, small, hot, etc. The QR literature contains many different proposals for qualitative representations of quantity. Given that, there is yet many aspects of quantities that are not well understood, and in this paper we address the following two fundamental questions –

What do our (cognitive) representations of quantities look like? Or, what representational machinery is needed for quantities?

How are these representations built with experience? Or, what are the distinctions that we (should) make?

These are questions about how cognition works, as well as about how the world is organized [as opposed to Bierwish, 1967]. Section 3 attempts to answer the former, and argues that our representations must contain symbolic reference points, and some informational equivalents of distributions. Section 4 tackles the second question above, and proposes a mechanism for finding the right symbolic reference points on the space values a quantity can take. Section 5 presents a plan for implementing and testing the ideas presented here. The last section presents a list of other relevant questions that we hope to gain insight about, from this investigation, but are not the direct focus of this research currently.

2 Background

This section presents relevant background in qualitative reasoning and structured models of retrieval, similarity and generalization – to both of which this theory will make potentially useful contributions – providing cognitively-sound qualitative representations, and extending the structured models to include effects of quantities, something currently ignored in those models.

Qualitative Representations of Quantity

One of the goals of qualitative reasoning research has been to understand human-like commonsense reasoning without resorting to the preciseness of models that are differential algebraic equations and parameters that are real-valued numbers. There is a substantial body of research in QR that has shown that one can, indeed, do a lot of powerful reasoning with less detailed and partial knowledge. Qualitative reasoning has explored representations of varying level of resolutions – status algebras (normal/abnormal); sign algebra (– , 0, +), which is the weakest representation that supports reasoning about continuity; quantity spaces, where we represent a quantity value by ordinal relationships with specially chosen points in the space; intervals and their fuzzy versions; order of magnitude representations; finite algebras [Forbus 2003]. The representations differ in the kind of distinctions that they allow us to make.

Our answer to the first question raised in the introduction is that the quantity space representation, augmented with distributional information, accounts for observations and existing evidence from psychology and linguistics. The current evidence does not completely prove or disprove this claim, and we feel that bridging this gap between QR and cognitive science will be a contribution to both fields.

Our answer to the second question is the first attempt to come up with a general theory of what distinctions to make [but see Sachenbacher and Struss (2000) for an answer to very similar questions in a more restricted domain]. Also, we find certain other distinctions than Sachenbacher and Struss (2000) didn’t anticipate prevalent in natural language. Clearly the representations one might have of quantities are a function of the reasoning task at hand (one might bother about the human body temperature if talking of shower water, which might not be of concern while reasoning about physical phenomena like freezing and evaporation), and also at the level of detail of the representations (an economist might see more distinctions than the one of rich and poor). There might not be any domain-/context-/detail-independent stable representations of quantity. The question, then is, given that we know the context, or maybe given a default context, how do we find the distinctions, and the what are the general substrates of these representations.

2.2 Structured Models of Retrieval, Similarity and General

The structure-mapping engine (SME) [Falkenhainer et al, 1989] is a computational model of structure-mapping theory [Gentner, 1983]. MAC/FAC [Forbus et al, 1995] is a model of similarity-based retrieval, that uses a computationally cheap, structure-less filter before doing structural matching. SEQL [Kuehne et al, 2000] provides a framework for making generalizations based on exposure to multiple exemplars.

In most symbolic knowledge representation frameworks, quantities are not handled adequately, if at all. Representing them as numbers does not go far in being useful, for example, our models of retrieval (MAC/FAC), similarity (SME) and generalization (SEQL) do not care much about quantities represented such. The way quantity are implicated in these processes –

Retrieval: Just as Red the symbol occurring in the probe might remind me of other red objects, the a bird with wing-surface-area of 0.272 sq.m. (that is the Great black-bucked gull, a large bird) should remind me of other large birds. One way to make that happen might be to abstract the numeric representation of wing-surface-area to a symbol, say, Large, then it will show up in content vectors, and contribute to the dot product.

Similarity: A model of similarity that is sensitive to quantities will explain how –

Quantity values can make things that have similar amount of structural overlap more or less similar. There are two questions here – how to compute similarity along a single dimension, and how to combine the similarity along different dimensions in computing overall similarity of two cases. Feature vector models answer the former by computing a difference, which does not explain how distances that are close, but across a landmark, are perceived to be more different than distances within the same qualitative region. For combining differences along more than one dimension, the feature vector models posit weights, and some weighted distance metric, but do not provide a principled way to find these weights.

To make estimate of an unknown quantity based on a similar known case.

Generalization: Key part of learning a new domain is acquiring a sense of quantity for different quantities. E.g., from a trip to the zoo, a kid probably has learnt something about sizes of animals, their shelters, etc. As we learn to cook, we get a sense of amounts of ingredients, cooking times, etc.

A large part of the above being unaccounted for in the current models, we feel, is poor representations of quantity. A symbolic and relational representation of the kind we propose here automatically makes these models more quantity-aware.

3 Cognitive Representational Machinery

In this section, we present and argue for a cognitively plausible representation of quantity. Of course, representations don’t arise in vacuum – they are molded by the kinds of reasoning tasks we perform with them, the underlying reality of how the things we are trying to represent are, and how we perceive that reality. We present these different kinds of constraints, which form the desiderata for what our representation must be able to do and account for. Based on these, we argue that our representational machinery for quantities must contain partially (or possibly totally) ordered symbolic reference points (quantity space), and distributional information about the quantity, or an informational equivalent thereof. We start with some concrete examples of quantity, talk about the various constraints, which is followed by a discussion of the proposed representation. An attempt is made to ground it in existing psychological and linguistic evidence.

The notion of quantity is usually employed to describe features that take on values that vary (or can be considered to vary) continuously over an ordered space (e.g., size, temperature, price). Depending upon the level of detail in representation, and the type of quantity , the operations that one can do with quantities that aren’t possible with nominal attributes are – ordinal comparisons, compute differences, and compute ratios/multiples.

Our knowledge about quantities is of various kinds – we talk of Expensive and Cheap things, we know that basketball players are usually Tall, we know that Boiling point of water is 100 degree Celsius. Below we present the various constraints that shape our knowledge about quantities.

3.1 Constraints

3.1.1 Reasoning Constraints: The three distinct kinds of reasoning tasks involving quantities are –

Comparison: These involve comparing two values on an underlying scale of quantity (or dimension ), e.g., Is John taller than Chris? We are constantly making such comparisons, and this is the most rudimentary manner in which we learn about quantities. Our knowledge of how the quantity varies (its distribution), and linguistic labels like Large and Small, are but a compressed record of large number of such comparisons.

Classification: These involve making judgments about whether a quantity value is equal to, less than or greater than a specific value , e.g., Is the water boiling?, Will this couch fit in the freight elevator?, Can I make the deadline?, Is he below the poverty line?, etc. Most of the classifications involve comparisons with interesting points on the space of values that a quantity can take, moving across which has consequences on other, different aspects of the object in concern. The metaphor of phase transitions describes many of such interesting points, although such transitions in everyday domains are not as sharply and well defined as in scientific domains (consider Poverty line versus Freezing point). We talk about this more later in this section.

Estimation: These involve inferring a quantitative/numeric value for a particular quantity, e.g., How tall is he? What is the mileage of your car? This the activity that has the strongest connection to quantitative scales – one can go a long way to account for the above two without resorting to numbers, but estimation involves mapping back to numbers [Subrahmanyam and Gelman, 1998; Linder, 1991]. A lot of times the knowledge of interesting points on the scale (these might provide a default anchor to adjust from, in the style of anchoring and adjustment [Tversky and Kahmenan, 1974]. Brown and Siegler (1993) proposed a framework for real-world quantitative estimation called the metrics and mappings framework. They make a distinction between the quantitative, or metric knowledge (which includes distributional properties of parameters), and ordinal information (mapping knowledge). Through a set of experiments they showed that the ways people revise and assimilate quantitative and ordinal information are quite different.

3.1.2 In-the-world constraints: The quantities that we identify, and the distinctions on the space of values that we make, reflect how the quantity varies in the real-world instances of its occurrence. To quote William James (1890), “The components of an absolutely changeless group of not-elsewhere-occurring attributes could never be discriminated. If all cold things were wet, and all wet things cold; is it likely that we should discriminate between coldness and wetness?” Bierwish (1967) disagrees, and says about dimensional adjectives (universal semantic markers in general) that they “do not represent properties of the surrounding world in the broadest sense, but rather certain deep seated properties of the human organism and the perceptual apparatus, properties which determine the way in which the universe is conceived, adapted and worked on.”

We believe that there are, indeed, properties of the underlying reality that influence our representation – the variance of quantities in the real-world instances of it, and the various causal/structural relationship between quantities. Phase transitions, for example, seem to be quite a property of the world – there is a certain point beyond which ice melts into liquid water .

Consider any real world object , most of the attributes that describe it are constrained, in the sense that they can not take on any arbitrary values, and the following are the two kinds of constraints –

Distributional Constraints: Most quantities have a range (a minimum and a maximum) and a distribution that determines how often a specific value shows up. E.g., the height of adult men might be between 4 and 10 ft, with most being around 5-6.5ft. References to Tall and Short men, then seems to be a reference to an underlying distribution of heights of people. A popular account of dimensional adjectives e.g., “Flamingo is a large bird” is that it establishes a comparison to an underlying categorical norm [Rips, 1980; but see Kennedy, 2003]. More than just the norm, we can usually talk about the low, medium, high for many quantities, which seems to be a qualitative summary of the distributional information. There is psychological evidence that establishes that we can and do accumulate distributions of quantities [Peterson and Beach, 1967; Malmi and Samson, 1983; Fried and Holoyak, 1984; Kraus et al, 1993; Ariely, 2001, among others]. Surprisingly, the next question of how we partition these distributions has not been raised at all. Coming up in section 4.2.

Structural Constraints: Besides the above, a quantity is also constrained by what values other quantities in the system take, its relationship with those other quantities, the causal theories of the domain; in general, the underlying structure of representation. Lets look at some examples –

It is generally true for all internal combustion engines (bikes, cars, planes, etc.) – that as the engine mass increases, the Brake Horse Power (BHP), Bore (diameter), Displacement (volume) increases; and the RPM decreases.

As the length of an animal increases, the length of the vocal chords increases. Thus, the larger the animal, the deeper its voice. Of course, that has implication for the entire sound producing/receiving apparatus of the animal, the distance its sound travels, etc.

The surface area of an animal determines the heat loss/exchange of gases/assimilation of food that it is capable of. Therefore, all spherical organisms are smaller than 1 mm in diameter, as the sphere is the shape with minimum surface area per unit volume. For animals that swim, it determines the drag, and for birds, the lift that it can generate. So, a change in surface area has repercussions on many aspects of the animal.

These constraints are very interesting, as they represent the underlying mechanism, or the causal story of the object, as they tie it to the other aspects of it. As we move along the space of values a quantity can take, it is possible that we transition into a region where the underlying causal story is different. These relational constraints can induce extremely important and interesting distinctions of quality on the space of quantity.

3.1.3 Psychological Constraints: The way in which we acquire knowledge about the world leads to making the following distinction between two classes of quantities –

Perceptual: e.g., brightness, loudness, size of things that we see, etc. We have a direct sensory measure of these attributes.

Conceptual, or Abstract: e.g., prices of things, mileage of cars, gross domestic products of countries, the size of countries, the GPA of a student, the clock speed of a computer. The distinction is that there is no direct sensory perception of these attributes.

Sometimes, though, it might not be clear whether a quantity’s underlying model in the person’s head is perceptual or conceptual, though, as one might perceptualize even the most abstract quantities – e.g. the size of a country as the size on the map, or the clock speed to the perceptual experience of speed of the experience of working on that computer. This paper focuses on the conceptual dimensions, where rich structured representations and higher-level cognitive processes might play a bigger part than that in perceptual dimensions.

3.2 Proposed Representation

Although our ability to perceive quantities seems to be continuous, our memory of them seems discrete. Based on the observations in section 3.2, here we propose that our representations must contain symbolic reference points, and distributional information.

3.2.1 Symbolic reference points : A partially, or possibly totally ordered set of symbolic reference points, which forms the quantity space [Forbus 1984]. Any value on the scale can then be represented via ordinal relationships to these symbolic reference points. These symbols are of two types –

Structural Limit Points: Symbols like Boiling Point and Poverty Line, that denote changes of quality, usually changes in the underlying causal story and many other aspects of the objects in concern.

Distributional Limit Points: Symbols like Large and Small, which might arise from distributional information about how that quantity varies.

We talk about this distinction in more detail in section 4. In language we find clumped together parts of the space of quantity, intervals, which we give names like Large, Medium, Small, etc; or recognize special points on the space like Poverty Line, Freezing point. Kennedy and McNally (1999) have made the Absolute/Relative distinction, where adjectives like Full, and Wet can be defined in an absolute fashion independent of context, as opposed to Tall. The following (incomplete) table charts different symbolic references to quantity –

Point	Interval

Absolute	Boiling Point, Full, Empty	Wet, Dry

Relative	Poverty Line	Tall, Warm, Expensive

It will be interesting to work out the mapping between these and the structural/distributional distinction proposed here.

QP theory [Forbus 1984] showed us that a lot of powerful reasoning could be done with a quantity space representation. It is the minimal representation that supports variable resolution. The symbolic and relational nature of this representation automatically makes it much more useful in our (structured/symbolic) representational framework, and we expect that it will go a long way in making our models of retrieval, similarity and generalization more quantity-aware.

There are some psychological results that provide some support for such a representation. A strong case for precisely the quantity space cannot be made, however we have some evidence that suggest that our representations do have a similar nature. There are two kinds of evidence –

Those that point to the cognitive significance of landmarks – categorical perception [Harnad, 1987], magnet effect in speech perception [Kuhl, 91], finer discriminations near the landmarks [Cech and Shoben, 1985].

Those that demonstrate the existence and effects of symbolic nature of these landmarks. The semantic congruity effect [Banks and Flora, 1977] is the fact that we are better and faster and judging the larger of two large things than the smaller of two large things. Part of the account from experiments involving adults learning novel dimension words, by Ryalls and Smith (2000), is the fact that in usage, we make statements like “X is larger than Y” more often than “Y is smaller than X”, if X and Y are both on the large end of the scale.

In our research till now, we haven’t been able to find enough evidence in the existing literature that argues for or against the existence of symbolic reference points and the quantity space representation. One of the goals of this project is to set this representation on firmer grounds, bridging evidence from psychology and linguistics. There is a large amount of unexplored and possibly relevant literature that might be brought to bear on this.

3.2.2 Distributional information: A distribution is a summary of how the quantity value varies. This is the part of the theory that needs to be worked out in more detail. If we take the existing psychological evidence to mean that we can accumulate distributions – the unanswered questions are –

What is the class of objects for which we compute distributions? As opposed to laboratory contexts, in the real-world we see rich objects rather than isolated quantities. So, do we have a distribution of all lengths, or length of vehicles, or length of cars, or length of sedans? This is similar to the question that Kennedy (2003) raises about how to determine the comparison class for an adjective.

How do the symbols on the space of quantity map on to these distributions?

3.3 Implications of the quantity space representation

Formally, an ordered set of points that partitions the space is equivalent to a set of ordered intervals, but interestingly, most of natural language seems to refer to intervals, while physics and scientific domains are more interested in the particular reference points. Kennedy (2000) is investigating very detailed issues of representation from a linguistic aspect. This study is complementary as we are looking for the particular/natural/cognitive distinctions, and we expect that both these efforts will gain from each other. There are some interesting questions –

Intervals versus points: Why is it that physical and scientific domains (and thus qualitative reasoning) finds the transitions, the points (freezing point, boiling point), more interesting; as opposed to language, where it seems that most of the references are to intervals (cold, hot), even when the transitions are sharp (e.g., wet/dry)?

Crisp versus soft transitions: The way scientific domains are setup, most of the transitions (freezing point, a liquid-solid phase transition) are usually crisp, but many of such transitions in other everyday domains seem to be softer. For example, consider tall/short, expensive/inexpensive, there does not seem to be a sharp transition point . We suspect that there might be a psychological distinction between the crisp and soft transitions – discrimination should worsen across the soft transitions (tall/short) as opposed to crisp transitions (wet/dry). Furthermore, there seem to be two kinds of ways in which these transitions are soft –

Spread: an interval as opposed to being a sharp point.

Degree of change: interesting changes (possibly in other aspects of the object than just the quantity in concern, e.g. being above and below the poverty line has an effect on other aspects of life) happen as we move across a transition, and these transitions could vary in the degree of change that happens.

Again, most of the classical transitions in scientific domains seem to be of little spread and high degree of change. The psychological and linguistic significance of the two parameters of the softness need to be explored.

4 Necessary and Relevant Distinctions

The last section argued for the quantity space representation – that the symbolic reference points provides a way to make the distinctions on the space of values of a quantity that we make. Here we address the second question raised at the beginning of this paper – which amounts to asking – how many symbols do we use, and where and how do they map on the space of quantity values ?

4.1 Structural Limit Points

Structural limit points are a generalization of the idea of limit points introduced in QP theory [Forbus 1984]. One should only make the necessary and relevant qualitative distinctions, QP theory advises us. In the domain of processes, QP theory provides the intuition for these distinctions: where things change, i.e., different processes and/or model fragments get de/activated, e.g., Freezing Point and Boiling Point of a liquid. Is there a general principle that provides these distinctions for more than just dynamical processes?

One can always partition the quantity space arbitrarily – so, one could have an ad hoc rule that said that we’ll always divide the space between the minimum and maximum into three parts – high, medium and low . That would mean that we are suggesting that there are some partitions that are more natural than others. These are those that correspond to the necessary and relevant qualitative distinctions . It is not easy to provide metrics for why they are more natural than others; below are some features of the natural partitions –

Just the right level of granularity for the kind of comparisons and reasoning one does with the knowledge, e.g., Freezing Point and Boiling Point might be fine for reasoning about physical behavior, but if one is talking about shower water, then more distinctions like Cold, Body Temperature, Warm and Scalding Hot might be more appropriate.

Predictive of other properties of the system, e.g., Poverty Line, Lower Class, Middle Class, Upper Class.

The structural constraints on quantities reflect a fundamental fact about the way things are in the world. Things in the world come in packages or bundles. For example, a “muscle car” has a powerful engine, is expensive, is designed for style and fun rather than safe, practical driving. In psychological literature, a similar notion is expressed by attribute co-variation or feature correlation. But there’s much more than that – these are not merely bundles of correlated attributes, but are structural bundles . The entities, and quantities associated with them, tied by relations and higher order relations constraining them, give rise to the structure therein. Processes (as in QP theory), are a special case of these structural bundles (where the key relationships are of causality and influence) for the class of dynamical physical systems. The necessary and relevant qualitative distinctions correspond to discontinuities in the underlying reality as captured by the structure in the representation .

Lets look at a few detailed examples – consider the dimension of size of dictionaries (as measured in number of pages, volume, or weight). There seem to be at least three meaningful distinctions of quality that might be projected on to size – pocket, table-top, and library-sized dictionaries. Why are they changes of quality? Because the underlying reason/story for these three types of dictionaries are quite different – the key aspect of the pocket dictionary is portability, and thus it has finer print, thinner pages, less detailed meaning, probably not much etymology and usage information, etc; while the key aspect of the library sized one is comprehensiveness, and thus it follows that it is larger, heavier, much higher number of entries and even arcane and obsolete words, etymologies, usage information, is well bound as it is big and thick, pages are tougher to stand more usage, etc. The table-top dictionary falls somewhere in between. On the dimension of size, thus, the distinctions of pocket, table-top and library-size, define interesting distinctions which have deep relationships to the underlying causal story, the underlying quality of dictionaries.

Consider people’s income. Poverty line, lower class, middle class and upper class define changes of quality on the space of income, as we expect that many other aspects of people – their lifestyle, the amount of time/money they spend on entertainment, education, the kind of vacations they have (or don’t), the family and social climates in which they live, their expectations and relationships to the rest of the social structure, etc. changes as we move across these interesting partitions of the scale of income.

These changes of quality in the above two examples are reminiscent of phase transitions in physics/thermodynamics – just as a lot of underlying properties and the relationships that tie them together change as we move across the phase transitions. The phase transition is a very useful metaphor. There are two types of phase transitions – first-order (sharp discontinuity, solid?liquid change, often these are accompanied by symmetry breaking, also called order-disorder transitions) and second-order (where one can continuously move from one phase to another, e.g., magnetization). Across a phase transition, different equations of state hold – in general, these transitions are identified by looking for changes of a global order parameter (which captures aspects of symmetry, e.g. magnetization of a ferromagnet) as a given control parameter (e.g., temperature) changes [See Sethna, 1992 for an introduction, and Gunton et al, 1983 and Gleiser, 1998 for more detailed explanation]. The structure of relationships is the analogue for symmetry, and the crisp/soft distinctions are analogous to first-/second-order transitions.

The partitions induced by structural differences, as in the above examples are the structural limit points. Starting from a set of exemplars, the structural limit points are the ordered transitions on the space of alignable quantities that correspond to structurally distinct clusters. The claim here is that these structural limit points are a major part of the natural partitions on most quantity space.

4.2 Distributional Limit Points

The importance of the structural limit points presented in the previous section is apparent – it is predictive of structural properties of the system, and thus quite useful in doing qualitative reasoning. Surprisingly, though, the language is contains many references to quantity which look very different from the structural limit points or the intervals they might induce – consider Large, Tall, Short, Expensive, etc. When we say a large flamingo, that seems to be reference to the distribution of sizes of flamingos and the fact that the particular flamingo we are looking at is larger than the norm, and is somewhere in the tail end of larger sizes in the distribution. Such distinctions like small, medium, large seem to be making cuts based on the distribution of values that the quantity takes, and the three most common distributions have different intuitions – an intuitive understanding of normal distribution might be that there are fewer short and tall people, then there is regular height people, and also that the range of tall and short is larger than the regular size. Similar interpretations can be made from the dashed lines drawn in Figure 2 for other types of distributions. The power law, or the Zipf distribution is extremely interesting, as it is extremely common – and a meaningful norm for such distributions can’t be defined.

Are there some systematic ways in which people make cuts on a distribution they have abstracted? There is little known about this. In appendix 1, we describe a very simple experiment that tries to look for this.

4.3 Implications

Most symbolic references to quantity have both a structural and distributional interpretation of them – so being Tall has structural consequences, for example, for a basketball player. An interesting question is two see the interactions between these two types of limit points. When do we chose to use structural limit points, and when distributional? The answer has got to do with the nature of the quantity. Some quantities are more causally central, i.e. deeply affect other aspects of the system, than others – compare horsepower of a car to size of the door handles. In the class of examples that we are looking at, there will be a tendency to describe a quantity purely using distributional information if –

The parameter doesn’t have deep causal connection to the rest of the system, or is not causally central (in terms of structured representation, has low systematicity) – height of poets as compared to height of basketball players.

There is not much of variation in the underlying structure (as far as is known in our representation) at all, e.g. size of male grown-up penguins.

The question of context sensitivity of our partitions of quantity might be captured by the representation of the class of examples in such a manner that we have all the context relevant aspects well represented in it [Mostek et al, 2000].

5 Next steps

To support, test and refine the ideas presented above, we ***. SEQL [Kuehne, 2000] will provide the basic framework for finding structural clusters. The main clear next steps are –

Next generation SEQL –

Heuristics for first cut distributional partitioning: The first problem is to bootstrap the process. To find the interesting structural limit points, we cannot ignore the numbers altogether, and thus we need to a do a rough, first-cut symbolic partitioning for the quantities. In the previous section, we mentioned distributional and structural constraints. This is where the distributional constraints play a role. Purely by looking at the distribution of a single quantity, one can divide it into ranges (e.g., low, medium, high population).

Extending category representations and SEQL to assimilate and include distributional information.

Building at least three different case libraries of examples rich in quantities and structured representations to test these ideas on. The CYC knowledgebase has a information about countries taken from the CIA World Factbook. There are about thirty quantities whose numeric values are known for most countries (area, population, birth rate, total labor force, gross domestic product, number of airports, etc.). Currently there isn’t much knowledge about the relationships between these parameters, we will be adding that. There are many other facts about different countries – like the organizations that it’s a member of, different political events in which it was a part, etc. that are also in the knowledge base. The first step is to build a case library of a large number of countries that has lots of quantities, and reasonably rich in structure. We had made an earlier attempt to build a case library of cars.

Once we have that, the goal is to see if we can use the above ideas to generate symbolic representations for the quantities. For testing the ‘goodness/natural-ness’ of these representations, we can –

Compare the representations thus generated to experts’ qualitative representations of the same quantities.

Incorporate these representations into the cases, and see if we get better retrieval and similarity measurements as compared to human subjects.

Build an analogical quantity estimator, which will estimate a value for a quantity by retrieving a strongly similar example for which it knows the value. This will be a part of the Back of the Envelope Reasoner [Paritosh and Forbus, 2003].

It is possible that we might run some experiments with human subjects in either the testing phase, or to confirm some of our intuitions. Chris Kennedy has shown an interest in this project, and some ideas here, and we might also do exploratory corpora analysis. As said earlier, another part of this project will involve ongoing literature survey across various disciplines to ground the theory in existing evidence.

6 More questions

The distributional landmarks and structural limit points provide a symbolic and structural representation of quantity that has the potential to be quite useful in our framework, making both steps of MAC/FAC more sensitive to quantities. It will provide for a principled way to partition the space of a quantity, tells us which quantities are more predictive (those that are more systematic) than others of structural properties, and a simple principle for combining differences along different quantities. There are a lot of things that are not explained, some of them are –

How do the dimensions come about to be in the first place? We will presuppose the existence and the conventional usage of the dimensions. Presumably, part of learning a domain is recognizing the dimensions in the first place (e.g., Clark, 1973, presents evidence for how big is easier than length and height, which are easier than width – “semantic feature hypothesis”).

Clearly, the worldview that there exists a stable partitioning for all quantities in the LTM isn’t true, and these representations might be highly contextual. This will be partly addressed by context sensitive dynamic representations.

How do we make comparisons and quantitative judgments that cut across different categories? For example, cheap, medium-range, and pricey gifts, will consist of things without a lot of structural similarity.

Where exactly are the symbols that make up the quantity space and the ordinal relations between them stored? If they are in the generalizations, then how do they help MAC/FAC for the exemplars? On the other hand, if they are with the exemplars, one will have to update them all in batch, as the structural limit points are dynamic, since they might change with the set of exemplars that one has seen .

Clearly, a full understanding of quantities is a major enterprise, but hopefully the ideas here will form a part of it.

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Appendix 1: Distributional Abstraction of Landmarks

There are N (say 30) circles of varying sizes. They are all alike, but for size. The subject is presented with this pile, and asked to divide it into smaller piles/groups.

They are free to chose the number of groups and size of each group (the number of circles in a group).

They are given the number of groups (e.g., 3, which could be thought of as the small, medium or large groups), but are free to chose the size of groups.

Now, one can do this, and vary the underlying distribution of the radius (or area) of the circles -- they could come from uniform, gaussian, exponential, power law, distributions. For example, if the sizes are linearly increasing between a MIN and MAX, then an equal split into N/3 circles per group might be meaningful; but if they come from a gaussian, which reflects in the fact that there are a larger number of medium-sized circles than there are of small or large. One can also try bi-modal distributions and so on. Statistically, the distribution provides some clue in partitioning. By looking at how people do in experiment A and B above, one can see how well people intuit the underlying distributions. Other key thing is a lot of real world dimensions are power-law like, e.g., there are many people with modest income, and few with arbitrarily high incomes. So, what we have done here is given people just one dimension, and we are seeing what kinds of partitioning they do.

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Smith, L. B., Cooney, N. and McCord, C. (1986). What is “High”? The development of reference points for “High” and “Low.” Child Development, 57, 583-602.

Sera, M. and Smith, L. B. (1987). Categorical and relative interpretations of “big” and “little.” Cognitive Development, 2, 89-112.

Lost and Found

A convenient and very powerful representation of quantities is using numbers (and the relevant units). A quantity is any of the examples above, and has at least three parts -- {value, unit, attribute }, for example my stipend might be {1102, USD/month, stipend}.

The proposed representation is a set of symbols on the quantity space that partition the space of quantity – sometimes the symbols correspond to points e.g., Poverty Line, Freezing Point, and sometimes to intervals e.g., Upper Class, Large. The symbols may be partially or totally ordered. Given this framework for representing quantities, we still have to figure out what and where are these special points on the space of quanti

But the quantity space representation seems to be more general than just processes – e.g., our notion of cheap, medium-range and expensive computers, or division of income groups into poverty line, lower class, middle class and upper class, etc. Is there a more general notion that will provide us a principled way of finding the necessary and relevant distinctions?

The origins of these partitions of the quantity space are very diverse, e.g. –

Personal experiences e.g., one’s notion of [Bland, Mild, Spicy].

(Social) conventions e.g., [Poverty Line, Lower Class, Middle Class, Upper Class], or [Short, Regular, Tall] when talking of people’s heights (this seems to be part social, part personal experiences).

Physics of the phenomenon - [Freezing Point, Boiling Point].

The problem, take two.

The bridge – “ability to perceive the dimensions seem to be continuous, but our memory of them seems discrete”

Two intertwined questions –

What will these representations look like that will make this bridge?

And how do these representations come about to be?

Quantities in analogical processes – there are two approaches –

Construct qualitative representations of quantity (from distributional and structural information, which has to be kept with the generalizations?), and add these to the descriptions, and match as normally.

Use differences in numerical properties, scaled via distributional information, to contribute to the structural evaluation score of a match.

The former makes more sense. Theoretically its appealing, uniform, and helps the MAC stage as well, which the latter will have a tough time dealing with.

Structural Bundles

Consider any real world object , most of the attributes that describe it are constrained, in the sense that they can not take on any arbitrary values. There seems to be two major classes of constraints on the values that a quantity can have –

Distributional Constraints: Most quantities have a range (a minimum and a maximum) and a distribution that determines how often a specific value shows up. E.g., the height of adult men might be between 4 and 10 ft, with most being around 5-6.5ft.

Structural Constraints: Besides the above, a quantity is also constrained by what values other quantities in the system take, its relationship with those other quantities, the causal theories of the domain; in general, the underlying structure of representation. Lets look at some examples –

This generally holds for all internal combustion engines (bikes, cars, planes, etc.). As the engine mass increases, the Brake Horse Power (BHP), Bore (diameter), Displacement (volume) increases; and the RPM decreases.

As the length of an animal increases, the length of the vocal chords increases. Thus, the larger the animal, the deeper its voice. Of course, that has implication for the entire sound producing/receiving apparatus of the animal, the distance its sound travels, etc.

The surface area of an animal determines the heat loss/exchange of gases/assimilation of food that it is capable of. Therefore, all spherical organisms are smaller than 1 mm in diameter, as the sphere is the shape with minimum surface area per unit volume. For animals that swim, it determines the drag, and for birds, the lift that it can generate. So, a change in surface area has repercussions on many aspects of the animal.

Different quantities vary in the degree to which they are structurally constrained – let us call a quantity that participate in more deeply nested parts of representation as more systematic than another which participates in less deeply nested parts of representation.

Find and cite compelling psychological evidence for the same.

No prior work which connects the distributions that accumulated to actual qualitative/ symbolic representations, in a principled, cognitively plausible fashion. Additional twist – how to handle power law distributions, which are pretty ubiquitous, but oft overlooked in related research.

The next section gives examples of what I mean by quantities. Section 3 presents the motivations driving the work. A symbolic partitioning of the space of quantity is presented as a representation for quantities which is discussed in section 4. Section 5 presents the principle underlying these natural partitions, namely, things in the world come in structural bundles. The last two sections talk about how to test these ideas, and their implications.

Many interesting insights can be found in this work – integrable/separable distinction, analytic/holistic perception, developmental trends – see Tighe and Shepp, 1983 for a collection of articles on these issues.

As opposed to attributes, which take on nominal values, e.g., race, sex, color, etc.

Note the distinction between dimension and quantity – wingspan of birds and length of cars are different quantities, but are the same dimension of length.

This is an age-old distinction – nominal, ordinal, interval and ratio scales; based on what arithmetic operations are supported. A ratio scale is also interval and ordinal, and an interval scale is also a ordinal scale, but the converse is not true.

Consider “The space telescope is longer than it is wide.” These cross-dimensional comparisons can get quite complicated to interpret, e.g., “The Sears tower is as tall as the San Francisco Bay Bridge is long” doesn’t really mean Height(Sears Tower) = Length(San Francisco Bay Bridge). This is outside the scope of this paper, see Kennedy (2001) for an analysis and implications of such comparisons.

Or corresponding judgments involving intervals.

The theory presented here is non-committal to whether these distinctions are in-the-world or in-the-head. We currently believe there are aspects of our representation that are both. Maybe we will know better at the end of this investigation.

Comic books, mythology, and fantasy, for example, has the freedom to relax this constraint – a character can be arbitrarily strong, large, small or be able to fly, even though the physical design of the character might not be able to support it.

This is related to the act of visualizing or our processes of imagery for even abstract notions.

Various different terms – landmarks, benchmarks and limit points, to name a few, have been used – we avoid any specific interpretation by calling them plain ‘symbolic reference points.’ In QR literature, there is a subtle but important distinction between landmarks [Kuipers 1986] and limit points, which amounts to landmarks being concrete instantiations of limit points.

Here might be lurking an answer to the previous question – in the case that the transitions are not sharp and well known between, say expensive and inexpensive, one might have a preference for intervals, which do not always involve making a commitment about the transition point.

The question being asked here is what and where are the distinctions to be made, not what to label those distinctions.

For example, the Fuzzy logic community does something in the same spirit.

Since what is necessary and relevant is context-specific, these distinctions do depend upon the context, and the theory should bear that out.

The idea of bundles isn’t new, Rosch has talked about it, and maybe it’s much older. But since their bundles were of flat attributes, from a feature vector, the explanatory power of those bundles, in my opinion, is far less than structural bundles.

The structure of relationships is an even more general notion than causality, spatial arrangement, connectivity.

What if there are none? For example, imagine a set of exemplars whose structured representations are identical, and the only difference is in the value of the quantities. We can only imagine this happening with poorly represented examples (as one is just beginning to learn about a domain, for example, one might think that all automobiles are exactly the same but for their sizes and colors) or artificial examples. This where the distributional constraints might provide the partitioning.

The contextual determination of the semantic congruity effect (Cech and Shoben, 1985) seems to be evidence for the fact that we do not have a stable partitioning of quantities into set categories in the long-term memory.

The experiment is described to give a better feel of the problem here, I’d like to gather evidence from literature for intuitions regarding how it will turn out to be, but no luck till now. I did ask Linda Smith, and she was unaware of any such experiment; and Dedre had earlier asked Satoru Suzuki, who had the same answer.

It is for example, important to distinguish between stipend, and per capita income of a country, even though the units are the same.

Comic books, mythology, and fantasy, for example, has the freedom to relax this constraint – a character can be arbitrarily strong, large, small or be able to fly, even though the physical design of the character might not be able to support it.

23

Carving nature at her joints: Representations of quantity and t

Praveen Paritosh

2004-02-09T16:57:00