C. Taylor

D. Friedlander

E. Stabler

G.M. Kobele

J. Riggle

R. Brooks

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UCLAPapersTitleInduction of Prototypes in a Robotic Setting Using Local Search MDLPermalinkhttps://escholarship.org/uc/item/9jz9s82rAuthorsG.M. KobeleJ. RiggleR. Brookset al.Publication Date2004 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaInduction of Prototypes in a Robotic Setting using Local Search MDLGregory M. Kobele∗ Jason Riggle∗ Richard Brooks†David Friedlander‡ Charles Taylor§ Edward Stabler∗AbstractCategorizing objects sets the stage for more ad-vanced interactions with the environment. MinimumDescription Length learning provides a framework inwhich to investigate processes by which concept learn-ing might take place. Importantly, the concepts so ac-quired can be viewed as having a prototype structure- the concepts may apply to one object better thanto another. We ground our discussion in a real-worldsetting - objects to categorize are sensor readings ofthe behaviours of two mobile robots.1 IntroductionNatural and artificial agents are confronted withthe problem of constructing models of their environ-ments, thereby imposing order on an otherwise chaoticstream of sensory data. An important step in this di-rection is to recognize when a novel sense datum issufficiently similar to a previously seen one to makea similar response adaptive. This is the problem ofcategorization - how does one decide which proper-ties of sense data are relevant, and which are acciden-tal. Obviously, treating all properties as relevant leadsto no generalization at all, and treating all proper-ties as accidental leads to maladaptive behaviour (un-less the environment is perverse). This problem hasbeen explored in various settings using Minimal De-scription Length (MDL) concept learning, where thenotions of concise encoding of the hypothesis and ofefficient encoding of the observed data impose compet-ing pressures (generalizing by minimizing theory sizewhile still being informative about the data), wherethe measures of theory size and empirical informa-tiveness can be adjusted to fit the domain. We havepreviously applied this framework both to the clas-∗Department of Linguistics, UCLA. Los Angeles, CA 90095†Distributed Systems Department, The Pennsylvania StateUniversity, State College, PA 16804‡Informatics Department, The Pennsylvania State Univer-sity, State College, PA 16804§Department of Organismic Biology, Evolution and Ecology,UCLA. Los Angeles, CA 90095sification of simple robotic behaviour [1], as well asto language evolution [4, 5]. In these previous stud-ies concepts (or languages) were represented as deter-ministic finite state automata, and percepts were fi-nite strings over a fixed finite alphabet. Our currentstudy extends the previous ones in two ways. First, weenrich the hypothesis space of our agents to includenon-deterministic automata. Although deterministicand non-deterministic machines are equivalent in ex-pressive power, enriching the agent’s hypothesis spacein this manner has repercussions for the results ofthe learning process. Second, we exploit a particu-lar property of the MDL setup, yielding a cognitivelyinteresting prototype structure of concepts which al-lows for degrees of membership in a category. Onenovel aspect of the prototype structure presented hereis that the degrees of category membership are a holis-tic and emergent property of the conceptual space.That is, whether a perception is a “good example”of a particular category depends on what other cate-gories are available. In our setting, concepts are repre-sented as finite automata, and perceptions as stringsover a fixed finite alphabet. Given a space of con-cepts C, a perception p is a good example of a cate-gory C ∈ C just in case p ∈ L(C) and for any otherC ′ ∈ C with p ∈ L(C ′), the cost of encoding p in Cis less than or equal to the cost of encoding p in C ′(enc(p, C) ≤ enc(p, C ′)).We present a simple example in which a learner isfaced with the task of categorizing the behaviour oftwo robots, given by means of sensors in the robots’environment. In the learning stage each set of sensordata is labeled as an instance of a concept (e.g. ’ran-dom walking’, ’wall following’, ’chasing’, . . . ). Ourrobot learner then extracts the salient structure fromeach training set, possibly generalizing its theory of thetraining sets to include as yet unseen sensor inputs.2 Minimum Description LengthThe basic idea of the MDL framework is that thebest hypothesis to adopt when confronted with a set ofdata is the one that best navigates between the charyb-dis of theory complexity and the scylla of faithfullnessto the data set. In other words, we want the sim-plest theory with the greatest predictive power. Herewe adopt the length of the binary encoding of a ma-chine as our measure of theory complexity, and thecost of specifying a parse of the input (an acceptingpath though the machine) as our measue of faithful-ness to the data. The intuition is that the cost ofencoding a string is the amount of information oneneeds to exactly reconstruct it. This intuition moti-vates the following definition: as there might be morethan one parse of a string through a given machine,we define the cost of encoding a string in a machine asthe cost of the least path through the machine readingthe string.Formally, we define the cost of the binary encodingof a machine M = 〈Q,Σ, S, F, δ〉, where Q is a finiteset of states, Σ is a finite vocabulary, S, F ⊆ Q are thestart and final states, respectively, and δ ⊆ Q×Σε →2Q is the transition function, to besize(M) = size(δ) + size(S) + size(F )wheresize(δ) = |δ| (2log2 |Q|+ log2(|Σ|+ 1))and for Q0 ⊆ Q,size(Q0) = |Q0| (log2 |Q|)Identifying a parse with a sequence of arcs π =a1a2 . . . an in M , we define the cost of a parse cost(π)to be the sum of the number of bits needed to specifythe path through the machine. Setting zi to be thenumber of arcs with the same source as ai,cost(π,M) = Σni=1ziand the cost of encoding a string σ in M is theamount of information needed to recover σ given M ;for Πσ the set of parses of σ in M ,enc(σ,M) = min({cost(π)|π ∈ Πσ})Given a set of training data T , we first constructthe prefix-tree acceptor P (T ) for T . Next, we con-struct the set of machines formed by merging twostates in P (T ), greedily keeping only those machinesM which minimize size(M) + Σσ∈T enc(σ,M). Wecontinue this process of merging states and discardingmachines which are suboptimal until we reach a fixedpoint.3 Behavior Recognition with TwoRobotsThe floor was divided into four rectangular sectionsand each section was assigned a letter. At any instant,the positions of the two robots can be represented by apair of letters. The position of the robots in a sectionwas determined by two sensors placed in the room.The sensor setup is shown in figure 1.Figure 1: Experimental SetupEach robotic configuration (i.e. pair of letters in-dicating the positions of the first and second robots)is taken as a primitive symbol in a larger alphabet (of16 symbols), into which the sensor readings are ulti-mately translated. Sensor readings were taken every.2 seconds. As the machines induced by our algorithmfor even a five minute time slice were too large to fitreadably in the space allotted, we give simplified ex-amples in the next sections.4 PrototypesPrototype theory [3] is a theory of concepts whichis intended to account for behavioural asymmetriesin identifying objects as falling under particular con-cepts, as well as regularities in ‘prototypicality judge-ments’ across individuals. A model of this type ofdata, should minimally provide an ordering relationacross entities participating in the concept. Oshersonand Smith [2] challenged the sufficiency of this mini-mal criterion of adequacy. They argued that fuzzy settheory [6] does not provide an adequate model of thedata, even though fuzzy set membership does providejust such an ordering over individuals. They pointout that the theory of concepts should be able to ac-count for the compositional nature of concepts - thatis, that some concepts are formed from others. Theyclaim that concept compositionality is not faithfullyreflected in Zadeh’s fuzzy set operators. As a simpleexample they chose the concept striped apple, whichthey took to be composed of the (more) atomic con-cepts of striped and apple via concept conjunction. Onthe assumption that a prototypical striped apple isneither a prototypical striped thing nor a prototypi-cal apple, this fact is not well-modeled using Zadeh’sfuzzy set intersection operator, which identifies the de-gree of membership of an object in the conjunction oftwo fuzzy sets with the least degree of membership ofthat object in the conjuncts.In the model presented herein, the cost of encod-ing a percept into a machine yields an ordering acrossparticipants of a concept, with p being a more pro-totypical M than p′ iff enc(p,M) < enc(p′,M). Ifwe take the standard operations of machine intersec-tion (cross-product of machines followed by removal ofdead states and arcs) and machine union (creation ofa new start state, and empty transitions to the startstates of each concept) to model the binary conceptualconnectives ‘and’ and ‘or’, we see that in the ‘or’ caseour model and the fuzzy set model (in which an ele-ment is in the union of two sets to the greater of thedegrees in which it is in each of the individual sets)coincide (recall that in the fuzzy set model, the higherthe degree of membership, the more prototypical, andin our model, the lower the cost of the encoding, themore prototypical). As shown below (in figures 2, 3and 4), the removal of dead states (and, more impor-tantly, of the arcs to them) after machine intersectioncan increase the goodness of fit of the conjoined con-cept to the instance. This allows for a prototypicalstriped apple to be a better example of the conceptstriped apple than of either striped or apple.Using the floor layout described in figure 1, figure2 represents the concept of a single robot remainingarbitrarily long in section a, then moving to b for twosensor readings, and finally possibly going to c andstaying there.Figure 3 represents the concept of a single robotremaining in a for two sensor readings, then possibly1a2b3cbFigure 2: machine M1 recognizes L1 = a∗bbc∗. Thestrings with the cheapest encoding (and highest pro-totypicality) are those in bbc∗.4 5a6b7ccaFigure 3: machine M2 recognizes L2 = aab∗c∗. Themost prototypical strings are in aac∗.moving to b and remaining there arbitrarily long, andfinally possibly going to c and staying there.Finally, figure 4 represents the concept which is theconjunction of the two previous ones; namely, thatof a single robot remaining in a for two sensor read-ings, then moving to b and staying there a like amountof time, and then possibly moving to c and stayingthere indefinitely. The observation aabb, which is ofthe robot staying in a for two turns before movingto b and staying there for two turns, participates inall three concepts, although it is cheapest to describe(and thus most prototypical) in M3 (see figure 5) asdesired.5 SummaryWe have presented a model of categorization whichresults in a natural ordering between instances of aconcept, as well as a way to determine which conceptbest fits a new instance. This latter point might beextremely valuable in practice, especially as having away to discriminate between concepts which apply tothe given instance may allow for a more adaptive re-sponse. The model also seems able to surmount someintuitive difficulties previous models of concepts hadin regard to the compositional nature of conceptualstructure.AcknowledgementsThis work was supported by the UCLA Center forEmbedded Network Sensors, the Defense Advance Re-search Projects Agency (DARPA), administered by14 15a3626b37cc16abFigure 4: machine M3 = M1 × M2 recognizes L3 =L1 ∩ L2 = aabbc∗. The most prototypical strings arethose in aabbc∗.EncodingM1 2M2 2M3 0Figure 5: A comparison between the costs of encodingthe string aabb in each machine.the Army Research Office under Emergent Surveil-lance Plexus MURI Award No. DAAD19-01-1-0504,and DARPA MURI award administered by the USAirforce No. F49620-01-1-0361. Any opinions, find-ings, and conclusions or recommendations expressedin this publication are those of the authors and do notnecessarily reflect the views of the sponsoring agencies.References[1] Yoosook Lee, Jason Riggle, Travis C. Collier, EdwardStabler, and Charles E. Taylor. Adaptive communica-tion among collaborative agents: Preliminary resultsfrom symbol grounding. In Masanori Sugisaka andHiroshi Tanaka, editors, Proceedings of the Eighth In-ternational Symposium on Artificial Life and Robotics,pages 149–155, 2003.[2] D. N. Osherson and E. E. Smith. On the adequacy ofprototype theory as a theory of concepts. Cognition,11:237–262, 1981.[3] E. Rosch. On the internal structure of perceptual andsemantic categories. In T. E. Moore, editor, Cognitivedevelopment and the acquisition of language. AcademicPress, New York, 1973.[4] Tracy Teal, Daniel Albro, Edward Stabler, andCharles E. Taylor. Compression and adaptation. InJ. Nicoud D. Floreano and F. Mondada, editors,ECAL99, pages 709–719, Berlin, 1999. Springer-Verlag.[5] Tracy K. Teal and Charles E. Taylor. Effects of acqui-sition on language evolution. Artificial Life, 6(2):129–143, Spring 2000.[6] L. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965.

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Induction of Prototypes in a Robotic Setting Using Local Search MDL

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Induction of Prototypes in a Robotic Setting Using Local Search MDL

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