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The Cognitive Symmetry Engine: An Active Approach to Knowledge Abstract
• Interaction knowledge; i.e., knowledge of the relation of self to the external world and the possible interac- Knowledge acquisition and representation is an integral part of an active cognitive system, and such systems arecontinuously updating their knowledge of their own embod- iment, the external world and the relation between the two.
Given a stream of control commands and sensory data, we
want to understand how the three types of knowledge are and sensory data, how can the three types of acquired. We propose that a core set of abstract symmetry knowledge (self, world, self-world) be achieved? theories constrain this process, and give an in-depth exam- We base the answer to this question on the following as- ple for the case of autonomous agent sensorimotor recon- • Control (action) precedes perception during learning.
1. Introduction
• Control results from perception during performance.
Knowledge representation has been identified as a centralissue for autonomous robot development. This involves • Experiential knowledge is embedded in control- perception loops by maintaining appropriate invari- • Symbolic knowledge is derived from experiential knowledge by finding appropriate invariants.
• Inter-robot sharing occurs at the symbolic level, but However, this implies a somewhat static view of knowledge, requires shared experiential knowledge.
whereas knowledge is just one aspect of a cognitive system.
Cognition is an active process which attempts to produce • Symmetry operators are key in all aspects of cognition, the appropriate response in a given context. Note that this is and in particular, for knowledge acquisition, represen- very different from the common framework of simply pro- ducing a scene description given a set of percepts.
Examples of previous work in this direction include Roy [12], Granlund [2], and Krueger et al. [7]. Roy proposes Self-knowledge; i.e., how the various actuators and a method to ground language in action and perception.
sensors of the robot platform are related and act on the His agent schemas interpret signs (i.e., provide a compu- tational semiotics) in order to guide action, and he posits • World knowledge; i.e., information about objects and specific processing channels that detect patterns.
properties of the world that exist independently of the schemas combine motor action and sensory processing to ground knowledge in terms of five types of projections of beliefs, sensory input, action, and categorizers and inten- with other agents. It is necessary to create new theories and tions. Granlund describes a cognitive computer vision ar- realizations for cognitive organization in complex, real-time chitecture that provides an interpretation by linking actions systems that consist of interacting domain specific agents, and percepts; that is, it relates percepts of objects to ac- each with rich internal state and complex actions in order tions on them. He further argues that a more symbolic to facilitate the construction of effectively organized cogni- structure is appropriate for communication and is abstracted tive infrastructure. The proposed technical basis for this is away from the specific motor-percept loop. This framework symmetry operators used in perception, representation and has strongly influenced our approach. Object Action Com- plexes (OACs) [7] are generalized from instantiated state The Domain Theory Hypothesis: Semantic cognitive con-
transition fragments (ISTFs) observed during low-level exe- tent may be effectively discovered by restricting controller cution. This allows the representation of actions and objects solutions to be models of specific symmetry theories intrin- in tight correspondence. All these approaches include ac- tion as a key element of the learning process and the result-ing knowledge representation. We demonstrate here how The Domain Theory predicates: (1) a representation of symmetry relations can constrain this learning process (and an innate theory and inference rules for the theory, (2) a could be exploited in any of these approaches) and apply perceptual mechanism to determine elements of a set and this to the acquisition of self-knowledge (sensorimotor re- operators on the set, (3) a mechanism to determine that the set and its operators are a model of the innate theory, and(4) mechanisms to allow the exploitation of the model in 2. Symmetry
Early on, Pierce [11] described an approach to learning We propose to explore the thesis that symmetry theory a model of the sensor set of an autonomous agent. Features provides key organizing principles for cognitive architec- are defined in terms of raw sensor data, and feature opera- tures. As described by Vernon et al. [13], cognition ”can tors are defined which map features to features. The goal is be viewed as a process by which the system achieves ro- to construct a perceptual system for this structure. One of bust, adaptive, anticipatory, autonomous behavior, entail- the fundamental feature operators is the grouping operator ing embodied perception and action.” Their survey consid- which assigns features to a group if they are similar. This ers two basic alternative approaches to cognition: cogni- work was extended to spatio-visual exploration in a series of tivist (physical symbol systems) and emergent (dynamical papers [8, 11]. For a detailed critique of Pierce’s work, see systems), where the cognitivist paradigm is more closely [4]. Olsson extended this work as well [9]. He used infor- aligned with disembodied symbol manipulation and knowl- mation theoretic measures for sensorimotor reconstruction, edge representation based on a priori models, and the emer- and no innate knowledge of physical phenomena nor the gent paradigm purports dynamic skill construction in re- sensors is assumed. Like Pierce, Olsson uses random move- sponse to perturbations to the embodiment. An important ments to build the representation and learns the effect of aspect of this discussion which concerns us here is that actions on sensors to perform visually guided movements.
raised by Krichmar and Edelman [6]: ”the system should The major contributions are the analysis of information the- be able to effect perceptual categorization: i.e., to organize oretic measures and motion flow. O’Regan and No¨e [10] unlabeled sensory signals of all modalities into categories use the term sensorimotor contingencies and give an algo- without a priori knowledge or external instruction.” We ad- rithm which can determine the dimension of the space of the dress this issue and propose that certain fundamental a priori environment by ”analyzing the laws that link motor outputs knowledge about symmetries is vital to this function.
to sensor inputs”; their mathematical formulation is elegant.
Our goal is to advance the state of the art in embod- ied cognitive systems. The requirement for cognitive abil- 3. Symmetry in Sensorimotor Recon-
ity is ubiquitous, and its achievement is an essential step struction
for autonomous mental development. At its root, a cog-nitive architecture is a structural commitment to processes Symmetry [15] plays a deep role in our understanding of and representations that permit adaptive control in an op- the world in that it addresses key issues of invariance, and as erating environment that cannot be modeled completely a noted by Viana [14]: “Symmetry provides a set of rules with priori. A cognitive agent optimizes its behavior to achieve which we may describe certain regularities among experi- an objective efficiently by finding models that resolve hid- mental objects.” By determining operators which leave cer- den state information and that help it to predict the future tain aspects of state invariant, it is possible to either identify under a variety of real-world situations. These processes similar objects or to maintain specific constraints while per- involve monitoring, exploration, logic, and communication forming other operations (e.g., move forward while main- taining a constant distance from a wall). A symmetry de- no known (predictable) relation between the actuation se- fines an invariant. The simplest invariant is identity. This quence and the sensor values, and (3) the simultaneous ac- can apply to an individual item, i.e., a thing is itself, or to tuation of multiple actuators confuses the relationship be- a set of similar objects. In general, an invariant is defined by a transformation under which one object is mapped to To better understand sensorimotor effects, a systems ap- proach is helpful. That is, rather than giving random control Invariants are very useful things to recognize, and we sequences and trying to decipher what happens, it is more propose that various types of invariant operators provide a effective to hypothesize what the actuator is (given limited basis for cognitive functions, and that it is also useful to choices) and then provide control inputs for which the ef- have processes that attempt to discover invariance relations fects are known. Such hypotheses can be tested as part of among sensorimotor data and subsequently processed ver- the developmental process. The basic types of control that can be applied include: none, impulse, constant, step, lin-ear, periodic, or other (e.g., random).
3.1. Symmetry Detection in Signals
Next, consider sensors. Some may be time-dependent (e.g., energy level), while others may depend on the envi- Assume a set of sensors, S = {Si, i = 1 . . . nS } each of ronment (e.g., range sensors). Thus, it may be possible to which produces a finite sequence of indexed sense data val- classify ideal (noiseless) sensors into time-dependent and ues, Sij where i gives the sensor index and j gives an or- time-independent by applying no actuation and looking to dinal temporal index, and a set of actuators, A = {Ai, i =1 . . . n see which sensor signals are not constant (this assumes the A} each of which has a finite length associated con- spatial environment does not change). Therefore, it may be trol signal, Aij, where i is the actuator index and j is a more useful to not actuate the system, and then classify sen- temporal ordinal index of the control values.
sors based on their variance properties. That is, in realistic We are interested in determining the similarity of senso- (with noise) scenarios, it may be possible to group sensors rimotor signals. Thus, the type of each sensor as well as the relation to motor control actions play a role. It is quitepossible that knowledge of the physical phenomenon that Consider Pierce’s sensorimotor reconstruction process.
stimulates a sensor may also be exploited to help determine If realistic noise models are included, the four types of sen- the structure of the sensor system and its relation to motor sors in his experiments (range, broken range, bearing and energy) can all be correctly grouped with no motion at all.
We suppose that certain 1D signal classes are important (This assumes some energy loss occurs to run the sensors.) and are known a priori to the agent (i.e., that there are pro- All this can be determined just using the equals symmetry cesses for identifying signals of these types). The basic sig- operator (identity) and the means and variances of the sen- nals are: (1) constant: y = a (for some fixed constant a), (2) linear: y = at + b (function of time index), (3) periodic:has period P and the most significant Fourier coefficientsC, (4) and Gaussian: sample from Gaussian distribution 3.3. Exploiting Actuation
with mean µ and variance σ2. Thus, a first level symmetry Of course, actuation can help understand the structure of is one that characterizes a single signal as belonging to one the sensorimotor system. For example, consider what can be determined by simply rotating a two-wheeled robot thathas a set of 22 range sensors arranged equi-spaced on a cir- 3.2. Sensorimotor Reconstruction
cle. Assume that the control signal results in a slow rotation The sensorimotor reconstruction process consists of the fol- parallel to the plane of robot motion (i.e., each range sen- lowing steps: (1) perform actuation command sequences, sor moves through a small angle to produce its next sample) (2) record sensor data, (3) determine sensor equivalence and rotates more than 2π radians. Then each range sensor classes, and (4) determine sensor-actuator relations. An ad- produces a data sequence that is a shifted version of each of ditional criterion is to make this process as efficient as pos- the others – i.e., there is a translation symmetry (of periodic signals) between each pair. The general problem is then: Olsson, Pierce and others produce sensor data by apply- ing random values to the actuators for some preset amount General Symmetry Transform Discovery
of time, and record the sensor sequences, and then look for Problem: Given two sensors, S1 and S2, with
similarities in those sequences. This has several problems: data sequences T1 and T2, find a symmetry op- (1) there is no guarantee that random movements will result in sensor data that characterizes similar sensors, (2) there is Symmetry-based Sensorimotor Recon-
Correctness Measure: Given (1) a set of sensors, {Si, i =
struction Algorithm
1 : n} (2) a correct grouping matrix, G, where G is an nby n binary valued matrix with G(i, j) = 1 if sensors S Using the symmetries described above, we propose the fol- j are in the same group and G(i, j) = 0 otherwise, and (3) H an n by n binary matrix which is the result of the Algorithm SBSG: Symmetry-based Sensor Grouping
grouping generator, then the grouping correctness measure Sensor Grouping with Noise (No actuation)
This algorithm assumes that sensors have an associated Assume that the sensors each have a statistical noise model.
noise. Note that this requires no actuation and assumes the The real-valued range sensors have Gaussian noise sampled environment does not change. Finally, the similarity test for the above algorithm depends on the agent embodiment.
binary-valued bearing sensors have salt and pepper noise Algorithm SBSR: Symmetry-based Sensorimotor Re-
where the correct value is flipped p% of the time.
construction
nally, the energy sensor has Gaussian noise also sampled from N (0, 1). (The broken range sensor returns a constant Based on this, the grouping correctness results are given in Figure 1. Sensor data sampling time was varied from 1 to 20 seconds for binary noise of 5%, 10% and 25%, and Gaussian variance values of 0.1, 1, and 10. Ten trials were run for each case and the means are shown in the figure.
As can be seen, perfect sensor grouping is achieved after This determines the relative distance (in actuation units) be- 20 seconds without any actuation cost. Previous methods tween sensors. E.g., for a set of equi-spaced range sensors, required driving both wheels for a longer time and they cost about 30ka/s more in energy than our method (ka/s is theactuation to sensing cost ratio).
4. Comparison to Pierce’s Work
A set of simulation experiments are described in Chapter 4 of Pierce’s dissertation [11]. The first involves a mobile agent with a set of range sensors, a power level sensor, and four compass sensors. The sensors are grouped and then a structural layout in 2D is determined. The second experi- ment concerns an array of photoreceptors. Here we examine the first experiment, and in particular, the group generator.
4.1. Symmetry-based Grouping Operator
Any simulation experiment should carefully state the ques- tions to be answered by the experiment and attempt to set up a valid statistical framework. In addition, the sensitivity of the answer to essential parameters needs to be examined.
Figure 1: Grouping Correctness vs. Number of Samples; The question to be answered is: Grouping Correctness:
left to right are for binary salt and pepper noise of 5%, 10%, What is the correctness performance of the proposed group- and 25%; curves for 0.1, 1.0, and 10.0 variance are given in ing generator? This requires a definition of correctness for performance and we propose the following (for more de-tails, see [4]): Sensor Grouping (Actuated)
Given a set of sensors that characterize the group opera-tion nature of an actuator (in this case rotation), the sensorscan be grouped based on the fact that similar sensors pro-duce data that has a translation symmetry along the tempo-ral axis. Figure 2 shows representative data for the rangeand compass sensors. The simple determination of a trans-lation symmetry between signals allows both grouping (i.e.,the signals match well at some time offset), and the angu-lar difference between the sensors (given by the toffset atwhich the symmetry occurs); toffset is proportional to theangle between the the sensors in terms of actuation units.
Figure 3: One of the 200 Static Images.
grouped the pixel signals separately from the microphone due to the difference in their variance properties.
Actuated Experiment
We also took a set of images by rotating the camera by onedegree for 360 degrees. Domain translation symmetry al- Figure 2: Sensor data showing translation symmetry: Row lows the identification of all the pixel signals along a row 1 shows sensors 1, 2, and 13; Row 2 shows compass sensors as similar to each other (i.e., they are all in the plane of the rotation). Due to the translation amount, the offset betweenthe signals is also discovered.
5. Physical Experiment
6. Conclusions and Future Work
We have performed experiments with physical sensors tovalidate the proposed approach. Data was taken for boththe static case (no actuation) and the actuated case (camera We propose symmetry theory as a basis for sensorimotor re- construction in embodied cognitive agents and have shownthat this allows the identification of structure with simpleand elegant algorithms which are very efficient and result in Unactuated Experiment
grounded self-knowledge. The exploitation of noise struc-ture in the sensors allows unactuated grouping of the sen- Two sensors were used in this experiment: a camera and a sors, and a simple one actuator rotation permits the recov- microphone. The camera was set up in an office and a se- ery of the spatial arrangement of the sensors. This method quence of 200 images was taken at a 10Hz rate. Figure 3 was shown to hold for physical sensors as well.
shows one of these images. The 25x25 center set of pixelsfrom the image comprise a set of 625 pixel signals each of Next steps include the acquisition of world knowledge length 200. An example trace and its histogram are given and self-world interaction knowledge. In addition, the ab- in Figure 4. As can be seen, this is qualitatively a Gaussian straction of this knowledge to allow sharing is also neces- sample. Figure 5 shows a 200 sequence signal of micro- sary. We intend to explore these in the framework of Object phone data, and its histogram which also looks Gaussian.
Acknowledgments
The application of our symmetry detectors classified This material is based upon work supported by the Na- all pixel and microphone signals as Gaussian signals, and tional Science Foundation under Grant No. 1021038.
Figure 4: Trace and Histogram of the 200 Pixel Values of Figure 5: Trace and Histogram of the 200 Amplitude Values References
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A Formal Definition of Object-Action Complexes and

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