WO2005067600A2 - Systems and methods for using statistical techniques to reason with noisy data - Google Patents

Systems and methods for using statistical techniques to reason with noisy data Download PDF

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WO2005067600A2
WO2005067600A2 PCT/US2005/000299 US2005000299W WO2005067600A2 WO 2005067600 A2 WO2005067600 A2 WO 2005067600A2 US 2005000299 W US2005000299 W US 2005000299W WO 2005067600 A2 WO2005067600 A2 WO 2005067600A2
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knowledge base
word
determining
knowledge
inferencing
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French (fr)
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WO2005067600A3 (en
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Rakesh Gupta
Mykel J. Kochenderfer
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Honda Motor Co Ltd
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Honda Motor Co Ltd
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    • GPHYSICS
    • G06COMPUTING OR CALCULATING; COUNTING
    • G06NCOMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N5/00Computing arrangements using knowledge-based models
    • G06N5/04Inference or reasoning models
    • GPHYSICS
    • G06COMPUTING OR CALCULATING; COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F40/00Handling natural language data
    • G06F40/20Natural language analysis
    • G06F40/237Lexical tools
    • GPHYSICS
    • G10MUSICAL INSTRUMENTS; ACOUSTICS
    • G10LSPEECH ANALYSIS TECHNIQUES OR SPEECH SYNTHESIS; SPEECH RECOGNITION; SPEECH OR VOICE PROCESSING TECHNIQUES; SPEECH OR AUDIO CODING OR DECODING
    • G10L15/00Speech recognition
    • G10L15/26Speech to text systems
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02PCLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10TECHNICAL SUBJECTS COVERED BY FORMER USPC
    • Y10STECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y10S707/00Data processing: database and file management or data structures
    • Y10S707/99941Database schema or data structure
    • Y10S707/99943Generating database or data structure, e.g. via user interface

Definitions

  • the present invention relates to automated reasoning. More particularly, the present invention relates to using statistical techniques in order to enable automated reasoning with noisy data.
  • Robots will soon be expected to accomplish tasks within their environments and to satisfy the perceived requests and desires of their users. Determining these desires and reasoning about how they can be accomplished requires common sense.
  • One of the main challenges in the field of robotics is making robots more intelligent by endowing them with common sense. In order for robots to have common sense, they need some amount of general knowledge to use as a basis. They can then reason from this basis to accomplish their tasks and interact with their world intelligently.
  • Logical reasoning a topic within the field of artificial intelligence, is concerned with using computers to perform logical reasoning.
  • Logical reasoning such as deduction, consists of drawing conclusions from facts.
  • Conclusions are formed by applying rules (e.g., implications) to facts. This is also known as inferencing.
  • Logical reasoning systems comprise implications, a basis of facts, and control mechanisms for applying implications to these facts.
  • facts and implications are stored in a "knowledge base.” Facts are established by, for example, user input (e.g., by typing in "The stove is hot") or sensor input (e.g., by placing a heat sensor near the stove).
  • Implications are established by, for example, user input (e.g., by typing in "An item in a refrigerator is cold") or machine learning.
  • an implication must require that its conclusion follow inevitably from the fact from which it is drawn.
  • implications can be applied to both facts (originally input into the system) and conclusions (generated by implications). Because of the interactions between facts, implications, and conclusions, it is important that the initially-input facts and implications be correct. If the facts or implications are false or inconsistent with one another, the resulting conclusions may be invalid.
  • Compilation of a knowledge base can comprise cleaning, generalizing, and compacting the data in the knowledge base. Compilation can decrease the amount of resources (time, memory, etc.) required by the reasoning process.
  • a robot can use the processed knowledge base to perform many different types of tasks in many different environments.
  • the robot is a mobile robot used in a home or office environment. Possible tasks include, for example, answering a query, determining a course of action that is designed to achieve a particular goal, and determining its own location.
  • Possible tasks include, for example, answering a query, determining a course of action that is designed to achieve a particular goal, and determining its own location.
  • Figure 1 illustrates a block diagram of a preferred embodiment of an apparatus for reasoning when noisy data is present.
  • Figure 2 illustrates a more detailed block diagram of the contents of the memory unit in Figure 1.
  • Figure 3 illustrates a method for reasoning when noisy data is present.
  • Figure 4 illustrates a method for processing a knowledge base.
  • Figure 5 illustrates a method for determining a location using Bayesian inferencing.
  • DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS [0018]
  • numerous specific details are set forth in order to provide a thorough understanding of the invention. It will be apparent, however, to one skilled in the art that the invention can be practiced without these specific details. In other instances, structures and devices are shown in block diagram form in order to avoid obscuring the invention.
  • This apparatus is specially constructed for the required purposes, or it comprises a general-purpose computer selectively activated or reconfigured by a computer program stored in the computer.
  • a computer program is stored in a computer readable storage medium, such as, but not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, magnetic or optical cards, or any type of media suitable for storing electronic instructions, and each coupled to a computer system bus.
  • Figure 1 illustrates a block diagram of a preferred embodiment of an apparatus for reasoning when noisy data is present.
  • Reasoning apparatus 100 preferably includes a processor 110, a main memory 120, a data storage device 130, and an input/output controller 180, all of which are communicatively coupled to a system bus 140.
  • Reasoning apparatus 100 can be, for example, a computer or a robot.
  • Reasoning apparatus 100 uses statistical reasoning to enable it to reason when noisy data is present.
  • Statistical reasoning unlike logical reasoning, includes the notion of uncertainty. This uncertainty is expressed through the use of statistics, such as latent semantic analysis, and probability theory, such as Bayesian networks.
  • Statistical reasoning has traditionally been used to work with large text documents in the fields of data retrieval and data mining.
  • Reasoning apparatus 100 makes the knowledge base unambiguously reflect consensus knowledge. Reasoning apparatus 100 can then reason from the knowledge base without creating invalid results. Statistical reasoning techniques have not been used in this area in the past. One possible reason for this is that in the past, the available data has not been dense enough to leverage statistical techniques.
  • Reasoning apparatus 100 also uses statistical methods to process queries and commands. This results in reasoning apparatus 100 being able to handle tasks that contain words that are not in the knowledge base.
  • Processor 110 processes data signals and comprises various computing architectures including a complex instruction set computer (CISC) architecture, a reduced instruction set computer (RISC) architecture, or an architecture implementing a combination of instruction sets. Although only a single processor is shown in Figure 1, multiple processors may be included.
  • CISC complex instruction set computer
  • RISC reduced instruction set computer
  • Main memory 120 stores instructions and/or data that are executed by processor 110.
  • the instructions and/or data comprise code for performing any and/or all of the techniques described herein.
  • Main memory 120 is preferably a dynamic random access memory (DRAM) device, a static random access memory (SRAM) device, or some other memory device known in the art.
  • DRAM dynamic random access memory
  • SRAM static random access memory
  • Data storage device 130 stores data and instructions for processor 110 and comprises one or more devices including a hard disk drive, a floppy disk drive, a CD-ROM device, a DND-ROM device, a DND-RAM device, a DND-RW device, a flash memory device, or some other mass storage device known in the art.
  • Network controller 180 links reasoning apparatus 100 to other devices so that reasoning apparatus 100 can communicate with these devices.
  • System bus 140 represents a shared bus for communicating information and data throughout reasoning apparatus 100.
  • System bus 140 represents one or more buses including an industry standard architecture (ISA) bus, a peripheral component interconnect (PCI) bus, a universal serial bus (USB), or some other bus known in the art to provide similar functionality.
  • ISA industry standard architecture
  • PCI peripheral component interconnect
  • USB universal serial bus
  • Additional components that may be coupled to reasoning apparatus 100 through system bus 140 include a display device 150, a keyboard 160, and a cursor control device 170.
  • Display device 150 represents any device equipped to display electronic images and data to a local user or maintainer.
  • Display device 150 is a cathode ray tube (CRT), a liquid crystal display (LCD), or any other similarly equipped display device, screen, or monitor.
  • Keyboard 160 represents an alphanumeric input device coupled to reasoning apparatus 100 to communicate information and command selections to processor 110.
  • Cursor control device 170 represents a user input device equipped to communicate positional data as well as command selections to processor 110.
  • Cursor control device 170 includes a mouse, a trackball, a stylus, a pen, cursor direction keys, or other mechanisms to cause movement of a cursor.
  • reasoning apparatus 100 includes more or fewer components than those shown in Figure 1 without departing from the spirit and scope of the present invention.
  • reasoning apparatus 100 may include additional memory, such as, for example, a first or second level cache or one or more application specific integrated circuits (ASICs).
  • ASICs application specific integrated circuits
  • reasoning apparatus 100 may be comprised solely of ASICs.
  • components may be coupled to reasoning apparatus 100 including, for example, image scanning devices, digital still or video cameras, or other devices that may or may not be equipped to capture and/or download electronic data to/from reasoning apparatus 100.
  • FIG. 2 illustrates a more detailed block diagram of the contents of the memory unit in Figure 1.
  • memory unit 120 comprises several code modules for reasoning when noisy data is present.
  • the code modules in memory unit 120 include a main program 200, knowledge base processing program 205, inferencing program 210, spell check module 215, stopword module 220, stemming module 225, latent semantic analysis module 230, WordNet module 235, statistics generation module 240, and knowledge base 245.
  • All code modules 205, 210, 215, 220, 225, 230, 235, 240, 245 are communicatively coupled to main program 200.
  • Main program 200 centrally controls the operation and process flow of reasoning apparatus 100, transmitting instructions and data to as well as receiving data from each code module 205, 210, 215, 220, 225, 230, 235, 240, 245. Details of its operation will be discussed below with reference to Figure 3.
  • Knowledge base processing program 205 processes a knowledge base in order to make it self-consistent and also compiles a knowledge base. Knowledge base processing program 205 accomplishes this by using spell check module 215, stopword module 220, stemming module 225, and latent semantic analysis module 230.
  • Knowledge base 245 stores a knowledge base comprising facts and implications. As implications are applied to facts, the resulting conclusions are also stored in knowledge base 245.
  • Spell check module 215 checks and corrects the spelling of words. Stopword module 220 removes particular words from a collection of words. Stemming module 225 maps word variants to the same word. For example, "cooking", “cooked”, and “cooks” would be mapped to "cook”. Latent semantic analysis module 230 performs latent semantic analysis on a collection of implications to find the most relevant implications. WordNet module 235 compacts data. WordNet module 235 also enables reasoning about words that do not exist in a knowledge base. Inferencing program 210 uses facts and implications stored in a knowledge base and generates conclusions through inferencing. Statistics generation module 240 generates statistics based on data in a knowledge base.
  • Code modules 200-245 will be further discussed below with reference to figures 3 and 4. 2.
  • Usage of Reasoning Apparatus [0040] As noted above, reasoning apparatus 100 can be used in many different contexts to perform many different tasks. Figure 3 illustrates a method for reasoning when noisy data is present. In one embodiment, this reasoning is based on common sense knowledge.
  • main program 200 begins 300. Main program 200 then loads
  • a statement is a relation based on a real-world object.
  • o is the object
  • p is a property of the object (an adjective or attribute).
  • one statement is (coffee, heated).
  • a statement can describe the current state of an object ("the coffee is heated”) or it can describe the desired state of an object ("the coffee should be heated”).
  • a statement can also represent an action ("do something to make the coffee heated”).
  • the variable I denotes the set of all desires implications in the knowledge base 245.
  • variable S denotes the set of all statements that are true in the current state of the world.
  • the data is loaded into a MySQLTM database from MySQL
  • the Statements, Causes, and Desires relations enable inferencing.
  • the Objects, Uses, Rooms, Proximity, and Locations relations are used to accomplish specific tasks. These tasks include goal achievement and parameter determination and will be further discussed below.
  • the data might have been communicated to reasoning apparatus 100 through the input/output controller 180.
  • the data might have been present in data storage device 130.
  • main program 200 processes 310 the knowledge base 245 using statistical methods.
  • One purpose for this processing is to make the knowledge base self-consistent.
  • One way to ensure that a knowledge base is self-consistent is to handcraft it, making sure that each piece of information is valid and checking that it does not conflict with other information in the knowledge base. Since a knowledge base needs to contain a lot of information in order to be useful, handcrafting one can take a lot of time. It would be preferable if the knowledge base could be made self-consistent without requiring a person to review each piece of information manually.
  • Another purpose for this processing is to compile the knowledge base, generalizing and compacting it so that it requires less storage space and reasoning can be performed on it more quickly.
  • Step 310 is detailed in Figure 4.
  • Figure 4 illustrates a method for processing a knowledge base.
  • knowledge base processing program 205 begins 400.
  • Knowledge base processing program 205 checks 405 and corrects the spelling of the words in the knowledge base 245 by using spell check module 215.
  • knowledge base processing program 205 removes 410 common words from the knowledge base 245, such as "a”, "an”, and "the”, by using stopword module 220.
  • knowledge base processing program 205 performs stemming 415 on the words in the knowledge base 245.
  • Steps 405, 410, and 415 are optional. They increase the value of the knowledge base by standardizing the data within it.
  • step 420 knowledge base processing program 205 groups 420 words of similar and related meanings together into notional families.
  • Notional families serve to cluster data and reduce data dimensionality in order to find the most relevant implications.
  • notional families identify a consensus in the knowledge base, and this consensus serves as the reasoning apparatus' common sense.
  • Notional families were first proposed in "The automatic derivation of information retrieval encodement from machine readable text" by H.P. Luhn, Information Retrieval and Machine Translation, 3(2), pp. 1021-1028, 1961, which is hereby incorporated by reference.
  • Latent semantic analysis is a statistical method that indexes phrases or documents to reflect topic similarity. Topic similarity is based on word co-occurrence, a valid indicator of topic relatedness. In latent semantic space, phrases can have high similarity even if they do not share any terms, so long as their terms are semantically similar according to the co-occurrence analysis. In other words, latent semantic analysis addresses the synonymy problem of people using different words to refer to the same concept. Latent semantic analysis results in a unique consequent statement for each antecedent statement and vice versa.
  • latent semantic analysis indexes implications leading to each consequent and then applies these indices via singular value decomposition to find the most relevant implications.
  • the time needed to perform singular value decomposition is exponential based on the rank of the statement by implication matrix and the number of singular values that are computed.
  • Latent semantic analysis is described further in "Indexing by Latent Semantic Analysis” by S.C. Deerwester, S.T. Dumais. T.K. Landauer, G.W. Furnas, and R.A. Harshman, Journal of the American Society of Information Science, 41(6), pp. 391-407, 1990; and "Foundations of Statistical Natural Language Processing” by CD. Manning and H. Schuetze, Chapter: Topics in Information Retrieval: Latent Semantic Indexing, 2001; both of which are hereby incorporated by reference.
  • Spectral clustering methods cluster points using eigenvectors of matrices derived from that data.
  • principal component analysis is to determine the components s ls s 2 , ..., s n that explain the maximum amount of variance possible by n linearly transformed components.
  • Spectral clustering is described further in "A Comparison of Spectral Clustering Algorithms" by D. Nerma and M.
  • knowledge base processing program 205 compiles 425 the knowledge base using WordNet module 235.
  • WordNet® from Princeton University, is a lexical database for the English language. Nouns, verbs, adjectives, and adverbs are organized into synonym sets. Each set represents one underlying lexical concept (the "root word”), and different relations link the sets.
  • Each root word is a hypernym of the related synonyms.
  • Hypernymy is a linguistic term for an is-a relationship. For example, since a fork is silverware, "silverware" is a hypernym of "fork.”
  • WordNet synonym sets involves the root word "can” (as a noun).
  • a first hypernym would be "tin can,” a second hypernym would be “toilet,” and a third hypernym would be “bathroom.”
  • Synonyms in the "tin can” group would include “milk can” and “oil can,” while synonyms in the "bathroom” group would include “restroom” and "washroom.”
  • WordNet module 235 generalizes knowledge in the knowledge base 245 by replacing synonyms with their hypernyms. WordNet module 235 then compacts the knowledge base by associating implications with hypernyms rather than with synonyms. Then, when a synonym is encountered, the hypernym can be determined and the hypernym' s implications can be used.
  • the knowledge base does not have any implications about "fork,” it can use implications about "silverware.” Without the WordNet synonym sets, a "fork” antecedent would not match any implications, and no consequent would be determined.
  • the WordNet synonym sets also save memory by storing each implication once (at the hypernym level), rather than storing each implication multiple times (at the synonym level). Also, better reasoning results can be obtained by first pruning irrelevant hypernyms from the WordNet synonym sets based on the real- world context. [0056] If the synonyms co-occurred in the knowledge base, then they would have already been clustered together by latent semantic analysis or a similar technique. WordNet has the added benefit of clustering synonyms together even if they do not occur in the knowledge base.
  • Step 425 is optional, but compiling the knowledge base enables more implications to be used, saves storage space, and increases the speed of the reasoning process.
  • knowledge base processing program 205 After knowledge base processing program 205 has compiled 425 the knowledge base 245 using WordNet module 235, knowledge base processing program 205 ends 430. This also ends the processing of the knowledge base 245 (step 310 in Figure 3). [0058] At this point, the knowledge base is self-consistent. It is preferably completely self-consistent, but in one embodiment, it is substantially self-consistent. In addition, there is a unique implication leading to each consequent, which means that there is a unique antecedent for each consequent. The next step is to actually use the reasoning apparatus 100. A person submits a command to the reasoning apparatus 100, and main program 200 receives 315 the command.
  • Reasoning apparatus 100 can process a command even if the command contains a word that is not in the knowledge base. For example, one query is "Where might I find a carafe?" The knowledge base has the following two pieces of information: 1) Bottles and refrigerators are found together; and 2) Refrigerators are found in the kitchen.
  • main program 200 uses WordNet module 235 to prune out irrelevant senses of "bottle" in the context of an indoor environment. From the remaining senses, main program 200 determines that "bottle” is a hypernym for "carafe.” Since "bottle” is a hypernym for "carafe," knowledge about bottles also applies to carafes. Thus, carafes and refrigerators are found together. Since refrigerators are found in the kitchen, so are carafes. [0060] After main program 200 has received 315 the command, it performs 320 the task specified by the command. After main program 200 has performed 320 the task, it ends 325.
  • Reasoning apparatus 100 can be used in many different contexts. The discussion below focuses on using reasoning apparatus 100 in conjunction with a mobile robot in an indoor home or office environment. However, reasoning apparatus 100 could also be used without a robot and in environments other than a home or office. Reasoning apparatus 100 can also be used to perform many different types of tasks. One task is to answer a query. In one situation, reasoning apparatus 100 may be able to answer the query using information already in the knowledge base. In another situation, answering a query may also require inferencing based on that information. 2.1 Sample Task: Goal Achievement [0062] Another task is to determine a course of action that is designed to achieve a particular goal.
  • This goal can be directly input by a user, or its desirability can be deduced by the reasoning apparatus 100 based on the knowledge base and inferencing. Once the goal is known, reasoning apparatus 100 can determine a course of action to perform in order to achieve the goal. The course of action would be determined by backward chaining implications starting at the goal situation.
  • variable D denotes the set of all statements that are explicitly desired.
  • variable D* denotes the set of all statements that are desired, either explicitly or by implication.
  • the variable C denotes the set of all statements that are capable of being directly accomplished by the robot.
  • the variable C* denotes the set of all statements that are capable of being accomplished by the robot, either directly or indirectly (i.e., by implication).
  • the first step is for reasoning apparatus 100 to determine a goal.
  • the robot touches a soda can and senses that it is warm.
  • Main program 200 then performs inferencing over "(soda-can, w ⁇ rm)"using inferencing program 210.
  • Inferencing program 210 checks whether "(soda-can, warm)" is the antecedent of any desires implications.
  • inferencing program 210 checks whether "(soda-can, chilled) ' " is the consequent of any causes implications. [0066] In one embodiment, inferencing program 210 uses the following action- selection algorithm. The algorithm uses state knowledge and common sense to determine a list of actions to take. Given the implications (I and F), the current state (S and D) and the capabilities of the robot (Q, the following algorithm computes the actions that the robot should take:
  • Inferencing program 210 can also determine goals and courses of action where more than one inferencing step is necessary. [0068] Once the course of action has been determined, the reasoning apparatus 100 can proceed to carry out those actions in order to achieve the goal (e.g., if the reasoning apparatus 100 were used in conjunction with a robot). Reasoning apparatus 100 directs the robot to perform these actions, the actions are performed, and the goal is achieved.
  • the knowledge base 245 would also contain information about objects themselves (the Objects relation) and how various objects are used (the Uses relation). The robot would use this information to determine how to accomplish various tasks.
  • the knowledge base 245 might also contain information about in what types of rooms particular objects are usually found (the Locations relation) and which objects are usually found together (the Proximity relation).
  • Reasoning apparatus 100 can also use its observations of the environment and its common sense knowledge about objects and locations to determine its own location. Location determination is helpful when, for example, reasoning apparatus 100 is used in conjunction with a mobile robot.
  • a related task is to label rooms and open areas in a home or office given 1) a two-dimensional map, 2) location knowledge, and 3) simulated objects and room labels.
  • Map structure can be generated using common sense constraints such as 1) an office is owned by an individual; 2) a gallery and a walkway are not owned by an individual; and 3) a reception area is located at an entrance to an office.
  • Map layout information such as labels of objects and rooms, can then be incorporated into sentences and modified map layouts to reflect sentential information. This is further discussed in "Reasoning with Analogical Representations" by K. Myers and K. Konolige, in Principles of Knowledge Representation and Reasoning: Proceedings of the Third International Conference (KR92) (B. Nebel, C. Rich, and W. Swartout, eds.), San Mateo, California, 1992, which is hereby incorporated by reference. Note that a map does not have to be human-generated; it can also be generated by laser data. Laser data can also be extracted from cameras to build a sparse map of the unknown environment.
  • Figure 5 illustrates a method for determining a location using Bayesian inferencing.
  • a robot is located in one room ( ⁇ ) of a set of rooms ( ⁇ ): ⁇ e ⁇ .
  • Reasoning apparatus 100 determines the type of the room based on 1) which objects are probably present in the room, 2) which of these objects is usually found in each type of room, and 3) how common these objects are in general. When these probabilities are used with Bayes' theorem, the type of room with the highest probability can be calculated.
  • reasoning apparatus 100 determines 505 which objects are in the room.
  • the robot collects knowledge (D) about objects in the room using one or more methods.
  • a mobile robot should be able to associate human terms with sensory data by receiving a running description of its immediate surroundings as it explores a new home or office.
  • human-provided labels could be room types (e.g., "This is a kitchen") or objects ("This is a chair”).
  • object recognition e.g., by taking a picture of the room and then processing the picture.
  • the robot determines whether certain objects (x) are present in the room. If the robot's knowledge were perfect, then it would know with 100% certainty which objects were present. However, each of the methods discussed above has a different inherent confidence value (level of certainty). While direct user input might have close to 100% certainty, speech recognition probably has less certainty, perhaps only 91%. This means that if speech recognition determines that there is a bed in the room, the probability that there is actually a bed in the room might be 91%. Since the level of certainty differs based on the method used, the probability that particular objects have been observed is the conditional probability distribution P(x,
  • reasoning apparatus 100 determines 510 which of these objects is usually found in each type of room. This is represented symbolically as P(x,
  • ⁇ ). One way to calculate this value is to count the number of entries in the knowledge base wherein a particular object x, is mentioned as being in a particular room w and then divide by the number of entries of wherein any object x is mentioned as being in that particular room w. This can be represented symbolically as P(x, I ⁇ ) C(x, , ⁇ ) I C( ⁇ ), where x, is a particular object and w is a particular room [0075] Reasoning apparatus 100 then determines 515 how common these objects are in general. This is represented symbolically as P(x,).
  • reasoning apparatus 100 determines 520 the location of the robot.
  • Vector x indicates the presence or absence of the objects. Each object , is associated with an index /. If the value of x at index / is 1, then object x t is present; if the value is 0, then object x t is not present. Since x contains only binary values, the computational complexity is linear in the number of possible objects.

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