Probabilistic Logic Networks in a Nutshell

Transcription

1 Probabilistic Logic Networks in a Nutshell Matthew Ikle 1 1 Adams State College, Alamosa CO Abstract. We begin with a brief overview of Probabilistic Logic Networks, distinguish PLN from other approaches to reasoning under uncertainty, and describe some of the main conceptual foundations and goals of PLN. We summarize how knowledge is represented within PLN and describe the four basic truth-value types. We describe a few basic firstorder inference rules and formulas, outline PLN s approach to handling higher-order inference via reduction to first-order rules, and follow this by a brief summary of PLN s handling of quantifiers. Since PLN was and continues to be developed as one of several major components of a broader and more general artificial intelligence project, we next describe the OpenCog project and PLN s roles within the project. Keywords: probabilistic logic, probabilistic networks, artificial general intelligence, PLN, OpenCog 1 Introduction: What is PLN? First introduced as a probabilistic reasoning system within the Webmind Artificial Intelligence project by Ben Goertzel and the late Jeff Pressing, the Probabilistic Logic Networks (PLN) system has evolved and grown considerably. PLN now serves as the probabilistic reasoning system within the open source OpenCog AI engine[7], which has replaced Webmind. The primary focus of PLN is to serve as a systematic, comprehensive, and pragmatic system to manage uncertainty: to handle and reason about imprecise, uncertain, incomplete, and inconsistent data, and reasoning involving uncertain conclusions. Perhaps one of PLN s most striking characteristics is its dual nature. Designed as part of a broader artificial intelligence system, PLN is very practical, encompassing heuristic approaches as necessary. At the same time, considerable effort has been made to ground as much of PLN as possible upon solid theoretical and mathematical foundations. A result of this duality is the following list of desired characteristics: PLN should enable uncertainty-savvy versions of all known varieties of logical reasoning: including, for instance, higher-order reasoning involving quantifiers, higher-order functions, and so forth; PLN should reduce to crisp theorem prover style behavior in the limiting case where uncertainty tends to zero; PLN should encompass inductive and abductive as well as deductive reasoning;

2 2 Probabilistic Logic Networks PLN should agree with probability theory in those reasoning case where probability in its current state of development provides solutions within reasonable calculational effort based on assumptions that are plausible in the context of real-world embodied software systems; PLN should gracefully incorporate heuristics not explicitly based on probability theory, in cases where probability theory, at its current state of development, does not provide adequate pragmatic solutions; PLN should provide scalable reasoning, in the sense of being able to carry out inferences involving at least billions of premises and carry out more intensive and accurate reasoning when the number of premises is fewer; PLN should easily accept input from, and send input to, natural language processing software systems. 2 Relationship of PLN to Other Uncertain Inference Engines It is clear that uncertain inference is hardly a new idea. What is new within PLN is its focus on bridging the theoretical and practical, and how it incorporates and integrates ideas from a variety of sources. PLN borrows heavily upon other approaches to uncertain inference and in many ways represents an amalgam of a large number of these ideas, including such standard approaches as Bayesian probability theory, and fuzzy logic, as well as from more unusual ideas including Pei Wang s Non-Axiomatic Reasoning System (NARS)[12], algorithmic information theory, and Walley s theory of imprecise probabilities[11]. One of the key differences between PLN and other approaches to probabilistic logic lies in PLN s foundation upon term logic. As we shall see later, this foundational choice allows one to reduce PLN s higher-order inference rules to more basic first-order rules. Overall, PLN owes the most to Pei Wang s NARS system and Walley s theory of imprecise probabilities. Pei Wang pioneered the use of uncertain term logic in his NARS system, and in large measure provided the motivation for the development of PLN. Indeed PLN began as part of a collaboration with Wang as an attempt to create a probabilistic analogue to NARS, though there remain many conceptual and mathematical differences between the two, and PLN has long ago diverged from these roots. Peter Walley s theory of imprecise probabilities provided motivation for the development of our indefinite probabilities approach. Essentially a hybridization of Walley s imprecise probabilities with Bayesian credible intervals, indefinite probabilities provide a general and mathematically sound method for calculating the weight-of-evidence underlying the conclusions of uncertain inferences. Moreover, both Walley s imprecise beta-binomial model and standard Bayesian inference can be mathematically viewed as limiting cases of the indefinite probability model. Of the wide array of uncertain inference methods, Bayes nets represent perhaps the most similar approach to PLN, although the graph structures themselves are quite dissimilar. While both methods succeed at embodying probability

3 Probabilistic Logic Networks 3 theory in a set of date structures and algorithms, PLN was designed with different purposes in mind. As a pragmatic approach with an eye towards interaction with an integrative artificial intelligence system, PLN was designed to interface with other cognitive processes and with other kinds of inference, including intensional inference, fuzzy inference, and higher-order inference using quantifiers, variables, and combinators. While PLN utilizes fuzzy set membership as the semantics for Member relationship truth-values, it maintains a clear distinction between uncertainty and partial membership. For many of the purposes commonly associated with fuzzy membership, PLN uses intensional probabilities, giving the advantage of keeping more things within a probabilistic framework. 3 Knowledge Representation within PLN Declarative knowledge representation within PLN is handled by a weighted labeled hypergraph called the Atomspace, which consists of multiple types of nodes and links, generally weighted with probabilistic truth values and attention values PLN is divided into first-order and higher-order sub-theories (FOPLN and HOPLN). These terms are used in a nonstandard way drawn from NARS. We develop FOPLN first, and then derive HOPLN therefrom. FOPLN is a term logic, involving terms and relationships (links) between terms. It is an uncertain logic, in the sense that both terms and relationships are associated with truth value objects, which may come in multiple varieties ranging from single numbers to complex structures like indefinite probabilities[3]. Terms may be either elementary observations, or abstract tokens drawn from a token-set T. 3.1 Core FOPLN Relationships Core FOPLN involves relationships drawn from the set: negation; Inheritance and probabilistic conjunction and disjunction; Member and fuzzy conjunction and disjunction. Elementary observations can have only Member links, while token terms can have any kinds of links. PLN makes clear distinctions, via link type semantics, between probabilistic relationships and fuzzy set relationships. Member semantics are usually fuzzy relationships (though they can also be crisp), whereas Inheritance relationships are probabilistic, and there are rules governing the interoperation of the two types. 3.2 Auxiliary FOPLN Relationships Beyond the core FOPLN relationships, FOPLN involves additional relationship types of two varieties. There are simple ones like Similarity, defined by Similarity A B We say a relationship R is simple if the truth value of R A B can be calculated in terms of the truth values of core FOPLN relationships between A and B.

4 4 Probabilistic Logic Networks There are also complex ones like IntensionalInheritance, which measures the extensional inheritance between the set of properties or patterns associated with one term and the corresponding set associated with another. 3.3 PLN Truth Values Truth-values come in four basic types. In order of increasingly information about the full probability distribution they are strength truth-values, which consist of single numbers; e.g., < s > or <.8 >. Usually strength values denote probabilities but this is not always the case. SimpleTruthValues, consisting of pairs of numbers. These pairs come in two forms: < s, w >, where s is a strength and w is a weight of evidence and < s, N >, where N is a count. Weight of evidence is a qualitative measure of belief, while count is a quantitative measure of accumulated evidence. IndefiniteTruthValues, which quantify truth-values in terms of an interval [L, U], a credibility level b, and an integer k (called the lookahead). IndefiniteTruthValues quantify the idea that after k more observations there is a probability b that the conclusion of the inference will appear to lie in [L, U]. See [3] for more details. DistributionalTruthValues, which are discretized approximations to entire probability distributions. This gradation of truth-value types serves several purposes. Strength and Simple truth values can be used when speed is of the essence or when one simply has little information. When accuracy is most important and when we have additional information concerning an Atom s full probability distribution, then Indefinite and Distributional truth values may be more pertinent. 3.4 PLN Rules and Formulas A distinction is made in PLN between rules and formulas. PLN logical inferences take the form of syllogistic rules, which give patterns for combining statements with matching terms. Examples of PLN rules include, but are not limited to, are the deduction ((A B) (B C) (A C)), induction ((A B) (A C) (B C)), abduction ((A C) (B C) (A C)), inversion rules ((A B) (B A)). Related to each rule is a formula which calculates the truth value resulting from application of the rule. As an example, suppose s A, s B, s C, s AB, and s BC represent the truth values for the terms A, B, C, as well the truth values of the relationships A B and B C, respectively. Then, under suitable conditions imposed upon these input truth values, the formula for the deduction rule is given by: s AC = s AB s BC + (1 s AB) (s C s B s BC ) 1 s B,

5 Probabilistic Logic Networks 5 where s AC represents the truth value of the relationship A C. This formula is directly derived from probability theory given the assumption that A B and B C are independent. Using a combination of probability theory and heuristics, PLN also effectively handles cases in which independence is not a valid assumption. 4 Higher-Order PLN Higher-order PLN (HOPLN) is defined as the subset of PLN that applies to predicates (considered as functions mapping arguments into truth values). It includes mechanisms for dealing with variable-bearing expressions and higherorder functions. A predicate, in PLN, is a special kind of term that embodies a function mapping terms or relationships into truth-values. HOPLN contains several relationships that act upon predicates including Evaluation, Implication, and several types of quantifiers. The relationships can involve constant terms, variables, or a mixture. PLN supports a variety of quantifiers, including traditional crisp and fuzzy quantifiers, plus the AverageQuantifier defined so that the truth value of AverageQuantifier X F (X) is a weighted average of F (X) over all relevant inputs X [3]. AverageQuantifier is used implicitly in PLN to handle logical relationships between predicates, so that e.g. the conclusion of the above deduction is implicitly interpreted as AverageQuantifier X Implication Evaluation is Fluffy X Evaluation is cat X 4.1 Reducing HOPLN to FOPLN In [3] it is shown that in principle, over any finite observation set, HOPLN reduces to FOPLN. The key ideas of this reduction are the elimination of variables via use of higher-order functions, and the use of the set-theoretic definition of function embodied in the SatisfyingSet operator to map function-argument relationships into set-member relationships. As an example, consider the Implication link. In HOPLN, where X is a variable Implication R 1 A X R 2 B X

6 6 Probabilistic Logic Networks may be reduced to Inheritance SatisfyingSet(R 1 A X) SatisfyingSet(R 2 B X) where e.g. SatisfyingSet(R 1 A X) is the fuzzy set of all X satisfying the relationship R 1 (A, X). 5 PLN and AI While PLN serves as a standalone system, recall that PLN grew out of a desire to build an uncertain reasoning module for use within a more general artificial intelligence framework. In order to completely understand many of the open research problems within PLN, it helps to understand the roles PLN plays within this larger context. To address this issue, we examine PLN from two additional viewpoints. First as PLN relates to intelligent agents, and then we will provide an overview of how PLN fits into the larger OpenCog framework. 5.1 SRAM Here we very briefly review a simple formal model of intelligent agents called SRAM, for Simple Realistic Agent Model. Following a theoretical framework developed by Legg and Hutter[8], we consider a class of active agents which observe and explore their environment and also take actions in it, which may affect the environment. The agent sends information to the environment and the environment sends signals to the agent. Agents can also experience rewards. To this framework, we add a set M of memory actions which allow agents to maintain memories (of finite size), and at each time step to carry out internal actions on their memories as well as external actions in the environment. We also introduce the notions of goals and consider the environment as sending goalsymbols to the agent along with regular observation-symbols. In this extended framework, an interaction sequence looks like m 1 a 1 o 1 g 1 r 1 m 2 a 2 o 2 g 2 r 2... where the m i s represent memory actions, the a i s represent external actions, the o i s represent observations, the g i s represent agent goals, and the r i s represent rewards. It is assumed that the reward r i provided to an agent at time i is determined by the goal function g i. If w is introduced as a single symbol to denote the combination of a memory action and an external action, and y is introduced as a single symbol to denote the combination of an observation, a goal and a reward, we can simplify this interaction sequence as w 1 y 1 w 2 y 2...

7 Probabilistic Logic Networks 7 Each goal function maps each finite interaction sequence I g,s,t = wy s:t with g s corresponding to g, into a value r g (I g,s,t ) [0, 1] indicating the value or raw reward of achieving the goal during that interaction sequence. The total reward r t obtained by the agent is the sum of the raw rewards obtained at time t from all goals whose symbols occur in the agent s history before t. The agent is represented as a function π which takes the current history as input, and produces an action as output. Agents need not be deterministic, an agent may for instance induce a probability distribution over the space of possible actions, conditioned on the current history. In this case we may characterize the agent by a probability distribution π(w t wy <t ). Similarly, the environment may be characterized by a probability distribution µ(y k wy <k ). Taken together, the distributions π and µ define a probability measure over the space of interaction sequences. Following Legg and Hutter, we will consider the class of environments that are reward-summable, meaning that the total amount of reward they return to any agent is bounded by 1. We will also use the term context to denote the combination of an environment, a goal function and a reward function. If the agent is acting in environment µ, and is provided with g t = g for the time-interval T = t {t 1,..., t 2 }, then the expected goal-achievement of the agent during the interval is V π µ,g,t t 2 r i t 1 where E is the space of computable, reward-summable environments. Next, we introduce a second-order probability distribution ν, which is a probability distribution over the space of environments µ. The distribution ν assigns each environment a probability. What is key in the above formalism is that this second-order probability distribution ties in nicely with the indefinite probabilities framework and allows us to ground a form of possible worlds semantics within experiential semantics. An agent, experiencing a single stream of perceptions, may use this to construct an ensemble of simulated possible worlds, which may then be used in various sorts of inferences using a commonplace idea in the field of statistics: subsampling, a form of bootstrapping. This notion ties in closely with SRAM, which considers a probability distribution over a space of environments which are themselves probability distributions. What a real agent has is actually a single series of remembered observations. But it can induce a hopeful approximation of this distribution over environments by subsampling its memory and asking: what would it imply about the world if the items in this subsample were the only things I d seen?

8 8 Probabilistic Logic Networks 5.2 PLN s Relationship to OpenCog Now we briefly describe the OCP (OCP) AGI architecture, implemented within the open-source OpenCog AI framework. OCP combines multiple AI paradigms such as uncertain logic, computational linguistics, evolutionary program learning and connectionist attention allocation in a unified architecture. Cognitive processes embodying these different paradigms interoperate together on a common neural-symbolic knowledge store called the Atomspace. The interaction of these processes is designed to encourage the self-organizing emergence of highlevel network structures in the Atomspace, including superposed hierarchical and heterarchical knowledge networks, and a self-model network enabling metaknowledge and meta-learning. The high-level architecture of OCP involves the use of multiple cognitive processes associated with multiple types of memory to enable an intelligent agent to execute the procedures that it believes have the best probability of working toward its goals in its current context. OCP handles low-level perception and action via an extension called OpenCogBot, which integrates a hierarchical temporal memory system, DeSTIN [1]. OCP s memory types are the declarative, procedural, sensory, and episodic memory types that are widely discussed in cognitive neuroscience [10], plus attentional memory for allocating system resources generically, and intentional memory for allocating system resources in a goal-directed way. Table 1 overviews these memory types, giving key references and indicating the corresponding cognitive processes, and also indicating which of the generic patternist cognitive dynamics each cognitive process corresponds to (pattern creation, association, etc.). 6 Conclusion and Future Research Directions The OpenCog software (with PLN as it core reasoning system) has been used for commercial applications in the area of natural language processing and data mining [5]. A collaboration between Novamente LLC and The Electric Sheep Company demonstrated an OpenCog-controlled virtual dog in a virtual world, that can learn new tricks via imitative and reinforcement learning[4] (see http: //novamente.net/example for some videos of these virtual dogs in action). More recently, a new project based at Hong Kong Polytechnic University called M-Lab will explore the creation of generally intelligent humanoid game characters, powered by OpenCog and M-Labs Lucid game engine, with the capability for simple English conversation and realistic human-like emotional dynamics. Once again, as part of this effort, PLN will play a pivotal role, supplying the core planning and inference mechanisms. As these projects proceed, it is clear that new challenges will arise and that PLN will encounter challenges and require alterations and additions. As a standalone system, many challenging problems still remain, most notably forward and backward chaining inference control.

9 Memory Specific Cognitive Processes Type Probabilistic Logic Networks (PLN) [3]; Declarative concept blending [2] MOSES (a novel probabilistic Procedural evolutionary program learning algorithm) [9] Episodic internal simulation engine [4] Attentional Economic Attention Networks (ECAN) [6] Intentional probabilistic goal hierarchy refined by PLN and ECAN, structured according to Psi Probabilistic Logic Networks 9 General Cognitive Functions pattern creation pattern creation association, pattern creation attention allocation,, association, credit assignment credit assignment, pattern creation Sensory Supplied by DeSTIN integration association, attention allocation, pattern creation, credit assignment Table 1. Memory Types and Cognitive Processes in OpenCog Prime. The third column indicates the general cognitive function that each specific cognitive process carries out, according to the patternist theory of cognition. References 1. Arel, I., Rose, D., Coop, R.: Destin: A scalable deep learning architecture with application to high-dimensional robust pattern recognition. Proc. AAAI Workshop on Biologically Inspired Cognitive Architectures (2009) 2. Fauconnier, G., Turner, M.: The Way We Think: Conceptual Blending and the Mind s Hidden Complexities. Basic (2002) 3. Goertzel, B., Ikle, M., Goertzel, I., Heljakka, A.: Probabilistic Logic Networks. Springer (2008) 4. Goertzel, B., Et Al, C.P.: An integrative methodology for teaching embodied nonlinguistic agents, applied to virtual animals in second life. In: Proc.of the First Conf. on AGI. IOS Press (2008) 5. Goertzel, B., Pinto, H., Pennachin, C., Goertzel, I.F.: Using dependency parsing and probabilistic inference to extract relationships between genes, proteins and malignancies implicit among multiple biomedical research abstracts. In: Proc. of Bio-NLP 2006 (2006) 6. Goertzel, B., Pitt, J., Ikle, M., Pennachin, C., Liu, R.: Glocal memory: a design principle for artificial brains and minds. Neurocomputing (Apr 2010) 7. Hart, D., Goertzel, B.: Opencog: A software framework for integrative artificial general intelligence. In: AGI. Frontiers in Artificial Intelligence and Applications, vol. 171, pp IOS Press (2008), agi/agi2008.html#hartg08 8. Legg, S., Hutter, M.: A formal measure of machine intelligence. In: Proc. of Benelaam, 2006 (2007) 9. Looks, M.: Competent Program Evolution. PhD Thesis, Computer Science Department, Washington University (2006)

10 10 Probabilistic Logic Networks 10. Tulving, E., Craik, R.: The Oxford Handbook of Memory. Oxford U. Press (2005) 11. Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman-Hall (1990) 12. Wang, P.: Non-Axiomatic Reasoning System: Exploring the Essence of Intelligence. Ph.D. thesis, Indiana University (1995)