An Alternative To Deduction

Deductive Representations are well defined, easily inspected, and precise. However, they are also brittle, inflexible and difficult do debug. We propose an alternative plausible representation which which is weaker than deduction. It can accept and reason with knowledge which is less precise, and there is no notion of global consistency, only local consistency of an explanation. Like connectionism and unlike deductive approaches this approach gains strength from empirical data. This research was supported by the National Science Foundation under grant NSF–IRI–87–19766 Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society 2 1. PI–EBL APPROACH There are many advantages to a reasoning model based on a declarative representation of world knowledge. Such a representation may be leveraged by many knowledge intensive mechanisms: it can be communicated to other agents, multiple models can be combined, parts of a model can be isolated and independently verified, etc. Because deduction is well understood it is natural to ground these models in a deductive framework. Requiring all knowledge to be expressed in a deductive form is restrictive, and causes systems to be very brittle. Connectionist approaches, on the other hand, are not based on such a restricted language, which is a source of their flexibility [Rumelhart86]. The symbolic/connectionist distinction may appear unimportant since the sources of knowledge potentially available are the same. However, research in connectionism is impeded by its representation––its very difficult to understand what abstract principles are revealed by a connectionist solution to a problem. The advantages of both approaches can be achieved by a reasoning paradigm based on an explicit world model (like symbolic approaches) using a non–deductive representation (like connectionist approaches). Research is facilitated by such a paradigm, high–level knowledge intensive mechanisms can be more easily constructed, and the principles underlying their success are much more available for inspection. In this paper we introduce a logic of plausible reasoning with these desired properties. As in distributed connectionist systems, our approach crucially involves learning through observations of the world. The machine learning paradigm used to support our logic of plausible reasoning is EBL. Explanation Base Learning is a learning mechanism which combines a model of the world with examples from the world to construct explanations which can be later used to reason about the world [DeJong86, Mitchell86]. Like other knowledge intensive mechanisms, it gains its strength from its explicit model. Much of EBL’s potential as a methodology has not been tapped, however, because the knowledge representation it uses is too restrictive. For example, EBL’s restriction to symbol level learning is due to the base representation, not an inherent restriction on the EBL paradigm. (Symbol level learning adds no new knowledge but simply makes some knowledge much easier to derive [Dietterich86].) EBL’s limited use of its training examples is also traceable to its base representation. In general the flexibility of any architecture depends, in large part, on the representation language it uses, no matter what high level task is being performed. We present a plausible representation and inferencing mechanism as an alternative to classical deduction. The flexibility of this representation is demonstrated by using the Plausible Inferencer as the explanative component for an Explanation Based Learner (PI–EBL). Through its flexible representation PI–EBL overcomes EBL’s deficiencies mentioned above, and acquires some advantages associated with both connectionism and traditional induction by greater use of its training examples. 2. MOTIVATION OF PLAUSIBLE DOMAIN THEORIES There are three important properties inherent in any model of a domain. We will use these properties to demonstrate the inadequacies of deductive knowledge representations. These properties also are used as a guide in the selection of an appropriate plausible representation. The first property, simplicity, is a measure of the amount of inference needed to derive a conclusion about the domain from the domain theory. This is often related to the syntactic simplicity of the domain theory. At times we will use this as a secondary definition of simplicity. The second property, coverage, is a measure of the number of conclusions about the domain which can be drawn from the domain theory. So a theory with low coverage makes no predictions about large parts of the domain. The third property, faithfulness, is a measure of the chances that a statement derivable from the domain theory is consistent with the domain itself. An unfaithful domain theory makes incorrect predictions about its doProceedings of the Thirteenth Annual Conference of the Cognitive Science Society 3 main. In understanding these terms it is helpful to notice the analogy: faithfulness to soundness and coverage to completeness. We avoid these more familiar terms in our discussion because their meaning within model theory [Genesereth87] is more constrained than the meaning intended in this paper. In particular we do not wish to constrain our domain theories to be theories which are modeled by their domain in the model theoretic sense. In fact, we are anticipating domain theories which are not even satisfiable (internally consistent); obviously if there is no model of our domain theory the domain itself could not be such a model. Simplicity, coverage, and faithfulness are all terms which describe how well a theory captures a domain. An optimal theory is one that maximizes all three aspects of the theory. Often this turns out to be a tradeoff, theories which maximizes one property do so at the expense of another. This is best demonstrated through example. Consider theories that describe things which are able to fly. One possibility is: Bird(x) Fly(x) . This is both very simple, and has pretty good coverage. Another possibility, Animate(x) Bird(x) Fly(x) , trades some of this simplicity for greater faithfulness. It takes more computational effort to derive Fly(x) , and the knowledge base is syntactically more complex, but it correctly handles the dead bird case. Adding HasFeathers(x) Bird(x) to the knowledge base will increase coverage since we now infer Fly(x) even in the case that we don’t know x is a bird, but only that it has feathers. This increase is at the expense of both simplicity, adding another inference step, and faithfulness, not all feathered things are birds. Faithfulness, simplicity, and coverage form a tradeoff: the optimal choice for this tradeoff is a compromise which does not place too much importance on any one of the three attributes; ignoring any aspect in favor of the others has diminishing returns after a point. Deductive frameworks are not sufficient as a knowledge representation mechanism. Their deficiency stems from their precision (a non–optimal simplicity/faithfulness tradeoff). Knowledge placed in a deductive framework must be valid––true for all possible examples. This is a problem since much of the knowledge an intelligent agent must deal with is true for most but not all examples, and such knowledge cannot be used in a deductive framework. The impossibility of representing real world knowledge in a deductive theory is noted by McCarthy as the qualification problem [Genesereth87, McCarthy69]. The qualification problem states that any universally quantified implication will need a large (infinite) number of preconditions to exclude all possible exceptions. The reader may be convinced by considering the implication: Bird(x) Fly(x) . Correcting this rule to handle exceptions would require adding Alive(x) , Penguin(x) , Sane(x) , etc. as conjuncts. Of course such a deductive theory is not practical. The traditional approach to this problem is to translate the original domain into a micro–world which is consistent with the original domain to a fixed level of detail. Because the micro–world does not reflect all details of the real world, it is finitely describable using a deductive theory, thus a deductive theory may be employed. The disadvantage to this approach is that the model is really a model of the micro–world not a model of the original domain. The agent using such a theory has no recourse if some important portion of the domain is not modeled in the micro–world. Thus, the micro–world must be complex enough to handle the most detailed example encountered by the agent. This forces the entire system to represent and reason using this detailed and complex theory for all examples, even those which can be handled using a much simpler theory. Adapting the theory to handle one additional special–case example could easily result in many previously tractable problems to becoming inaccessible. Furthermore, simple theories which make different assumptions (based on different micro–worlds) cannot be directly combined because the resulting theory would likely be internally inconsistent. This is a stifling restriction: different micro–worlds (consistent micro–theories) will be useful in reasoning about different problems, and Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society 4 combining inconsistent micro–theories into a single consistent micro–world will result in a theory which is too precise. It is too precise because the closure of these micro–theories contain many facts inconsistent with other micro–theories, and the combined theory must be consistent with only one of these choices. Because the combined theory is deductive it must specify exactly how its component theories interacted, even when the interactions are uninteresting parts of the deductive closure. Using a micro–world forces faithfulness to be traded for simplicity even when the faithfulness gained only disambiguates cases which are unimportant in practice. No deductive chain of inference no matter how complex may be inconsistent with any other over the entire theory. Requiring the theory to be internally consistent without gaining consistency with the world gains us little power at the great expense of scaleability. An agent using a deductive knowledge base cannot accept any knowledge about the world until all its interactions with the closure of the agents knowledge are precisely specified. This restriction is unacceptable. Avoiding this restriction and its fixed level of detail forces us to abandon the micro–world approach. Non–monotonic theories provide a mechanism for reasoning with incomplete knowledge [McCarthy86, Reiter80]. McCarthy’s Abnormal(x) predicates allows one set of rules which derives a conclusion to override another rule. Like deductive theories, however, the sentences entailed by a nonmotonic theory is still precisely defined without any examples being needed to disambiguate inconsistencies. The whole notion of circumscription makes precise the closure of knowledge. In order for a representation to be an effective alternative to the micro–world approach, it cannot be deductive; it must incorporate a means of trading between faithfulness, simplicity, and coverage. This means it cannot allow the closing of the knowledge base. Instead the world must be left open. Inferred knowledge should be used as suggestions to be empirically verified, rather than theorems with are necessarily true. 3. REPRESENTATION OF PLAUSIBLE DOMAIN THEORIES Plausible domain theories are syntactically similar to a set of implications. Semantically they are quite different, however, since the theories are not deductive. The theories must suggest possible relationships between concepts without entailing them. Semantically the plausible domain theory may be interpreted as deductive implications with missing preconditions. The missing preconditions are collectively called the implicit context of the implication. The implicit context is the set of additional constraints sufficient to guarantee that the plausible implication deductively holds. The preconditions of a plausible implication are not sufficient for concluding their consequent. Thus we refer to plausible implications as influents to emphasize the idea that the consequent is influenced, but not entailed, by the preconditions. Asserting an influent specifies that its preconditions are relevant to determining its consequent in some contexts*. An influent also specifies the direction of influence between the preconditions and consequent, and they may be positively or negatively related. These influents provide the basis for representing PI–EBL domain theories. 4. PLAUSIBLE INFERENCING To build explanations from a plausible domain theory (a set of influents) we need an inference mechanism. Influents may be combined with other influents in a way analogous to deductive implications. The standard mechanism of unifying a precondition of one rule with the consequent of another is not sufficient alone however. An additional inference rule allows combining several influents with the same consequent into an influent set. Thus, it is possible to conjecture a complete explana* The influent determines its consequent as in relevance logics while allowing the axiom set to remain open as intuitionistic logics. Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society