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Formalizing Access Limited Reasoning

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Formalizing Access Limited Reasoning ALL Formalizing Access Limited Reasoning y J M Crawford Benjamin Kuip ers Department of Computer Sciences The University of Texas At Austin Austin Texas jccsutexasedu kuip erscsutexasedu May Abstract AccessLimited Logic ALL is a language for knowledge representation which formalizes the access Where a classical deductive metho d or logic limitations inherent in a networkstructured knowledge base programming language would retrieve all assertions that satisfy a given pattern an accesslimited logic retrieves all assertions reachable by following an available access path The complexity of inference is thus indep endent of the size of the knowledgebase and dep ends only on its lo cal connectivity AccessLimited Logic though incomplete still has a well dened semantics and a weakened form of completeness Socratic Completeness which guarantees that for any query which is a logical consequence of the knowledgebase there exists a series of queries after which the original query will succeed This chapter presents an overview of ALL and sketches the pro ofs of its So cratic Completeness and p olynomial time complexity Intro duction It has long b een a guiding principle in work on semantic networks that by imp osing a network structure on large knowledge bases one can increase the eciency of reasoning Intuitively in a semantic network related concepts are lo cated close together in the network and thus search and inference can b e guided by the structure of the knowledgebase However formalisms for semantic networks have generally treated the semantic network notation as a variant of predicate calculus and have regarded the access limitations inherent indexing mechanisms In AccessLimited Logic ALL we incorp orate in a network to b e an extralogical these access limitations directly into the logic One b enet of this approach is that we can assess the impact of access limitations on the completeness and complexity of reasoning ALL is in fact not complete but it is So cratically Complete that is for any query which is a logical consequence of the knowledgebase there exists a series of queries after which the original query will succeed Further the complexity of inference in of a knowledgebase and dep ends only on the size of the accessible p ortion ALL is indep endent of the size of the knowledgebase This work has taken place in the Qualitative Reasoning Group at the Articial Intelligence Lab oratory The University of Texas at Austin Research of the Qualitative Reasoning Group is supp orted in part by NSF grants IRI IRI and IRI by NASA grants NAG and NAG and by the Texas Advanced Research Program under grant no This pap er app eared in John Sowa Ed Principles of Semantic Networks San Mateo CA Morgan Kaufmann pp y Supp orted in part by a fellowship from GTE OVERVIEW OF ACCESSLIMITED LOGIC Reasoning is hard If a knowledge representation language is as expressive as rstorder predicate calculus then the problem of deciding what an agent implicitly knows ie what an agent could logically deduce from its knowledge is unsolvable Bo olos Jerey Thus a sound and decidable knowledge representation and reasoning system must either give up expressive p ower or use a weak inference system with an incomplete set of deduction rules or articial resource limits eg b ounds on the numb er of applications of modus ponens However such inference systems tend to b e dicult to describ e semantically and tend to place unnatural reasoning ability Levesque limits on an agents As an example of nontrivial inference consider the following problem from Wylie In a certain bank the p ositions of cashier manager and teller are held by Brown Jones and Smith though not necessarily resp ectively The teller who was an only child earns the least Smith who married Browns sister earns more than the manager What p osition do es each man ll A p erson lo oking at such a problem cannot come up with a solution immediately though the p ositions follow from a fairly small amount of commonsense knowledge ab out families partial orders and coreference A certain amount of conscious thought is required one has to ask oneself just the right questions Similarly we do not exp ect our knowledgerepresentation system to b e able to solve such a problem immediately since intuitively we exp ect it to b e able to reason only ab out as well as a p erson could reason without conscious thought We do however exp ect it to b e able to solve such a problem after b eing given an appropriate set of leading questions If we ask If Smith were the manager then how could he earn more than the manager If Smith were the teller then how could he earn more than the manager If Brown were the teller then how could he have a sister then one can see immediately that Smith must b e the cashier Brown the manager and Jones the teller Similarly a knowledgerepresentation system after b eing ask by the user or heuristically generating such a series of questions should b e able to determine which man holds which p osition We have translated this problem into ALL and given it along with appropriate commonsense knowledge to our lisp implementation Algernon Inference in Algernon is incomplete and Algernon fails initially to solve the problem However after we ask it the questions given ab ove it is able to determine which p osition each man lls 1 More abstractly ALL has an imp ortant prop erty we call Socratic Completeness for any query of a prop osition which is a consequence in predicate calculus of the knowledgebase there exists a preliminary query after which the original query will succeed ALL also has a more technical weakened completeness prop erty which we call Partitional Completeness Roughly Partitional Completeness says that if all facts and rules needed to prove a query are lo cated close enough see section to the query then it will succeed immediately The rest of this chapter is organized as follows In section we discuss our general approach to knowledge representation Section gives an overview of the formalization of ALL and the pro ofs of So cratic and Partitional Completeness We outline the argument for the p olynomial time complexity of ALL in section In section we briey discuss related work and section is our conclusion Overview of AccessLimited Logic In the broadest sense the study of knowledgerepresentation is the study of how to represent knowledge in such a way that the knowledge can b e used by a machine From this vague denition we can conclude that a knowledgerepresentation system must have the following prop erties 1 We have since discovered that the idea of So cratic Completeness is also used in Powers where it is referred to as So cratic Adequacy OVERVIEW OF ACCESSLIMITED LOGIC It must have a reasonably compact syntax It must have a well dened semantics so that one can say precisely what is b eing represented It must have sucient expressive p ower to represent human knowledge It must have an ecient p owerful and understandable reasoning mechanism It must b e usable to build large knowledgebases It has proved dicult however to achieve the third and forth prop erties simultaneously Our approach in ALL b egins with the well known mapping b etween atomic prop ositions in predicate stands in relation r to the ob ject b calculus and slots in frames the atomic prop osition that the ob ject a can b e written logically as r a b or expressed in frames by including ob ject b in the r slot of ob ject a Hayes a P P a b values f b g We refer to the pair ha r i as a frameslot Thus r a b is equivalent to saying that the value b is in the 2 frameslot ha r i The frames directly accessible from a frameslot are those which app ear in the frameslot Extending this idea we dene an access path in a network of frames as a sequence of frames such that each is directly accessible from a frameslot of its predecessor It is useful to generalize this denition and allow access paths to branch on all values in the frameslots A sequence of prop ositions denes an access path if any variable app earing as the rst argument to a prop osition has app eared previously in the sequence op erationally this means that retrieval always accesses a known frameslot For example Johns parents sister can b e expressed in ALL as the path entJ ohn x sister x y par This denes an access path from the frame for J ohn to the frames for J ohns parents found by lo oking in the frameslot hJ ohn par enti to J ohns parents sisters F rom access paths we build the inference rules of ALL A rule is always asso ciated with a particular slot in the network Backward chaining ifneeded rules are written in the form the structure of and is discussed b elow and applied when a value for the slot is needed Forward chaining ifadded rules and applied when a new value for the slot is inserted In either case the are written in the form antecedent of a rule must dene an access path b eginning with the slot the rule is asso ciated with For example using the access path ab ove we can write the ifneeded rule aunt J ohn y par entJ ohn x sister x y But we cannot write the logically equivalent rule auntJ ohn y sister x y par entJ ohn x 3 since the antecedent do es not dene an access path 2 Slots in ALL contain only frames and rules dened b elow 3 The restriction to access paths limits the syntax of ALL but is not a fundamental limit on its expressive p ower since one could always add a new constant and make it the rst argument to every predicate This would amount to making the entire knowledgebase a single frame THE LOGICAL
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