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Negation and Proof by Contradiction in Access-Limited Logic

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Negation and Proof by Contradiction in Access-Limited Logic In Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI-91) Cambridge, MA: AAAI/MIT Press, 1991. Negation and Pro of by Contradiction in Logic AccessLimited J M Crawford B J Kuip ers Department of Computer Sciences The University of Texas At Austin Austin Texas jccsutexasedu kuip erscsutexasedu Abstract of a tractable and So cratically Completeness inference metho d for a language in which complete inference is AccessLimited Logic ALL is a language for complete Co ok The heart known to b e NP knowledge representation which formalizes the ac of the extension is the addition of classical negation cess limitations inherent in a network structured ie not negation by failure to ALL The addition of Where a deductive metho d such knowledgebase negation is imp ortant b ecause it allows one to express as resolution would retrieve all assertions that sat negative and disjunctive information F urther the ad isfy a given pattern an accesslimite d logic re dition of negation allows us to augment inference by trieves all assertions reachable by following an dus ponens the only inference metho d required in mo available access path a deductive database with inference by reductio ad ab In this pap er we extend previous work to include surdum the ability to make assumptions and reason negation disjunction and the ability to make as by contradiction sumptions and reason by contradiction We show So cratic Completeness guarantees that for any fact remains So cratically that the extended ALL neg which is a logical consequence of the knowledgebase Complete thus guaranteeing that for any fact ther e exists a series of preliminary queries and assump which is a logical consequence of the knowledge tions after which a query of the fact will succeed Since base there exists a series of preliminary queries complete inference in rst order logic without existen and assumptions after which a query of the fact tial quantication is known to b e NP complete the will succeed and computationally tractable We ating such series must b e NP com problem of gener show further that the key factor determining the plete Ho wever we will show that the key factor de computational dicult y of nding such a series of termining the diculty of nding such a series is the preliminary queries and assumptions is the depth en any depth of assumption nesting in the series Giv of assumption nesting We thus demonstrate the knowledgebase and any fact if there exists a series existence of a family of increasingly p owerful in of preliminary queries and assumptions after which a ference metho ds parameterized by the depth of query of the fact will succeed and the series only nests assumption nesting ranging from incomplete and assumptions to depth n then the series can b e found tractable to complete and intractable in time prop ortional to the size of the knowledgebase raised to the nth p ower This result is particularly signicant since we exp ect commonsense reasoning to tro duction In require large knowledgebases but relatively shallow P ast work in ALL Crawford Kuip ers assumption nesting has shown So cratic Completeness and computational It has b een apparent at least since the work of tractability for a language with expressive p ower equiv that a knowledge representation language Wo o ds alent to that of a deductive database In this pa sucient to supp ort the building of large knowledge p er we extend the expressive p ower of ALL to that bases must have a clear semantics Without a clear of rst order logic without existential quantication semantics one can never b e sure exactly what a given and thus for the rst time demonstrate the existence expression represents which deductions should follow This work has taken place in the Qualitative Reason from it or how it compares to an expression in a dif ing Group at the Articial Intelligence Lab oratory The ferent knowledge representation language Exp erience ersity of Texas at Austin Research of the Qualita Univ with formally sp ecied knowledge representation sys tive Reasoning Group is supp orted in part by the Texas tems has revealed a tradeo b etween the expressive under grant no Advanced Research Program p ower of knowledge representation systems and their NSF grants IRI and IRI and by NASA If for exam grant NAG computational complexity Levesque 2 describ ed op erationally ple a knowledge representation language is as expres In ALL deductions which sive as rstorder predicate calculus then the prob follow simply by rule chaining are obvious and can lem of deciding what an agent could logically deduce b e made by a single op eration The deductive p ower of from its knowledge is unsolvable Bo ALL on a single op eration is thus roughly equivalent to olos Jerey 3 that of a deductive database There are several p ossible solutions to this problem As will b e seen b elow the more complex deductions are those which require restrict the expressive p ower of the representation reasoning by contradiction or equivalently reasoning language describ e the reasoning ability of the sys by cases tem with an op erational semantics rather than a mo del Our approach in ALL b egins with the well known give up completeness but theoretic semantics or mapping b etween atomic prop ositions in predicate cal guarantee So cratic Completeness culus and slots in frames the atomic prop osition that Unfortunately there are problems with each of these the ob ject a stands in relation r to the ob ject b can b e solutions If one is interested in expressing the full written logically as r a b or expressed in frames by range of human commonsense knowledge then re including ob ject b in the r slot of ob ject a Hayes stricting the expressive p ower of the representation lan guage is not an option An op erational semantics may a suce to describ e the reasoning ability of the system r but it do es not dene the semantics of statements in r a b values f b g the representation language can attempt to give a mo del theoretic semantics One for the representation language and an op erational se ALL allows forwardchaining and backwardchaining mantics to reasoning However reasoning will then b e inference rules Antecedents of ALL rules must al incomplete with resp ect to the semantics of the rep ways dene access paths through the network of frames 1 resentation language That is there will b e cases Access paths are sequences of predicates which can in which according to the mo del theoretic semantics b e evaluated by retrieving values from known slots of h j f but the system cannot derive f from T h T known frames For example in ALL one could write Thus the meaning ascrib ed to statements by the mo del the backwardchaining rule theoretic semantics will not always match their mean aunt J ohn y par entJ ohn x sister x y ing to the system The third solution So cratic Completeness do es But one cannot write the logically equivalent rule guarantee that the mo del theoretic semantics describ es auntJ ohn y sister x y par entJ ohn x the p otential reasoning ability of the system and thus together with soundness guarantees that the mean since evaluation of the antecedent would require a ing ascrib ed to statements by the mo del theoretic se global search of the knowledgebase for pairs of frames mantics is the same as that ascrib ed to statements by 4 in a sister relation The use of access paths allows the system However So cratic Completeness is a weak ALL op erations to b e p erformed in time prop ortional prop erty and can b e satised by any system of infer to the size of the accessible p ortion of the knowledge ence with a complete set of pro of rules base Crawford Kuip ers In ALL we use a mo del theoretic semantics to de In order to add negation to ALL we b egin by adding scrib e the p otential reasoning ability of the system r If we for each relation r an additional relation by guaranteeing So cratic Completeness but uses an made no other changes this would give us a language op erational semantics to describ e the b ehavior of the with the expressive p ower to represent negation but system on individual op erations In a sense it is not we would have lost So cratic Completeness In order surprising that complete inference with resp ect to a to maintain So cratic Completeness we have to con mo del theoretic semantics is intractable the mo del nect r with r We do this by adding the notion of an theoretic semantics was not intended to describ e which inconsistent knowledgebase and the ability to make inferences are easy to make or are useful in a given con 2 A mo del theoretic semantics may well not b e appropri text The mo del theoretic semantics describ es which ate since obviousness often dep ends as much on the form inferences are p ermissible and is thus an appropriate of knowledge as its meaning eg p is obvious from q and description of the p otential long term reasoning ability p but is not so obvious from q and :p !:q q ! of the system We exp ect a single op eration to p ermit 3 ALL allows b oth backward and forward chaining rules only deductions which are ob vious The b ehavior If one uses only backward chaining rules or only forward of the system on single op erations can thus b est b e chaining rules then the deductive p ower of ALL on a single alent to that of a deductive database op eration is equiv 1 4 One could achieve completeness by giving up tractabil The restriction to access paths limits the syntax of ity however if a system is intractable then in practice this ALL but is not a fundamental
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