cs774 (Prasad)L7Negation1 Negation by Failure

Motivation p(X) :- q(X). q(a). ?-q(a). true ?-p(a). true ?-q(b). false ?-p(b). false ‘false’ really corresponds to ‘cannot prove’. That is, ‘true’ and ‘false’ are responses to the question : “Is it a theorem/logical consequence?” cs774 (Prasad)L7Negation2

Closed World Assumption The database is complete with respect to positive information. That is, all positive atomic consequences are provable. Failure to prove goal G can be interpreted as evidence that G is false, or that negation of G (that is, ~G) is true. cs774 (Prasad)L7Negation3

‘Negation as failure’ operator (in the query) p(X) :- q(X). q(a). ?- q(a). true ?- p(a). true ?- q(b). false ?- \+ q(b). true ?- \+ p(b). true cs774 (Prasad)L7Negation4

Nonmonotonic Reasoning p(X) :- q(X). q(a). q(b). ?-q(b). true ?-p(b). true Previous conclusions (e.g., \+ q(b), \+ p(b), etc ) overturned/overridden when new facts are added. This is in stark contrast with classical logics, where the set of theorems grows monotonically with the axioms. cs774 (Prasad)L7Negation5

Monotonic vs non-monotonic entailment cs774 (Prasad)L7Negation6

‘Negation as failure’ in a rule p(X) :- q(X), \+ r(X). q(a). r(a). q(b). ?-p(a). false ?-p(b). true ?-q(c). false ?-p(c). false ?- \+ p(c). true cs774 (Prasad)L7Negation7

Negation : Theory vs Practice p :- \+ p. ?-p. Computation: infinite loop Translation into logic: {p} Recursion through negation results in a computation that does not fail finitely. cs774 (Prasad)L7Negation8

Negation : Theory vs Practice p(X) :- p(s(X)). ?-p(a). Computation: infinite loop ?- \+ p(a). Computation: infinite loop Ideally, the latter should succeed because p(a) is not provable from the input, but the Prolog query \+ p(a) loops, as p(a) does not fail finitely. cs774 (Prasad)L7Negation9

Negation Meta-predicate: Simulation using Cut not(p) :- p, !, fail. not(p). Informally, if p succeeds, then not(p) fails. Else, not(p) succeeds. p :- \+ q(X). Informally, p succeeds if there is no x such that q(x) succeeds. p fails if there is some x such that q(x) succeeds. cs774 (Prasad)L7Negation10 Negation : Meaning

Example: Correct use of \+ Hotel is full if there are no vacant rooms. Room 13 is vacant. Room 113 is vacant. hotelFull :- \+ vacantRoom(X). vacantRoom(13). vacantRoom(113). ?- hotelFull. No, because there are vacant rooms. –Note that some implementations will complain about variables inside negated goals, as explained later. cs774 (Prasad)L7Negation11

Example: Incorrect use of \+ X is at home if X is not out. Sue is out. John is Sue’s husband. home(X):- \+ out(X). out(sue). husband(john,sue). ?- home(john). True. ?- home(X). False. –Even though John is at home, it is not extracted. –I.e., the query is equivalent to “Is everyone out?” cs774 (Prasad)L7Negation12

Example: Characteristics of \+ student(bill). married(joe). unmarriedStudent(X):- \+ married(X), student(X). ?- unmarriedStudent(X). False. bachelor(X):- student(X), \+ married(X). ?- bachelor(X). X = bill Negated goals do not generate bindings. cs774 (Prasad)L7Negation13

Recursion through Negation Revisited p :- \+ q. q :- \+ p. Logically equivalent to: p v q Ambiguity Minimal models {p} and {q}. cs774 (Prasad)L7Negation14

Stratified Negation cs774 (Prasad)L7Negation15 p1 :- p2, \+ q. … p2 :- p1, \+ r2. q1 :- q2, q3, \+ r1. … q4. r1 :- r2. r2.

Stratified Negation Syntactic restriction for characterizing “good programs” –What is the purpose? Associate unique meaning –How is it obtained? Mutual recursion among same level predicates Negated predicates / atoms must contain lower level predicates No recursion through negation cs774 (Prasad)L7Negation16

Example: Common-sense Reasoning fly(X) :- bird(X). bird(X) :- eagle(X). bird(tweety). eagle(toto). ?- fly(tweety). True. ?- fly(toto). True. Monotonic reasoning? Birds fly. cs774 (Prasad)L7Negation17

Example: Common-sense Reasoning (Exceptions) fly(X) :- bird(X), \+ abnormal(X). abnormal(X) :- penguin(X). bird(X) :- penguin(X). penguin(tweety). ?- fly(tweety). False. Non-monotonic reasoning Typically, birds fly. –Infer abnormality only if it is provable. –Otherwise, assume normal. cs774 (Prasad)L7Negation18