Prediction-hardness of acyclic conjunctive queries

AbstractA conjunctive query problem is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. In this paper, a tuple, a conjunctive query and a database in relational database theory are regarded as a ground atom, a nonrecursive function-free definite clause and a finite set of ground atoms, respectively, in inductive logic programming terminology. An acyclic conjunctive query problem is a conjunctive query problem with acyclicity. Concerned with the acyclic conjunctive query problem, in this paper, we present the hardness results of predicting acyclic conjunctive queries from an instance with a j-database of which predicate symbol is at most j-ary. Also we deal with two kinds of instances, a simple instance as a set of ground atoms and an extended instance as a set of pairs of a ground atom and a description. We mainly show that, from both a simple and an extended instances, acyclic conjunctive queries are not polynomial-time predictable with j-databases (j⩾3) under the cryptographic assumptions, and predicting acyclic conjunctive queries with 2-databases is as hard as predicting DNF formulas. Hence, the acyclic conjunctive queries become a natural example that the equivalence between subsumption-efficiency and efficient pac-learnability from both a simple and an extended instances collapses

Complexity in the Case against Accuracy Estimation

Some authors haverepeatedl pointed out that the use of the accuracy, inparticulq for comparingclgN -,::LN is not adequate. The main argument concerns some assumptions ofsel-I 11 vall,N or correctnessunderlnes the use of this criterion. In this paper, we study the computational burden of the accuracy's replacy'sN forbuil:I# and comparingclaringNqP using 13 the framework of Inductive Logic Programming.Replamming is investigated in three ways: complIIIL of the accuracy with anadditional requirement,replrement of the accuracy with 15 bi-criterionrecentl introduced fromstatistical decision theory: the Receiver Operating Characteristicanalisti andrepl,I'NG# of the accuracy by asingl criterion. We prove very hard 17 resul, for most of thepossibl repllONG##I A #rstresul shows thataltNq': the arbitrary multraryNIII' ofcl-IPq appears to betotalq uselq# "Arbitrary" is to be taken in its broadest 19 meaning, inparticul# exponential The second point is the sudden appearance of the negative resuli which is not a function of the criteria's demands. The third point is theequivalNGin 21 di#culN of al these di#erent criteria. In contrast, thesingl accuracy's optimization appears to be tractabl in this framework. 23 c 2002Publ-LL: byEl-L:-O Science B.V. 1. I936361108 An essential task of Machine Learning (ML) and Data Mining (DM) systems is relIII to cl#L-#NGPIOIN ThisbasicalO consists in giving the most accurate answer 27 #TelN +33-596-72-73-64; fax: +33-596-72-73-62

Some lower bounds for the Computational Complexity of Inductive Logic Programming

The field of Inductive Logic Programming (ILP), which is concerned with the induction of Horn clauses from examples and background knowledge, has received increased attention over the last time. Recently, some positive results concerning the learnability of restricted logic programs have been published. In this paper we review these restrictions and prove some lower-bounds of the computational complexity of learning. In particular, we show that a learning algorithm for i2-determinate Horn clauses (with variable i) could be used to decide the PSPACE-complete problem of Finite State Automata Intersection, and that a learning algorithm for 12-nondeterminate Horn clauses could be used to decide the NP-complete problem of Boolean Clause Satisfiability (SAT). This also shows, that these Horn clauses are not PAC-learnable, unless RP = NP = PSPACE. Keywords: Inductive Logic Programming, PAC-Learning. 1 Introduction Most success within the field of Machine Learning has been achiev..