Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Generalising the Interaction Rules in Probabilistic Logic Arjen Hommersom and Peter J. F. Lucas Institute for Computing and Information Sciences Radboud University Nijmegen Nijmegen, The Netherlands {arjenh,peterl}@cs.ru.nl Abstract probabilistic logic hand in hand such that the resulting logic supports this double, logical and probabilistic, view. The re- The last two decades has seen the emergence of sulting class of probabilistic logics has the advantage that it many different probabilistic logics that use logi- can be used to gear a logic to the problem at hand. cal languages to specify, and sometimes reason, In the next section, we first review the basics of ProbLog, with probability distributions. Probabilistic log- which is followed by the development of the main methods ics that support reasoning with probability distri- used in generalising probabilistic Boolean interaction, and, butions, such as ProbLog, use an implicit definition finally, default logic is briefly discussed. We will use prob- of an interaction rule to combine probabilistic ev- abilistic Boolean interaction as a sound and generic, alge- idence about atoms. In this paper, we show that braic way to combine uncertain evidence, whereas default this interaction rule is an example of a more general logic will be used as our language to implement the interac- class of interactions that can be described by non- tion operators, again reflecting this double perspective on the monotonic logics. We furthermore show that such probabilistic logic. The new probabilistic logical framework local interactions about the probability of an atom is described in Section 3 and compared to other approaches can be described by convolution. The resulting ex- in Section 4. The achievements of this research are reflected tended probabilistic logic supports non-monotonic upon in Section 5. reasoning with probabilistic information. 2 Preliminaries 1 Introduction In this section, probabilistic logic and a method for combin- The last two decades has seen the emergence of many dif- ing probabilistic evidence is described. We also briefly review ferent probabilistic logics, such as Markov logic [Richardson concepts from default logic. ] [ and Domingos, 2006 , probabilistic Horn clause logic Poole, 2.1 Probabilistic Logic: Syntax and Semantics 1993], and ProbLog [Kimmig et al., 2010], that use logical languages to specify, and sometimes reason, with probabil- There are many different proposals in the scientific literature ity distributions. Probabilistic logics are based on first-order for probabilistic logics. In this paper we focus on the class logic, and they share the advantage of first-order logic in com- of logics where probabilistic reasoning and logical reason- parison to propositional logic in that they can be looked upon ing go hand in hand. ProbLog is a typical example of such a as more expressive knowledge-representation languages with language [Kimmig et al., 2010]. ProbLog was specifically de- considerable modelling power, which for probabilistic logics signed as a basic language to which other probabilistic logical is extended to areas where uncertainty is encountered. languages can be compiled [De Raedt et al., 2008]; thus, the However, despite their generality, all of these languages results from this paper apply to a whole range of languages. are based on different semantic principles. Some of the prin- It is assumed that the reader is familiar with logic program- ciples, while taken for granted, should actually be considered ming terminology, where a program consists of Horn clauses as part of the design choices. When different choices would of the form B ← A ,...,A . have been made, a different probabilistic logic would have 1 n emerged because of the close interaction between the logical where B, Ai are atoms of the form p(t1,...,tm), with tk aspects and probabilistic aspects of a probabilistic logic. terms.Ifn =0, then the Horn clause is called a fact,ifB In the research described in the present paper, we introduce is ⊥ (false), then the clause is called a query; otherwise, the the theoretical foundation of a whole class of probabilistic clause is called a rule. logics, starting from a very general and unrestrictive proba- The ProbLog language extends logic programming by al- bilistic logic, the above-mentioned ProbLog, and show that it lowing that probabilities are attached to facts, then called la- is possible to obtain a new, but related, probabilistic logic by belled facts. Let F denote the set of labelled facts, then each changing the way probabilistic evidence is combined. We do element of F has the form this by redesigning the probabilistic and logical basis of the p :: f. 912 also abbreviated to pf , where pf = P (f) ∈ [0, 1] has the flu and pneumonia. Thus, according to Equation (1) we get: meaning of a probability, and f conforms to the syntax of an P ( )=P ( )P ( ) atom. The meaning of an atom in terms of probability theory T fever flu pneumonia is that of a set of random variables. For example, + P (flu)(1 − P (pneumonia)) +(1− P (flu))P (pneumonia) 0.4::parent(X). =0.7 · 0.2+0.3 · 0.2+0.7 · 0.8 specifies the collection of random variables parent(X), for =0.14 + 0.06 + 0.56 ( ) each ground instance of parent X , obtained by applying =0.76 substitutions Θ to parent(X). For example, parent(X)θ = parent(john) with θ = {john/X}, where θ ∈ Θ. The result- As is well known, this probabilistic result is identical to what ing ground atoms are called logical facts. would have been obtained by one of the most popular ways to In addition to labelled facts, a ProbLog program consists of model the interaction of (conditionally) independent events: rules, constituting the background knowledge B of the pro- the so-called noisy-OR [Pearl, 1988]. With the noisy-OR one gram. Now, let T = F ∪ B be a ProbLog program and let models an uncertain, disjunctive interaction between events. Θf be the set of all possible substitutions associated with the As the noisy-OR is based on logical disjunction, just one of logical fact f. Then, LT is defined as the subset-maximal set the 16 binary Boolean operators, one can imagine that there of logical facts that can be added to B applying the set of might be other, equally sound, ways to combine probabilis- substitutions Θf to the fact f, for each f ∈ F . The ProbLog tic evidence. However, in order to be able to combine such program then defines a joint probability distribution on the evidence, one needs an algebraic method to combine such logical facts LT . Let L ⊆ LT , then: information. A general, algebraic way to combine probabilis- tic information is available from basic probability theory al- PT (L)= pf (1 − pf ) . though it is rarely used to model Boolean interaction. In the fθ∈L f θ ∈LT \L following, we briefly review the necessary basics from prob- ability theory, with convolution as a special case, and then Let q now be any query to the ProbLog program, then it investigate how ideas from probabilistic logic, Boolean in- holds that teraction and default logic can be merged to obtain a more expressive probabilistic logic, which we call probabilistic in- PT (q)= P (q | L)PT (L), (1) teraction logic, or ProbIL for short. L⊆LT where 2.2 Probabilistic Boolean Interaction In the following, a probability mass function of a random 1 ∃θ : B ∪ L qθ P (q | L)= if variable X is referred to by fX ; P denotes the associated 0 otherwise probability distribution. A classical result from probability L P (q | L)= theory that is useful when studying sums of variables is the and ‘ ’ indicates logical entailment. Each with [ 1 q following well-known theorem (cf. Grimmett and Stirzaker, is called an explanation of . 2001]). The semantics of ProbLog is called the distribution seman- tics; it has been borrowed from PRISM [Sato, 1995]. Basi- Theorem 1. Let f be a joint probability mass function of the cally, in the distribution semantics all facts are assumed to be random variables X and Y , such that X +Y = z. Then it P (X + Y = z)=f (z)= f(x, z − x) mutually independent. However, this does not imply that it holds that X+Y x . is impossible to encode probabilistic dependences: all depen- dences are defined at the logical level. This allows defining Proof. See [Grimmett and Stirzaker, 2001]. any joint probability distribution. The distribution semantics X Y also has particular consequences for obtaining probabilistic If and are independent, then, in addition, the follow- interactions between facts, as illustrated by the following ex- ing corollary holds. ample. Corollary 1. Let X and Y be two independent random vari- ables, then it holds that Example 1. Consider the following (trivial) ProbLog pro- T gram that represents some causal, medical knowledge: P (X + Y = z)=fX+Y (z) 0.7 :: flu. = P (X = x)P (Y = z − x) 0.2 :: pneumonia. x fever ← flu. = fX (x)fY (z − x) (2) fever ← pneumonia. x In this program flu and pneumonia are two independent ran- The probability mass function fX+Y is in that case called dom variables according to the distribution semantics. We the convolution of fX and fY , and it is commonly denoted as now wish to compute the probability PT (fever). Note that fX+Y = fX ∗fY . The convolution theorem is very useful, as fever can be explained from either flu, pneumonia, or both sums of independent random variables occur very frequently 913 in probability theory and statistics. It can also be applied re- Thus, the example demonstrates that the noisy-OR can be cursively, i.e., described quite naturally by convolution.