Probabilistic Rule Induction for Transparent CBR Under Uncertainty

We introduce a new approach to probabilistic logic programming in which probabilities are defined over a set of possible worlds. More precisely, classical program clauses are extended by a subinterval of [0,1] that describes a range for the conditional probability of the head of a clause given its body. We then analyze the complexity of selected probabilistic logic programming tasks. It turns out that probabilistic logic programming is computationally more complex than classical logic programming, More precisely, the tractability of special cases of classical logic programming generally does not carry over to the corresponding special cases of probabilistic logic programming. Moreover, we also draw a precise picture of the complexity of deciding and computing tight logical consequences in probabilistic reasoning with conditional constraints in general. We then present linear optimization techniques for deciding satisfiability and computing tight logical consequencesof probabilistic logic programs. These techniques are efficient in the special case in which we have little relevant purely probabilistic knowledge. We finally show that probabilistic logic programming under certain syntactic and semantic restrictions is closely related to van Emden's quantitative deduction, and thus has computational properties similar to calssical logic programming. Based on this result, we present an efficient approximation technique for probabilistic logic programming.

Probabilistic Inductive Logic Programming (PILP) systems extend ILP by allowing the world to be represented using probabilistic facts and rules, and by learning probabilistic theories that can be used to make predictions. However, such systems can be inefficient both due to the large search space inherited from the ILP algorithm and to the probabilistic evaluation needed whenever a new candidate theory is generated. To address the latter issue, this work introduces probability estimators aimed at improving the efficiency of PILP systems. An estimator can avoid the computational cost of probabilistic theory evaluation by providing an estimate of the value of the combination of two subtheories. Experiments are performed on three real-world datasets of different areas (biology, medical and web-based) and show that, by reducing the number of theories to be evaluated, the estimators can significantly shorten the execution time without losing probabilistic accuracy.

Probabilistic Logic Programming extends Logic Programming by enabling the representation of uncertain information by means of probability theory. Probabilistic Logic Programming is at the intersection of two wider research fields: the integration of logic and probability and Probabilistic Programming. Logic enables the representation of complex relations among entities while probability theory is useful for model uncertainty over attributes and relations. Combining the two is a very active field of study. Probabilistic Programming extends programming languages with probabilistic primitives that can be used to write complex probabilistic models. Algorithms for the inference and learning tasks are then provided automatically by the system. Probabilistic Logic programming is at the same time a logic language, with its knowledge representation capabilities, and a Turing complete language, with its computation capabilities, thus providing the best of both worlds. Since its birth, the field of Probabilistic Logic Programming has seen a steady increase of activity, with many proposals for languages and algorithms for inference and learning. Foundations of Probabilistic Logic Programming aims at providing an overview of the field with a special emphasis on languages under the Distribution Semantics, one of the most influential approaches. The book presents the main ideas for semantics, inference, and learning and highlights connections between the methods. Many examples of the book include a link to a page of the web application http://cplint.eu where the code can be run online.

The basics of logic programming, as embodied by Prolog, are generally well-known in the programming language community. However, more advanced techniques, such as tabling, answer subsumption and probabilistic logic programming fail to attract the attention of a larger audience. The cause for the community's seemingly limited interest lies with the presentation of these features: the literature frequently focuses on implementations and examples that do little to aid the understanding of non-experts in the field. The key point is that many of these advanced logic programming features can be characterised in more generally known, more accessible terms. In my research I try to reconcile these advanced concepts from logic programming (Tabling, Answer subsumption and probabilistic programming) with concepts from functional programming (effects, monads and applicative functors).

SummaryWe present the web application ‘cplinton SWI‐Prolog for SHaring that allows the user to write (SWISH)' Probabilistic Logic Programs and submit the computation of the probability of queries with a web browser. The application is based on SWISH, a web framework for Logic Programming. SWISH is based on various features and packages of SWI‐Prolog, in particular, its web server and its Pengine library, that allow to create remote Prolog engines and to pose queries to them. In order to develop the web application, we started from the PITA system, which is included incplint, a suite of programs for reasoning over Logic Programs with Annotated Disjunctions, by porting PITA to SWI‐Prolog. Moreover, we modified the PITA library so that it can be executed in a multi‐threading environment. Developing ‘cplinton SWISH’ also required modification of the JavaScript SWISH code that creates and queries Pengines. ‘cplinton SWISH’ includes a number of examples that cover a wide range of domains and provide interesting applications of Probabilistic Logic Programming. By providing a web interface tocplint, we allow users to experiment with Probabilistic Logic Programming without the need to install a system, a procedure that is often complex, error prone, and limited mainly to the Linux platform. In this way, we aim to reach out to a wider audience and popularize Probabilistic Logic Programming. Copyright © 2015 John Wiley & Sons, Ltd.

Combining probabilistic logic programming with the power of maximum entropy

We examine the meaning and the complexity of probabilistic logic programs that consist of a set of rules and a set of independent probabilistic facts (that is, programs based on Sato's distribution semantics). We focus on two semantics, respectively based on stable and on well-founded models. We show that the semantics based on stable models (referred to as the "credal semantics") produces sets of probability measures that dominate infinitely monotone Choquet capacities; we describe several useful consequences of this result. We then examine the complexity of inference with probabilistic logic programs. We distinguish between the complexity of inference when a probabilistic program and a query are given (the inferential complexity), and the complexity of inference when the probabilistic program is fixed and the query is given (the query complexity, akin to data complexity as used in database theory). We obtain results on the inferential and query complexity for acyclic, stratified, and normal propositional and relational programs; complexity reaches various levels of the counting hierarchy and even exponential levels.

Lifted inference has been proposed for various probabilistic logical frameworks in order to compute the probability of queries in a time that depends on the size of the domains of the random variables rather than the number of instances. Even if various authors have underlined its importance for probabilistic logic programming (PLP), lifted inference has been applied up to now only to relational languages outside of logic programming. In this paper we adapt Generalized Counting First Order Variable Elimination (GC-FOVE) to the problem of computing the probability of queries to probabilistic logic programs under the distribution semantics. In particular, we extend the Prolog Factor Language (PFL) to include two new types of factors that are needed for representing ProbLog programs. These factors take into account the existing causal independence relationships among random variables and are managed by the extension to variable elimination proposed by Zhang and Poole for dealing with convergent variables and heterogeneous factors. Two new operators are added to GC-FOVE for treating heterogeneous factors. The resulting algorithm, called LP2for Lifted Probabilistic Logic Programming, has been implemented by modifying the PFL implementation of GC-FOVE and tested on three benchmarks for lifted inference. A comparison with PITA and ProbLog2 shows the potential of the approach.

Inductive Logic Programming: Issues, results and the challenge of Learning Language in Logic

Probabilistic logic programming offers a powerful way to describe and evaluate structured statistical models. To investigate the practicality of probabilistic logic programming for structure learning in bioinformatics, we undertook a simplified bacterial gene-finding benchmark in PRISM, a probabilistic dialect of Prolog. We evaluate Hidden Markov Model structures for bacterial protein-coding gene potential, including a simple null model structure, three structures based on existing bacterial gene finders and two novel model structures. We test standard versions as well as ADPH length modeling and three-state versions of the five model structures. The models are all represented as probabilistic logic programs and evaluated using the PRISM machine learning system in terms of statistical information criteria and gene-finding prediction accuracy, in two bacterial genomes. Neither of our implementations of the two currently most used model structures are best performing in terms of statistical information criteria or prediction performances, suggesting that better-fitting models might be achievable. The source code of all PRISM models, data and additional scripts are freely available for download at: http://github.com/somork/codonhmm. Supplementary data are available at Bioinformatics online.

We present a probabilistic logic programming framework that allows the representation of conditional probabilities. While conditional probabilities are the most commonly used method for representing uncertainty in probabilistic expert systems, they have been largely neglected by work in quantitative logic programming. We define a fixpoint theory, declarative semantics, and proof procedure for the new class of probabilistic logic programs. Compared to other approaches to quantitative logic programming, we provide a true probabilistic framework with potential applications in probabilistic expert systems and decision support systems. We also discuss the relationship between such programs and Bayesian networks, thus moving toward a unification of two major approaches to automated reasoning.

Recursive loops in a logic program present a challenging problem to the PLP (Probabilistic Logic Programming) framework. On the one hand, they loop forever so that the PLP backward-chaining inferences would never stop. On the other hand, they may generate cyclic influences, which are disallowed in Bayesian networks. Therefore, in existing PLP approaches, logic programs with recursive loops are considered to be problematic and thus are excluded. In this article, we propose a novel solution to this problem by making use of recursive loops to build a stationary dynamic Bayesian network. We introduce a new PLP formalism, called a Bayesian knowledge base . It allows recursive loops and contains logic clauses of the form A ← A 1 ,…, A l , true , Context , Types , which naturally formulate the knowledge that the A i s have direct influences on A in the context Context under the type constraints Types . We use the well-founded model of a logic program to define the direct influence relation and apply SLG-resolution to compute the space of random variables together with their parental connections. This establishes a clear declarative semantics for a Bayesian knowledge base. We view a logic program with recursive loops as a special temporal model, where backward-chaining cycles of the form A ← … A ← … are interpreted as feedbacks. This extends existing PLP approaches, which mainly aim at (nontemporal) relational models.

Many real world domains require the representation of a measure of uncertainty. The most common such representation is probability, and the combination of probability with logic programs has given rise to the field of Probabilistic Logic Programming (PLP), leading to languages such as the Independent Choice Logic, Logic Programs with Annotated Disjunctions (LPADs), Problog, PRISM, and others. These languages share a similar distribution semantics, and methods have been devised to translate programs between these languages. The complexity of computing the probability of queries to these general PLP programs is very high due to the need to combine the probabilities of explanations that may not be exclusive. As one alternative, the PRISM system reduces the complexity of query answering by restricting the form of programs it can evaluate. As an entirely different alternative, Possibilistic Logic Programs adopt a simpler metric of uncertainty than probability.Each of these approaches—general PLP, restricted PLP, and Possibilistic Logic Programming—can be useful in different domains depending on the form of uncertainty to be represented, on the form of programs needed to model problems, and on the scale of the problems to be solved. In this paper, we show how the PITA system, which originally supported the general PLP language of LPADs, can also efficiently support restricted PLP and Possibilistic Logic Programs. PITA relies on tabling with answer subsumption and consists of a transformation along with an API for library functions that interface with answer subsumption. We show that, by adapting its transformation and library functions, PITA can be parameterized to PITA(IND, EXC) which supports the restricted PLP of PRISM, including optimizations that reduce non-discriminating arguments and the computation of Viterbi paths. Furthermore, we show PITA to be competitive with PRISM for complex queries to Hidden Markov Model examples, and sometimes much faster. We further show how PITA can be parameterized to PITA(COUNT) which computes the number of different explanations for a subgoal, and to PITA(POSS) which scalably implements Possibilistic Logic Programming. PITA is a supported package in version 3.3 of XSB.

In this paper, we present a revision strategy of revising a conditional probabilistic logic program (PLP) when new information is received (which is in the form of probabilistic formulae), through the technique of variable forgetting. We first extend the traditional forgetting method to forget a conditional event in PLPs. We then propose two revision operators to revise a PLP based on our forgetting method. By revision through forgetting, the irrelevant knowledge in the original PLP is retained according to the minimal change principle. We prove that our revision operators satisfy most of the postulates for probabilistic belief revision. A main advantage of our revision operators is that a new PLP is explicitly obtained after revision, since our revision operator performs forgetting a conditional event at the syntax level.

This paper is concerned with problems that arise when submitting large quantities of data to analysis by an Inductive Logic Programming (ILP) system. Complexity arguments usually make it prohibitive to analyse such datasets in their entirety. We examine two schemes that allow an ILP system to construct theories by sampling from this large pool of data. The first, “subsampling”, is a single-sample design in which the utility of a potential rule is evaluated on a randomly selected sub-sample of the data. The second, “logical windowing”, is multiple-sample design that tests and sequentially includes errors made by a partially correct theory. Both schemes are derived from techniques developed to enable propositional learning methods (like decision trees) to cope with large datasets. The ILP system CProgol, equipped with each of these methods, is used to construct theories for two datasets—one artificial (a chess endgame) and the other naturally occurring (a language tagging problem). In each case, we ask the following questions of CProgol equipped with sampling: (1) Is its theory comparable in predictive accuracy to that obtained if all the data were used (that is, no sampling was employed)?s and (2) Is its theory constructed in less time than the one obtained with all the data? For the problems considered, the answers to these questions is “yes”. This suggests that an ILP program equipped with an appropriate sampling method could begin to address problems satisfactorily that have hitherto been inaccessible simply due to data extent.