Incremental learning of event definitions with Inductive Logic Programming
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Abstract
Event recognition systems rely on knowledge bases of event definitions to infer occurrences of events in time. Using a logical framework for representing and reasoning about events offers direct connections to machine learning, via Inductive Logic Programming (ILP), thus allowing to avoid the tedious and errorprone task of manual knowledge construction. However, learning temporal logical formalisms, which are typically utilized by logicbased event recognition systems is a challenging task, which most ILP systems cannot fully undertake. In addition, eventbased data is usually massive and collected at different times and under various circumstances. Ideally, systems that learn from temporal data should be able to operate in an incremental mode, that is, revise prior constructed knowledge in the face of new evidence. In this work we present an incremental method for learning and revising eventbased knowledge, in the form of Event Calculus programs. The proposed algorithm relies on abductive–inductive learning and comprises a scalable clause refinement methodology, based on a compressive summarization of clause coverage in a stream of examples. We present an empirical evaluation of our approach on real and synthetic data from activity recognition and city transport applications.
Keywords
Incremental learning Abductive–Inductive Logic Programming Event Calculus Event recognition1 Introduction
The growing amounts of temporal data collected during the execution of various tasks within organizations are hard to utilize without the assistance of automated processes. Event recognition (Etzion and Niblett 2010; Luckham 2001; Luckham and Schulte 2008) refers to the automatic detection of event occurrences within a system. From a sequence of lowlevel events (for example sensor data) an event recognition system recognizes highlevel events of interest, that is, events that satisfy some pattern. Event recognition systems with a logicbased representation of event definitions, such as the Event Calculus (Kowalski and Sergot 1986), are attracting significant attention in the event processing community for a number of reasons, including the expressiveness and understandability of the formalized knowledge, their declarative, formal semantics (Paschke 2005; Artikis et al. 2012) and their ability to handle rich background knowledge. Using logic programs in particular, has an extra advantage, due to the close connection between logic programming and machine learning in the field of Inductive Logic Programming (ILP) (Lavrač and Džeroski 1993; Muggleton and De Raedt 1994). However, such applications impose challenges that make most ILP systems inappropriate.
Several logical formalisms which incorporate time and change employ nonmonotonic operators as a means for representing commonsense phenomena (Mueller 2006). Negation as Failure (NaF) is a prominent example. However, most ILP learners cannot handle NaF at all, or lack a robust NaF semantics (Sakama 2000; Ray 2009). Another problem that often arises when dealing with events, is the need to infer implicit or missing knowledge, for instance possible causes of observed events. In ILP the ability to reason with missing, or indirectly observable knowledge is called nonObservational Predicate Learning (nonOPL) (Muggleton 1995). This is a task that most ILP systems have difficulty to handle, especially when combined with NaF in the background knowledge (Ray 2006). One way to address this problem is through the combination of ILP with Abductive Logic Programming (ALP) (Denecker and Kakas 2002; Kakas and Mancarella 1990; Kakas et al. 1993). Abduction in logic programming is usually given a nonmonotonic semantics (Eshghi and Kowalski 1989) and in addition, it is by nature an appropriate framework for reasoning with incomplete knowledge. The combination of ILP with ALP has a long history in the literature (Ade and Denecker 1995). However, only recently has it brought about systems such as XHAIL (Ray 2009), TAL (Corapi et al. 2010) and ASPAL (Corapi et al. 2012; Athakravi et al. 2013) that may be used for the induction of eventbased knowledge.
The above three systems which, to the best of our knowledge, are the only ILP learners that address the aforementioned learnability issues, are batch learners, in the sense that all training data must be in place prior to the initiation of the learning process. This is not always suitable for eventoriented learning tasks, where data is often collected at different times and under various circumstances, or arrives in streams. In order to account for new training examples, a batch learner has no alternative but to relearn a hypothesis from scratch. The cost is poor scalability when “learning in the large” (Dietterich et al. 2008) from a growing set of data. This is particularly true in the case of temporal data, which usually come in large volumes. Consider for instance data which span a large period of time, or sensor data transmitted at a very high frequency.
An alternative approach is learning incrementally, that is, processing training instances when they become available, and altering previously inferred knowledge to fit new observations. This process, also known as Theory Revision (Wrobel 1996), exploits previous computations to speedup the learning, since revising a hypothesis is generally considered more efficient than learning it from scratch (Biba et al. 2006; Esposito et al. 2000; Cattafi et al. 2010). Numerous theory revision systems have been proposed in the literature, however their applicability in the presence of NaF is limited (Corapi et al. 2008). Additionally, as historical data grow over time, it becomes progressively harder to revise knowledge, so that it accounts both for new evidence and past experience. The development of scalable algorithms for theory revision has thus been identified as an important endeavour (Muggleton et al. 2012). One direction towards scaling theory revision systems is the development of techniques for reducing the need for reconsulting the whole history of accumulated experience, while updating existing knowledge.
This is the direction we take in this work. We build upon the ideas of nonmonotonic ILP and use XHAIL as the basis for a scalable, incremental learner for the induction of event definitions in the form of Event Calculus theories. XHAIL has been used for the induction of action theories (Sloman and Lupu 2010; Alrajeh et al. 2010, 2011, 2012, 2009). Moreover, in Corapi et al. (2008) it has been used for theory revision in an incremental setting, revising hypotheses with respect to a recent, userdefined subset of the perceived experience. In contrast, the learner we present here performs revisions that account for all examples seen so far. We describe a compressive “memory” structure, which reduces the need for reconsulting past experience in response to a revision. Using this structure, we propose a method which, given a stream of examples, a theory which accounts for them and a new training instance, requires at most one pass over the examples in order to revise the initial theory, so that it accounts for both past and new evidence. We evaluate empirically our approach on real and synthetic data from an activity recognition application and a transport management application. Our results indicate that our approach is significantly more efficient than XHAIL, without compromising predictive accuracy, and scales adequately to large data volumes.
The rest of this paper is structured as follows. In Sect. 2 we present the Event Calculus dialect that we employ, describe the domain of activity recognition that we use as a running example and discuss abductive–inductive learning and XHAIL. In Sect. 3 we present our proposed method. In Sect. 4 we discuss some theoretical and practical implications of our approach. In Sect. 5 we present the experimental evaluation, and finally in Sects. 6 and 7 we discuss related work and draw our main conclusions.
2 Background
We assume a firstorder language as in Lloyd (1987) where not in front of literals denotes NaF. We define the entailment relation between logic programs in terms of the stable model semantics (Gelfond and Lifschitz 1988)—see “Appendix 1” for details on the basics of logic programming used in this work. Following Prolog’s convention, predicates and ground terms in logical formulae start with a lower case letter, while variable terms start with a capital letter.
2.1 The Event Calculus
The Event Calculus (Kowalski and Sergot 1986) is a temporal logic for reasoning about events and their effects. It is a formalism that has been successfully used in numerous event recognition applications (Paschke 2005; Artikis et al. 2015; Chaudet 2006; Cervesato and Montanari 2000). The ontology of the Event Calculus comprises time points, i.e. integers or real numbers; fluents, i.e. properties which have certain values in time; and events, i.e. occurrences in time that may affect fluents and alter their value. The domainindependent axioms of the formalism incorporate the common sense law of inertia, according to which fluents persist over time, unless they are affected by an event. We call the Event Calculus dialect used in this work Simplified Discrete Event Calculus (SDEC). It is a simplified version of the Discrete Event Calculus, a dialect which is equivalent to the classical Event Calculus when time ranges over integer domains (Mueller 2008).
The basic predicates and axioms of SDEC
Predicate  Meaning 

happensAt(E, T)  Event E occurs at time T 
initiatedAt(F, T)  At time T a period of time for which fluent F holds is initiated 
terminatedAt(F, T)  At time T a period of time for which fluent F holds is terminated 
holdsAt(F, T)  Fluent F holds at time T 
Axioms  
\(\begin{array}{lr} \mathsf{holdsAt }(F,T+1) \leftarrow \\ \qquad \mathsf{initiatedAt }(F,T)\\ \end{array}\)  \(\begin{array}{l} \mathsf{holdsAt }(F,T+1) \leftarrow \\ \qquad \mathsf{holdsAt }(F,T),\\ \qquad \mathsf{not }\ \mathsf{terminatedAt }(F,T) \\ \end{array}\) 
An annotated stream of lowlevel events
Narrative  Annotation 

\(\cdots \)  \(\cdots \) 
happensAt \((inactive(id_1),999)\)  not holdsAt \((moving(id_1,id_2),999)\) 
happensAt \((active(id_2),999)\)  
holdsAt \((coords(id_1,201,432),999)\)  
holdsAt \((coords(id_2,230,460),999)\)  
holdsAt \((direction(id_1,270),999)\)  
holdsAt \((direction(id_2,270),999)\)  
happensAt \((walking(id_1),1000)\)  not holdsAt \((moving(id_1,id_2),1000)\) 
happensAt \((walking(id_2),1000)\)  
holdsAt \((coords(id_1,201,454),1000)\)  
holdsAt \((coords(id_2,230,440),1000)\)  
holdsAt \((direction(id_1,270),1000)\)  
holdsAt \((direction(id_2,270),1000)\)  
happensAt \((walking(id_1),1001)\)  holdsAt \((moving(id_1,id_2),1001)\) 
happensAt \((walking(id_2),1001)\)  
holdsAt \((coords(id_1,201,454),1001)\)  
holdsAt \((coords(id_2,227,440),1001)\)  
holdsAt \((direction(id_1,275),1001)\)  
holdsAt \((direction(id_2,278),1001)\)  
\(\cdots \)  \(\cdots \) 
Lowlevel events are represented in SDEC by streams of ground happensAt/2 atoms (see Table 2), while highlevel events and other domain knowledge are represented by ground holdsAt/2 atoms. Streams of lowlevel events together with domainspecific knowledge will henceforth constitute the narrative, in ILP terminology, while knowledge about highlevel events is the annotation. Table 2 presents an annotated stream of lowlevel events. We can see for instance that the person \(id_1\) is inactive at time 999, her (x, y) coordinates are (201, 432) and her direction is \(270^{\circ }\). The annotation for the same time point informs us that \(id_1\) and \(id_2\) are not moving together. Fluents express both highlevel events and input information, such as the coordinates of a person. We discriminate between inertial and statically defined fluents. The former should be inferred by the Event Calculus axioms, while the latter are provided with the input.
2.2 Abductive–inductive learning
Given a domain description in the language of SDEC, the aim of machine learning addressed in this work is to derive the DomainSpecific Axioms, that is, the axioms that specify how the occurrence of lowlevel events affect the truth values of fluents that represent highlevel events, by initiating or terminating them. Thus, we wish to learn initiatedAt/2 and terminatedAt/2 definitions from positive and negative examples.
Inductive Logic Programming (ILP) refers to a set of techniques for learning hypotheses in the form of clausal theories, and is the machine learning approach that we adopt in this work. An ILP task is a triplet ILP(B, E, M) where B is some background knowledge, M is some language bias and \(E = E^{+}\cup E^{}\) is a set of positive (\(E^{+}\)) and negative (\(E^{}\)) examples for the target predicates, represented as logical facts. The goal is to derive a set of nonground clauses H in the language of M that cover the examples w.r.t. B, i.e. \(B\cup H \vDash E^{+}\) and \(B\cup H \nvDash E^{}\).
Henceforth, the term “example” encompasses anything known true at a specific time point. We assume a closed world, thus anything that is not explicitly given is considered false (to avoid confusion, in the tables throughout the paper we state both positive and negated annotation atoms). An example is either positive or negative based on the annotation. For instance the examples at times 999 and 1000 in Table 2 are negative, while the example at time 1001 is positive.
Learning event definitions in the form of domainspecific Event Calculus axioms with ILP requires nonObservational Predicate Learning (nonOPL) (Muggleton 1995), meaning that instances of target predicates (initiatedAt/2 and terminatedAt/2) are not provided with the supervision, which consists of holdsAt/2 atoms. A solution is to use Abductive Logic Programming (ALP) to obtain the missing instances. An ALP task is a triplet ALP(B, A, G) where B is some background knowledge, G is a set of observations that must be explained, represented by ground logical atoms, and A is a set of predicates called abducibles. An explanation \({\varDelta }\) is a set of ground atoms from A such that \(B\cup {\varDelta } \vDash G\).
2.2.1 The XHAIL system
XHAIL is an abductive–inductive system that constructs hypotheses in a threephase process. Given an ILP task \( ILP(B,E,M) \), the first two phases return a ground program K, called Kernel Set of E, such that \(B\cup K\vDash E\). The first phase generates the heads of K’s clauses by abductively deriving from B a set \({\varDelta }\) of instances of head atoms, as defined by the language bias, such that \(B\cup {\varDelta } \vDash E\). The second phase generates K, by saturating each previously abduced atom with instances of body atoms that deductively follow from \(B\cup {\varDelta }\). The language bias used by XHAIL is mode declarations (see “Appendix 1” for a formal account).
By construction, the Kernel Set covers the provided examples. In order to find a good hypothesis, XHAIL thus searches in the space of theories that subsume the Kernel Set. To this end, the latter is variabilized, i.e. each term that corresponds to a variable, according to the language bias, is replaced by an actual variable. The variablized Kernel Set \(K_v\) is subject to a syntactic transformation of its clauses, which involves two new predicates \( try/3 \) and \( use/2 \).
For each clause \(C_i \in K_v\) and each body literal \(\delta _{i}^{j}\in C_i\), a new atom \( v(\delta _{i}^{j}) \) is generated, as a special term that contains the variables that appear in \(\delta _{i}^{j}\). The new atom is wrapped inside an atom of the form \( try(i,j,v(\delta _{i}^{j})) \). An extra atom \( use(i,0) \) is added to the body of \(C_i\) and two new clauses \( try(i,j,v(\delta _{i}^{j})) \leftarrow use(i,j), \delta _{i}^{j} \) and \( try(i,j,v(\delta _{i}^{j})) \leftarrow \mathsf{not }\ use(i,j) \) are generated, for each body literal \(\delta _{i}^{j} \in C_i\).
Hypothesis generation by XHAIL for Example 1
Input  
Narrative  Annotation 
\( \mathsf{happensAt }(abrupt(id_1), \ 1). \)  \( \mathsf{holdsAt }(fighting(id_1,id_2), \ 1). \) 
\( \mathsf{happensAt }(walking(id_2), \ 1). \)  \( \mathsf{not }\ \mathsf{holdsAt }(fighting(id_3,id_4), \ 1). \) 
\( \mathsf{not }\ \mathsf{holdsAt }(close(id_1,id_2,23), \ 1). \)  \( \mathsf{not }\ \mathsf{holdsAt }(fighting(id_1,id_2), \ 2). \) 
\( \mathsf{happensAt }(abrupt(id_3), \ 2). \)  \( \mathsf{not }\ \mathsf{holdsAt }(fighting(id_3,id_4), \ 2). \) 
\( \mathsf{happensAt }(abrupt(id_4), \ 2). \)  \( \mathsf{not }\ \mathsf{holdsAt }(fighting(id_1,id_2), \ 3). \) 
\( \mathsf{holdsAt }(close(id_3,id_4,23), \ 2). \)  \( \mathsf{holdsAt }(fighting(id_3,id_4), \ 3). \) 
Mode declarations  Background knowledge: SDEC (Table 1) 
Phase 1 (Abduction):  
\({\varDelta }_ 1 = \{ \mathsf{initiatedAt }(fighting(id_3,id_4), \ 2) ,\)  
\( \mathsf{terminatedAt }(fighting(id_1,id_2), \ 1) \} \)  
Phase 2 (Deduction):  
Kernel Set K:  Variabilized Kernel Set \(K_v\): 
\( \mathsf{initiatedAt }(fighting(id_3,id_4), \ 2) \ \leftarrow \)  \( \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \) 
\(\quad \mathsf{happensAt }(abrupt(id_3), \ 2), \)  \(\quad \mathsf{happensAt }(abrupt(X), \ T), \) 
\( \quad \mathsf{happensAt }(abrupt(id_4), \ 2), \)  \(\quad \mathsf{happensAt }(abrupt(Y), \ T), \) 
\(\quad \mathsf{holdsAt }(close(id_3,id_4,23), \ 2). \)  \(\quad \mathsf{holdsAt }(close(X,Y,23), \ T). \) 
\( \mathsf{terminatedAt }(fighting(id_1,id_2), \ 1) \ \leftarrow \)  \( \mathsf{terminatedAt }(fighting(X,Y), \ T) \ \leftarrow \) 
\(\quad \mathsf{happensAt }(abrupt(id_1), \ 1), \)  \(\quad \mathsf{happensAt }(abrupt(X), \ T), \) 
\(\quad \mathsf{happensAt }(walking(id_2), \ 1), \)  \(\quad \mathsf{happensAt }(walking(Y), \ T), \) 
\(\quad \mathsf{not }\ \mathsf{holdsAt }(close(id_1,id_2,23), \ 1). \)  \(\quad \mathsf{not }\ \mathsf{holdsAt }(close(X,Y,23), \ T). \) 
Phase 3 (Induction):  
Program \(U_{K_v}\) (Syntactic transformation of \(K_v\)):  
\( \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \)  \( \mathsf{terminatedAt }(fighting(X,Y), \ T) \ \leftarrow \) 
\(\quad use(1,0),try(1,1,vars(X,T)), \)  \(\quad use(2,0),try(2,1,vars(X,T)), \) 
\(\quad try(1,2,vars(Y,T)), \)  \(\quad try(2,2,vars(Y,T)), \) 
\(\quad try(1,3,vars(X,Y,T)). \)  \(\quad try(2,3,vars(X,Y,T)). \) 
\( try(1,1,vars(X,T)) \ \leftarrow \)  \( try(2,1,vars(X,T)) \ \leftarrow \) 
\(\quad use(1,1),\mathsf{happensAt }(abrupt(X), \ T). \)  \(\quad use(2,1),\mathsf{happensAt }(abrupt(X), \ T). \) 
\( try(1,1,vars(X,T)) \ \leftarrow \mathsf{not }\ use(1,1).\)  \( try(2,1,vars(X,T)) \ \leftarrow \mathsf{not }\ use(1,1).\) 
\( try(1,2,vars(Y,T)) \ \leftarrow \)  \( try(2,2,vars(Y,T)) \ \leftarrow \) 
\(\quad use(1,2),\mathsf{happensAt }(abrupt(Y), \ T). \)  \(\quad use(2,2),\mathsf{happensAt }(walking(Y), \ T). \) 
\( try(1,2,vars(X,T)) \ \leftarrow \mathsf{not }\ use(1,2).\)  \( try(2,2,vars(Y,T)) \ \leftarrow \mathsf{not }\ use(2,2).\) 
\( try(1,3,vars(X,Y,T)) \ \leftarrow \)  \( try(2,3,vars(X,Y,T)) \ \leftarrow \) 
\(\quad use(1,3),\mathsf{holdsAt }(close(X,Y,23), \ T). \)  \(\quad use(2,3),\mathsf{not }\ \mathsf{holdsAt }(close(X,Y,23), \ T). \) 
\( try(1,3,vars(X,T)) \ \leftarrow \mathsf{not }\ use(1,3).\)  \( try(2,3,vars(X,Y,T)) \ \leftarrow \mathsf{not }\ use(2,3).\) 
Search:  Abductive Solution: 
\( ALP(\mathsf {SDEC}\cup U_{K_v},\{use/2\},Narrative\cup Annotation )\)  \({\varDelta }_2 = \{use(1,0),use(1,3), \) 
\(\qquad \quad use(2,0),use(2,2) \}\)  
Output hypothesis  
\( \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \)  \( \mathsf{terminatedAt }(fighting(X,Y), \ T) \ \leftarrow \) 
\(\quad \mathsf{holdsAt }(close(X,Y,23), \ T). \)  \(\quad \mathsf{happensAt }(walking(Y), \ T). \) 
Example 1
Table 3 presents the process of hypothesis generation by XHAIL. The input consists of a set of examples, a set of mode declarations (omitted for simplicity) and the axioms of SDEC as background knowledge. The annotation says that \( fighting \) between persons \(id_1\) and \(id_2\) holds at time 1 and it does not hold at times 2 and 3, hence it is terminated at time 1. Respectively, fighting between persons \(id_3\) and \(id_4\) holds at time 3 and does not hold at times 1 and 2, hence it is initiated at time 2. XHAIL obtains these explanations for the holdsAt/2 literals of the annotation abductively, using the head mode declarations as abducibles. In its first phase, it derives the two ground atoms in \({\varDelta }_1\) (Phase 1, Table 3). In its second phase, XHAIL forms a Kernel Set (Phase 2, Table 3), by generating one clause from each abduced atom in \({\varDelta }_1\), using this atom as head, and body literals that deductively follow from \(\mathsf {SDEC}\cup {\varDelta }_1\) as the body of the clause.
The Kernel Set is variabilized and the third phase of XHAIL functionality concerns the actual search for a hypothesis. This search is biased by minimality, i.e. preference towards hypotheses with fewer literals. A hypothesis is thus constructed by dropping as many literals and clauses from \(K_v\) as possible, while correctly accounting for all the examples. The syntactic transformation on \(K_v\) (see also Sect. 2.2.1) results in the defeasible program \(U_{K_v}\).
Literals and clauses necessary to cover the examples are selected from \(U_{K_v}\) by means of abducing a set of use / 2 atoms, as explanations of the examples, from the ALP task presented in Phase 3 of Table 3. \({\varDelta }_2\) from Table 3 is a minimal explanation for this ALP task. \( use(1,0) \) and \( use(2,0) \) correspond to the head atoms of the two \(K_v\) clauses, while \( use(1,3) \) and \( use(2,2) \) correspond respectively to their third and second body literal. The output hypothesis in Table 3 is constructed by these literals, while all other literals and clauses from \(K_v\) are discarded. \(\square \)
XHAIL provides an appropriate framework for learning Event Calculus programs. A major drawback however is that it scales poorly, partly because of the increased computational complexity of adbuction, which lies at the core of its functionality, and partly because of the combinatorial complexity of learning whole theories, which may result in an intractable search space. In what follows, we use the XHAIL machinery to develop a novel incremental algorithm that scales to large volumes of sequential data, typical of eventbased applications.
3 ILED: incremental learning of event definitions
A hypothesis H is called incomplete if it does not account for some positive examples and inconsistent if it erroneously accounts for some negative examples. Incompleteness is typically treated by generalization, e.g. addition of new clauses, or removal of literals from existing clauses. Inconsistency is treated by specialization, e.g. removal of clauses, or addition of new literals to existing clauses. Theory revision is the process of acting upon a hypothesis, in order to change the examples it accounts for. Theory revision is at the core of incremental learning settings, where examples are provided over time. A learner induces a hypothesis from scratch, from the first available set of examples, and treats this hypothesis as a revisable background theory in order to account for new examples.
Definition 1 provides a concrete account of the incremental setting we assume for our approach, which we call incremental learning of event definitions (ILED).
Definition 1
(Incremental learning) We assume an ILP task \( ILP(\mathsf {SDEC},\mathcal {E},M) \), where \(\mathcal {E}\) is a database of examples, called historical memory, storing examples presented over time. Initially \(\mathcal {E} = \emptyset \). At time n the learner is presented with a hypothesis \(H_n\) such that \(\mathsf {SDEC} \cup H_n \vDash \mathcal {E}\), in addition to a new set of examples \(w_n\). The goal is to revise \(H_n\) to a hypothesis \(H_{n+1}\), so that \(\mathsf {SDEC} \cup H_{n+1} \vDash \mathcal {E}\cup w_n\).
A main challenge of adopting a full memory approach is to scale it up to a growing size of experience. This is in line with a key requirement of incremental learning where “the incorporation of experience into memory during learning should be computationally efficient, that is, theory revision must be efficient in fitting new incoming observations” (Langley 1995; Di Mauro et al. 2005). In the stream processing literature, the number of passes over a stream of data is often used as a measure of the efficiency of algorithms (Li et al. 2004; Li and Lee 2009). In this spirit, the main contribution of ILED, in addition to scaling up XHAIL, is that it adopts a “singlepass” theory revision strategy, that is, a strategy that requires at most one pass over \(\mathcal {E}\) in order to compute \(H_{n+1}\) from \(H_{n}\).

Addition of new clauses.

Refinement of existing clauses, i.e. replacement of an existing clause with one or more specializations of that clause.
Figure 1 illustrates the revision process with a simple example. New clauses are generated by generalizing a Kernel Set of the incoming window, as shown in Fig. 1, where a terminatedAt/2 clause is generated from the new window \(w_n\). To facilitate refinement of existing clauses, each clause in the running hypothesis is associated with a memory of the examples it covers throughout \(\mathcal {E}\), in the form of a “bottom program”, which we call support set. The support set is constructed gradually, from previous Kernel Sets, as new example windows arrive. It serves as a refinement search space, where the single clause in the running hypothesis \(H_n\) is refined w.r.t. the incoming window \(w_n\) into two specializations. Each such specialization is constructed by adding to the initial clause one antecedent from the two support set clauses which are presented in Fig. 1. The revised hypothesis \(H_{n+1}\) is constructed from the refined clauses and the new ones, along with the preserved clauses of \(H_n\), if any.
ILED’s support set can be seen as the Sset in a version space (Mitchell 1979), i.e. the space of all overlyspecific hypotheses, progressively augmented while new examples arrive. Similarly, a running hypothesis of ILED can be seen as an element of the Gset in a version space, i.e. the space of all overlygeneral hypotheses that account for all examples seen so far, and need to be further refined as new examples arrive.
There are two key features of ILED that contribute towards its scalability: First, reprocessing of past experience is necessary only in the case where new clauses are generated by a revision, and is redundant in the case where a revision consists of refinements of existing clauses only. Second, reprocessing of past experience requires a single pass over the historical memory, meaning that it suffices to revisit each past window exactly once to ensure that the output revised hypothesis \(H_{n+1}\) is complete & consistent w.r.t. the entire historical memory. These properties of ILED are due to the support set, which we next present in detail. A proof of soundness and the singlepass revision strategy of ILED is given in Proposition 3, “Appendix 2”. The Pseudocode of ILED’s strategy is provided in Algorithm 3, “Appendix 2”.
3.1 Support set
In order to define the support set, we use the notion of mostspecific clause. Given a set of mode declarations M, a clause C in the mode language \(\mathcal {L}(M)\) (see “Appendix 1” for a formal definition) is mostspecific if it does not \(\theta \)subsume any other clause in \(\mathcal {L}(M)\). \(\theta \)subsumption is defined below.
Definition 2
(\(\theta \)subsumption) Clause C \(\theta \)subsumes clause D, denoted \(C \preceq D\), if there exists a substitution \(\theta \) such that \(head(C)\theta = \ head(D)\) and \(body(C)\theta \subseteq body(D)\), where \( head(C) \) and \( body(C) \) denote the head and the body of clause C respectively. Program \(\Pi _1\) \(\theta \)subsumes program \(\Pi _2\) if for each clause \(C\in \Pi _1\) there exists a clause \(D\in \Pi _2\) such that \(C\preceq D\).
Intuitively, the support set of a clause C is a “bottom program” that consists of mostspecific versions of the clauses that disjunctively define the concept captured by C. A formal account is given in Definition 3.
Definition 3
The support set of clause C is thus defined as the set consisting of one bottom clause (Muggleton 1995) per each example \(e \in cov_{\mathcal {E}}(C) \), i.e. one mostspecific clause D of \(\mathcal {L}(M)\) such that \(C \preceq D\) and \(\mathsf {SDEC} \cup D \vDash e\). Assuming no length bounds on hypothesized clauses, each such bottom clause is unique^{2} and covers at least one example from \( cov_{E}(C) \); note that since the bottom clauses for a set of examples in \( cov_{\mathcal {E}}(C) \) may coincide (i.e. be \(\theta \)subsumption equivalent – they \(\theta \)subsume each other), a clause D in \( C.supp \) may cover more than one example from \( cov_{\mathcal {E}}(C) \). Proposition 1 highlights the main property of the structure. The proof is given in “Appendix 2”.
Proposition 1
Let C be a clause in \(\mathcal {L}(M)\). \( C.supp \) is the most specific program of \(\mathcal {L}(M)\) such that \( cov_{\mathcal {E}}(C.supp)=cov_{\mathcal {E}}(C) \).
Proposition 1 implies that clause C and its support set \( C.supp \) define a space \(\mathcal {S}\) of specializations of C, each of which is bound by a mostspecific specialization, among those that cover the positive examples that C covers. In other words, for every \(D \in \mathcal {S}\) there is a \( C_s \in C.supp \) so that \(C \preceq D \preceq C_s\) and \(C_s\) covers at least one example from \( cov_{\mathcal {E}}(C) \). Moreover, Proposition 1 ensures that space \(\mathcal {S}\) contains refinements of clause C that collectively preserve the coverage of C in the historical memory. The purpose of \( C.supp \) is thus to serve as a search space for refinements \(R_{C}\) of clause C for which \( C\preceq R_C \preceq C.supp \) holds. Since such refinements preserve C’s coverage of positive examples, clause C may be refined w.r.t. a window \(w_n\), avoiding the overhead of retesting the refined program on \(\mathcal {E}\) for completeness. However, to ensure that the support set can indeed be used as a refinement search space, one must ensure that \( C.supp \) will always contain such a refinement \(R_C\). This proof is provided in Proposition 2, “Appendix 9”.
Knowledge for Example 2
Window \(w_1\)  
Narrative  Annotation 
happensAt(\( active(id_1),10 \)).  not holdsAt(\( fighting(id_1,id_2),10 \)). 
happensAt(\( abrupt(id_2),10 \)).  holdsAt(\( fighting(id_1,id_2),11 \)). 
holdsAt(\( close(id_1,id_2,23),10 \)).  
Kernel Set  Variabilized Kernel Set 
\(\mathsf{initiatedAt }(fighting(id_1,id_2),10) \leftarrow \)  \(K_1 = \mathsf{initiatedAt }(fighting(X,Y),T) \leftarrow \) 
\(\qquad \qquad \mathsf{happensAt }(active(id_1),10)\),  \(\qquad \qquad \mathsf{happensAt }(active(X),T)\), 
\(\qquad \qquad \mathsf{happensAt }(abrupt(id_2),10)\)  \(\qquad \qquad \mathsf{happensAt }(abrupt(Y),T)\), 
\(\qquad \qquad \mathsf{holdsAt }(close(id_1,id_2,23),10)\)  \(\qquad \qquad \mathsf{holdsAt }(close(X,Y,23),T)\). 
Running Hypothesis  Support Set 
\(C = \mathsf{initiatedAt }(fighting(X,Y),T) \leftarrow \)  \( C.supp = \{K_1\} \) 
\(\qquad \qquad \mathsf{happensAt }(active(X),T).\)  
Window \(w_2\)  
Narrative  Annotation 
happensAt(\( active(id_1),20 \)).  not holdsAt(\( fighting(id_1,id_2),20 \)). 
happensAt(\( kicking(id_2),20 \)).  holdsAt(\( fighting(id_1,id_2),21 \)). 
holdsAt(\( close(id_1,id_2,23),20 \)).  
Kernel Set  Variabilized Kernel Set 
\(\mathsf{initiatedAt }(fighting(id_1,id_2),20) \leftarrow \)  \(K_2 = \mathsf{initiatedAt }(fighting(X,Y),T) \leftarrow \) 
\(\qquad \qquad \mathsf{happensAt }(active(id_1),20)\),  \(\qquad \qquad \mathsf{happensAt }(active(X),T)\), 
\(\qquad \qquad \mathsf{happensAt }(kicking(id_2),20)\)  \(\qquad \qquad \mathsf{happensAt }(kicking(Y),T)\), 
\(\qquad \qquad \mathsf{holdsAt }(close(id_1,id_2,23),20)\)  \(\qquad \qquad \mathsf{holdsAt }(close(X,Y,23),T)\). 
Running Hypothesis  Support Set 
Remains unchanged  \( C.supp = \{K_1,K_2\} \) 
Window \(w_3\)  
Narrative  Annotation 
happensAt(\( active(id_1),30 \)).  not holdsAt(\( fighting(id_1,id_2),30 \)). 
happensAt(\( walking(id_2),30 \)).  not holdsAt(\( fighting(id_1,id_2),31 \)). 
not holdsAt(\( close(id_1,id_2,23),30 \)).  
Revised Hypothesis  Support Set 
\(C_1 = \mathsf{initiatedAt }(fighting(X,Y),T) \leftarrow \)  \( C_1.supp = \{K_1,K_2\} \) 
\(\qquad \qquad \mathsf{happensAt }(active(X),T),\)  
\(\qquad \qquad \mathsf{holdsAt }(close(X,Y,23),T).\) 
Example 2
Consider the annotated examples and running hypothesis related to the fighting highlevel event from the activity recognition application shown in Table 4. We assume that ILED starts with an empty hypothesis and an empty historical memory, and that \(w_1\) is the first input example window. The currently empty hypothesis does not cover the provided examples, since in \(w_1\) fighting between persons \(id_1\) and \(id_2\) is initiated at time 10 and thus holds at time 11. Hence ILED starts the process of generating an initial hypothesis. In the case of an empty hypothesis, ILED reduces to XHAIL and operates on a Kernel Set of \(w_1\) only. The variabilized Kernel Set in this case will be the singleclause program \(K_1\) presented in Table 4, generated from the corresponding ground clause. Generalizing this Kernel Set yields a minimal hypothesis that covers \(w_1\). One such hypothesis is clause C shown in Table 4. ILED stores \(w_1\) in \(\mathcal {E}\) and initializes the support set of the newly generated clause C as in line 3 of Algorithm 1, by selecting from \(K_1\) the clauses that are \(\theta \)subsumed by C, in this case, \(K_1\)’s single clause.
Window \(w_2\) arrives next. In \(w_2\), fighting is initiated at time 20 and thus holds at time 21. The running hypothesis correctly accounts for that and thus no revision is required. However, \( C.supp \) does not cover \(w_2\) and unless proper actions are taken, property (i) of Proposition 1 will not hold once \(w_2\) is stored in \(\mathcal {E}\). ILED thus generates a new Kernel Set \(K_2\) from window \(w_2\), as presented in Table 4, and updates \( C.supp \) as shown in lines 7–11 of Algorithm 1. Since C \(\theta \)subsumes \(K_2\), the latter is added to \( C.supp \), which now becomes \( C.supp = \{K_1,K_2\}\). Now \( cov_{\mathcal {E}}(C.supp) = cov_{\mathcal {E}}(C) \), hence in effect, \( C.supp \) is a summarization of the coverage of clause C in the historical memory.
is such a refinement \(R_C\), since it does not cover the negative example in \(w_3\) and subsumes \( C.supp \). ILED however is biased towards minimal theories, in terms of the overall number of literals and would prefer the more compressed refinement \(C_1\), shown in Table 4, which also rejects the negative example in \(w_3\) and subsumes \( C.supp \). Clause \(C_1\) replaces the initial clause C in the running hypothesis. The hypothesis now becomes complete and consistent w.r.t. \(\mathcal {E}\). Note that the hypothesis was refined by local reasoning only, i.e. reasoning within window \(w_3\) and the support set, avoiding costly lookback in the historical memory. The support set of the new clause \(C_1\) is initialized (line 5 of Algorithm 1), by selecting the subset of the support set of its parent clause that is \(\theta \)subsumed by \(C_1\). In this case \( C_1 \preceq C.supp = \{K_1,K_2\} \), hence \( C_1.supp = C.supp \). \(\square \)
The support set of a clause C is a compressed enumeration of the examples that C covers throughout the historical memory. It is compressed because each variabilized clause in the set is expected to encode many examples. In contrast, a ground version of the support set would be a plain enumeration of examples, since in the general case, it would require one ground clause per example. The main advantage of the “lifted” character of the support set over a plain enumeration of the examples is that it requires much less memory to encode the necessary information, an important feature in largescale (temporal) applications. Moreover, given that training examples are typically characterized by heavy repetition, abstracting away redundant parts of the search space results in a memory structure that is expected to grow in size slowly, allowing for fast search that scales to a large amount of historical data.
3.2 Implementing revisions
Syntactic transformations performed by ILED
GeneralizationTransformation  RefinementTransformation 
Input: A variabilized Kernel set \(K_{v}\)  Input: A running hypothesis \(H_n\) 
For each clause \( D_i = \alpha _{i} \leftarrow \delta _{i}^{1},\ldots ,\delta _{i}^{n}\in F_v \):  For each clause \( D_i\in H_n \): 
Add an extra atom \( use(i,0) \) to the body of \(D_i\)  For each clause \({\varGamma }_ {i}^{j} \in D_i.supp \) 
and replace each body literal \(\delta _{i}^{j}\) with a new  Generate one clause 
atom of the form \( try(i,j,v(\delta _{i}^{j})) \), where \( v(\delta _{i}^{j}) \)  \( \alpha _i \leftarrow body(D_i) \wedge \mathsf{not }\ exception(i,j,v(\alpha _{i})) \) 
contains the variables that appear in \(\delta _{i}^{j}\).  where \(\alpha _i\) is the head of \(D_i\) and \( v(\alpha _i) \) 
Generate two new clauses of the form  contains its variables. Generate one clause 
\( try(i,j,v(\delta _{i}^{j})) \leftarrow use(i,j),\delta _{i}^{j} \) and  \( exception(i,j,v(a_i)) \leftarrow use(i,j,k),\mathsf{not }\ \delta _{i}^{j,k} \) 
\( try(i,j,v(\delta _{i}^{j})) \leftarrow \mathsf{not }\ use(i,j) \) for each \(\delta _{i}^{j}\).  for each body literal \(\delta _{i}^{j,k}\) of \({\varGamma }_{i}^{j}\). 
Algorithm 2 presents the revision function of ILED. The input consists of SDEC as background knowledge, a running hypothesis \(H_n\), an example window \(w_n\) and a variabilized Kernel Set \(K_{v}^{w_n}\) of \(w_n\). The clauses of \(K_{v}^{w_n}\) and \(H_n\) are subject to the GeneralizationTansformation and the RefinementTransformation respectively, presented in Table 5. The former is the transformation discussed in Sect. 2.2.1, that turns the Kernel Set into a defeasible program, allowing the construction of new clauses. The RefinementTransformation aims at the refinement of the clauses of \(H_n\) using their support sets. It involves two fresh predicates, \( exception/3 \) and use / 3. For each clause \(D_i \in H_n\) and for each of its support set clauses \({\varGamma }_{i}^{j} \in D_i.supp\), one new clause \( head(D_i) \leftarrow body(D_i) \wedge \mathsf{not }\ exception(i,j,v(head(D_i))) \) is generated, where \( v(head(D_i)) \) is a term that contains the variables of \( head(C_i) \). Then an additional clause \( exception(i,j,v(head(D_i))) \leftarrow use(i,j,k)\wedge \mathsf{not }\ \delta _{i}^{j,k} \) is generated, for each body literal \(\delta _{i}^{j,k} \in {\varGamma }_{i}^{j}\).
The syntactically transformed clauses are put together in a program \(U(K_{v}^{w_n},H_n)\) (line 1 of Algorithm 2), which is used as a background theory along with SDEC. A minimal set of use / 2 and use / 3 atoms is abduced as a solution to the abductive task \({\varPhi }\) in line 2 of Algorithm 2. Abduced \( use/2 \) atoms are used to construct a set of \( NewClauses \), as discussed in Sect. 2.2.1 (line 5 of Algorithm 2). These new clauses account for some of the examples in \(w_n\), which cannot be covered by existing clauses in \(H_n\). The abduced \( use/3 \) atoms indicate clauses of \(H_n\) that must be refined. From these atoms, a refinement \(R_{D_i}\) is generated for each incorrect clause \(D_i \in H_n\), such that \(D_i \preceq R_{D_i} \preceq D_i.supp\) (line 6 of Algorithm 2). Clauses that lack a corresponding use / 3 atom in the abductive solution are retained (line 7 of Algorithm 2).
The intuition behind refinement generation is as follows: Assume that clause \(D_i\in H_n\) must be refined. This can be achieved by means of the extra clauses generated by the RefinementTransformation. These clauses provide definitions for the exception atom, namely one for each body literal in each clause of \(D_i.supp\). From these clauses, one can satisfy the exception atom by satisfying the complement of the corresponding support set literal and abducing the accompanying \( use/3 \) atom. Since an abductive solution \({\varDelta }\) is minimal, the abduced use / 3 atoms correspond precisely to the clauses that must be refined.
Clause refinement by ILED
Input  
Narrative  Annotation 
\( \mathsf{happensAt }(abrupt(id_1), \ 1). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_1,id_2),1).\) 
\( \mathsf{happensAt }(inactive(id_2), \ 1). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_3,id_4),1).\) 
\( \mathsf{holdsAt }(close(id_1,id_2,23), \ 1). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_1,id_2),2).\) 
\( \mathsf{happensAt }(abrupt(id_3), \ 2). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_3,id_4),2).\) 
\( \mathsf{happensAt }(abrupt(id_4), \ 2). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_1,id_2),3).\) 
\( \mathsf{not }\ \mathsf{holdsAt }(close(id_3,id_4,23), \ 2). \)  \(\mathsf{not }\ \mathsf{holdsAt }(fighting(id_3,id_4),3).\) 
Running hypothesis  Support set 
\( C = \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \)  \( C_s^1 = \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \) 
\(\qquad \quad \mathsf{happensAt }(abrupt(X), \ T). \)  \(\qquad \quad \mathsf{happensAt }(abrupt(X), \ T), \) 
\(\qquad \quad \mathsf{happensAt }(abrupt(Y), \ T), \)  
\(\qquad \quad \mathsf{holdsAt }(close(X,Y,23), \ T) .\)  
\( C_s^2 = \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \)  
\(\qquad \quad \mathsf{happensAt }(abrupt(X), \ T), \)  
\(\qquad \quad \mathsf{happensAt }(active(Y), \ T), \)  
\(\qquad \quad \mathsf{holdsAt }(close(X,Y,23), \ T) .\)  
Refinement transformation:  
From \(C_{s}^1:\)  From \(C_{s}^2:\) 
\(\mathsf{initiatedAt }(fighting(X,Y), \ T) \leftarrow \)  \(\mathsf{initiatedAt }(fighting(X,Y), \ T) \leftarrow \) 
\(\qquad \mathsf{happensAt }(abrupt(X), \ T),\)  \(\qquad \mathsf{happensAt }(abrupt(X), \ T),\) 
\(\qquad \mathsf{not }\ exception(1,1,vars(X,Y,T)).\)  \(\qquad \mathsf{not }\ exception(1,2,vars(X,Y,T)).\) 
\(exception(1,1,vars(X,Y,T)) \leftarrow \)  \(exception(1,2,vars(X,Y,T)) \leftarrow \) 
\(\qquad use(1,1,2),\mathsf{not }\ \mathsf{happensAt }(abrupt(Y),T).\)  \(\qquad use(1,2,2),\mathsf{not }\ \mathsf{happensAt }(active(Y),T).\) 
\(exception(1,1,vars(X,Y,T)) \leftarrow \)  \(exception(1,2,vars(X,Y,T)) \leftarrow \) 
\(\qquad use(1,1,3),\mathsf{not }\ \mathsf{holdsAt }(close(X,Y,23),T).\)  \(\qquad use(1,2,3), \mathsf{not }\ \mathsf{holdsAt }(close(X,Y,23),T).\) 
Minimal abductive solution  Generated refinements 
\({\varDelta } \, = \{use(1,1,2),use(1,1,3),use(1,2,2)\} \)  \( \mathsf{initiatedAt }(fighting(X,Y), \ T)\ \leftarrow \) 
\(\quad \mathsf{happensAt }(abrupt(X), \ T), \)  
\(\quad \mathsf{happensAt }(abrupt(Y), \ T), \)  
\(\quad \mathsf{holdsAt }(close(X,Y,23), \ T) .\)  
\( \mathsf{initiatedAt }(fighting(X,Y), \ T) \ \leftarrow \)  
\(\quad \mathsf{happensAt }(abrupt(X), \ T), \)  
\(\quad \mathsf{happensAt }(active(Y), \ T). \) 
Example 3
Table 6 presents the process of ILED’s refinement. The annotation lacks positive examples and the running hypothesis consists of a single clause C, with a support set of two clauses. Clause C is inconsistent since it entails two negative examples, namely \( \mathsf{holdsAt }(fighting(id_1,id_2), \ 2) \) and \( \mathsf{holdsAt }(fighting(id_3,id_4), \ 3) \). The program that results by applying the RefinementTransformation to the support set of clause C is presented in Table 6, along with a minimal abductive explanation of the examples, in terms of use / 3 atoms. Atoms use(1, 1, 2) and use(1, 1, 3) correspond respectively to the second and third body literals of the first support set clause, which are added to the body of clause C, resulting in the first specialization presented in Table 6. The third abduced atom use(1, 2, 2) corresponds to the second body literal of the second support set clause, which results in the second specialization in Table 6. Together, these specializations form a refinement of clause C that subsumes \( C.supp \). \(\square \)
Minimal abductive solutions imply that the running hypothesis is minimally revised. Revisions are minimal w.r.t. the length of the clauses in the revised hypothesis, but are not minimal w.r.t. the number of clauses, since the refinement strategy described above may result in refinements that include redundant clauses: Selecting one specialization from each support set clause to generate a refinement of a clause is suboptimal, since there may exist other refinements with fewer clauses that also subsume the whole support set, as Example 2 demonstrates. To avoid unnecessary increase of the hypothesis size, the generation of refinements is followed by a “reduction” step (line 8 of Algorithm 2). The ReduceRefined function works as follows. For each refined clause C, it first generates all possible refinements from \( C.supp \). This can be realized with the abductive refinement technique described above. The only difference is that the abductive solver is instructed to find all abductive explanations in terms of use / 3 atoms, instead of one. Once all refinements are generated, ReduceRefined searches the revised hypothesis, augmented with all refinements of clause C, to find a reduced set of refinements of C that subsume \( C.supp \).
4 Discussion
An additional tradeoff for efficiency is that not all of ILED’s revisions are fully evaluated on the historical memory. For instance, selecting a particular clause in order to cover a new example, may result in a large number of refinements and an unnecessarily lengthy hypothesis, as compared to one that may have been obtained by selecting a different initial clause. On the other hand, fully evaluating all possible choices over \(\mathcal {E}\) requires extensive inference. Thus simplicity and compression of hypotheses in ILED have been sacrificed for efficiency.
In ILED, a large part of the theorem proving effort that is involved in clause refinement reduces to computing subsumption between clauses, which is a hard task. Moreover, just as the historical memory grows over time, so do (in the general case) the support sets of the clauses in the running hypothesis, increasing the cost of computing subsumption. However, as in principle the largest part of a search space is redundant and the support set focuses only on its interesting parts, one would not expect that the support set will grow to a size that makes subsumption computation less efficient than inference over the entire \(\mathcal {E}\). In addition, a number of optimization techniques have been developed over the years and several generic subsumption engines have been proposed (Maloberti and Sebag 2004; Kuzelka and Zelezny 2008; Santos and Muggleton 2010), some of which are able to efficiently compute subsumption relations between clauses comprising thousands of literals and hundreds of distinct variables.
5 Experimental evaluation
In this section, we present experimental results from two realworld applications: Activity recognition, using real data from the benchmark CAVIAR video surveillance dataset,^{3} as well as large volumes of synthetic CAVIAR data; and City Transport Management (CTM) using data from the PRONTO^{4} project.
Part of our experimental evaluation aims to compare ILED with XHAIL. To achieve this aim we had to implement XHAIL, because the original implementation was not publicly available until recently (Bragaglia and Ray 2014). All experiments were conducted on a 3.2 GHz Linux machine with 4 GB of RAM. The algorithms were implemented in Python, using the Clingo^{5} Answer Set Solver (Gebser et al. 2012) as the main reasoning component, and a Mongodb^{6} NoSQL database for the historical memory of the examples. The code and datasets used in these experiments can be downloaded from https://github.com/nkatzz/ILED.
5.1 Activity recognition
In activity recognition, our goal is to learn definitions of highlevel events, such as fighting, moving and meeting, from streams of lowlevel events like walking, standing, active and abrupt, as well as spatiotemporal knowledge. We use the benchmark CAVIAR dataset for experimentation. Details on the CAVIAR dataset can be found in Artikis et al. (2010).
CAVIAR contains noisy data mainly due to human errors in the annotation (List et al. 2005; Artikis et al. 2010). Thus, for the experiments we manually selected a noisefree subset of CAVIAR. The resulting dataset consists of 1000 examples (that is, data for 1000 distinct time points) concerning the highlevel events moving, meeting and fighting. These data, selected from different parts of the CAVIAR dataset, were combined into a continuous annotated stream of narrative atoms, with time ranging from 0 to 1000.
In addition to the real data, we generated synthetic data based on the manuallydeveloped CAVIAR event definitions described in Artikis et al. (2010). In particular, streams of lowlevel events were created randomly and were then classified using the rules of Artikis et al. (2010). The generated data consists of approximately \(10^5\) examples, which amounts to 100 MB of data.
The synthetic data is much more complex than the real CAVIAR data. This is due to two main reasons: First, the synthetic data includes significantly more initiations and terminations of a highlevel event, thus much larger learning effort is required to explain it. Second, in the synthetic dataset more than one highlevel event may be initiated or terminated at the same time point. This results in Kernel Sets with more clauses, which are hard to generalize simultaneously.
5.1.1 ILED versus XHAIL
The purpose of this experiment was to assess whether ILED can efficiently generate hypotheses comparable in size and predictive quality to those of XHAIL. To this end, we compared both systems on real and synthetic data using tenfold cross validation with replacement. For the real data, 90 % of randomly selected examples, from the total of 1000 were used for training, while the remaining 10 % was retained for testing. At each run, the training data were presented to ILED in example windows of sizes 10, 50, 100. The data were presented in one batch to XHAIL. For the synthetic data, 1000 examples were randomly sampled at each run from the dataset for training, while the remaining data were retained for testing. Similar to the real data experiments, ILED operated on windows of sizes of 10, 50, 100 examples and XHAIL on a single batch.
Comparison of ILED and XHAIL
ILED  XHAIL  

\(G = 10\)  \(G = 50\)  \(G = 100\)  \(G = 900\)  
Real CAVIAR data  
Training time (s)  34.15 (\(\pm \)6.87)  23.04 (\(\pm \)13.50)  286.74 (\(\pm \)98.87)  1560.88 (\(\pm \)4.24) 
Revisions  11.2 (\(\pm \)3.05)  9.1 (\(\pm \)0.32)  5.2 (\(\pm \)2.1)  \(\) 
Hypothesis size  17.82 (\(\pm \)2.18)  17.54 (\(\pm \)1.5)  17.5 (\(\pm \)1.43)  15 (\(\pm \)0.067) 
Precision  98.713 (\(\pm \)0.052)  99.767 (\(\pm \)0.038)  99.971 (\(\pm \)0.041)  99.973 (\(\pm \)0.028) 
Recall  99.789 (\(\pm \)0.083)  99.845 (\(\pm \)0.32)  99.988 (\(\pm \)0.021)  99.992 (\(\pm \)0.305) 
Synthetic CAVIAR data  
Training time (s)  38.92 (\(\pm \)9.15)  33.87 (\(\pm \)9.74)  468 (\(\pm \)102.62)  21429 (\(\pm \)342.87) 
Revisions  28.7 (\(\pm \)9.34)  15.4 (\(\pm \)7.5)  12.2 (\(\pm \)6.23)  \(\) 
Hypothesis size  143.52 (\(\pm \)19.14)  138.46 (\(\pm \)22.7)  126.43 (\(\pm \)15.8)  118.18 (\(\pm \)14.48) 
Precision  55.713 (\(\pm \)0.781)  57.613 (\(\pm \)0.883)  63.236 (\(\pm \)0.536)  63.822 (\(\pm \)0.733) 
Recall  68.213 (\(\pm \)0.873)  71.813 (\(\pm \)0.756)  71.997 (\(\pm \)0.518)  71.918 (\(\pm \)0.918) 
Concerning the size of the produced hypothesis, the results show that in the case of real CAVIAR data, the hypotheses constructed by ILED are comparable in size with a hypothesis constructed by XHAIL. In the case of synthetic data, the hypotheses returned by both XHAIL and ILED were significantly more complex. Note that for ILED the hypothesis size decreases as the window size increases. This is reflected in the number of revisions that ILED performs, which is significantly smaller when the input comes in larger batches of examples. In principle, the richer the input, the better the hypothesis that is initially acquired, and consequently, the less the need for revisions in response to new training instances. There is a tradeoff between the window size (thus the complexity of the abductive search) and the number of revisions. A small number of revisions on complex data (i.e. larger windows) may have a greater total cost in terms of training time, as compared to a greater number of revisions on simpler data (i.e. smaller windows). For example, in the case of window size 100 for the real CAVIAR data, ILED performs 5 revisions on average and requires significantly more time than in the case of a window size 50, where it performs 9 revisions on average. On the other hand, training times for windows of size 50 are slightly better than those obtained when the examples are presented in smaller windows of size 10. In this case, the “unit cost” of performing revisions w.r.t a single window are comparable between windows of size 10 and 50. Thus the overall cost in terms of training time is determined by the total number of revisions, which is greater in the case of window size 10.
Concerning predictive quality, the results indicate that ILED’s precision and recall scores are comparable to those of XHAIL. For larger input windows, precision and recall are almost the same as those of XHAIL. This is because ILED produces better hypotheses from larger input windows. Precision and recall are smaller in the case of synthetic data for both systems, because the testing set in this case is much larger and complex than in the case of real data.
5.1.2 ILED scalability
5.2 City transport management
Clauses (5)–(8) define low driving quality. Essentially, driving quality is said to be low when the driving style is unsafe and the vehicle is nonpunctual. Driving quality is defined in terms of highlevel events (we omit the definition of driving style to save space). Therefore, the bodies of the clauses defining driving quality include initiatedAt/2 and terminatedAt/2 literals.
5.2.1 ILED versus XHAIL
In this experiment, we tried to learn simultaneously definitions for all target concepts, a total of nine interrelated highlevel events, seven of which are level1, one is level2 and one is level3. The total number of lowlevel events is eleven, while for both highlevel and lowlevel events, their negations are considered during learning. According to the employed language bias, each highlevel event must be learnt, while at the same time it may be present in the body of another highlevel event in the form of a (potentially negated) holdsAt/2, initiatedAt/2, or terminatedAt/2 predicate.
Comparative performance of ILED and XHAIL on selected subsets of the CTM dataset each containing 20 examples
ILED  XHAIL  

\(G = 5\)  \(G = 10\)  \(G = 20\)  
Training time (h)  1.35 (\(\pm \)0.17)  1.88 (\(\pm \)0.13)  4.35 (\(\pm \)0.2) 
Hypothesis size  28.32 (\(\pm \)1.19)  24.13 (\(\pm \)2.54)  24.02 (\(\pm \)0.23) 
Revisions  14.78 (\(\pm \)2.24)  13.42 (\(\pm \)2.08)  \(\) 
Precision  63.344 (\(\pm \)5.24)  64.644 (\(\pm \)3.45)  66.245 (\(\pm \)3.83) 
Recall  59.832 (\(\pm \)7.13)  61.423 (\(\pm \)5.34)  62.567 (\(\pm \)4.65) 
ILED took on average 1–2 h to complete the learning task, for windows of 5 and 10 examples, while XHAIL required more than 4 h on average to learn hypotheses from batches of 20 examples. Compared to activity recognition, the learning setting requires larger Kernel Set structures that are hard to reason with. An average Kernel Set generated from a batch of just 20 examples consisted of approximately 30–35 clauses, with 60–70 literals each.
Like the activity recognition experiments, precision and recall scores for ILED are comparable to those of XHAIL, with the latter being slightly better. Unlike the activity recognition experiments, precision and recall had a large diversity between different runs. Due to the complexity of the CTM dataset, the constructed hypotheses had a large diversity, depending on the random samples that were used for training. For example, some highlevel event definitions were unnecessarily lengthy and difficult to be understood by a human expert. On the other hand, some level1 definitions could, in some runs of the experiment, be learnt correctly even from a limited amount of data. Such definitions are fairly simple, consisting of one initiation and one termination rule, with one body literal in each case.
This experiment demonstrates several limitations of learning in large and complex applications. The complexity of the domain increases the intensity of the learning task, which in turn makes training times forbidding, even for small amount of data such as 20 examples (approximately 150 atoms). This forces one to process small sets of examples at a time, which in complex domains like CTM, results in overfitted theories and rapid increase in hypothesis size.
5.2.2 Learning with hierarchical bias
ILED with hierarchical bias
ILED  

\(G = 10\)  \(G = 50\)  \(G = 100\)  
Level1  
Training time (min)  4.46–4.88  5.78–6.44  6.24–6.88 
Revisions  2–11  2–9  2–9 
Hypothesis size  4–18  4–16  4–16 
Precision (%)  100  100  100 
Recall (%)  100  100  100 
Level2  
Training time (min)  8.76  9.14  9.86 
Revisions  24  17  17 
Hypothesis size  31  27  27 
Precision (%)  100  100  100 
Recall (%)  100  100  100 
Level3  
Training time (min)  5.78  6.14  6.78 
Revisions  6  5  5 
Hypothesis size  13  10  10 
Precision (%)  100  100  100 
Recall (%)  100  100  100 
The experimental setting was therefore as follows: Starting from the level1 target events, we processed the whole CTM dataset in windows of 10, 50 and 100 examples with ILED. Each highlevel event was learnt independently of the others. Once complete definitions for all level1 highlevel events were constructed, they were added to the background knowledge. Then we proceeded with learning the definition of the single level2 event (see Fig. 3). Finally, after successfully constructing the level2 definition, we performed learning in the toplevel of the hierarchy, using the previously constructed level1 and level2 event definitions as background knowledge. We did not attempt a comparison with XHAIL because it is not able to operate on the entire dataset.
Table 9 presents the results. For level1 events, scores are presented as minimum–maximum pairs. For instance, the training times for level1 events with windows of 10 examples, range from 4.46 to 4.88 min. Levels 2 and 3 have just one definition each, therefore Table 9 presents the respective scores from each run. Training times, hypotheses sizes and overall numbers of revisions are comparable for all levels of the event hierarchy. Level1 event definitions were the easiest to acquire, with training times ranging approximately between 4.50 and 7 min. This was expected since clauses in level1 definitions are significantly simpler than level2 and level3 ones. The level2 event definition was the hardest to construct with training times ranging between 8 and 10 min, while a significant number of revisions was required for all window granularities. The definition of this highlevel event (drivingStyle) is relatively complex, in contrast to the simpler level3 definition, for which training times are comparable to the ones of level1 events.
The largest parts of training times were dedicated to checking an already correct definition against the part of the dataset that had not been processed yet. That is, for all target events, ILED converged to a complete definition relatively quickly, i.e. in approximately 1.5–3 min after the initiation of the learning process. From that point on, the extra time was spent on testing the hypothesis against the new incoming data.
Window granularity slightly affects the produced hypothesis for all target highlevel events. Indeed, the definitions constructed with windows of 10 examples are slightly larger than the ones constructed with larger window sizes of 50 and 100 examples. Notably, the definitions constructed with windows of granularity 50 and 100, were found concise, meaningful and very close to the actual handcrafted rules that were utilized in PRONTO.
6 Related work
A thorough review of the drawbacks of stateoftheart ILP systems with respect to nonmonotonic domains, as well as the deficiencies of existing approaches to learning Event Calculus programs can be found in Ray (2009), Otero (2001, 2003) and Sakama (2005, 2001). The main obstacle, common to many learners which combine ILP with some form of abduction, like PROGOL5 (Muggleton and Bryant 2000), ALECTO (Moyle 2003), HAIL (Ray et al. 2003) and IMPARO (Kimber et al. 2009), is that they cannot perform abduction through negation and are thus essentially limited to Observational Predicate Learning.
TAL (Corapi et al. 2010) is a top–down nonmonotonic learner which is able to solve the same class of problems as XHAIL. It obtains a top theory by appropriately mapping an ILP problem to a corresponding ALP instance, so that solutions to the latter may be translated to solutions for the initial ILP problem. Recently, the main ideas behind TAL were employed in the ASPAL system (Corapi et al. 2012), an inductive learner which relies on Answer Set Programming as a unifying abductive–inductive framework.
In Athakravi et al. (2013) the methodology behind TAL and ASPAL have been ported into a learner that constructs hypotheses progressively, towards more scalable learning. To address the fact that ASPALs top theory grows exponentially with the length of its clauses, RASPAL, the system proposed in Athakravi et al. (2013), imposes bounds on the length of the top theory. Partial hypotheses of specified clause length are iteratively obtained in a refinement loop. At each iteration of this loop, the hypothesis obtained from the previous refinement step is further refined by dropping or adding literals or clauses, using theory revision as described in Corapi et al. (2008). The process continues until a complete and consistent hypothesis is obtained. The main difference between RASPAL and our approach is that in order to ensure soundness, RASPAL has to process all examples simultaneously. At each iteration of its refinement loop, all examples are taken into account repeatedly, in order to ensure that the revisions account for all them. Therefore, the grounding bottleneck that RASPAL faces is expected to persist in domains that involve large volumes of sequential data, typical of temporal applications, as the ones that we address in this work. This is because even by imposing a small initial maximum clause length to RASPAL, in order to constrain the search space, with a sufficient amount of data the resulting ground program will still be intractable, if the data is processed simultaneously. In contrast, ILED is able to break the dataset in smaller data windows and process them in isolation, while ensuring soundness. By properly restricting window size, so that the unit cost of learning/revising from a single window is acceptable, ILED scales to large volumes of data, since the cost of theory revision grows as a linear function of the number of example windows in the historical memory.
The combination of ILP with ALP has recently been applied to metainterpretive learning (MIL), a learning framework where the goal is to obtain hypotheses in the presence of a metainterpreter. The latter is a higherorder program, hypothesizing about predicates or even rules of the domain. Given such background knowledge and a set of examples, MIL uses abduction w.r.t. the metainterpreter to construct firstorder hypotheses. MIL can be realized both in Prolog and in Answer Set Programming, and it has been implemented in the METAGOL system (Muggleton et al. 2014). MIL is an elegant framework, able to address difficult problems like predicate invention and mutually recursive programs. However, it has a number of important drawbacks. First, its expressivity is limited, as MIL is currently restricted to dyadic Datalog, i.e. Datalog where the arity of each predicate is at most two. Second, given the increased computational complexity of higherorder reasoning, scaling to large volumes of data is a potential bottleneck for MIL.
CLINT (De Raedt and Bruynooghe 1994) is a seminal abductive–inductive theory revision system. It employs two revision operators: A generalization operator that adds new clauses/facts to the theory and a specialization operator, that retracts incorrect clauses from the theory. To generate new clauses, CLINT uses “starting clauses”, i.e. variabilized, mostspecific clauses that cover a positive example, which are then maximally generalized to obtain a good hypothesis clause. CLINT also uses abduction in order to explain some examples, by deriving new ground facts which are simply added to the theory. Abduction and induction are independent and complementary, i.e. one is used when the other fails to cover an example, contrary to ILED, where the two processes are tightly coupled, allowing to handle nonOPL. Additionally, CLINT is restricted to Horn logic.
In Duboc et al. (2009), the theory revision system FORTE (Richards and Mooney 1995) is enhanced by PROGOL’s bottom set construction routine and mode declarations, towards a more efficient refinement operator. In order to refine a clause C, FORTE_MBC (the resulting system), uses mode declarations and inverse entailment to construct a bottom clause from a positive example covered by C. It then searches for antecedents within the bottom clause. As in the case of ILED, the constrained search space results in a more efficient clause refinement process. However FORTE_MBC (like FORTE itself) learns Horn theories and does not support nonObservational Predicate Learning, thus it cannot be used for the revision of Event Calculus programs. In addition, it cannot operate on an empty hypothesis (i.e. it cannot induce a hypothesis from scratch).
INTHELEX (Esposito et al. 2000) learns/revises Datalog theories and has been used in the study of several aspects of incremental learning, such as order effects (Di Mauro et al. 2004, 2005) and concept drift (Esposito et al. 2004). In Biba et al. (2006) the authors present an approach towards scaling INTHELEX by associating clauses in the theory at hand with examples they cover, via a relational schema. Thus, when a clause is refined, only the examples that were previously covered by this clause are checked. Similarly, when a clause is generalized, only the negative examples are checked again. The scalable version of INTHELEX presented in Biba et al. (2006) maintains alternative versions of the hypothesis at each step, allowing it to backtrack to previous states. In addition, it keeps in memory several statistics related to the examples that the system has already seen, such as the number of refinements that each example has caused, a “refinement history” of each clause, etc.
Several limitations make INTHELEX inappropriate for inducing/revising Event Calculus programs. First, the restriction of its input language to Datalog limits its applicability to richer, relational event domains. For instance, complex relations between entities cannot be easily expressed in INTHELEX. Second, the use of background knowledge is limited, excluding for instance auxiliary clauses that may be used for spatiotemporal reasoning during learning time. Third, although INTHELEX uses abduction for the completion of imperfect input data, it relies on Observational Predicate Learning, meaning that it is not able to reason with predicates which are not directly observable in the examples.
7 Conclusions
We presented an incremental ILP system, ILED, for constructing event recognition knowledge bases in the form of Event Calculus theories. ILED combines techniques from nonmonotonic ILP and in particular, the XHAIL algorithm, with theory revision. It acquires an initial hypothesis from the first available piece of data, and revises this hypothesis as new data arrive. Revisions account for all accumulated experience. The main contribution of ILED is that it scalesup XHAIL to large volumes of sequential data with a timelike structure, typical of eventbased applications. By means of a compressive memory structure that supports clause refinement, ILED has a scalable, singlepass revision strategy, thanks to which the cost of theory revision grows as a tractable function of the perceived experience. In this work, ILED was evaluated on an activity recognition application and a transport management application. The results indicate that ILED is significantly more efficient than XHAIL, without compromising the quality of the generated hypothesis in terms of predictive accuracy and hypothesis size. Moreover, ILED scales adequately to large data volumes which XHAIL cannot handle. Future work concerns mechanisms for handling noise and concept drift.
Footnotes
 1.
 2.
The bottom clause relative to an example can be large, or even infinite. To constrain its size, several restrictions are imposed on the language, such as a maximum clause length, or a maximum variable depth. We refrain from assuming extra language bias related to clause length and instead, for the purposes of this work, we assume a finite domain and impose no particular bounds on clause length. In such context, the bottom clause of an example e is unique and results from the ground mostspecific clause that covers e, by properly replacing terms with variables, as indicated by the mode declarations.
 3.
 4.
 5.
 6.
 7.
Notes
Acknowledgments
This work was partly funded by the EU Project SPEEDD (FP7 619435). We would like to thank the reviewers of the Machine Learning Journal for their valuable comments.
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