Meta-interpretive learning as metarule specialisation

Abstract

In Meta-interpretive learning (MIL) the metarules, second-order datalog clauses acting as inductive bias, are manually defined by the user. In this work we show that second-order metarules for MIL can be learned by MIL. We define a generality ordering of metarules by $\theta$-subsumption and show that user-defined sort metarules are derivable by specialisation of the most-general matrix metarules in a language class; and that these matrix metarules are in turn derivable by specialisation of third-order punch metarules with variables quantified over the set of atoms and for which only an upper bound on their number of literals need be user-defined. We show that the cardinality of a metarule language is polynomial in the number of literals in punch metarules. We re-frame MIL as metarule specialisation by resolution. We modify the MIL metarule specialisation operator to return new metarules rather than first-order clauses and prove the correctness of the new operator. We implement the new operator as TOIL, a sub-system of the MIL system Louise. Our experiments show that as user-defined sort metarules are progressively replaced by sort metarules learned by TOIL, Louise’s predictive accuracy and training times are maintained. We conclude that automatically derived metarules can replace user-defined metarules.

1 Introduction

Table 1 Metarule ordering by $\theta$-subsumption.

Meta-Interpretive Learning (MIL) (Muggleton et al., 2014; Muggleton & Lin, 2015) is a recent approach to Inductive Logic Programming (ILP) (Muggleton & de Raedt, 1994) capable of learning logic programs with recursive clauses and invented predicates from examples, background knowledge and a declarative bias, the “metarules”. Metarules are datalog clauses with variables quantified over predicate symbols that are therefore second-order clauses.

Metarules are often interpreted as language bias or “clause templates” in earlier work, particularly outside the MIL literature, but they are in truth second-order background knowledge used in mixed-order resolution with the first-order background knowledge to derive the first-order clauses of a hypothesis that explains the training examples. Resolution with second-order metarules is made decidable thanks to their encapsulation into first-order definite clauses, e.g. a metarule $P(x,y) \leftarrow Q(x,y)$ is encapsulated as a definite clause $m(P,x,y) \leftarrow m(Q,x,y)$.

In the MIL literature, metarules are typically defined by a user according to intuition or knowledge of a problem domain. A common criticism of MIL is the dearth of principled approaches for metarule selection (see e.g. Cropper & Tourret, 2018). In this work, we formalise MIL as metarule specialisation by resolution and thereby provide a principled approach to learning new metarules from examples, background knowledge and maximally general metarules.

In particular, we show that user-defined, fully-connected sort metarules commonly used in the MIL literature can be derived automatically by specialisation of maximally general second-order matrix metarules; and that matrix metarules can be themselves derived by specialisation of third-order punch metarules. This specialisation can be performed by a standard MIL clause construction operator modified to return metarules rather than first-order clauses. Predicate invention, performed in MIL by resolution between metarules, is equivalent to a derivation of new metarules with arbitrary numbers of literals. Additionally, specialisation of third-order, punch metarules imposes no restriction on the arities of literals of the derived metarules. These two capabilities combined finally liberate MIL from the confines of the $H^2_2$ language fragment of metarules with up to two body literals of arity at most 2, that is almost exclusively used in the literature and that is expressive but difficult to use in practice. Table 1 illustrates punch, matrix and sort metarules and their generality relations, highlighting the subsumption ordering of second-order metarules with arbitrary numbers of literals of arbitrary arities. We illustrate learning metarules outside $H^2_2$ with a worked example in Sect. 7.2. We present an example of learning in $H^2_2$ in Appendix B.

While the number of specialisations of third order punch metarules can grow very large, they can be derived efficiently by Top Program Construction (TPC) (Patsantzis & Muggleton, 2021), a polynomial-time MIL algorithm that forms the basis of the MIL system Louise (Patsantzis & Muggleton, 2019a). We implement metarule learning by TPC in Louise as a new sub-system called TOIL.

We make the following contributions:

• We define a generality ordering of third- and second-order metarules and first-order clauses, by $\theta$-subsumption.

• We prove that metarules and first-order clauses are derivable by specialisation of more general metarules.

• We redefine MIL as metarule specialisation by SLD-resolution.

• We propose a modified MIL specialisation operator to return metarules rather than first-order clauses and prove its correctness.

• We prove that sets of metarule specialisations are polynomial-time enumerable.

• We implement our modified operator as a sub-system of Louise called TOIL.

• We verify experimentally that when user-defined metarules are replaced by metarules learned by TOIL, Louise maintains its predictive accuracy at the cost of a small increase in training times.

In Sect. 2 we discuss relevant earlier work. In Sect. 3 we give some background on MIL. In Sect. 4 we develop the framework of MIL as metarule specialisation and derive our main theoretical results. In Sect. 5 we describe TOIL. In Sect. 6 we compare Louise’s performance with user-defined and TOIL-learned metarules. We summarise our findings in Sect. 7 and propose future work.

2 Related work

Declarative bias in clausal form, similar to metarules, is common in machine learing. Emde et al. (1983) propose the use of Horn clauses to declare transitivity, conversity, or parallelism relations between binary predicates, however these early “metarules” are first-order definite clauses with ground predicate symbols. A related approach of “rule models”, having variables in place of predicate symbols, is developed in later systems METAXA.3 (Emde 1987), BLIP (Wrobel 1988) and MOBAL (Kietz and Wrobel 1992; Morik 1993). More recent work in ILP and program synthesis uses metarules as templates to restrict the hypothesis search space, for example the work by Evans and Grefenstette (2018) or Si et al. (2018), and Si et al. (2019) which draw inspiration from MIL but use metarules only as templates.

By contrast, metarules are used in MIL not as “templates” with “blanks” to be “filled in”, but as second-order formulae with the structure of datalog clauses that are resolved with first-order background knowledge to refute examples and derive the first-order clauses of a hypothesis. This use of metarules in reasoning is unique to MIL, which includes our present work.

Our work improves on the earlier use of metarules in MIL and ILP in several ways. Firstly, we extend the concept of metarules to include third-order metarules with variables quantified over the set of atoms. Such metarules are too general to be used efficiently as clause templates and are best understood within a generality framework that relates them to second-order metarules. Secondly, we define such a generality ordering by $\theta$-subsumption over third- and second-order metarules and first-order definite clauses. Kietz and Wrobel (1992) define a generality ordering between “rule models” as a special case of $\theta$-subsumption (Plotkin, 1972) restricted to variable instantiation. Our framework instead extends the full definition of $\theta -$subsumption to metarules. Si et al. (2018) also extend $\theta$-subsumption to metarules without restriction, but maintain two disjoint generality orders: one for metarules; and one for first-order clauses. Our work is the first to consider third-order metarules and place third- and second-order metarules and first-order clauses in a single ordering. Thirdly, we formalise the description of MIL as metarule specialisation by resolution and thus explain clause construction and predicate invention in MIL as the application of a metarule specialisation operator.

Our main contribution is the modification of the MIL specialisation operator to derive new metarules by specialisation of more general metarules, thus learning second-order theories. Previous work on MIL requires metarules to be defined manually, by intuition or domain knowledge, but our approach learns metarules from examples and first-order background knowledge. Automatic selection of metarules for MIL by logical reduction of a metarule language is studied by Cropper and Muggleton (2015) and Cropper and Tourret (2018). Unlike this earlier work, our approach learns new metarules rather than selecting from a user-defined set and does not suffer from the irreducibility of metarule languages to minimal sets reported by Cropper and Tourret (2018). Our approach naturally reduces each metarule language to a minimal set of most-general metarules from which all other metarules in the same language can be derived. Further, in our approach, the number of literals in third-order metarules suffices to define a metarule language. Finally, previous work on MIL has remained restricted to the $H^2_2$ language, of datalog metarules with at most two body literals of arity at most 2, which is expressive but difficult to use in practice.¹ Our approach is capable of learning metarules without restriction to the numbers of literals or their arities.

Our work is comparable to recent work by Cropper and Morel (2021), on the system Popper, and Si et al. (2019) on the system ALPS, both of which are capable of learning inductive bias, the latter in the form of metarules. Both of those systems employ generate-and-test approaches guided by $\theta$-subsumption. Our approach differs to Popper and ALPS in that it is an application of SLD resolution to second- and third-order clauses, rather than a generate-and-test approach.

Our implementation extends the Top Program Construction algorithm (Patsantzis & Muggleton, 2021) that avoids an expensive search of a potentially large hypothesis space and instead constructs all clauses that entail an example with respect to background knowledge. We extend this earlier work with the ability to construct second-order clauses without compromising efficiency.

3 Background

3.1 Logical notation

In this section, we extend the terminology established in Nienhuys-Cheng and de Wolf (1997) with MIL-specific terminology for second- and third-order definite clauses and programs. We only describe the salient terms in the nomenclature.

We define a language of first- and higher-order logic programs composed of clauses, themselves composed of terms, as follows. ${\mathcal{P}}, {\mathcal{F}}, {\mathcal{C}}, {\mathcal{A}}$ are disjoint sets of predicate symbols, function symbols, constants and atoms, respectively. Variables are quantified over (the elements of) a set. A variable is a term. A constant is a term. If F is a function symbol or a variable quantified over ${\mathcal {F}}$, and $t_1, \ldots, t_n$ are terms, then $F(t_1, \ldots, t_n)$ is a term, and n is the arity of F. If P is a predicate symbol or a variable quantified over ${\mathcal {P}}$, and $t_1, \ldots, t_n$ are terms, then $P(t_1, \ldots, t_n)$ is an atomic formula, or simply atom, and n is the arity of P. Arities are natural numbers. Comma-separated terms in parentheses are the arguments of a term or an atom. Terms with $n > 0$ arguments are functional terms, or simply functions. Constants are functions with 0 arguments.

A literal is an atom, or the negation of an atom, or a variable quantified over ${\mathcal {A}}$, or the negation of a variable quantified over ${\mathcal {A}}$. A clause is a set of literals, interpreted as a disjunction. A logic program, or simply program, is a set of clauses, interpreted as a conjunction. A clause is Horn if it has at most one positive literal. A Horn clause is definite if it has exactly one positive literal, otherwise it is a Horn goal. A literal is datalog (Ceri et al., 1989) if it contains no functions other than constants. A Horn clause is datalog if it contains only datalog literals and each variable in a positive literal is shared with at least one negative literal. A logic program is definite if it contains only definite clauses and datalog if it contains only datalog clauses.

Terms without variables are ground. Atoms with only ground terms are ground. Clauses with only ground atoms are ground. Ground terms, atoms and clauses are 0’th-order. Variables quantified over ${\mathcal {C}}$ are first-order, variables quantified over ${\mathcal {P}}$ or ${\mathcal {F}}$ are second-order, and variables quantified over ${\mathcal {A}}$ are third-order. Functions and atoms with order k arguments are order k. Clauses with order k literals are order k. Programs with order k clauses are order k. A non-ground order k term, atom or clause, or an order k program is also order $k-1$.

We denote a variable X quantified over a set S as $\exists _{\in S} X$ or $\forall _{\in S} X$.

A substitution of variables in a clause C is a finite set $\vartheta = \{x_1/t_1, \ldots, x_n/t_n\}$ mapping each variable $x_i$ in C to a term, atom, or symbol $t_i$. $C\vartheta$, the application of $\vartheta$ to C, replaces each occurrence of $x_i$ with $t_i$ simultaneously.

In keeping with logic programming convention, we will write clauses as implications, e.g. the definite clause ${A \vee \lnot B \vee \lnot C}$ will be written as $A \leftarrow B, C$ where the comma, “,”, indicates a conjunction; and refer to the positive and negative literals in a clause as the “head” and “body” of the clause, respectively.

3.2 Meta-interpretive learning

MIL is an approach to ILP where programs are learned by specialisation of a set of higher-order metarules. Metarules are defined in the MIL literature as second-order definite datalog clauses with existentially quantified variables in place of predicate symbols and constants (Muggleton & Lin, 2015). In Sect. 4 we introduce third-order metarules, with variables quantified over ${\mathcal {A}}$ as literals.

A system that performs MIL is a Meta-Interpretive Learner, or MIL-learner (with a slight abuse of abbreviation to allow a natural pronunciation). Examples of MIL systems are Metagol (Cropper & Muggleton, 2016b), Thelma (Patsantzis & Muggleton, 2019b), Louise (Patsantzis & Muggleton, 2019a) and Hexmil (Kaminski et al., 2018). A MIL-learner is given the elements of a MIL problem and returns a hypothesis, a logic program constructed as a solution to the MIL problem. A MIL problem is a sextuple, ${\mathcal {T}} = \langle E^+, E^-, B, {\mathcal {M}}, I, {\mathcal {H}} \rangle$ where (a) $E^+$ is a set of ground atoms and $E^-$ is a set of negated ground atoms of one or more target predicates, the positive and negative examples, respectively; (b) B is the background knowledge, a set of definite clause definitions with datalog heads; (c) ${\mathcal {M}}$ is a set of second-order metarules; (d) I is a set of additional symbols reserved for invented predicates not defined in B or $E^+$; and (e) ${\mathcal {H}}$ is the hypothesis space, a set of hypotheses.

Each hypothesis H in ${\mathcal {H}}$ is a set of datalog clauses, a definition of one or more target predicates in $E^+$, and may include definitions of one or more predicates in I. For each $H \in {\mathcal {H}}$, if $H \wedge B \models E^+$ and $\forall e^- \in E^-: H \wedge B \not \models e^-$, then H is a correct hypothesis.

The set, ${\mathcal {L}}$, of clauses in all hypotheses in ${\mathcal {H}}$ is the Hypothesis Language. For each clause $C \in {\mathcal {L}}$, there exists a metarule $M \in {\mathcal {M}}$ such that $M \wedge B \wedge E^+ \models C$, i.e. each clause in ${\mathcal {L}}$ is an instance of a metarule in ${\mathcal {M}}$ with second-order existentially quantified variables substituted for symbols in ${\mathcal {P}}$ and first-order existentially quantified variables substituted for constants in ${\mathcal {C}}$. ${\mathcal {P}}$ and ${\mathcal {C}}$ are populated from the symbols and constants in $B, E^+$ and I.

Typically a MIL learner is not explicitly given ${\mathcal {H}}$ or ${\mathcal {L}}$, rather those are implicitly defined by ${\mathcal {M}}$ and the constants ${\mathcal {C}}$ and symbols ${\mathcal {P}}$.

A substitution of the existentially quantified variables in a metarule M is a metasubstitution of M. By logic programming convention, a substitution of universally quantified variables is denoted by a lower-case letter, $\vartheta , \sigma , \omega$. We follow the convention and also denote metasubstitutions with capital Greek letters, $\varTheta , \varSigma , \varOmega$, etc. $\vartheta \varTheta$ denotes the composition of the substitution $\vartheta$ and metasubstitution $\varTheta$. For brevity, we refer to the composition of a substitution and metasubstitution as meta/substitution. To help the reader distinguish a meta/substitution from its application to a metarule we note a meta/substitution as, e.g. $\vartheta \varTheta /M$, whereas $M\vartheta \varTheta$ is the result of applying $\vartheta \varTheta$ to M.

MIL learners construct clauses in H during refutation-proof of a set of positive examples by SLD-resolution (Nienhuys-Cheng & de Wolf, 1997) with B and ${\mathcal {M}}$. Resolution is performed by a meta-interpreter designed to preserve the metasubstitutions of metarules in a successful proof by refutation, while discarding the substitutions of universally quantified variables to avoid over-specialisation. Metasubstitutions applied to their corresponding metarules are first-order definite clauses. In a “second pass” negative examples are refuted by the same meta-interpreter with B, H and ${\mathcal {M}}$, and any clauses in H found to entail a negative example are either removed from H (Louise) or replaced on backtracking (Metagol).

Resolution between second-order metarules and first-order clauses is made decidable by encapsulation, a mapping from metarules with existentially and universally quantified second- and first-order variables, to definite clauses with only universally quantified, first order variables. For example, the metarule $\exists P,Q \forall x,y: P(x,y) \leftarrow Q(x,y)$ is encapsulated as the first-order definite clause $\forall P,Q,x,y: m(P,x,y) \leftarrow m(Q,x,y)$. Encapsulation further maps each predicate symbol $p \in {\mathcal {P}}$ to a new constant $p \in {\mathcal {C}}$.²

4 Framework

4.1 Metarule languages

Table 2 “Canonical” set of fully-connected $H^2_2$ metarules

In this section we introduce a formal notation for sets of metarules.

Definition 1

A metarule language ${\mathscr {M}}^{l}_{a}$ is a set of metarules and their instances where l is a natural number or an interval over the natural numbers, denoting the number of literals in clauses in ${\mathscr {M}}^{l}_{a}$, and a is a natural number, or an interval over the natural numbers, or a sequence of natural numbers, denoting the arities of literals in clauses in ${\mathscr {M}}^{l}_{a}$.

When the arity term, a, in ${\mathscr {M}}^{l}_{a}$ is a sequence, a total ordering is assumed over the literals in each clause in ${\mathscr {M}}^{l}_{a}$ such that (a) the positive literal is ordered before any negative literals, (b) negative literals are ordered by lexicographic order of the names of their symbols and variables and (c) literals with the same symbol and variable names are ordered by ascending arity.

Example 1

${\mathscr {M}}^{3}_{2}$ is the language of metarules and first-order definite clauses with exactly three literals each of arity exactly 2; ${\mathscr {M}}^{[2,3]}_{\,[1,2]}$ is the language of metarules and first-order definite clauses with 2 or 3 literals of arities between 1 and 2; and ${\mathscr {M}}^{3}_{\langle 1,2,3 \rangle }$ is the language of metarules and first-order definite clauses having exactly three literals, a positive literal of arity 1 and two negative literals of arity 2 and arity 3, in that order.

The arities of literals in third-order metarules may not be known or relevant, in which case the arity term may be omitted from the formal notation of a language.

Example 2

${\mathscr {M}}^3$ is the language of third-order metarules, second-order metarules and first-order definite clauses with exactly 3 literals. ${\mathscr {M}}^{[1,5]}$ is the language of third- and second-order metarules and first-order definite clauses with 1 to 5 literals.

Of special interest to MIL is the ${\mathscr {M}}^{[2,3]}_{2}$ language of fully-connected (see Definition 2) second-order metarules with one function symbol which is decidable when ${\mathcal {P}}$ and ${\mathcal {C}}$ are finite (Muggleton & Lin, 2015) and second-order variables are universally quantified by first-order encapsulation. We denote this language exceptionally as $H^2_2$ in keeping with the MIL literature. The set of 14 second-order $H^2_2$ metarules defined in Cropper and Muggleton (2015) are given in Table 2.

4.2 Metarule taxonomy

Table 3 Second-order matrix metarules in the $H^2_2$ language

Table 4 Third-order punch metarules in the $H^2_2$ language

Metarules found in the MIL literature, such as the ones listed in Table 2 are typically fully-connected. Definition 2 extends the definition of fully-connected metarules in Cropper and Muggleton (2015) to encompass first-order clauses.

Definition 2

(Fully-connected datalog) Let M be a second-order metarule or a first-order definite clause. Two literals $l_i, l_j \in M$ are connected if they share a first-order term, or if there exists a literal $l_k \in M$ such that $l_i, l_k$ are connected and $l_j, l_k$ are connected. M is fully connected iff each literal l in M appears exactly once in M and l is connected to every other literal in M.

Fully-connected metarules are specialised, in the sense that all their universally quantified variables are shared between their literals; existentially quantified variables may also be shared. Accordingly, fully-connected metarules can be generalised by replacing each instance of each of their variables with a new, unique variable of the same order and quantification as the replaced variable. Applying this generalisation procedure to the metarules in Table 2 we obtain the metarules in Table 3.

We observe that each metarule in Table 3 is a generalisation of a metarule in the $H^2_2$ language listed in Table 2 and so the metarules in Table 3 are the most-general metarules in $H^2_2$. Further, those most-general $H^2_2$ metarules can themselves be generalised to the third-order metarules in Table 4 by replacing each of their literals with a third-order variable. This observation informs our definition of three taxa of metarules and a total ordering by $\theta$-subsumption of third- and second-order metarules and their first-order instances.

4.3 Punch, sort and matrix metarules

In the following sections we employ a moveable type metaphor for the elements of our metarule taxonomy. In typeset printing, first a punch of a glyph is sculpted in relief in steel. The punch is used to emboss the shape of the glyph in copper, creating a matrix. The copper matrix is filled with molten soft metal to form a cast of the glyph called a sort and used to finally imprint the glyph onto paper. Thus each “level” of type elements “stamps” its shape onto the next.

Accordingly, in our taxonomy of metarules, a third-order metarule is a punch metarule denoted by $\breve{M}$, a most-general metarule in a second-order language is a matrix metarule denoted by $\mathring{M}$, and a fully-connected second-order metarule is a sort metarule, denoted by $\dot{M}$. As a mnemonic device the reader may remember that a “wider” accent denotes higher generality.

Definition 3

(Punch metarules) A punch metarule $\breve{M}$ is a third-order definite clause of the form: $\exists _{\in {\mathcal {A}}} A_1, A_2, \ldots, A_n: \{A_1 \vee \lnot A_2 \vee \ldots \vee \lnot A_n\}$.

Definition 4

(Matrix metarules) A matrix metarule $\mathring{M}$ (not to be confused with the linear algebra, or first-order logic concepts of a matrix) is a second-order definite clause of the form: $\exists _{\in {\mathcal {P}}} \tau , \exists _{\in {\mathcal {C}}} \sigma , \forall _{\in {\mathcal {C}}} \rho : \{L_1 \vee \lnot L_2 \vee \ldots \vee \lnot L_n\}$ where $\tau , \sigma , \rho$ are disjoint sets of variables, each $L_i \in \mathring{M}$ is a second-order atom $P_i(v_{i1}, \ldots, v_{im})$, $P_i \in \tau , v_{i1}, \ldots, v_{im} \in \sigma \cup \rho$ and none of $P_i, v_{i1}, \ldots, v_{im}$ is shared with any other literal $L_k \in \mathring{M}$.

Definition 5

(Sort metarules) A sort metarule $\dot{M}$ (not to be confused with the logic programming concept of a sort) is a second-order definite clause of the form: $\exists _{\in {\mathcal {P}}} \tau , \exists _{\in {\mathcal {C}}} \sigma , \forall _{\in {\mathcal {C}}} \rho : \{L_1 \vee \lnot L_2 \vee \ldots \vee \lnot L_n\}$ where $\tau , \sigma , \rho$ are disjoint sets of variables, each $L_i \in \dot{M}$ is a second-order atom $P_i(v_{i1}, \ldots, v_{im})$, $P_i \in \tau , v_{i1}, \ldots, v_{im} \in \sigma \cup \rho$ and at least one $L_i \in \dot{M}$ shares a variable in $\tau \cup \sigma \cup \rho$ with at least one other literal $L_k \in \dot{M}$.

Note 1

Sort metarules are not necessarily fully-connected, rather fully-connected metarules are a sub-set of the sort metarules. The metarules typically used in the MIL literature, such as the 14 Canonical $H^2_2$ metarules in Table 2 are fully connected sort metarules.

We define metarules as sets of literals according to Sect. 3.1. As discussed at the end of that section we will write clauses in the logic programming convention, as implications, and this also applies to metarules. Additionally, in keeping with MIL convention we will write metarules concisely without quantifiers instead denoting quantification by means of capitalisation: uppercase letters for existentially quantified variables, lower-case letters for universally quantified variables.

Thus, we will write a punch metarule $\exists _{\in {\mathcal {A}}}P,Q,R: \{P \vee \lnot Q \vee \lnot R\}$ as an implication $P \leftarrow Q, R$, a matrix metarule $\exists _{\in {\mathcal {P}}} P,Q,R, \forall _{\in {\mathcal {C}}}x,y,z,u,v,w: \{P(x,y) \vee \lnot Q(z,u) \vee \lnot R(v,w)\}$ as an implication $P(x,y) \leftarrow Q(z,u), R(v,w)$ and a sort metarule $\exists _{\in {\mathcal {P}}}P, Q, R$, $\exists _{\in {\mathcal {C}}}X, \forall _{\in {\mathcal {C}}}x,y,z: \{P(x,y) \vee \lnot Q(x,z) \vee \lnot R(X)\}$ as an implication $P(x,y) \leftarrow Q(x,z), R(X)$.

Defining metarules as sets of clauses facilitates their comparison in terms of generality, while denoting them as implications, without quantifiers, makes them easier to read and closely follows their implementation in MIL systems as Prolog clauses (with encapsulation).

4.4 Metarule generality order

We extend $\theta$-subsumption between clauses, as defined by Plotkin (1972), to encompass metarules with existentially quantified variables:

Definition 6

(Meta-subsumption) Let C be a metarule or a first-order definite clause and D be a metarule or a first-order definite clause. $C \preceq D$ (read C subsumes D) iff $\exists \vartheta , \varTheta : C \vartheta \varTheta \subseteq D$ where $\vartheta$ is a substitution of the universally quantified variables in C and $\varTheta$ is a metasubstitution of the existentially quantified variables in C.

Lemma 1

(3rd-order subsumption) Let $\breve{M}$ be a punch metarule in the language ${\mathscr {M}}^{l}$ and $\mathring{M}$ be a matrix metarule in the language ${\mathscr {M}}^{k}_{a}$. Then, $\forall a: l \le k \rightarrow \breve{M} \preceq \mathring{M}$.

Proof

Let $\breve{M} = \{A_1 \vee \lnot A_2 \ldots \vee \lnot A_l\}$ and $\mathring{M} = \{L_1 \vee \lnot L_2 \ldots \vee \lnot L_k\}$. Assume a total ordering over the literals in $\breve{M}$ and $\mathring{M}$ as described in Sect. 4.2. Let $\vartheta = \emptyset$ and let $\varTheta$ be the metasubstitution that maps each $A_i \in \breve{M}$ to each $L_i \in \mathring{M}$. While $l \le k$, $\exists \vartheta , \varTheta$ and $\breve{M} \vartheta \varTheta \subseteq \mathring{M}$. $\square$

Lemma 2

(2nd-order subsumption) Let $\mathring{M}$ be a matrix metarule in the language ${\mathscr {M}}^{l}_{a}$ and $\dot{M}$ be a sort metarule in the language ${\mathscr {M}}^{k}_{b}$, where a, b are two integers or two sequences of integers having the same first element. Then $l \le k \rightarrow \mathring{M} \preceq \dot{M}$ iff $a = b$ or a is a subsequence of b.

Proof

Let $P_1, \ldots, P_l$ be the existentially quantified variables and $v_1, \ldots, v_n$ the universally quantified variables in $\mathring{M} \in {\mathscr {M}}^{l}_{a}$. Let $Q_1, \ldots, Q_k$ be the existentially quantified variables and $u_1, \ldots, u_m$ the universally quantified variables in $\dot{M} \in {\mathscr {M}}^{k}_{b}$. Assume a total ordering over the literals in $\mathring{M}$ and $\dot{M}$ as described in Sect. 4.2. Let $\vartheta$ be the substitution that maps each $v_i$ to $u_i$ and $\varTheta$ the metasubstitution that maps each $P_j$ to $Q_j$. While $l \le k$, and either $a \le b$ or a is a subsequence of b, $\exists \vartheta , \varTheta$ and $\mathring{M} \vartheta \varTheta \subseteq \dot{M}$. $\square$

Lemma 3

(1st-order subsumption) Let $\dot{M}$ be a sort metarule in the language ${\mathscr {M}}^{l}_{a}$ and C be a first-order clause in the language ${\mathscr {M}}^{k}_{b}$, where a, b are two integers or two sequences of integers having the same first element. Then $l \le k \rightarrow \dot{M} \preceq C$ iff $a = b$ or a is a subsequence of b.

Proof

Let $P_1, \ldots, P_l$ be the existentially quantified variables and $v_1, \ldots, v_n$ be the universally quantified variables in $\dot{M} \in {\mathscr {M}}^{l}_{a}$. Let $Q_1, \ldots, Q_k$ be the predicate symbols and constants in $C \in {\mathscr {M}}^{k}_{b}$ and $t_1, \ldots, t_m$ be the first-order terms in C. Assume a total ordering over the literals in $\dot{M}$ and C as described in Sect. 4.2.. Let $\vartheta$ be the substitution that maps each $v_i$ to $t_i$ and $\varTheta$ be the metasubstitution that maps each $P_j$ to $Q_j$. While $l \le k$, and either $a \le b$ or a is a subsequence of b, $\exists \vartheta , \varTheta$ and $\dot{M} \vartheta \varTheta \subseteq C$. $\square$

Corollary 1

Let ${\mathscr {M}}^{l}_{a}$ be a metarule language. There exists a unique, minimal set of matrix metarules ${\mathcal {M}}^* = \{\mathring{M}_1, \ldots, \mathring{M}_n\} \subseteq {\mathscr {M}}^{l}_{a}$ such that for each sort metarule $\dot{M} \in {\mathscr {M}}^{l}_{a}$ $\exists \mathring{M}_i \in {\mathcal {M}}^*: \mathring{M}_i \preceq \dot{M}$. Each $\mathring{M}_i \in {\mathcal {M}}^*$ can be derived by replacing each variable in any single $\dot{M} \in {\mathscr {M}}^{l}_{a}$ subsumed by $\mathring{M}_i$ with a new, unique variable.

4.5 Metarule specialisation

We now show how first-order clauses and second-order metarules can be derived by specialisation of more general metarules. We define two ways to specialise a metarule or a clause: by variable substitution or introduction of new literals.

Note 2

In the following definitions and theorem, let $M_1, M_2$ be two metarules, or a metarule and a definite clause.

Definition 7

(V-specialisation) Let $\vartheta , \varTheta$ be substitutions of the universally and existentially quantified variables, respectively, in $M_1$ such that $M_1 \vartheta \varTheta = M_2$. Then $M_1 \vartheta \varTheta$ is a variable specialisation, or v-specialisation, of $M_1$, and $M_2$ is derivable from $M_1$ by v-specialisation, or $M_1 \vdash _v M_2$.

Definition 8

(L-specialisation) Let L be a set of literals such that $M_1 \cup L = M_2$. Then $M_1 \cup L$ is a literal specialisation, or l-specialisation, of $M_1$, and $M_2$ is derivable from $M_1$ by l-specialisation, or $M_1 \vdash _l M_2$.

Definition 9

(VL- Specialisation) Let $M_1 \vartheta \varTheta$ be a v-specialisation of $M_1$, $M_1 \cup L$ be an l-specialisation of $M_1$, and $M_1 \vartheta \varTheta \cup L = M_2$. Then $M_1 \vartheta \varTheta \cup L$ is a variable and literal specialisation, or vl-specialisation, of $M_1$, and $M_2$ is derivable from $M_1$ by vl-specialisation, or $M_1 \vdash _{vl} M_2$.

Theorem 1

(Metarule specialisation) $M_1 \preceq M_2 \rightarrow M_1 \vdash _{vl} M_2$.

Proof

If $M_1 \preceq M_2$ then: a) $\exists \vartheta , \varTheta : M_1 \vartheta \varTheta \subseteq M_2$ and b) $\exists L: M_1 \vartheta \varTheta \cup L = M_2$. (a) follows directly from Definition 6. (b) follows from (a) and the subset relation: if $M_1 \vartheta \varTheta \subseteq M_2$ then $\exists L \in M_2: M_2 \setminus L = M_1 \vartheta \varTheta$ and $M_1 \vartheta \varTheta \cup L = M_2$. By Definitions 7 and 8, $M_1 \vartheta \varTheta$ is a v-specialisation of $M_1$ and $M_1 \vartheta \varTheta \cup L$ is an l-specialisation of $M_1 \vartheta \varTheta$. Therefore, $M_1 \vdash _v M_1 \vartheta \varTheta \vdash _l M_1 \vartheta \varTheta \cup L = M_2$ and so $M_1 \vdash _{vl} M_2$. $\square$

Observation 1

There are two special cases of (a) in the proof of Theorem 1 with respect to the set of literals L: either $M_1 \vartheta \varTheta = M_2$ or $M_1 \vartheta \varTheta \subset M_2$. In the case where $M_1 \vartheta \varTheta = M_2$, $L = \emptyset$. Otherwise, $L \ne \emptyset$.

4.6 MIL as metarule specialisation

In this section we explain MIL as vl-specialisation of metarules.

Algorithm 1 lists the MIL specialisation operator used in a MIL meta - interpreter to construct new clauses by refutation of a literal, $\lnot e$. MIL systems implement Algorithm 1 idiosyncratically: In Metagol $B^* = B \cup H$ and $\lnot e$ is refuted with B, H or ${\mathcal {M}}$ successively (Cropper & Muggleton, 2016a); in Louise $B^* = B \cup E^+$ and $\lnot e$ is refuted with $B^* = B \cup E^+$ and ${\mathcal {M}}$ simultaneously (Patsantzis & Muggleton, 2021).

Initially, e is a positive example in $E^+$ and if refutation of $\lnot e$ succeeds the returned clause $M\varTheta$ is a clause in the definition of a target predicate.

If refutation fails, each atom in $body(M\sigma \varSigma )$ in line 4 in Procedure CONSTRUCT becomes the input literal $\lnot e$ and is refuted recursively by resolution with $B^* \cup {\mathcal {M}}$, until $\square$ is derived. In that case, $M\varTheta$ is a clause in the definition of an invented predicate, thus predicate invention in MIL is achieved by resolution between metarules. Given that metarules do not have predicate symbols, when $\lnot body(M\sigma \varSigma )$ is successfully refuted, the existentially quantified second-order variable P in $head(M\varTheta )$ remains free. Hence, a new predicate symbol in I is substituted for P.³

If e is in $E^-$ and refutation succeeds, $M\varTheta$ is inconsistent and must be replaced in, or discarded from H.

We observe that Algorithm 1 returns vl-specialisations of metarules in ${\mathcal {M}}$.

Theorem 2

(MIL as metarule specialisation) Let $e, B^*, {\mathcal {M}}$ be as in Algorithm 1, M be a fully-connected sort metarule selected in line 2 of Procedure CONSTRUCT and $M\varTheta =$ CONSTRUCT$(\lnot e, B^*, {\mathcal {M}})$. $M\varTheta$ is a vl-specialisation of M.

Proof

Let $\vartheta = L = \emptyset$. By Definition 9$M\vartheta \varTheta \cup L = M \varTheta$ is a vl-specialisation of M. $\square$

4.7 Implicit l-specialisation

Table 5 $M_3$ is an l-specialisation and self-resolvent of $M_1$. $C_1,C_2$ are instances of $M_1$. $C_3$ is an instance of $M_3$ and resolvent of $C_1, C_2$

Theorem 2 states that Procedure CONSTRUCT returns vl-specialisations of metarules when the set of introduced literals, L, is empty. This is a special case of vl-specialisation. What about the general case, when $L \ne \emptyset$? We conjecture that it is not necessary to explicitly construct such non-empty l-specialisations, because the v/specialisations returned by Procedure CONSTRUCT suffice to reconstruct non-empty l/specialisations by resolution. Suppose $M_1, M_2 \in {\mathcal {M}}$ and $C_1, C_2$ are non-empty v/specialisations of $M_1, M_2$, respectively, such that $\exists e \in E^+, \not \exists e \in E^-: \{\lnot e\} \cup \{C_1, C_2\} \cup B^* \vdash \square$. We assume that $C_1, C_2$ can resolve with each other and that $\{\lnot e\} \cup B^* \setminus \{e\} \cup \{C_{i \in \{1,2\}}\} \nvdash \square$. Then, there exists a resolvent $C_3$ of $C_1, C_2$ such that $\exists e \in E^+, \not \exists e \in E^-: \{\lnot e\} \cup \{C_3\} \cup B^* \vdash \square$. This follows from the Resolution Theorem (Robinson, 1965). Moreover, there exists a metarule $M_3$ that is a resolvent of $M_1, M_2$ and such that $C_3$ is a vl-specialisation of $M_3$ where the set of introduced literals, L, is not empty. If so, it should not be necessary to explicitly derive $M_3$ and $C_3$, given $M_1, M_2$ and Procedure CONSTRUCT. Cropper and Muggleton (2015) prove our conjecture for the $H^2_2$ language. We leave a more general proof for future work. Table 5 illustrates the concept of such occult specialisations.

4.8 Metarule specialisation by MIL

Algorithm 1 learns first-order definite clauses. Our motivation for this work is to learn metarules that can replace user-defined metarules. User-defined metarules are fully-connected sort metarules and chosen so that if M is a user-defined metarule and $M\varTheta$ is a vl-specialisation of M returned by Procedure CONSTRUCT in Algorithm 1, then $\exists e^+ \in E^+: M\varTheta \wedge B^* \models e^+$. Therefore, to replace user-defined metarules with automatically derived metarules, we must automatically derive fully-connected sort metarules having vl-specialisations that entail one or more positive examples in $E^+$ with respect to $B^*$.

We achieve this goal by modifying Algorithm 1, as Algorithm 2, to generalise the substitutions of both universally and existentially quantified variables in metarules. Such substitutions are fully-ground by successful resolution with $B^*$ therefore, in order to produce metarules rather than first-order clauses, we must replace the ground terms in those substitutions with new variables. We propose Procedure LIFT in Algorithm 3 to perform this “variabilisation” operation.

Lemma 4

(Fully-connected lifting) Let $\vartheta \varTheta /M$ be a meta/substitution of a punch or matrix metarule M and $M\vartheta \varTheta$ be a fully-connected definite clause. The application of LIFT$(\vartheta \varTheta )$ to M, M.LIFT$(\vartheta \varTheta )$, is a fully-connected sort metarule.

Proof

Procedure LIFT replaces each occurrence of a ground term with the same variable throughout $\vartheta \varTheta$ so that if two literals $l_i,l_k \in M\vartheta \varTheta$ share a ground term, $\{l_i,l_k\}.$ LIFT$(\vartheta \varTheta ) \in M.$ LIFT$(\vartheta \varTheta )$ share a variable. Therefore $M\vartheta \varTheta$ is fully-connected iff M.LIFT$(\vartheta \varTheta )$ is fully-connected. $\square$

Lemma 5

(Lifting subsumption) Let $\vartheta \varTheta /M$ be a meta/substitution of a punch or matrix metarule M. Then $M \preceq M.$ LIFT$(\vartheta \varTheta ) \preceq M\vartheta \varTheta$.

Proof

$M \preceq M.$ LIFT$(\vartheta \varTheta )$ by Definition 6. Construct a meta/substitution $\sigma \varSigma$ by mapping each $w_i$ in $v_i/w_i \in \,$ LIFT$(\vartheta \varTheta )$ to $t_i$ in $v_i/t_i \in \vartheta \varTheta$. $\exists \sigma \varSigma : M.$ LIFT$(\vartheta \varTheta )\sigma \varSigma = M \vartheta \varTheta$, therefore M.LIFT$(\vartheta \varTheta ) \preceq M \vartheta \varTheta$. $\square$

Example 3

Let $M = P(x,y) \leftarrow Q(z,u)$, $\vartheta \varTheta = \{P/p, Q/q, x/a, y/b, z/a, u/b\}$. Then: $M \vartheta \varTheta = p(a,b) \leftarrow q(a,b)$, LIFT$(\vartheta \varTheta ) =$ $\{P/P_1, Q/Q_1, x/x_1, y/y_1, z/x_1$, $u/y_1\}$, M.LIFT$(\vartheta \varTheta ) =$ $P_1(x_1,y_1) \leftarrow Q_1(x_1,y_1)$, $\sigma \varSigma = \{P_1/p, Q_1/q, x_1/a, y_1/b \}$ and M.LIFT$(\vartheta \varTheta ) \sigma \varSigma = p(a,b) \leftarrow q(a,b) = M \vartheta \varTheta$.

Theorem 3

(Soundness) Let $e, B^*, {\mathcal {M}}$ be as in Algorithm 2. If $M' =$ VL-SPECIALISE $(\lnot e, B^*, {\mathcal {M}})$ then $M'$ is a fully-connected sort metarule and $\exists \varSigma / M': M'\varSigma \wedge B^* \models e$.

Proof

Assume Theorem 3 is false. Then, $M' =$ VL-SPECIALISE$(\lnot e, B^*, {\mathcal {M}})$ and (a) $M'$ is not a fully-connected sort metarule, or (b) $\not \exists \varSigma /M': M'\varSigma \wedge B^* \models e$. In Procedure VL-SPECIALISE $M' = M.$ LIFT$(\vartheta \varTheta )$ is returned iff (c) M is the punch or matrix metarule selected in line 2, (d) $\exists \vartheta , \varTheta /M: M\vartheta \varTheta \cup B^* \cup \{\lnot e\} \vdash _{SLD} \square$ iff $M\vartheta \varTheta \wedge B^* \models e$ and (e) $M \vartheta \varTheta$ is a fully-connected definite clause. By Lemma 5, if (c) and (d) hold then $M' \preceq M \vartheta \varTheta$ because $\exists \sigma \varSigma : M' \sigma \varSigma = M \vartheta \varTheta$. Therefore if (c) and (d) hold then $M'\sigma \varSigma \wedge B^* \models e$ and $M' \varSigma \wedge B^* \models e$. By Lemma 4, if (e) holds then $M'$ is a fully-connected sort metarule. Therefore, either $M' \ne$ VL-SPECIALISE$(\lnot e, B^*, {\mathcal {M}})$ and (a), (b) are false, or Theorem 3 is true. This refutes the assumption and completes the proof. $\square$

4.9 Cardinality of metarule languages

In this section we turn our attention to the cardinalities of metarule languages and show that they are polynomial in the number of punch metarule literals (Table 6).

Table 6 Cardinality of metarule languages with at most k body literals according to Lemmas 6–10 and Theorem 4. k: maximum number of body literals in punch metarules; a: cardinality of the set, ${\mathcal {A}}_{\mathring{M}}$, of matrix metarule literals; n: (constant) number of existentially and universally quantified variables in sort metarule literals; c: cardinality of the set, ${\mathcal {C}}$, of constants; p: cardinality of the set, ${\mathcal {P}}$, of predicate symbols

Definition 10

(Clause length) Let C be a metarule or a first-order definite clause. The length of C is the number of literals in C.

Note 3

In the following lemmas and proofs, metarules that differ only in the names of their variables are considered identical.

Lemma 6

(Number of punch metarules) The number of punch metarules of length in [1, k ] is k.

Proof

There exist k n-tuples of third-order variables for $n \in [1,k]$. Exactly one definite clause can be formed from each such n-tuple (see Note 3). $\square$

Example 4

Suppose $k = 3$. The set of n-tuples of third-order variables in punch metarules of length 1 to k is $\{\{P\}, \{P,Q\}, \{P,Q,R\}\}$. Exactly one definite clause can be formed from each such n-tuple: $\{P\}$, $\{P \vee \lnot Q\}$ and $\{P \vee \lnot Q \vee \lnot R\}$.

Lemma 7

(Number of matrix metarules) Let ${\mathcal {A}}_{\mathring{M}} \subseteq {\mathcal {A}}$ be the set of matrix metarule atoms and $a = |{\mathcal {A}}_{\mathring{M}}|$. The number of matrix metarules of length k is at most $k(a^k/k!)$.

Proof

Let $\{A_1,\ldots,A_k: A_i \in {\mathcal {A}}_{\mathring{M}}\}$ be the k-tuple of atoms in a matrix metarule of length k. The number of such k-tuples is the number of subsets of ${\mathcal {A}}_{\mathring{M}}$ of length k, called the k-combinations of ${\mathcal {A}}_{\mathring{M}}$, which is equal to the binomial coefficient $\left( {\begin{array}{c}a\\ k\end{array}}\right)$, which is at most $a^k/k!$ for $1 \le k \le a$ (Cormen et al., 2001). Note that if a is less than k a matrix metarule of length k cannot be formed because matrix metarule literals must be distinct atoms. For each such k-tuple, T, exactly k definite clauses can be formed by taking one atom in T as the single positive literal, in turn. $\square$

Example 5

Suppose $k = 3$, ${\mathcal {A}}_{\mathring{M}} = \{P(x,y), Q(z,u,v), R(w)\}$. The set, T, of 3-tuples of atoms in ${\mathcal {A}}_{\mathring{M}}$ is $\{ \{P(x,y), Q(z,u,v)$, $R(w) \} \} = \{{\mathcal {A}}_{\mathring{M}}\}$. The set of definite clauses formed by taking each atom in a 3-tuple in T as a positive literal in turn is: $\{\{P(x,y)$, $\lnot Q(z,u,v)$, $\lnot R(w)\}$, $\{\lnot P(x,y)$, Q(z, u, v), $\lnot R(w)\}$, $\{\lnot P(x,y)$,$\lnot Q(z,u,v)$, $R(w)\}\}$.

Lemma 8

(Number of sort metarules) Let e be the number of existentially quantified first- and second-order variables, and u be the number of universally quantified first-order variables, in all sort metarules of length k in the language ${\mathscr {M}}^{k}_{b}$. Let $n = e + u$. The number of sort metarules in ${\mathscr {M}}^{k}_{b}$ is less than $(2n-1)^n/ n!$.

Proof

Let $\{P_1, \ldots, P_e, v_1, \ldots, v_u\}$ be the multiset⁴ with cardinality $n = e + u$ of existentially quantified first- and second-order variables $P_i$ and universally quantified first-order variables $v_j$ each with multiplicity 1 or more. Let S be the set of all such multisets. Let $S'$ be the set of multisets in S each containing existentially quantified second-order variables of total multiplicity k, existentially quantified first-order variables of total multiplicity $e-k$, and universally quantified first-order variables with total multiplicity u at least one of which has multiplicity between 2 and u. Multiplicities of variables in elements of $S'$ are constrained by Definition 5 and the cardinality of elements of $S'$ is $n = e + u$ therefore $S'$ is the set of multisets of variables in sort metarules of length k in the language ${\mathscr {M}}^{k}_{b}$. The cardinality of S is equal to the multiset coefficient $\left( \!\left( {\begin{array}{c}n\\ n\end{array}}\right) \!\right)$ which is equal to the binomial coefficient $\left( {\begin{array}{c}2n-1\\ n\end{array}}\right)$ (Stanley, 2011). S necessarily includes elements not in $S'$, for example multisets containing no universally quantified variables with multiplicity 2. Therefore $|S'| < |S|$ and so $|S'| < \left( {\begin{array}{c}2n-1\\ n\end{array}}\right)$. $(2n-1)^n/n!$ is an upper bound for $\left( {\begin{array}{c}2n-1\\ n\end{array}}\right)$ for $1 \le n \le 2n-1$ which is always the case, therefore $|S'| < (2n-1)^n/n!$. $\square$

Example 6

Suppose $k = 3$, n = 9. The set, S, of n-multisets of existentially and universally quantified variables is $\{$ $\{P, P, P, x$, x, x, x, $x, x\}$, $\{P, P, P, x, x, x, x, x, y, \}$, $\{P, P, P, x, x, x, x, x, z\}$, $\ldots \}$ etc. The set, $S'$, of n-multisets of existentially and universally quantified variables in sort metarules of length 3 in the language ${\mathscr {M}}^{3}_{2}$ is $\{ \{P, P, P, x, x, x, x, x, x\}, \{P, P, P$, $y, y, x, x, x, x\}, \{P, P, P, y, x, y, x, x, x\}, \{P, P, P$, $y, x, x, y, x, x\}, \ldots \}$ etc.

Lemma 9

(Number of metasubstitutions) Let $p = {\mathcal {P}}$, $c = |{\mathcal {C}}|$ and let n be as in Lemma 8. The number of metasubstitutions of sort metarules of length k is less than $p^kc^n$.

Proof

Let h be the number of predicate symbols in the heads and b the number of predicate symbols in literals in the body, of all metasubstitutions of a sort metarule of length k, and let e be as in Lemma 8. Let $\{H,B_1,\ldots,B_{k-1},c_1,\ldots,c_{e-k}\}$ be the e-tuple of a predicate symbol H substituting the existentially quantified second-order variable in the head literal, predicate symbols $B_i$ substituting existentially quantified second-order variables in the body literals, and constants $c_j$ substituting existentially quantified first-order variables in all literals, in a sort metarule of length k. There exist $hb^{k-1}c^{e-k}$ such e-tuples. $hb^{k-1}$ is at most $p^k$ as when all symbols in ${\mathcal {P}}$ are of target predicates. $c^{e-k}$ is always less than $c^n$ because at least k existentially-quantified variables in a sort metarule of length k must be second-order. Therefore, $hb^{k-1}c^{e-k}$ is less than $p^kc^n$. $\square$

Example 7

Suppose $k = 3, e = 4$, $H \in \{p\}$, $B_i \in \{q,r\}$, ${\mathcal {C}}$ $= \{a,b,c\}$. The set of 4-tuples of predicate symbols and constants in metasubstitutions of sort metarules of length 3 with one existentially quantified first-order variable is: $\{ \{p,q,q,a\}$, $\{p,q,q,b\}$, $\{p,q,q,c\}$, $\{p,q$,$r,a\}$, ..., $\{p,r,r,a\}$, $\{p,r,r,b\}$, $\{p,r,r,c\}\}$.

Observation 2

Lemma 9 is a refinement of earlier results by Lin et al. (2014); Cropper and Tourret (2018) who calculate the cardinality of the set of metasubstitutions of a single sort metarule as $p^k$ (or $p^3$ for $H^2_2$ metarules). Our result takes into account, firstly the restriction that only symbols in $E^+$ and I can be substituted for second-order variables in the heads of sort metarules, and secondly the possible metasubstitution of existentially quantified first-order variables by constants, neither of which is considered in the earlier results.

Lemma 10

(Number of ground clauses) Let n, c be as in Lemmas 8and 9. The number of ground substitutions of the universally quantified variables in a sort metarule is less than $c^n$.

Proof

Let $\{v_1,\ldots,v_c\}$ be the c-tuple of constants substituted for u universally quantified variables. There are $c^u$ such u-tuples. In a ground substitution of the universally quantified variables in a sort metarule, $c^u$ is always less than $c^n$ because $n = e+u$, where e is as in Lemma 8, and there are exactly $k > 0$ existentially quantified second-order variables in a sort metarule of length k, therefore $e > 0$. $\square$

Example 8

Let $u = 3,\; {\mathcal {C}} = \{a,b,c\}$. The set of 3-tuples of constants substituted for 3 universally quantified variables is $\{\{a,a,a\}$, $\{a,a,b\}$, $\{a,a,c\}$, $\{a,b,a\}$, $\ldots, \{c,c,b\}, \{c,c,c\} \}$.

Note 4

While we have derived exact results in the proofs of Lemmas 9 and 10 we have chosen to state these two Lemmas in terms of upper bounds in the interest of simplifying notation, particularly the notation of Theorem 4.

Theorem 4

(Cardinality of metarule languages) Let k, a, n, p, c be as in Lemmas 6–10. The number of vl- specialisations of punch metarules of length in [1,  k] is at most:

$$\begin{aligned} \sum _{i=1}^{k} \dfrac{ia^i (2n-1)^n p^ic^{2n}}{i!n!} \end{aligned}$$

Proof

By Lemma 6 there are k punch metarules in the language $M^{k}_{}$. The cardinality of the set of vl/specialisations of the k punch metarules in the language $M^{k}_{}$ is the sum of the cardinalities of the sets of vl/specialisations of punch metarules in each language $M^{i}_{}$, where $i \in [1,k]$.

Let $i \in [1,k]$. By Lemma 7 there exist at most $i(a^i/i!)$ matrix metarule specialisations of a punch metarule with i body literals. By Lemma 8, there exist fewer than $(2n-1)^n/n!$ sort metarule specialisations of each such matrix metarule. By Lemma 9 there exist fewer than $p^kc^n$ metasubstitutions of each such sort metarule. By Lemma 10 there exist fewer than $c^n$ ground first-order clause specialisations of each such metasubstitution.

Thus, the cardinality of the set of vl-specialisations of punch metarules in the language $M^{k}_{}$ is at most the sum for all $i \in [1,k]$ of the product $i(a^i/i!) ((2n-1)^n/ n!) p^ic^nc^n$. We may rewrite this product as the fraction $\frac{ia^i (2n-1)^n p^ic^{2n}}{i!n!}$. $\square$

Corollary 2

Each metarule language ${\mathscr {M}}^{l}_{a}$ is enumerable in time polynomial to the number of literals in the most general metarule in ${\mathscr {M}}^{l}_{a}$, i.e. l.

5 Implementation

We have created a prototype, partial implementation of Algorithms 2 and 3 in Prolog, as a new module added to Louise.⁵ The implementation is partial in that it performs only v-specialisation of punch and matrix metarules, but not l-specialisation. For clarity, we will refer to this new module as TOIL (an abbreviation of Third Order Inductive Learner). We now briefly discuss TOIL but leave a full description for future work, alongside a complete implementation.⁶

We distinguish punch metarule specialisation in TOIL as TOIL-3 and matrix metarule specialisation as TOIL-2. Both sub-systems are implemented as variants of the Top Program Construction algorithm in Louise. Each subsystem takes as input a MIL problem with punch or matrix metarules, respectively for TOIL-2 and TOIL-3, instead of sort metarules, and outputs a set of sort metarules.

According to line 5 of Procedure VL-SPECIALISE in Algorithm 2, both sub-systems test that a ground instance $M \vartheta \varTheta$ of an input metarule M is fully/connected before passing it to their implementation of Procedure LIFT. To do so, TOIL-2 maintains a “substitution buffer”, S, of tuples $c \mapsto k$ where each c is a constant and each k is the number of first-order variables in M substituted by c. If, when line 5 is reached, S includes any tuples where $k = 1$, $M \vartheta \varTheta$ is not fully/connected. S is first instantiated to the constants in an input example. When a new literal L of M is specialised, the first-order variables in L are first substituted for constants in S, ensuring that L is connected to literals earlier in M. L is then resolved with $B^*$. If resolution succeeds, a “look-ahead” heuristic, listed in Algorithm 4, attempts to predict whether the now fully-ground L allows a fully/connected instantiation of M to be derived. If so, S is updated with the new constants derived during resolution and the new counts of existing constants. If not, the process backtracks to try a new grounding of L.

TOIL-3 restricts instantiation of punch metarule literals to the set ${\mathcal {A}}_{\mathring{M}, B}$, of matrix metarule literals unifiable with the heads of clauses in $B^*$ (${\mathcal {A}}_{\mathring{M}, B}$ is generated automatically by TOIL-3). Because atoms in ${\mathcal {A}}_{\mathring{M}, B}$ are non-ground, it is not possible to apply the look-ahead heuristic employed in TOIL-2; TOIL-3 only uses the substitution buffer to ensure derived metarules are fully/connected.

Theorem 4 predicts that metarule languages are enumerable in polynomial time, but generating an entire metarule language is still expensive—and unnecessary. To avoid over-generation of metarule specialisations, TOIL limits the number of attempted metarule specialisations, in three ways: (a) by sub-sampling, i.e. training on a randomly selected sample of $E^+$; (b) by directly limiting the number of metarule specialisation attempts; and (c) by a cover-set procedure that removes from $E^+$ each example entailed by the last derived specialisation of an input metarule, before attempting a new one.

TOIL cannot directly derive sort metarules with existentially quantified first-order variables. These must be simulated by monadic background predicates representing possible theory constants, e.g., pi(3.14), e(2.71), g(9.834), c(300000), etc.

We leave a formal treatment of the properties of the connectedness constraints and specialisation limits described above to the aforementioned future work.

6 Experiments

A common criticism of the MIL approach is its dependence on user-defined metarules. In this Section we show experimentally that automatically derived fully-connected sort metarules can replace user-defined fully-connected sort metarules as shown in Sect. 4.8, thus addressing the aforementioned criticism. We formalise our motivation for our experiments as Experimental Hypotheses 1 and 2.

Experimental Hypothesis 1

Metarules learned by TOIL can replace user/defined metarules without decreasing Louise’s predictive accuracy.

Experimental Hypothesis 2

Metarules learned by TOIL can replace user/defined metarules without increasing Louise’s training time.

6.1 Experiment setup

We conduct a set of metarule replacement experiments where an initial set, ${\mathcal {M}}$, of user-defined, fully-connected sort metarules are progressively replaced by metarules learned by TOIL⁷.

Each metarule replacement experiment proceeds for $k = |{\mathcal {M}}| + 1$ steps. Each step is split into three separate legs. We repeat the experiment for $j = 10$ runs at the end of which we aggregate results. Each leg is associated with a new set of metarules: ${\mathcal {M}}_1$, ${\mathcal {M}}_2$ and ${\mathcal {M}}_3$ for legs 1 through 3, respectively. At the start of each run we initialise ${\mathcal {M}}_1$ to ${\mathcal {M}}$, and ${\mathcal {M}}_2$, ${\mathcal {M}}_3$ to $\emptyset$. At each step i after the first, we select, uniformly at random and without replacement, a new user-defined metarule $M_i$ and set ${\mathcal {M}}_1 = {\mathcal {M}}_1 \setminus \{M_i\}$, leaving $k-i$ metarules in ${\mathcal {M}}_1$. Thus, at step $i = 1$, ${\mathcal {M}}_1 = {\mathcal {M}}$ while at step $i = k$, ${\mathcal {M}}_1 = \emptyset$. In each step i we train TOIL-2 and TOIL-3 with a set of matrix or punch metarules (described in the following section), respectively, then we replace all the metarules in ${\mathcal {M}}_2$ with the output of TOIL-2 and replace all the metarules in ${\mathcal {M}}_3$ with the output of TOIL-3 (in other words, we renew ${\mathcal {M}}_2$ and ${\mathcal {M}}_3$ in each step). Then, in leg 1 we train Louise with the metarules in ${\mathcal {M}}_1$ only; in leg 2 we train Louise with the metarules in ${\mathcal {M}}_1 \cup {\mathcal {M}}_2$; and in leg 3 we train Louise with the metarules in ${\mathcal {M}}_1 \cup {\mathcal {M}}_3$. As to examples, at each step i we sample at random and without replacement 50% of the examples in each of $E^+$ and $E^-$ as a training partition and hold the rest out as a testing partition. We sample a new pair of training and testing partitions in each leg of each step of each run of the experiment and perform a learning attempt with Louise on the training partition. We measure the accuracy of the hypothesis learned in each learning attempt on the testing partition, and the duration of the learning attempt in seconds. We measure accuracy and duration in two separate learning attempts for each leg. In total we perform (10 runs * $|{\mathcal {M}}|$ steps * 3 legs * 2 measurements) distinct learning attempts, each with a new randomly chosen training and testing partition. We set a time limit of 300 sec. for each learning attempt. If a learning attempt exhausts the time limit we calculate the accuracy of the empty hypothesis on the testing partition. Finally, we return the mean and standard error of the accuracy and duration for the learning attempts at the same step of each leg over all 10 runs.

We run all experiments on a PC with 32 8-core Intel Xeon E5-2650 v2 CPUs clocked at 2.60 GHz with 251 Gb of RAM and running Ubuntu 16.04.7.

6.2 Experiment datasets

Table 7 Dataset summary. |B|: number of predicates defined in background knowledge. $|{\mathcal {M}}|$: starting number of sort metarules

We reuse the datasets described in Patsantzis and Muggleton (2021). These comprise: (a) Grid World, a grid-world generator for robot navigation problems; (b) Coloured Graph, a generator of fully-connected coloured graphs where the target predicate is a representation of the connectedness relation and comprising four separate datasets with different types of classification noise in the form of misclassified examples (false positives, false negatives, both kinds and none); and (c) M:tG Fragment, a hand-crafted grammar of the Controlled Natural Language of the Collectible Card Game, “Magic: the Gathering” where examples are strings entailed by the grammar. Table 7 summarises the MIL problem elements of the three datasets. We refer the reader to Patsantzis and Muggleton (2021) for a full description of the three datasets.

Instead of the metarules defined in the experiment datasets we start each experiment by initialising ${\mathcal {M}}$ to the set of 14 $H^2_2$ metarules in Cropper and Muggleton (2015), listed in Table 2, which we call the canonical $H^2_2$ set. We replace them with specialisations of the matrix metarules Meta-dyadic and Meta-monadic from Table 3 and the punch metarules TOM-3 for M:tG Fragment or TOM-2, TOM-3, from Table 4 otherwise. We limit over-generation in metarule specialisation as described in Sect. 5 by limiting metarule specialisation attempts to 1 for M:tG Fragment; and sub-sampling 50% of $E^+$ at the start of each metarule learning attempt, for Grid World and Coloured Graph.

Our configuration of the three experimental datasets is identical to that in Patsantzis and Muggleton (2021) with the exception of the Grid World dataset, which we configure to generate a grid world of dimensions $3 \times 3$. The resulting learning problem is trivial, but hypotheses learned with metarules derived by TOIL for worlds of larger dimensions tend to be extremely large (hypothesis cardinalities upwards of 6000 clauses are logged in preliminary experiments), consuming an inordinate amount of resources during evaluation. By comparison, Patsantzis and Muggleton (2021) report a hypothesis of 2567 clauses for a $5 \times 5$ world (as in our preliminary experiment). This observation indicates that future work must address over-generation by TOIL. Still, the size of the learned hypotheses serves as a stress test for our implementation.

6.3 Experiment results

Fig. 1

Experiment results measuring accuracy. User-defined sort metarules are progressively reduced throughout the experiment. Red circles: no replacement. Blue triangles: user-defined sort metarules are replaced by specialisations of matrix metarules derived by TOIL-2. Magenta crosses: user-defined sort metarules are replaced by specialisations of punch metarules derived by TOIL-3. Brown dotted line: baseline (empty hypothesis). x-axis: number of remaining sort metarules; y axis: accuracy on testing partition (50% of $E^+,E^-$). Error bars: standard error

Fig. 2

Experiment results measuring running times. Lines, axes and error bars as in Fig. 1. No baseline for timing experiments

Figure 1 lists the results of the experiments measuring predictive accuracy. We immediately observe that in the two legs of the experiment where user-defined metarules are replaced by learned metarules, marked by “TOIL-2” and “TOIL-3”, Louise’s accuracy is maintained, while it degrades in the leg of the experiment where metarules are reduced without replacement, marked “No replacement”. These results support Experimental Hypothesis 1.

Figure 2 lists the results of the six experiments measuring training times. We observe that training times for the “No replacement” leg of the experiments decrease as the number of user-defined metarules decreases, but remain more or less constant for the other two legs as removed metarules are replaced by metarules learned by TOIL-2 and -3. In the M:tG Fragment dataset, all but a single metarule in the canonical set, the Chain metarule, are redundant. TOIL-2 and -3 only learn the Chain metarule in their two legs of the experiment and so, as redundant metarules are removed and not replaced, training times decrease in all three legs of that experiment. These results support Experimental Hypothesis 2.

6.3.1 Learned metarules

Table 8 Metarules learned by TOIL-3 for the Coloured Graph - False Positives experiment collected from logging output during execution; these are exactly the 14 Canonical $H^2_2$ metarules in Table 2

During our experiments, the metarules learned by TOIL are logged to the command line of the executing system. We could thus examine and will now discuss examples of the metarules learned during execution.

For the M:tG Fragment dataset, we observed that TOIL-2 and -3 both learned a single metarule, the Chain metarule listed in Table 2. The target theory for M:tG Fragment is a grammar in Definite Clause Grammar form (Colmerauer, 1978; Kowalski, 1974), where each clause is indeed an instance of Chain. For this dataset, TOIL was able to learn the set of metarules that would probably also be chosen by a user.

For the Grid World dataset, TOIL-2 and -3 both learned a set of 22 $H^2_2$ metarules including the canonical set and 4 metarules with a single variable in the head literal, e.g. $P(x,x)\leftarrow Q(x,y), R(x,y)$ or $P(x,x)\leftarrow Q(x,y), R(y,x)$ that are useful to represent solutions of navigation tasks beginning and ending in the same “cell” of the grid world. Such metarules may be seen as over-specialisations, but they are fully-connected sort metarules which suggests that the constraints imposed on metarule specialisation to ensure only fully-connected metarules are returned, described in Sect. 5, are correctly defined.

Table 8 lists metarules learned by TOIL-2 and -3 for the Coloured Graph - False Positives dataset. The learned metarules are exactly the set of 14 Canonical $H^2_2$ metarules in Table 2. Cropper and Muggleton (2015) show that the 14 canonical $H^2_2$ metarules are reducible to a minimal set including only Inverse and Chain, therefore returning the entire canonical set is redundant. This is an example of the over-generation discussed in Sect. 5, that TOIL attempts to control by limiting the number of attempted metarule specialisations. Logical minimisation by Plotkin’s program reduction algorithm, as described by Cropper and Muggleton (2015), could also be of help to reduce redundancy in an already-learned set of metarules, although TOIL may be overwhelmed by over-generation before reduction has a chance to be applied. In any case, over-generation is a clear weakness of our approach and must be further addressed by future work.

7 Conclusions and future work

7.1 Summary

We have presented a novel approach for the automatic derivation of metarules for MIL, by MIL. We have shown that the user-defined fully-connected second-order sort metarules used in the MIL literature can be derived by specialisation of the most-general second-order matrix metarules in a language class, themselves derivable by specialisation of third-order punch metarules with literals that range over the set of second-order literals. We have shown that metarule languages are enumerable in time polynomial to the number of literals in punch metarules. We have defined two methods of metarule specialisation, v- and l- specialisation and shown that they are performed by MIL. We have proposed a modification of the MIL clause construction operator to return fully connected second-order sort metarules, rather than first-order clauses and proved its correctness. We have partially implemented the modified MIL operator as TOIL, a new sub-system of the MIL system Louise, and presented experiments demonstrating that metarules automatically derived by TOIL can replace user-defined metarules while maintaining predictive accuracy and training times.

7.2 Future work

Table 9 This table illustrates the use of metarules as a data structure for transferring generalisations between problems

The major practical limitations of our approach are the incomplete state of its implementation and its over-generation of metarule specialisations.

Our prototype implementation of Algorithms 2 and 3 in TOIL is only capable of v-specialisation. Work is under way to complete the implementation with the capability for l-specialisation. In Sect. 5, we have left a formal treatment of TOIL to the time this work is complete. Further work would improve the look-ahead heuristic in Algorithm 4 and our ability to limit attempted metarule specialisations to reduce over-generation. In general, we do not know of a good, principled (as in non-heuristic) and efficient approach to derive just enough metarules to solve a problem, without deriving too many and over-generalising.

Conversely to over-generation, TOIL also exhibits a tendency to produce metarules that are over-specialised to the examples in a MIL problem, a form of over-fitting. This seems to be a limitation of TOIL’s look-ahead heuristic listed in Algorithm 4 and used to ensure learned metarules are fully-connected. Future work should look for a principled approach to replace this heuristic, also.

TOIL-3 is capable of learning new metarules with literals of arbitrary arities, as illustrated in Table 9. We haven’t demonstrated this important ability with experiments. Additionally, we have not presented any empirical results measuring training times for TOIL itself—only for Louise. Theoretical results in Sect. 4.9 predict that learning metarules should be time-consuming, especially for larger metarule languages, and we have observed this while executing the experiments in Sect. 6, although more so for TOIL-3 than TOIL-2.

Our theoretical framework described in Sect. 4 extends $\theta$-subsumption to metarules. It remains to be seen if the related frameworks of relative subsumption and relative entailment (Nienhuys-Cheng & de Wolf, 1997) can also be extended to metarules. v- and l-specialisation seem to be related to Shapiro’s refinement operators (Shapiro, 2004), a point also made about metarules in general by Cropper and Muggleton (2015) but we haven’t explored this relation in this work.

Corollary 1 suggests that classes of learning problems can be solved by the same sets of metarules, as long as suitable solutions belong to the same metarule language. This observation introduces the possibility of transferring generalisations, in the form of learned metarules, across learning problems or problem domains thus in a sense forming analogies, a capability poorly represented in modern machine learning—and, in general, AI/systems (Mitchell, 2021). Such a capability would however rely on a method to determine the relevance of metarules to a problem; currently, no such method is known. Table 9 illustrates the transfer of learned metarules as analogies between problems.

Future work should test the accuracy of the metarules learned by TOIL with other systems that use metarules, besides Louise, for example Metagol (Cropper & Muggleton, 2016b), Popper (Cropper & Morel, 2021) and ALPS (Si et al., 2019).

Appendices

Appendix A: Predicate invention in MIL—full description

In Sect. 4.6 we have given a simplified description of Algorithm 1 omitting the recursive resolution step that takes place during predicate invention. We have done this to simplify the description of the algorithm and to isolate the specialisation operation that is the primary subject of Sect. 4.6. Algorithm 1 is accurate as long as predicate invention is not required. In this Appendix, Algorithm 5 is a more complete description of Algorithm 1 that includes recursion and predicate invention. Similarly, Algorithm 6 is a more complete description, including the predicate invention step, of Algorithm 2. In our implementation of TOIL the propagation of meta/substitution $\vartheta \varTheta$ in line 11 of Algorithms 5, 6 is handled by the Prolog engine.

Appendix B: An example of metarule specialisation

Table 10 Example of learning with metarules learned by TOIL

Table 10 illustrates the use of TOIL to learn metarules for Louise. In table section (A) the elements of a MIL problem are defined. In table section (B) a set of matrix metarules ${\mathcal {M}}_1$ and a set of punch metarules ${\mathcal {M}}_2$ are defined, each with a single member. In row (c) TOIL-2 learns a new fully-connected sort metarule from the elements of the MIL problem in table section (A) and the matrix metarule in ${\mathcal {M}}_1$. In row (d) TOIL-3 learns a new fully-connected sort metarule from the elements of the MIL problem in table section (A) and the punch metarule in ${\mathcal {M}}_2$. Note that both sub-systems of TOIL learn the same fully-connected punch metarule (the $H^2_2$ Chain metarule, listed in Table 2).

In rows (e) and (f) Louise is given the elements of the MIL problem in table section (A) and the metarule learned by TOIL-2 and TOIL-3, and learns the hypothesis starting at row (f). Note that this is a correct hypothesis constituting a grammar of the context-free $a^nb^n$ language.

It is interesting to observe that the program learned by Louise includes a definition of an invented predicate, $\$1$, in row (f). This is despite the fact that our implementation of TOIL does not perform predicate invention and so has not learned any metarules that require predicate invention to be learned. In the MIL problem in Table 10 a single metarule is sufficient to learn a correct hypothesis and this metarule can be learned without predicate invention, even though the correct hypothesis starting in (f) cannot, itself, be learned without predicate invention. This observation suggests that even the current, limited version of TOIL that cannot perform predicate invention, may be capable of learning a set of metarules that is sufficient to learn a correct hypothesis, when given to a system capable of predicate invention, like Louise (or Metagol).

The observation about predicate invention in the previous paragraph further highlights the generality of metarules and suggests the existence of a class of learning problems that can be solved with a number of metarules much smaller than the number of clauses in their target theory, a subject for further study.