Inverse entailment and progol

Mode declarations are a popular language bias.

“Mode declarations (Muggleton 1995) are a popular language bias (Blockeel and De Raedt 1998; Srinivasan 2001; Ray 2009; Corapi et al. 2010, 2011; Athakravi et al. 2013; Ahlgren and Yuen 2013; Law et al. 2014).”

Atom struggles to learn recursive programs because it needs examples of both the base and step cases of a recursive program.

“However, because it builds on Progol Muggleton (1995), and thus employs inverse entailment, Atom struggles to learn recursive programs because it needs examples of both the base and step cases of a recursive program.”

Many ILP systems are not guaranteed to learn optimal solutions, where optimal typically means the smallest program or the program with the minimal description length.

“Many ILP systems (Muggleton 1995; Srinivasan 2001; Blockeel and De Raedt 1998; Ray 2009) are not guaranteed to learn optimal solutions, where optimal typically means the smallest program or the program with the minimal description length.”

Many popular ILP systems, such as FOIL, Progol, TILDE, and Aleph struggle to learn recursive programs.

“Indeed, many popular ILP systems, such as FOIL (Quinlan 1990), Progol (Muggleton 1995), TILDE (Blockeel and De Raedt 1998), and Aleph (Srinivasan 2001) struggle to learn recursive programs.”

Many ILP systems allow a user to constrain the hypothesis space though clause constraints.

“Many ILP systems allow a user to constrain the hypothesis space though clause constraints (Muggleton 1995; Srinivasan 2001; Blockeel and De Raedt 1998; Ahlgren and Yuen 2013; Law et al. 2014).”

Some approaches combine top-down and bottom-up search.

“Some approaches combine the two (Muggleton, 1995; Srinivasan, 2001; Cropper, 2022).”

Some ILP systems, such as Progol, allow for clauses as examples.

“Some systems, such as Progol (Muggleton, 1995), allow for clauses as examples.”

Many systems employ the covering algorithm whereby hypotheses are constructed iteratively clause by clause.

“The above paragraph refers only to generality orders on single clauses, as many systems employ the covering algorithm whereby hypotheses are constructed iteratively clause by clause (Quinlan, 1990; De Raedt & Dehaspe, 1997; Muggleton, 1995; Blockeel & De Raedt, 1998; Srinivasan, 2001).”

Most ILP approaches use an Occamist bias to find the smallest hypothesis, measured in terms of the number of clauses, literals, or description length.

“Most approaches use an Occamist bias to find the smallest hypothesis, measured in terms of the number of clauses (Muggleton et al., 2015), literals (Law et al., 2014), or description length (Muggleton, 1995).”

A common limitation is that many systems rely on bottom clause construction.

“A common limitation is that many systems rely on bottom clause construction (Muggleton, 1995), which we discuss in detail in Section 6.1.”

Aleph is based on Progol.

“Aleph is based on Progol.”

Aleph is based on inverse entailment and uses bottom clause construction to restrict the hypothesis space.

“Aleph is based on inverse entailment (Muggleton, 1995) and uses bottom clause construction to restrict the hypothesis space.”

An ILP system can trivially construct the bottom clause for each example.

“For instance, an ILP system can trivially construct the bottom clause (Muggleton, 1995) for each example.”

Progol and Aleph could learn higher-order programs such as f(A,B):-map(A,B,f1) given appropriate mode declarations, but because they do not support predicate invention, they would be unable to invent f1/2.

“Given appropriate mode declarations (Muggleton 1995) for higher-order predicates such as map/3 , Progol and Aleph could learn higher-order programs such as f(A,B):-map(A,B,f1) . However, because Progol and Aleph do not support predicate invention, they would be unable to invent the predicate f1/2 in the above example.”