43 terms

Formulas S entails formula F iff...

all models of S also make F true

a clause

a conjunction (?) of literals

a definite clause

a clause with exactly one non-negated atom: h :- b,b,b.

atom

predicate and terms p(a,b, ...)

term

constant, variable or functor

literal

atom or negated atom

substitution

a mapping from variables to constants

grounding substitution of atom a

a substitution that maps all variables of a to constants

fact

a clause without body (may contain variables !)

program

a set of clauses

background theory or program

a set of background facts

sequential covering algorithm

learn one rule, remove the data it covers, repeat

refinement operator

operator that creates candidate atoms to add to a clause

marginalisation

computing the probability (the marginal) of random variables given some evidence p(a | e)

most probable explanation (MPE)

most likely joint state of the random variables given the evidence

weighted model counting

"counting the number of satisfying model, according to their probability", for marginalisation

weighted max-sat

"maximizing the satisfaction criteria"

weighted boolean formula

"the possible proofs derived from a program"

learning from interpretation

"?"

learning from entailment

"?"

theory in conjunctive normal form (CNF)

a conjunction of disjunctions of literals, i.e. of clauses

propositional logic

a logic where terms are either constants or functors (but not variables)

Herbrand base of a FOL first-order logic theory

all possible ground atoms constructed with the predicates, functors and constants

Herbrand interpretation, or (possible) world

assignment of a truth value to all (ground) atoms in the Herbrand base (consistent with the theory or not)

model of a theory

an Herbrand interpretation (or world) consistent with the theory (all formulas/clauses are true)

conjunction

a formula with AND

disjunction

a formula with OR

normal clause, or rule

universally quantified expression of the form h :- b,b,b.

normal program

a set of normal clauses

definite program

negation-free logic program

LHM Least Herbrand Model of a logic program

the smallest of all Herbrand Models. It contains all ground atoms entailed by the logic program (and no undefined ones)

well-founded semantics

a semantic where atoms can be true, false or undefined ("bottom")

closed world assumption

everything that is not implied to be true is assumed to be false

intensional probabilistic fact

probabilistic clause used to define the domains of the variables of a predicate

probabilistic atom

an atom that unifies with a ground probabilistic fact (opposite: derived atom)

derived atom

an atom that unifies with the head of some rule (opposite: probabilistic atom)

range-restricted clause

all variables in the head should appear in a positive literal in the body

atomic choice

"the choice to include (or not) a ground probabilistic fact (in a possible world)"

total choice

any subset of the set of all ground probabilistic fact

Example of Least Herbrand Model

FOL theory {a ← b} has 3 models {¬a,¬b}, {a,¬b},{a, b}.

LP {a :- b} has only one model {¬a,¬b}

LP {a :- b} has only one model {¬a,¬b}

ProbLog program

ground probalistic fact + intensional probabilistic facts + logic program (for derived atoms)

well-formed formula

all variables in the formula are quantified

A set of formulas S is inconsistent iff...

A set of formulas S has no model