Raisonnement Automatisé: Principes Et Applications (partie I: Logique Du Premier Ordre) 1 First Order Logic

2009 This document contains the first part of the M2R course RAPA: Automated Reasoning: Principle and Applications. It presents the basis of first-order logic and automated deduction: syntax, semantics, transformation into clausal form, unification and the Resolution calculus (with selection functions and atom ordering). Some basic properties of the Resolution calculus are also investigated (w.r.t. complexity and termination). This document is self-contained but additional references are provided for the interested reader. More details and additional explanations can be found in [5, 6]. [8] is an advanced textboook on the Resolution calculus and the Handbook of Automated Reasoning [12] covers the main lines of research in this field. First-order logic (FOL) is a formal language for expressing properties. Propo-sitional logic allows one to express basic statements (s.t. " Paris is a town " or " Berlin is a town " or " Paris is the capital of France ") and to combine them with logical connectives: ¬ (not), ∨ (or), ∧ (and), ⇒ (implies) and ⇔ (equivalence). First-order logic extends this language by using predicate symbols and quantifi-cation over individuals. For instance, the property " to be a town " may be expressed by a predicate symbol Town, which can be applied to different individuals: Town(Paris), Town(Berlin),. .. Using quantification, it is possible to express the property: " all countries have a capital " : ∀x[Country(x) ⇒ ∃yCapital(y, x)] (meaning: " for every x, if x is a country, then there exists a y such that y is the capital of x "). However, it is not possible in first order logic to express quantification over sets of individuals or over functions. For instance, the induction principle is not expressible in first-order logic: ∀P [P (0) ∧ ∀xP (x) ⇒ P (succ(x))] ⇒ ∀xP (x) is not a sentence of FOL, due to the quantification over the sets of natural numbers P. Similarly, the property ∀f ∃xf (x) = x (every function has a fixpoint) is not expressible in FOL.