Thelanguage ofpredicate calculus asdefinedinLecture11did not consider variables orquantifiers. Thislectureisconcerned with first-orderlogic, whichincorporatesquantification overobjectsusing variables as placeholders. Higher-order logics allow for quantification over functions or relations.
The universal quantifier ∀ can be interpreted as conjunction, with a possibly infinite number of conjuncts. In finite domains, universal quantification and conjunction are equivalent: e.g., in the finite domain D = {sky, mountains}, the following formula holds: ∀x blue(x)↔ blue(sky)∧ blue(mountains). Butininfinitedomains,universalquantification extends conjunction: e.g., the following formula holds in the infinite domain D = Z: ∀x odd(x)∨ even(x).
Analogously, the existential quantifier ∃ can be interpreted as disjunction, with a possibly infinite number of disjuncts. As above, existential quantification and disjunction are equivalent in finite domains: e.g., in domain D = {sky, grass}, the following formula holds: ∃x green(x)↔ green(sky)∨ green(grass). But in infinite domains, existential quantification extends disjunction: e.g., the following formula holds in the infinite domain D = N: ∃x, ∀y ¬succ(x, y).
First-order logic generalizes predicate calculus with variables and quantifiers. Like propositional logic and predicate calculus, a set of syntactic rules govern the construction of terms and formulas in first-orderlogic. Thealphabet of first-orderlogicextendsthealphabet ofpredicatecalculuswith the quantification symbols {∀, ∃} and the variables {x, y, z, . . .}.
The following definition extends the inductive definition of the terms of predicate calculus to first-order logic:
• variables x, y, z, . . . are terms
Similarly, the following definition extends the inductive definition of the formulas of predicate calculus to first-order logic:
• if x is a variable and φ is a formula, then
The semantics of first-order logic associate meanings with formulas by extending the notion of interpretationinpredicate calculus(andpropositionallogic)to adomain-mapping-assignment triple �D, M, α�, where α mapsvariablestoobjectsinthedomain. In first-orderlogic,themappingfrom termsto objectsisdefined(inductively) asfollows:
• for all variables x, I[x]= α(x)
M
– I[f(t1,...,tn)]= fM(I[t1],..., I[tn])
The definition of entailment in first-order logic extends that of the predicate calculus. For atomic formulas P (t1,...,tn), as before,
• I|= P (t1,...,tn)iff(I[t1],..., I[tn])∈ P M
For complext first-order formulas other than those that involve quantification, the definition is as it was before: e.g., �I,α�|= φ ∨ψ iff �I,α�|= φ or I|= ψ. In the remaining cases, the definition is as follows:
Here I{d/x} is an abbreviation for �D, M, α{d/x}�, where the assignment α{d/x} maps x to d, but otherwise agrees with α: i.e.,
α(y) ify �= x
α{d/x}(y)=
d otherwise Itisstraightforward toextend thenotionsof satisfiability, validity, and unsatisfiability to first-order logic.
Example: Given the infinite alphabet A = {add, mult, =, 0, 1, <, x1 ,x2,...}, consider the interpretation N, where the domain D = N, the nonvariable symbols in A are interpreted as usual, and α(xn) =2n, for all n ∈ N. Under this interpretation, φ ≡ mult(x1, add(x2,x3)) = x4 signifies 2(4+6) =8. Of course, N �|= φ. On the other hand, N|= mult(x1,x2)= x4.
A substitution is a function from variables to terms: i.e., σ = {t1/x1,...,tn/xn}, where each xi is a variable and each ti is a term. The set of variables {x1,...,xn}is the domain of σ; the set of terms {t1,...,tn}is the range of σ.
Before we can define the substitution operation, that is, the result of applying the substitution σ to the formula φ (abbreviatedφ|σ), we needsome terminology:
Now, the substitution operation is subject to the following two constraints:
Note that variable renaming does not change the meaning of formulas: e.g., ∃yy + y = 0 ≡∃zz + z = 0. Hence, to preserve meaning, first rename any bound variables whose names appear in the range of a substitution: e.g., φ ′ ≡∃zz + z = x|{y/x}, which reduces to ∃zz + z = y.
These constraints ensure soundness: i.e., no change in meaning. This property is stated formally in the following theorem, known as the Substitution Lemma.
Theorem: For all formulas φ, I|= φ|{t1 /x1 ,...,tn /xn} iff I{t1/x1,...,tn/xn}|= φ.
Given two substitutions σ and τ, sequential substitution στ is achieved via composition of substitutions, defined as follows: φ|στ =(φ|σ)|τ. For example, if σ = {f(y)/x} and τ = {b/y}, then R(x, y)|στ = R(f(b),b). But if θ = {f(y)/x, b/y}, then R(x, y)|θ = R(f(y),b)�R(f(b),b). Note
= that if σ binds all free variables to ground terms, then e|σσ′ = e|σ, for all substitutions σ ′ and all expressions e: e.g., if σ = {f(0)/x, 0/y}, then R(x, y)|στ = R(f(0), 0).
A variant of Herbrand’s theorem holds for universal logic, that is, first-order logic formulas with universal quantifiers only. But Herbrand’s theorem does not hold for formulas of first-order logic with existential quantifiers.
Counterexample: Consider the pair of sentences ∃xP (x) and ¬P (a). These sentences are sat-
M
isfiable: e.g., D = {◦, ⋄}, a= ◦, and P M = {⋄}. But these sentences do not have a Herbrand model. The Herbrand universe A = {a}and the Herbrand base B = {P (a)}. Thus, there are only two Herbrand models: P = ∅ and P = A. But neither of these models satisfy thepairof sentences.
In other words, there is no known semantic technique for deciding logical entailment. This is not surprising, as logical entailment in first-order logic is not decidable, meaning there is no effective procedure(of any complexity),and thereneverwillbe,thatcananswerthequestion “isformula φ entailed by a knowledge base KB?”
However, logical entailment in first-order logic is semi-decidable. That is, there do exist effective procedures that correctly conclude that a formula φ is entailed by a knowledge base KB when in fact thisis the case; butwhenKB �|= φ, suchprocedures need not terminate. The only semi-decision procedures known for first-order logic are proof-theoretic.
Additional rules of natural deduction that extend the rules for propositional logic to quantified formulas Table 1. The following restrictions apply:
Gentzen’s Soundness Theorem First-order logic is sound. G¨odel’s Completeness Theorem First-order logic is complete. Natural deduction is a sound and complete proof theory for first-order logic.
GENERALIZED MODUS PONENS(GMP)
Introduction Elimination
φ|{a/x} ∀xφ
(∀I) (∀E)
∀xφ φ|{t/x}
[φ|{a/x}]
.
.
.
.
φ|{t/x} ∃xφ ψ
(∃I) (∃E)
∃xφ ψ
Table1: Rules ofNaturalDeductionforQuantifiedFormulas.
if τ|σ = φi|σ, then
φ1 ∧... ∧φm → ψχ1 ∧... ∧χn → τ
(GMP)
(φ1 ∧... ∧φi−1 ∧χ1 ∧... ∧χn ∧φi+1 ∧... ∧φm → ψ)|σ
Theorem: GMP is sound. Theorem: GMP is complete for Horn databases. Remark: Not all knowledge bases are convertible to Horn databases.
GENERALIZED RESOLUTION(R)
if τj|σ = φi|σ, then
φ1 ∧... ∧φm → ψ1 ∨... ∨ψn χ1 ∧... ∧χk → τ1 ∨... ∨τl
(R)
(φ1 ∧... ∧φi−1 ∧χ1 ∧... ∧χk ∧φi+1 ∧... ∧φm → ψ1 ∨... ∨ψn ∨τ1 ∨... ∨τj−1 ∨τj+1 ∨... ∨τl)|σ
Theorem: Resolution is sound. Theorem: Resolution is refutation complete for normal form knowledge bases. Theorem: All knowledge bases are convertible to normal form knowledge bases.