Logic Programming and Artificial Intelligence
9 February 2010
LPA Problem Sheet 4
Due: 10:15, 17 February
1. [6 marks] Consider the first-order predicate calculus with the following lexicon:
Predicate symbols: P(with arity 2), and Q(with arity 1);
Function symbols: f(with arity 2) and a(with arity 0);
Variables: x,yand z.
In each of the following formulas draw lines to indicate which quantifier (if any) binds every
occurrence of every variable. Indicate which of these formulas are sentences. (Eg. In ∀x(P(x)→
∀x∀yQ(x, y, x)) you would connect the first occurrence of xto the second occurrence of x, and
you would connect the third, fourth and fifth occurrences of x. You would also connect the two
occurrences of y.)
(a) ∀xP (x, x)
(b) ∃x∀x(Q(x)→Q(x))
(c) ∀x(Q(x)→ ∀xQ(x))
(d) ∀x((Q(x)→ ∀xQ(x)) →Q(x))
(e) ∀x((∃yQ(x)) →Q(y))
(f) ∃x¬P(f(x, y ), a)
2. [17 marks] Let zero be an individual symbol, sbe a function symbol of arity one, and LE be
a predicate symbols of arity two. Let gbe the value assignment that maps xto 1 and yto 2.
Let Ibe a model whose domain is the natural numbers and whose assignment maps
zero to 0
sto λx.x + 1
LE to λx, y.(TRUE if x≤y, FALSE otherwise)
Give the values of each of the following:
(a) [[s(x)]]I,g
(b) [[s(x)]]I,g[y7→7]
(c) [[s(x)]]I,g[x7→7]
(d) [[LE (x, y)]]I,g
(e) [[∃x LE (x, y)]]I,g
(f) [[s(s(s(zero)))]]I
(g) [[LE (zer o, zero)]]I
(h) [[s(z ero) = zero →LE(s(z ero), z ero)]]I
(i) [[∃x z ero =x]]]I
(j) [[∀x∃y y =s(x)]]I
(k) [[∀y∃x y =s(x)]]I
(l) [[∀x∀y LE (x, y)∨LE (y, x)]]I
(m) [[(∀x∀y LE (x, y)) ∨(∀x∀y LE (y, x))]]I
(n) [[∀x∃y LE (x, y)]]I
(o) [[∀y∃x LE (x, y)]]I
(p) [[∃x∀y LE (x, y)]]I
(q) [[∃y∀x LE (x, y)]]I
3. [6 marks] Specify a model that falsifies the sentence (∀x∃yP (x, y)) →(∃y∀xP (x, y )).
4. [12 marks] State whether each of the following claims holds and use the semantic definition of
FOPC to support your answer rigorously.
1
