Due Monday, April 9, at 1 p.m.
(30%) 1. (i) Let A be a w and assume that the only connectives appearing in A are among
, , (i.e., , dont appear). Let A be obtained from A by replacing each propositional variable p appearing in A by
Due Wednesday, May 30 at 1 p.m.
(40%) 1. Show that
x(P (x) Q(x) (xP (x) xQ(x)
x(P (x) xP (x)
(c) cfw_P (x), y (P (y ) zQ(z )
P (x) y (x = y P (y )
x(A B ) xA xB
Here P, Q are unary relation symbols; A, B arbitr
Due Monday, May 21 at 1 p.m.
(10%) 1. Find formulas in prenex normal form logically equivalent to the following (here P, Q, R
are relation symbols).
(a) x[zP (x, y, z ) y (Q(y ) R(z )]
(b) xy [R(z ) z w(P (x, u, z ) uQ(u, w)]
Due Monday, May 14 at 1 p.m.
(20%) 1. Call a set X N eventually periodic if there is n0 N and p N, p 1 (a period)
such that for all n n0 ,
n X i n + p X.
Show that every eventually periodic set is rst-order denable in the structure
Due Monday, April 2 at 1 p.m.
(40%) 1. (a) Prove the correctness of the following algorithm for recognizing when a given string
S is a P -w:
If S = s1 s2 . . . sn , compute w(sn ), w(sn ) + w(sn1 ), . . . , w(sn ) + w(sn1 ) + + w(s1
Due Monday, April 23 at 1 p.m.
(35%) 1. Study the attached handout describing an algorithm for transforming a w A to a cnf
formula B , so that
A is satisable i B is satisable,
and prove the correctness of this algorithm.
(30%) 2. (i
Due Monday, May 7 at 1 p.m.
(20%) 1. Prove that a proper initial segment of a formula in rst-order logic is not a formula.
(10%) 2. (a) Write a sentence A6 in the language with no non-logical symbols (but recall that
= is available)
Due Monday, April 16 at 1 p.m.
(35%) 1 . (i) Prove that cfw_, , cfw_, are not complete.
(ii) Prove that |, are the only complete binary connectives.
(iii) Prove that, If. . . , then. . . , else. . . is not complete, but if we add t
Due Thursday, April 30 at 1 p.m.
IN ALL PROBLEMS BELOW YOU CANNOT USE THEOREM 1.11.5 IN THE NOTES
(since the exercises below form parts of the proof of that theorem). ALSO FORMULA
OR WFF MEANS WELL-FORMED FORMULA IN PROPOSITIONAL LO
Due Monday, June 4 at 1 p.m.
Note that all problems below are not starred. So you can work with other students for
each problem (but of course you should write up your own paper).
(50%) 1. Consider Turing machines on the