CS 476 Homework #2 Due 10:45am on 9/21
Note: Answers to the exercises listed below should be handed to the instructor in hardcopy and in typewritten
form (latex formatting preferred) by the deadline mentioned above. You should also email to ([email protected])
the Maude code for the exercises requiring that.
1. Solve Ex2.2 in Lecture 2.
2. The theory of groups has one sort, say Group, a constant 1, a unary function ( )−1and a binary function ◦
and has equations:
•x◦1 = x
•x◦(y◦z) = (x◦y)◦z
•x◦(x)−1= 1
Use the rules of inference of Lecture 4 to give a detailed formal proof of each of the following theorems of group
theory:
•(∀x) 1 ◦x=x
•(∀x) (x)−1◦x= 1
•(∀x, y) (x◦y)−1=y
−1◦x
−1
3. Solve Exercise 89 (pg. 122) in Peter ¨
Olveczky’s Lecture Notes. Include your solution and evaluation of some
4. Consider the following module, defined for computing the smallest element of a multiset of numbers:
fmod NAT-MSET-MIN is
sorts Nat NatMSet .
subsort Nat < NatMSet .
op 0 : -> Nat [ctor] .
op s : Nat -> Nat [ctor] .
op _ _ : NatMSet NatMSet -> NatMSet [assoc comm ctor] .
op _<_ : Nat Nat -> Bool .
op min : NatMSet -> Nat .
vars N M : Nat .
var S : NatMSet .
eq 0 < s(N) = true .
eq s(N) < 0 = false .
eq s(N) < s(M) = N < M .
eq min(N N S) = min(N S) .
ceq min(N M S) = min(N S) if N < M .
ceq min(N M) = N if N < M .
eq min(N) = N .
endfm
This module has two functions that are not completely defined, that is, such that when invoked on some ground
terms do not reduce to constructors. (Note: for the sort Bool, the constructors are of course true and false).
Do the following:
1
