Truth value assignment (or just truth assignment):
- Gives a value (0 or 1) to each propositional variable.
- Value can be extended to all formulas using those variables through the
truth tables, by working "inside-out" (example).
- A truth assignment "sa
- We have shown that a language is accepted by a FSA iff there is a R.E. for it.
- A language is called regular iff it is denoted by some R.E., or equivalently,
iff it is accepted by a FSA (deterministic or non-deterministic).
- This pro
Proving Logical Equivalences
Prove or disprove each of the logical equivalences below, without
using truth tables:
a) p -> (q /\ r) LEQV (p -> q) /\ (p -> r)
answer: p -> (q /\ r)
LEQV ~p \/ (q /\ r)
LEQV (~p \/ q) /\ (~p \/ r)
Notation for expressing concepts precisely, and formalism for reasoning.
"Propositions" are statements, which are either true or false,
as opposed to other kinds of English sentences (e.g., commands, questions).
Example: "David works h
- Let PV be a set of propositional variables. The set of propositional formulas
PF is the smallest set such that:
. any variable in PV is in PF.
. if p1 and p2 are in PF then so do the following expressions: ~p1, (p1 /\ p2),
Structures, valuations, interpretations:
- A "structure" S for a first-order language L consists of:
. a domain D (nonempty)
. for each constant symbol c of L, an element c^S of D
. for each n-ary predicate symbol A of L, a relation A^S over D^n
. if L co
Termination: Let E = l + 1 - i. E is definitely an integer because l and i
are both integers. Sine f <= l by the precondition: E_0 = l + 1 - i_0 =
l + 1 - f >= 1, and since i_k <= l at the begining of each iteration of the
RecBinSearch. Recurrence for worst-case running time of
if n = 1
T(n) = cfw_
cfw_ 7 + max cfw_ T(ceil(n/2), T(floor(n/2) + 1 if n > 1
1. Repeated substitutions. Since this is used only to get a guess, we
are allowed to make simplif
- Regular expressions describe sets of strings using a small number of
- The set of regular expressions (regexps) over an alphabet S is defined
as follows, assuming that S does not contain symbols "cfw_" and "e":
Test Preparation Questions
1- Let L be the set of strings over cfw_0,1 that contain both 01 and 10
(a) Daw the diagram of a DFSA that accepts L
(b) Give a regular expression that denotes L.
Answer: for part (a) one way is to have q0 -0-> q1
Example 2: Prove that for any natural numbers n >= 1, if x is a real
number such that 1 + x > 0 then (1 + x)^n >= 1+nx.
. S(n) can be defined as "if x is a real number and 1 + x > 0 then
(1 + x)^n >= 1 + nx".
. We want to prove S(n), for all n >= 1. (Note