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Copyright © 2006-2014 - Curt Hill
Using Propositional Logic
Several applications
Introduction• Section 1.2 of the text considers
several topics relevant to propositional logic– Translation of English to logical
propositions– Systems Specifications– Web searches– Logic puzzles– Digital Logic Circuits
• Some of these will be covered here
Copyright © 2006-2014 - Curt Hill
Translation
• It is often needed to take English language statements and create propositions or compound propositions from them
• However there are issues• Most natural languages are
ambiguous, while mathematical notation should not be
Copyright © 2014 Curt Hill
Process
• Look for any statements that could have a true or false value– The more the better– Generate a key of letters and their
statements
• Look for the connectives– The main are negation, conjunction,
disjunction and conditional
• Put the final statement together
Copyright © 2006-2014 - Curt Hill
Example• Consider the following text:
Neither the butler nor the maid did it. That leaves the chauffeur the cook. The chauffeur was at the airport at the time of the murder. The cook is the only one without an alibi. The heiress was murdered by poison. It is logical to conclude the cook did it.
• Now lets find the propositions
Copyright © 2006-2014 - Curt Hill
Propositions
• Butler murdered heiress, b• Maid murdered heiress, m• Chauffeur murdered heiress, h• Cook murdered heiress, c• Chauffeur was at airport, a• Cook would have opportunity to
poison someone, p
Copyright © 2006-2014 - Curt Hill
Assertions
• The given assertions are:– ¬b– ¬m– b m h c– a
• The chauffeur at the airport implies that he could not murder the heiress– a ¬h
Copyright © 2006-2014 - Curt Hill
Result
• The implied assertion is the starting point: b m h c• The assertion states that one or more
of these must be true
• We know that b,m,h are false• Thus c must be true• That p is true only strengthens the
result
Copyright © 2006-2014 - Curt Hill
Specifications• Their has been considerable interest in
formal specifications for programs and systems of programs– The idea is that English is too ambiguous– We specify the state of a program at the
beginning and end– Properly done it is not ambiguous– Not completely caught on due to the need
for personnel training
• Proving program correctness uses similar approach
• You may see this again
Copyright © 2006-2014 - Curt Hill
Searches• Google has become a dominant
player due to its web searching capabilities
• The general notion of a search is that you are looking for a web page that contains certain terms connected by logical connectives
• Since the great quantity of pages, we need to be able to restrict our search more than enlarge it
Copyright © 2006-2014 - Curt Hill
Logic Puzzles• There are a variety of puzzles that
force to use logic to solve• The book is partial to Knights and
Knaves• You are on an island with only two
types of inhabitants– Knights always tell the truth– Knaves always lie
• You meet two or three of them and from what they say determine which kind they are
Copyright © 2006-2014 - Curt Hill
Example
• You meet two people A and B• A says the two of us are knights• B says A is a knave• What is the truth of the matter?
Copyright © 2006-2014 - Curt Hill
Solving these
• Let A and B be variables– True means the person is a knight
and the negation the person is knave
• Then we set up a truth table and see which, if any, is consistent
• We may also reason about this without a truth table
Copyright © 2006-2014 - Curt Hill
Table
• A says the two of us are knights• B says A is a knave
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ATwo of us are knights
BA is a knave
T T
T F
F T
F F
Table Filled In
Copyright © 2006-2014 - Curt Hill
AThe two of us are knights
BA is a knave
T T F
T F F
F T T
F F F
Digital Logic Circuits
• This is the age of digital logic• All of our computers and computer
controlled devices implement Boolean logic
• A full coverage of this needs to await CSci 370 but here is just a teaser
• How would we add two binary numbers using gates?
• Consider the next six slidesCopyright © 2006-2014 - Curt Hill
Copyright 2005 Curt Hill
Gates• At the lowest level the building blocks of
computers are gates or switches• A CPU is a collection of gates• The fact that we can implement these in
a rather straightforward matter makes the construction of computers possible
• Typical gates can be constructed with just a transistor or few diodes
• From there we will see that things like an adder can be constructed from gates
Copyright 2005 Curt Hill
Gate Symbols
• We use a variety of symbols to diagram gate networks
• NOT• AND• OR• NAND (Not And)• NOR (Not Or)• Among others
Copyright © 2005-2007 Curt Hill
Half Adder
A
B Sum
Carry
AND
XOR
Copyright © 2005-2007 Curt Hill
Full Adder Picture
A In
B In
CarryIn
Half Adder
Half AdderSum
Carry
Sum Out
CarryOut
Sum
Carry
Copyright © 2005-2007 Curt Hill
Notes on Full Adder
• The previous diagram showed a one bit full adder
• N of these will be cascaded into an N bit adder
• The adds will be done in parallel• The carrys will cause the main
delays– As the carry propagates down the line
Copyright © 2005-2007 Curt Hill
A Four Bit Adder
Carryin
A
BSumout
A
BSumout
A
BSumout
A
BSumout
Carryout
Exercises
• 1.2• 5, 13, 17, 21, 35, 41
Copyright © 2006-2014 - Curt Hill
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