Computing Fundamentals 1Lecture 0

Lecturer: Patrick Brownehttp://www.comp.dit.ie/pbrowne/Compfund1/compfun1.htm

Room K308Based on Chapter 0.

A Logical approach to Discrete Math By David Gries and Fred B. Schneider

Leibniz: Logician, Mathematician

• “It is unworthy of excellent men to lose hours like slaves in the labour of calculations which could safely be relegated to anyone else if machines were used.” Leibniz 1646-1716.

• Some ideas from Leibniz are used in the course text.

Leibniz: Logician, Mathematician

• Leibniz identified the main components that make automated reasoning possible:

• A language for expressing ideas precisely

• Rules of calculation for manipulating ideas in the language

• The need to mechanize the calculations.• http://en.wikipedia.org/wiki/Gottfried_Leibniz

Computing Fundamentals 1

• This module presents theoretical aspects of computer science which are necessary to support and enhance other modules on the course. In particular the topics covered on this module will be required in computer technology, databases, software engineering, programming, GIS. Artificial Intelligence, and algorithms.

Computing Fundamentals 1 • Logic: Propositional calculus, truth tables, logical equivalence,

logical argument, predicate calculus, simple proofs. • Set Theory: Algebra of sets, power sets, cardinality, Venn diagrams,

programming using sets.• Relations: Types, representations, equivalence, partial order,

relational database theory.• Functions: The graph of a function, properties, composition,

functions in programming languages.• Boolean Algebra: Basic laws, simplification of expressions,

application to switching circuits.• Number Systems: Binary, octal, decimal, hexadecimal, simple

binary arithmetic. • Supporting software: The above topics will be supported by

software tools including functional and logic based languages.

Computing Fundamentals 1

• The aim of this module is to provide the student with the theoretical foundations for other modules on the programme.

Computing Fundamentals 1

• Main ReferenceA Logical approach to Discrete Math

By David Gries and Fred B. Schneider

Publisher: Springer; 1 edition (1993)

ISBN: 0387941150

Computing Fundamentals 1

• Useful Reading• Seymour Lipschutz, Essential Computer

Mathematics, Schaum's Outline series, 1987, ISBN 0-07-0379990-4.

• Winfred Karl Grassmann and Jean-Paul Tremblay , 1996, Logic and Discrete Mathematics; A Computer Science Perspective, Prentice Hall 1996, ISBN: 0-13-501206-6.

• Seymour Lipschutz and Marc Lars Lipson, 1997, Discrete Mathematics, Schaum's Outline series, ISBN 0-07-038045-7.  

Mathematics & Logic

• Mathematics can be used to represent, or model the world. Math provides a precise, concise representation of quantities and relationships. As Information Scientists we are required to reason correctly to reach conclusions, and hence we need a knowledge of formal logic.

Formalizing Common Sense

• The following pair of signs are displayed at the foot of an escalator. What do they mean? They have exactly the grammatical structure.

Shoes

Must Be

Worn

Dogs

Must Be

Carried

Formalizing Common Sense

• If I tell you: “The vase is in the box”

• Can you deduce by logic the following?

• “Hence the box is not inside the vase”

• Are the following well formed logical statements?

• I am standing and I am sitting.

• I am standing and I am running.

Syntax and Semantics

• What do the following symbols mean?The number six

VI

3 + 3

The smallest whole number greater that five.

The integer successor of five.

The positive square root of 36.

Syntax and Semantics

• We need to separate symbols from meaning.

• What do the following symbols mean?

2^7 = 1282^6 = 642^5 = 322^4 = 162^3 = 82^2 = 42^1 = 22^0 = 1

10^2 = 10010^1 = 1010^0 = 1

Computing Powers in CafeOBJ

mod! POWER { protecting(NAT) op _^_ : Nat Nat -> Nat vars X I : Nat eq X ^ 0 = s 0 . – s is successor function eq X ^ I = X * (X ^ ( p I)) . }• -- Open the module and reduce expression• open POWER red (2 ^ 7) + (2 ^ 5) + (2 ^ 4) + (2 ^ 1) + (2 ^ 0) .

(179):NzNat

Syntax and Semantics

Syntax refers to the structure of expressions, or rules for putting symbols together to form expressions.Semantics refers to the meaning of expressions or how they are evaluated.

Syntax and Semantics

Syntax refers to the structure of expressions, or rules for putting symbols together to form expressions.Semantics refers to the meaning of expressions or how they are evaluated.

What is computing?

Computer Science is no more about computers than astronomy is about telescopes. Attributed to Dijkstra

Logic

• Logic is the glue that binds together methods of reasoning, in all domains. Proof methods have their basis in formal logic. An interpretation assigns meaning to the operators, constants, and variables of a logic (model theory). From Gries & Schneider[1]

• Logic is the study of reasoning; it is specifically concerned with whether reasoning is correct. Logic focuses on the relationship among statements as opposed to the content of any particular statement. From Johnsonbaugh[2]

• Logic as Theory of Science: People can hold scientific theories. A scientific theory consists of a multiplicity of acts of knowing, of verifying and falsifying, validating and calculating. Logic is a theory which seeks to determine the conditions which must be satisfied by a collection of acts if it is to count as a science. Logic is a ‘theory of science’. Barry Smith[3]

Logic and Language

• Logic takes into account syntactically well-formed sentences and studies whether they are semantically correct.

• There are sentences that are syntactically correct but not from the semantic point of view. For instance "The big house is green" is both syntactically and semantically correct (provided the house under consideration is green), while "The big house is small" is contradictory as a sentence and thus semantically incorrect.

Logic and Language

• Keep in mind the distinction between formal logic on this course and the intuitive everyday logic (informal)

• Further we often use informal logic to reason about formal logic. For example, we can say the a particular logic is not suited to a particular task.

Logic and Language

• Consider two English sentences:– A Mother is a Person.– Ellen is a Mother.

• What does “is a” mean in each sentence?

• Two possible mathematical meanings:– A Mother is a Person: Mother Person⊆– Ellen is a Mother: Ellen ∊ Mother

• What is the relation between Ellen and Person?

Logic and Language

• We will start with a logic called equational logic• We will use the CafeOBJ language, which is based on

equational logic. We will also use CafeOBJ as a functional programming language and as a theorem prover.

• The role of CafeOBJ on this course is to provide a logically based language that can be used to represent the mathematical concepts such as logic itself, sets, functions, relations and even programs (considering programs as mathematical objects). At the end of the course you should be able to read and understand small CafeOBJ proofs and programs.

Mathematical Models

• Mathematical models are more concise, precise, and rigorous than informal English descriptions.

• A mathematical model allows us compute solutions.

• A mathematical model allows computers and humans to reason about the problems in a mechanical way. This allows us to:– manipulate expressions;– prove properties from and about expressions;– obtain new results from know facts or expressions

Mathematical Models

• There are rules that allow use to perform syntactic manipulations without regards for semantics.

• Rigour does not mean complexity.• Rigour is defined as “strict precision or exactness”.• Rigour usually leads to simplicity of proof and

extensibility. • With a mathematical based language, like CafeOBJ, one

can specify or represent , reason, prove, and compute• A value is, in contrast a computation does something. • A value has a value type, a computation has

computation type.

Equational logic

• Equational logic (EL) can be used as a tool to reason about systems. EL is based on equality and Leibniz’s rule for substituting equals for equals.

• You should be familiar with EL from school. For example. Mary has twice as many apples as John, can be written as: m = 2 * j

• There are many values of m and j that make the equation true (e.g. m=4 and j=2, or m=6 and j=3).

Equational logic

• Mary has twice as many apples as John. Mary throws half her apples away, because they were rotten, and John eats one of his. Mary still has twice as many apples as John. How many apples did Mary and John have initially?(0.1) m = 2 * j and [EQ1] m/2 = 2 * (j – 1) [EQ2]

• There are now fewer values of m and j that make these two equations true (m=4 and j=2).

• Are these equations consistent? Are there values of m and j that make one equation true and the other false? It happens that these two equations have a common solution.

The Apple0 problem 1

mod APPLES0 {[ Apples ]var M : Apples -- Mary's applesvar J : Apples -- Johns applesops 0 1 2 : -> Apples ops * quo : Apples Apples -> Apples eq [eq1]: M = J * 2 .eq [eq2]: M quo 2 = 2 * (J – 1) . }• The above specification represents the mini-world of Mary and John’s apples. As it

stands it does not compute a solution. • we have symbol;

– * representing multiplication operation– quo representing division operation (quotient)– 0, 1, 2 representing numbers– M,J variables representing number of Apples

• But these as yet do not have their normal computable interpretation. You could say that at the moment the symbols have an intended interpretation. Initially we focus representation not computation.

• See Apples1.mod and Apples2.mod on web page.

Equations represent facts about John’s and Mary’s apples

The Apple0 problem 1

• We can rewrite the following equations in terms of J only:

eq [eq1]: M = J * 2 .eq [eq2]: M quo 2 = 2 * (J – 1) .

• Giving:eq [eq1] : 2 * J quo 2 = 2 * (J - 1) .

The Apple problem 1

mod! APPLES1 {[ Apples ]var M : Apples -- Mary's applesvar J : Apples -- Johns applesops 0 1 2 : -> Apples op _quo_ : Apples Apples -> Apples {assoc }op _*_ : Apples Apples -> Apples {assoc }op _-_ : Apples Apples -> Apples {assoc }

eq [eq1] : 2 * J quo 2 = 2 * (J - 1) .eq [eq2] : 2 * (J - 1) = 2 * J - 2 . eq [eq3] : 2 * J quo 2 = J . -- what this equation mean?}

To test if an equation, say [eq1], is true:open APPLES1reduce 2 * J quo 2 == 2 * J - 2 . – true

A quotient is the result of a division. For example, dividing 6 by 3, the quotient is 2. In this case we do not have a numeric value for J. Order of equations matters.

This equality only holds for one value of J, it is not generally true.

Equations represent some of the true facts about John’s apples

The Computing the Apple problem

mod! APPLES2 {pr(INT)op solution : Int -> Intvar M : Int -- Mary's applesvar J : Int -- John’s appleseq solution(J) = if (2 * J quo 2 == 2 * (J - 1)) then J

else solution(J + 1) fi . }Mary’s apples not used to compute J.

Pre-condition, Post-condition

• A pre-condition of a program (or algorithm) statement is an assertion about the program variables in a state in which the statement must be executed, and a post-condition is an assertion about the state in which it may terminate. A reasonable precondition for the Apples2 program is that the first guess should be equal or greater than zero (no negative number of apples)

More Examples

• In English: we wish approximate • In Math:

– Let b approximate (therefore b2 approx. n)– Precondition: 0 <= n (root of negative undefined)– Postcondition: b2 <= n < (b+1)2

– The postcondition corresponds to computing the largest integer (b) that is at most . It is what we would expect to hold after the computation.

• These conditions do not tell us how to compute.• With maths can model, reason, and compute

n

n

n

More Examples

Here is an equation in CafeOBJ that will allow us to compute an approximation of the square root of a number.

eq approx(B, N) = if ((B * B <= N) and (N < (B + 1) * (B + 1)) ) then B else approx(B + 1, N) fi . (see ROOTN1.MOD on web page)• The Postcondition: b2 <= n < (b+1)2 can be written

as two conditions (b2 <= n) and (n <(b+1)2

nnn

Systematic Syntactic manipulation

From e=mc2 show e/c2=mPrecondition 0<c2

e = m*c2

= <Divide both sides by c2> e/c2 = (m*c2)/c2

= <Associativity of / and *> e/c2 = m*(c2/c2)= <(c2/c2) = 1 > e/c2 = m*1 <m*1 = m > e/c2=m

Transitivity: if a=b and b=c then we can conclude that a=c

Equality is transitive, hence we can conclude e=mc2 is equivalent e/c2=m

Einstein showed a mass–energy equivalence. Matter can be turned into energy, and energy into matter. Here we focus on syntactic manipulation disregarding the scientific meanings of the symbols in physics.

Mathematical expressions

• Syntax of mathematical expressions are constructed from constants, variables, and operators:

• A constant such as 123 is an expression• A variable such as y is an expression1. • An operator such as + used in expression

– 123 + y

• Brackets can be used to aggregate expressions e.g. 2 * (3 + 4) .

Mathematical expressions

• Syntax refers to the structure of expressions, or rules for putting symbols together to form expressions.

• Semantics refers to the meaning of expressions or how they are evaluated.

• We are interested in syntax, semantic, and how they are related.

Evaluation

• State is a list of variables with associated values.

• Evaluation of an expression E in a state is performed by replacing all variables in E by their values in the state and then computing the value of the resulting expression. For example:– Expression x – y + 2 – State (x,5),(y,6)– Gives 5 – 6 + 2– Evaluates to 1

Programming & Logic

• The basic components of current languages are:– Data types e.g. Integers, String, Polygon.– Variables to refer to data types e.g. a = 2;– Operations on data types e.g. area(polygon)– Control structures e.g. sequence, iteration, and

conditions.– Logic is an important part of programming, but it is

often implicit and external to the language. Some languages like Prolog or SQL are quite close to logic.

– In many cases a CafeOBJ program has a one-to-one relation with its mathematical or logical meaning.

CafeOBJ• The CafeOBJ system and documentation can be downloaded

from: http://www.ldl.jaist.ac.jp/cafeobj/• You should download the latest version.• The site contains installation instructions for Linux, Mac OS X,

and Windows• There are excellent CafeOBJ notes by Kokichi Futatsugi at:• http://www.jaist.ac.jp/~kokichi/class/i613-0712/• Kazuhiro OGATA has written an excellent set of notes on functional

programming using CafeOBJ • http://www.jaist.ac.jp/~ogata/lecture/i217/• Takahiro Seino has written an excellent introduction to CafeOBJ as

a theorem prover.• http://www.jaist.ac.jp/~t-seino/lectures/cafeobj-intro/en/index.html• The CafeOBJ manual is useful but a bit too detailed for our

needs

Installing CafeOBJ • For this course you should install CafeOBJ on your U:

drive in college and on your own machine.• For Windows 7 download from• https://bitbucket.org/tswd/cafeobj/downloads/cafeobj-

1.5.0-win32.zip• Unzip the file into a folder of your choice, say U:\ComFun1. A folder called

U:\CompFun1\cafeobj-1.5 will be made. • Click on CafeOBJ.exe in that new folder.• Or at command prompt• cd U:\CompFun1\cafeobj-1.5 and type CafeOBJ

Staring CafeOBJ

• After installing CafeOBJ there are typically four things that you need to do to work on a file:1. In a command prompt, start CafeOBJ, type CafeOBJ2. Set CafeOBJ to point to files in your current working

directory (or folder), type: cd yourDir3. Load the file you wish to work on: in workingFile4)Open the module you wish to working with: open ModuleName