Logic Seminar 1

Introduction

24.10.2005.

Slobodan Petrović

Introduction

• It has long been man’s ambition to find a general decision procedure to prove theorems.

• This desire dates back to Leibniz (1646-1716).• It was revived by Peano in the beginning of the 20th

century and by Hilbert's school in the 1920s. • A very important theorem was proved by Herbrand in

1930: he proposed a mechanical method to prove theorems.

• Unfortunately, his method was very difficult to apply since it was extremely time consuming to carry out by hand.

Introduction

• With the invention of digital computers, logicians regained interest in mechanical theorem proving.

• In 1960, Herbrand’s procedure was implemented by Gilmore on a digital computer.

• A more efficient procedure was proposed by Davis and Putnam.

Introduction

• A major breakthrough in mechanical theorem proving was made by J. A. Robinson in 1965.

• He developed a single inference rule, the resolution principle, which was shown to be highly efficient and very easily implemented on computers.

• Since then, many improvements of the resolution principle have been made.

Introduction

• Mechanical theorem proving has been applied to many areas, such as program analysis, program synthesis, deductive question-answering systems, problem-solving systems, and robot technology.

• In the field of computer security, it has been applied in protocol analysis.

Introduction

• There are many points of view from which we can study symbolic logic.

• Traditionally, it has been studied from philosophical and mathematical orientations.

• We are interested in the applications of symbolic logic to solving intellectually difficult problems.

• We want to use symbolic logic to represent problems and to obtain their solutions.

Introduction

• A simple example. • Assume that we have the following facts:

– F1

: If it is hot and humid, then it will rain.

– F2

: If it is humid, then it is hot.

– F3

: It is humid now.

• The question is: Will it rain?

• Let P, Q, and R represent “It is hot,” “It is humid,” and “It will rain,” respectively.

Introduction

• We shall use to represent “and” and to represent “imply”.

• Then, the three facts are represented as:– F1: P Q R

– F2: Q P

– F3: Q.

• Thus, English sentences have been translated into logical formulas.

Introduction

• It can be shown that whenever F1, F2, and F3 are true, the formula– F4: R

• is true.

• Therefore, we say that F4 logically follows from F1, F2, and F3.

• That is, it will rain.

Introduction

• Example. We have the following facts:– F1: Confucius is a man.– F2: Every man is mortal.

• To represent F1 and F2, we need a concept of predicate.

• We may let P(x) and Q(x) represent “x is a man” and “x is mortal,” respectively.

• We also use (x) to represent “for all x”.

Introduction

• We can now represent the facts by logical expressions:– F1: P(Confucius)

– F2: (x)(P(x)Q(x))

• From F1 and F2, we can logically deduce:

– F3: Q(Confucius)

• which means that Confucius is mortal.

Introduction

• In the examples, we essentially had to prove that a formula logically follows from other formulas.

• We call a statement that a formula logically follows from other formulas a theorem.

• A demonstration that a theorem is true, i.e. that a formula logically follows from other formulas, is called a proof of the theorem.

• The problem of mechanical theorem proving is to consider mechanical methods for finding proofs of theorems.