1. Logic Seminar 1 Introduction 24.10.2005. Slobodan Petrović

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

7. Introduction A simple example. Assume that we have the following facts: – F 1 : If it is hot and humid, then it will rain. – F 2 : If it is humid, then it is hot. – F 3 : 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.

8. Introduction We shall use  to represent “and” and  to represent “imply”. Then, the three facts are represented as: –F 1 : P  Q  R –F 2 : Q  P –F 3 : Q. Thus, English sentences have been translated into logical formulas.

9. Introduction It can be shown that whenever F 1, F 2, and F 3 are true, the formula –F 4 : R is true. Therefore, we say that F 4 logically follows from F 1, F 2, and F 3. That is, it will rain.

10. Introduction Example. We have the following facts: –F 1 : Confucius is a man. –F 2 : Every man is mortal. To represent F 1 and F 2, 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”.

11. Introduction We can now represent the facts by logical expressions: –F 1 : P(Confucius) –F 2 : (  x)(P(x)  Q(x)) From F 1 and F 2, we can logically deduce: –F 3 : Q(Confucius) which means that Confucius is mortal.

12. 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.