The Evolution of Mechanical Theorem Proving: From Herbrand to Modern Applications

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: • 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.

8. 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.

9. 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.

10. 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”.

11. 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.

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, iscalled a proof of the theorem. • The problem of mechanicaltheorem proving is to consider mechanical methods for finding proofs of theorems.