Teorías en los Hábitos de estudio

The first two sections of this chapter gave an overview of significant historical figures and events in the devel-opment of mathematical logic, foundational mathematics and proof theory during the 19^th and 20^th centuries.

These developments have contributed to the inception and growth of the field of automated reasoning. Notable consequences of the developments in logic and mathematics included the establishment of first-order logic as the logical system of choice and the formulation of axiomatic set theory.

In the final section of this chapter we outlined the origin of the field of automated theorem proving, the discovery of the resolution proof procedure and the advent of interactive theorem provers. Finally, the integrated development environment (IDE) Rodin — the hybrid theorem proving environment used in our research to evaluate the OTTERheuristics — was introduced.

Chapter 3

Deductive Systems

We have seen in Chapter 2 that the idea of mechanised proof can be traced back to the work of Gottfried Leibniz in the seventeenth century. However, interest in mechanised proof remained dormant for approximately 200 years before it was revived in the latter half of the nineteenth century. Developments in mathematics and formal logic contributed to new fields of enquiry such as proof theory and deductive reasoning. Numerous deductive systems were devised and studied for both propositional logic and first-order logic. These systems were a necessary precursor of automated reasoning which gained prominence with the advent of digital computing.

The current chapter is the first of two chapters expanding on Chapter 2 to provide a technical introduction to deductive systems, and axiomatic set theory (Chapter 4). In Section 3.1 we present a brief overview of decision procedures and the reasons for preferring deductive proof over e.g., the evaluation of truth value assignments (a semantic approach). The section is concluded with a definition of deductive systems and some remarks on problems that arise from mechanised deduction.

The Gentzen system G is presented in Section 3.2 as an example of a simple deductive system. Example 3.1 shows a derivation within the Gentzen system G and illustrates the basic principle that underpins all of deductive reasoning: if all premises are true and the rules of the deductive system applied, then every conclusion that is reached must necessarily be true too.

When a deductive system is implemented in an automated reasoner, it becomes part of the reasoner’s infer-ence mechanism. Two inferinfer-ence mechanisms are discussed in subsequent sections. Arguably the most signif-icant inference mechanism is resolution - at least in terms of the historical development of the field (Plaisted 2015). Resolution (as well as a number of adaptations) is discussed in Section 3.3 and term rewriting is dis-cussed in Section 3.4.

Finally, the chapter is concluded with a summary.

3.1 Decision Procedures versus Deductive Systems

A decision problem is a question in some formal system with a yes-or-no answer. The answer will typically depend on the values of some input parameters. An example of a decision problem in mathematics is the question, “Given the integers x and y, does x leave a remainder of 1 when dividing y?”

An algorithm that solves a particular decision problem is called a decision procedure. By definition, a decision procedure must (1) terminate after a finite number of steps and (2) produce a yes-or-no answer for the decision problem. In our example, an effective decision procedure to decide if x divides y with a remainder of 1 is long division.

In mathematical logic, typical decision problems include:

1. Is formula A satisfiable?

2. Is formula A valid?

3. Given a set of formulas U and an arbitrary formula A, is A ∈ U?

4. Given a set of formulas U and an arbitrary formula A, is A a logical consequence of U?

We now turn our attention to decision problem 4. In order to proceed with the discussion, consider the following definitions of theory from Ben-Ari (2004, p. 27):

Definition 3.1 (Theory). A set of formulas T is a theory iff it is closed under logical consequence. T is closed under logical consequence iff for all formulas A, if T  A then A ∈ T . The elements of T are

called theorems. 

Definition 3.2 (Theory of U). Let U be a set of formulas. T (U) = {A | U  A} is called the theory of U. The formulas of U are called axioms and the theory T (U) is axiomatisable.  From these definitions we see that every element of T (U) is a logical consequence of the set of axioms U.

Decision problem (4) can therefore be stated as follows: Given a set of formulas U and an arbitrary formula A, is A ∈ T (U)?

Assume U = {A1, A2, . . . , An}. Then A ∈ T (U) iff a decision procedure for validity answers ‘yes’ on the formula A1∧ A₂ ∧ · · · ∧ An⇒ A.

For finite U, the method of truth tables is a decision procedure for the propositional calculus¹. For example, let U = {p ⇒ q} and A = ¬ p ∨ q. Then we see that (p ⇒ q) ⇒ (¬ p ∨ q) is valid, i.e. A ∈ T (U), because every row of its truth table evaluates to T:

1A truth table for some formula A requires a row for each combination of possible assignments of truth values to the atoms in A.

This results in exponential growth in the number of rows, because n atoms can be assigned truth values in 2ⁿdifferent ways. Better performant decision procedures had been proposed by 1960, e.g., see Davis and Putnam (1960, p. 201).

p q p⇒ q ¬ p ∨ q (p ⇒ q) ⇒ (¬ p ∨ q)

T T T T T

T F F F T

F T T T T

F F T T T

However, there are several drawbacks when relying on decision procedures (Ben-Ari 2004, p. 43):

1. The set U can be infinite. For example, in the axiomatisation of set theory, the Axiom Schema of Com-prehension specifies that an infinite collection of subset formulas are axioms (see Section 4.3 below);

2. Some theories do not have decision procedures like the propositional calculus, e.g., a first-order theory that includes the Peano axioms; and

3. Since a decision procedure produces only a yes-or-no answer, no intermediate results can be obtained.

Deductive proof is an alternative to decision procedures for semantic concepts such as interpretation and consequence. Instead of dealing with semantics, one can choose a set of axioms and a set of syntactic rules for deducing new formulas from the axioms. Since these rules can be applied mechanically, deductive systems can be implemented in computer programs and the rules of the deductive system included in an inference mechanism. In essence, an automated theorem prover is an automated deductive system.

Deductive systems are therefore central to the research reported on in this dissertation. We give the defini-tion of a deductive system from (Ben-Ari 2004, p. 43):

Definition 3.3 (Deductive system). A deductive system is a set of axioms and a set of rules of inference.

A proof in a deductive system is a sequence of sets of formulas such that each element is either an axiom or it can be derived from previous elements of the sequence using a rule of inference.

If {A} is the last element of the sequence, A is a theorem, the sequence is a proof of A, and A is provable,

denoted by ` A. 

Seeing that deduction is purely syntactical, some of the problems that arise from relying on decision proce-dures disappear (Ben-Ari 2004, p. 44):

1. Even if the set of axioms is infinite, only a finite number of axioms will appear in any proof;

2. Every proof consists of a finite sequence of sets of formulas; and

3. Once a theorem is proved, it can be used in other proofs just like an axiom.

Unfortunately, deduction itself introduces new problems. For instance, since deduction is defined in terms of syntactic formula manipulation, efficient systematic search procedures may not exist. In fact, in most

deduc-tive systems, any axiom can be used at any point in the proof sequence. Deducdeduc-tive proof attempts are, therefore, often hindered by the combinatorial explosion in the search space (see Section 2.3).

In the next section we introduce the Gentzen system G to illustrate the notion of proof in a deductive system.