### Abstract

The course is an introduction to model theory: a branch of
mathematics that deals with the classification of structures by
means of logical formulas. Mathematical structures (groups, rings,
fields, ordered sets, lattices, and others) can be classified
according to the logical formulas that are true in them. Conversely,
given some logical formulas, one wants to find classes of structures
in which the formulas are true. We call those structures `models' of
the formulas; moreover, when S is the class of all models of a set F
of formulas, we say that F is a `theory' of S; hence, the name of
model theory. Historically, mathematicians have used models all the
time, they just did not build a theory around it. For example, to
demonstrate that Euclid's fifth postulate does not follow from the
rest, one can take a sphere: a (non-standard) model of all the
postulates except the fifth. Model theory however is relatively
young, it did not exist before the second half of the 20th Century.

In addition, this course provides the foundations of software
specification and formal verification of systems from the
perspective of the work on algebraic specification. It also
introduces some basic concepts necessary for the design of an
algebraic-specification language. One important characteristic of
the present course is the use of many-sorted first-order structures
(instead of single-sorted structures), which consist of collections
of sets (of data values) together with functions and relations over
those sets. Each such structure can be regarded as a model of a
concrete software system. Therefore, we can learn a lot about
software systems by analyzing their model theory. This abstraction
corresponds to the view that the correctness of the input/output
behaviour of a software system takes precedence over other
properties such as efficiency.

Lecture
notes (previous version)

### Classes (via Zoom)

- May 7 (16:00~18:00) Lecture 1 - pdf video
- May 14 (16:00~18:00) Lecture 2 - pdf video
- May 21 (16:00~18:00) Lecture 3 - pdf video