**Mathematics** is the study of quantity, structure, space, and change. Mathematicians seek out patterns,^{ } formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.

The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter.

**Fields of mathematics are**:

From these areas discrete mathematics is closest to computer languages and source code analysis.

### Discrete mathematics

Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. This includes, on the computer science side, computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Typical kinds of discrete mathematics are shown:

### Relations between Discrete Mathematics and Symbolic Analysis

In fact, symbolic analysis is one part of mathematics (in Finnish Ruohonen). However, that kind of mathematical symbolic analysis is one kind of analysis principle typical for mathematics like symbolic differentiation, symbolic integration etc. That kind of features can be found in modern mathematical tools like Mathematica 7.

From the software point-of-view, in Symbolic Analysis (Laitila) there is an atomistic model to describe source code and its behavior. That models obey the pure rules of graph theory, where all tools intented for graphs are useful. Furthermore, the theory for simulating these graph elements (see AHO-objects, e.g symbols) is pure theory of computation. Simulating branches with unknown conditional values lead to a problem of combinations, a state explosion (combinatorial explosion) problem. However, there is no connections from the atomistic symbolic model to cryptography.

**In summary, the framework of the atomistic model is rather close to the theory of discrete mathematics.**

As a conculusion, it is then reasonable *to ask whether there is a gap between mathematical formulation like albebra and formalism adapted from source code (programming languages) to be expressed in the symbolic atomistic model.*

## There is no Gap

Mathematics is a set of theories based on their type systems, where selected symbols are connected with carriers (symbolic clauses) and constants as operations (see figure below).

In programming languages the operations are expressions in the grammar (see Java grammar). From the side of automata theory each operation has been executed by a register machine or a similar automaton, which is a subset of the universal machine (see more).

Modern computer languages contain the features to allow programming any mathematical functions (except some demanding, very specific vector and array operations).

They are Turing strong (see more). The **Church–Turing thesis** states that any function that is computable by an algorithm is a computable function.

Progamming languages are an extension fo traditional mathematics where there are definitions, loops, conditionals (e.g. paths) and some other types of clauses. Formalism for Pascal has been proved to be downwards compatible to mathematics for decades ago, and deducible.

In book Symbolic Analysis we show that Java can partially be simulated by SAM. It can create an output tape (Turing machine) for any algorithm even though some symbols are unknown. Therefore execution of the Symbolic Abstract Machine, SAM, can directly be mapped from the mathematical side e.g. equations to the software side symbols. There is no gap between these two sides, because all the semantics can be expressed in Symbolic language and all symbols will be executed by Symbol:run-invocation.

Some links:

- Computability theory: http://en.wikipedia.org/wiki/Computability_theory

- Church-Turing thesis: http://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis

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