Update

algorithm.md

@@ -1,6 +1,6 @@
 # Algorithm
 
-Algorithm (from the name of Persian mathematician Muhammad ibn Musa al-Khwarizmi) is an exact step-by-step description of how to solve some type of a problem. Algorithms are basically what [programming](programming.md) is all about: we tell [computers](computer.md), in very exact ways (with [programming languages](programming_language.md)), how to solve problems -- we write algorithms. But algorithms don't have to be just computer programs, they are simply exact instruction for solving problems. Although maybe not as obvious, [mathematics](math.md) is also a lot about creating algorithms because it strives to give us exact instructions for solving problems -- a mathematical formula usually tells us what we have to do to compute something, so in a way it is an algorithm too.
+Algorithm (from the name of Persian mathematician Muhammad ibn Musa al-Khwarizmi) is an exact step-by-step description of how to solve some type of a problem. Algorithms are basically what [programming](programming.md) is all about: we tell [computers](computer.md), in very exact ways (with [programming languages](programming_language.md)), how to solve problems -- we write algorithms. But algorithms don't have to be just computer programs, they are simply exact, simple instruction for solving problems. Although maybe not as obvious, [mathematics](math.md) is also a lot about creating algorithms because it strives to give us exact instructions for solving problems -- a mathematical formula usually tells us (not explicitly but in an implied way) what we have to do to compute something, so in a way it is an algorithm too. A "series of steps" is a [good enough](good_enough.md) definition of algorithm for everyday use, however for [computer scientists](compsci.md) there is a much more precise, mathematical definition, which states that algorithm is that which can be represented by a [Turing machine](turing_machine.md).
 
 Cooking recipes are commonly given as an example of a non-computer algorithm, though they rarely contain branching ("if condition holds then do...") and loops ("while a condition holds do ..."), the key features of algorithms. The so called wall-follower is a simple algorithm to get out of any [maze](maze.md) which doesn't have any disconnected walls: you just pick either a left-hand or right-hand wall and then keep following it. Long division of numbers which students are taught at school is also an algorithm. You may write a crazy algorithm basically for any kind of problem, e.g. for how to clean a room or how to get a [girl](woman.md) to bed, but it has to be **precise** so that anyone can execute the algorithm just by blindly following the steps; if there is any ambiguity, it is not considered an algorithm; a vague, imprecise "hint" on how to find a solution (e.g. "the airport is somewhere in this general direction.") we rather call a [heuristic](heuristic.md). Heuristics are useful too and they may be utilized by an algorithm, e.g. to find a precise solution faster, but from programmer's point of view algorithms, the PRECISE ways of finding solutions, are the basics of everything.
 
@@ -12,11 +12,11 @@ Algorithms are mostly (possibly [not always](declarative.md), depending on exact
 - **selection** (branches, *if-then-else*): Two branches (blocks of instructions) preceded by a condition; the first branch is executed only if the condition holds, the second ("else") branch is executed only if the condition doesn't hold (e.g. "If user password is correct, log the user in, otherwise print out an error."). The second branch is optional (it may remain empty).
 - **iteration** (loops, repetition): Sequence of steps that's repeated as long as certain condition holds (e.g. "As long as end of file is not reached, read and print out the next character from the file.").
 
-Note: in a wider sense algorithms may be expressed in other (mathematically equivalent) ways than sequences of steps (non-[imperative](imperative.md) ways, see [declarative languages](declarative.md)), even mathematical equations are often called algorithms because they *imply* the steps towards solving a problem. But we'll stick to the common narrow meaning of algorithm given above.
+Note: as already said above, in a wider sense algorithms may be expressed in other (mathematically equivalent) ways than sequences of steps (non-[imperative](imperative.md) ways, see [declarative languages](declarative.md)), even mathematical equations may be called algorithms. But now we'll stick to the common narrow meaning of algorithm given above.
 
 Additional constructs can be introduced to make programming more comfortable, e.g. [subroutines/functions](function.md) (kind of small subprograms that the main program uses for solving the problem), [macros](macro.md) (shorthand commands that represent multiple commands) or [switch](switch.md) statements (selection but with more than two branches). Loops are also commonly divided into several types such as: counted loops, loops with condition and the beginning, loops with condition at the end and infinite loops (`for`, `while`, `do while` and `while (1)` in [C](c.md), respectively) -- in theory there can only be one generic type of loop but for convenience programming languages normally offer different "templates" for commonly used loops. Similarly to mathematical equations, algorithms make use of [variables](variable.md), i.e. values which can change and which have a specific name (such as *x* or *myVariable*).
 
-Practical programming is based on expressing algorithms via [text](text.md), but visual programming is also possible: [flowcharts](flowchart.md) are a way of visually expressing algorithms, you have probably seen some. [Decision trees](decision_tree.md) are special cases of algorithms that have no loops, you have probably seen some too. Even though some languages (mostly educational such as [Snap](snap.md)) are visual and similar to flow charts, it is not practical to create big algorithms in this way -- serious programs are written as a text in [programming languages](programming_language.md).
+Practical programming is based on expressing algorithms via [text](text.md) in [programming languages](programming_language.md), but visual programming is also possible: [flowcharts](flowchart.md) are a way of visually expressing algorithms, you have probably seen some. [Decision trees](decision_tree.md) are special cases of algorithms that have no loops, you have probably seen some too. Even though some languages (mostly educational such as [Snap](snap.md)) are visual and similar to flow charts, it is not practical to create big algorithms in this way -- serious programs are written as a text in [programming languages](programming_language.md).
 
 ## Example