Guide

1. Syntax
2. Introduction

• 2B. Relations
• 2C. Operators
• 2D. Data
• 2E. Tutorial

1. Intermediary

• 3B. Functions
• 2C. Operators

1. Optimizations

1. Syntax

Syntax

|Syntax |Output --- | --- | |#str|size(str) |t.x|`table.get(t,x)`| |`t['x']`|`table.get(t,x)`| |t.x:2|`table.set(t,x,2)`| |`:f(2)`|functor| --

|Types | Data --- | --- | |String|"str" |Number|1, 2.5 |Relation|`rel(x) true;` |Functor|`Tuple(x)` |List|[1,'2']

Operators

--- | --- | |and,or| |if,when,choose| |not,once|

Arithmetic Operators

--- | |+ - * / |`< > <= >= = !=`

Conversion (casting)

--- | |str |num, real

Lexicon

Syntax: a language's grammar.

2. Using the Language

Installation

Before doing anything, you may check that you've installed the language. By opening the language with the -v flag, you should see installation version number and confirm.

Open a command-line and type,

$ cosmos -v
Cosmos 0.16

If the language is installed, you'll see something akin to this.

Interpreter

A good way to try out a new language is by opening the interpreter, if one is available, and making statements in it. This can be done with the -i flag.

We'll assume you have already downloaded the language.

$ cosmos -i
> x=1
| x = 1
> x=1 or 2=x
| x = 1
| x = 2
> str='hello'
| str = 'hello'
> l=[1,2,3]
| l=[1,2,3]

This way, you should get a better idea of how the language works. Whenever you see a new kind of statement in this guide, you may use the interpreter to try it on.

Writing a file

Make a file hello.co with the content,

print('hello world')

A program that writes "hello world" on the screen is one of the simplest programs you can do. This can be done with a print command.

The file can be loaded with the -l flag.

$ cosmos -l hello
'hello world'`

2. From First Principles

Statements

A simple way to make a statement in Cosmos is to use equality, aka the = operator.

x=1
y=z
2.5=z

These are three kinds of statements you can make.

x=1
print('hello world')
io.writeln('hello world')

The first uses the relation of equality.

Then, the built-in print relation is used to write 'hello world'.

Finally, a writeln relation is taken from a module io to write 'hello world'.

Joining statements --

x=1
y=2 and x=1

Statements can be joined together with and.

Statements from separate lines are implictly joined.

x=1
y=2

... is the same as ...

x=1 and y=2

Data types --

Two basic types of data we've seen so far are Number and String.

Number includes 1, 2 and 2.5.

Unlike numbers, strings refer to a piece of text surrounded by double or single quotation in code.

String includes 'hello world' and "2".

Remember that 2 is a Number but '2' is a String.

Comments

Arithmetics

x = num(2+1*4/2)

The special num (or real) function computes the result of the mathematical operations, passing the result to x.

| x=1+2
print(x) //1+2

This is a technically correct program and saying that the value of x is 1+2 is an answer. Though, it may not be the one we want. Thus,

| x = num(2+1)
print(x) //3

A logic program may sometimes,

1. Apply a bunch of numerical operations, and,
2. Evaluate the result of those operations with num or real, sometimes at the end of the program. This is the "solving" step. x = 'hello'+' world'

   Different data types, such as strings, may have their own interpretation of addition. We may use addition to "add" two strings together, for example.

   print(str(x)) //'hello world'

   This is akin to casting in procedural languages. A value like 1+2 has a different type than Number. It's stored as-is before being solved by the language's arithmetic system.

Constraint Arithmetics

Cosmos is one of the few existing languages to have CLP as the default arithmetic system.

$ cosmos -i
> x=1+y
| x = 1+y
> x=num(1+y)
| x = _124
| y = _123
> x>5
| x > 5

Why?

As this is a rather unique decision, we should probably explain.

A regular language would have x>5 or y+1 simply fail. In those cases, it doesn't know the value of x or y. It wouldn't then be able to solve those equations.

As a logic language, they're simply taken as true until proven wrong. It's the same for equality, e.g. x=y would also give an error in a regular language. A logic language has constraints that are solved when needed.

This is commonly called Constraint Logic Programming (CLP). Cosmos has CLP for Reals[1] as its default arithmetic.[2] This also means that it uses floating-point numbers.

There are other known systems. However, they would not work well as the default arithmetics. CLP(FD) is limited to integers which would mean users couldn't use floating-point numbers at all.

As the default arithmetics, we wanted a system that would, (1) work under normal circumnstances, at the least, and (2) not be limited to that; if possible, explore and make use of logic programming. If it weren't possible to use numbers with decimals, it would clearly not work under the circumnstances in which procedural arithmetics do, as they do use them, thus breaking (1).

However, one can use other systems by calling the host language, and using a system from them. Furthermore, the current system may be changed by modifying core.pl.

[1] As far as we know, CLP for Reals goes back to Prolog III. See http://prolog-heritage.org/en/ph30.html.

[2] It should be rather be asked why a logic language would not use CLP and instead present a classical, procedural system as the default one! Even though Prolog III has it, see [1]. It's practically asking users to make incorrect programs, logically speaking.

2B. Relations

What is a relation?

Cosmos™ is one of the few remaining logic programming languages. It's no surprise therefore that there's some connection.

Logic itself concerns with statements that may be true or false, such as,

x equals 2
The double of 2 is 4.
Socrates is a human.
It's raining.

What these statements have in common is that they each have a relation. This is more evident when written in logic notation,

x = 2
double(2, 4)
human('socrates')
rain().

The relationship in double(2, 4) is double, which we defined beforehand. Naturally, this statement is true.

Out of those, _=_ is distinguishable enough that it doesn't need to be in logic notation. Generally, arithmetics stay in arithmetic notation.

Logic and function notation

double(4,x)
x=double(4)

Consider these two statements. Although the meaning is the same, one of them uses double as a relation and the other uses it as a function.

Custom Statements

Let's codify the intuition that "the double of x is y".

rel double(x,y)
        y=x*2

double(2,z)

We've seen statements that use built-in relations like print and _=_ but this is the first time we define our own relation. Specifically,

1. We define a relation using the keyword rel.
2. We then "call" upon that same relation. A fine rule to understand the second step is to substitute the statement for the definition.

   A statement like double(2,z) is, by our definition, z=2*2, once we've substituted x by 2 and y by z.

   In other words, the result is that z now equals 2*2.

   Though double is not particularly useful, relations in general can single out pieces of code allowing for later use.

   As we'll see later, relations can be kept in a module.

Whitespace

Any major "structure" such as rel and if are delimited by an increase in whitespace (generally, comprised of four spaces or a single tab character).

rel(true)
        false

The code that comprises the if- and else-parts are evident through this method.

An error is issued if there's any inconsistency. For example, if you chose the first indent to be four spaces but the second to be three.

As long as you make a consistent rule, i.e. four spaces or one tab-only, there should be no issues.

[1] As suggested by the programming language Python™, true statements can be used as placeholders.

2C. Operators

and/or

In a logic language, these operators are more important than they may first seem. They are not only logical operators, they describe the flow of a logic language.

We've already seen them in use.

> x=1 and 2=y
| x = 1 and y = 2
> x=1 or 2=x
| x = 1
| x = 2

Logic/declarative interpretation

A and B: Both A and B are true.

A or B: Either A, B or both are true.

More about truth --

As long as a valid interpretation for the statements holds, the interpreter will reply with an "answer".

It's possible there's no answer to be found, in which case the interpreter might not reply at all or give false as an answer.

> x=1 and x=2
| false

The above statement is clearly a contradiction, thus untrue. The two statements can't be both true at once.

Any other answer besides false is implictly a true answer, even if the interpreter doesn't say so.

No matter what the interpreter outputs, it's to be taken as true until said otherwise.

> true
| true
> false
| false

The statements true and false both give a true and a false answer, respectively.

The Procedural interpretation --

Cosmos is a language based on logic. Still, it runs on a computer. As such, there has to be a procedural (i.e. a machine-based) explanation on how the program runs.

We may call it the logic-procedural or simply procedural interpretation. This would be the procedural explanation on how our logic program is run.

Procedural interpretation

A and B: A and B are both evaluated, in sequence.

A or B: A is evaluated. If that fails, B is evaluated.

We might say it's a logic program that's run by a procedural "logical engine".

Let's say we have the statement `x=1 and x=2`.

A more complicated example --

`x=1 or x=2` essentially states that x may be 1 or 2. Which is to say, we don't know what the value is!

But what about `(x=1 or x=2) and x!=1`?

rel p(x)
    x=1 or x=2
    
rel main()
        p(x)
        x!=1
        io.writeln(x) //2

Logic and procedural interpretations --

Naturally, we have said that x is 1 or 2. Since we then state that x is not 1, it's only possible for it to be 2. This is the logical interpretation of the program.

The program will then call io.writeln to write 2 on the screen.

1. As p(x) is called, first x=1 is evaluated.
2. However, we then evaluate x!=1. A contradiction is found.
3. However, we can still go back. Our program then backtracks to p(x), and evaluates x=2.
4. This time there are no contradictions.
5. It prints succesfully. This is a proccess that, by the way, would not exist in a truly procedural language. A procedural program has no concept of backtracking. That is, it has no or. It's as if only and is used.

   It's because this is a logic language that we must ensure our program works correctly.

Limitations

This brings us to the well-known limitations of LP.

1. Infinite loops Let's consider,

   rel p(x)
       p(x)
       
   p(x)

   The program will call p(x), which will call p(x). This is an example of recursion. This is an example of an infinite loop, as the program will run forever.

   The problem with this is that even correct logical problems may fall into an infinite loop.

2. Performance <!--The other issue happens when you try to make a program.-->

A Counterpoint

A possible counterpoint is that-- every language has the same problem, that is, they too can fall into infinite loops.

Couple that with the fact this occurs when you're doing recursion. Generally, if your program relies on a relation calling itself it's pretty obvious that it's doing so, and you have a hint that you should be looking out for such things.

The second point is a result of combining a query solver with a programming language. If you simply want the solver to answer a query, does it matter if it takes a few seconds (or less) longer? If you want to use it as a fast programming language, shouldn't

The programmer then wants to instead use it as a blazing fast programming language, while at the same time expecting it to be fast without doing any work on optimizing it- which is not how it works in any language.

Even if it's not ideal, knowing how the program works is often necessary to optimize it in any language.

2C. Extra Operators

Operators

Adding to our repertoire, we have if and not.

Logically speaking, this lets us represents all kinds of statements in the form,

`If X, then Y. not X.`

Which are common in logic and show up in practical programming.

`If it's raining, then someone will bring an umbrella. If someone is a human, they're mortal. (aka All humans are mortals) It's not raining.

_if_
--

if(true) print('condition is true') else print('condition is not true')

We may again open the interpreter to try if-statements.

$ cosmos -i > if(x!=1) x=2; | x=2 > if(x!=1) x=2 or x=1; | x=2 | x=1 > if(x!=1) x=2; | x=2

The body of the if-stm will be evaluated when the condition, e.g. x=1, is true.

> if(false) x=2 else x=1; | x=1 > if(x=1) x=1 else x=0; | x=1 | x=0

It's possible for an if-statement to have an else-clause. If the condition is true, the if-part holds, otherwise the else-part holds.

In the latter example, the interpreter tried out both possibilities and gave two answers.

_elseif_
--

if(x=1) y=2 elseif(x=1) y=2 else true

Naturally, this is just a way to write,

_Logical interpretation_

You may think of an if-statement as equivalent to,

if(A) B else C; <==> (A and B) or (not A and C)

An if-statement without else is,

if(A) B; <==> (not A) or B

Though this may not be its exact implementation.

_not_
--

> not x=2 | x!=2 > not 5<1 | x=1 | x=0 > not x<=2 | x>2

As you may have noted, _not_ simply negates a statement.

_Logical interpretation_

The negation of A is simply false when A holds, and true when A doesn't.

Complex Conditionals
--

It's recommended keeping the conditions for an _if_ or _not_ to simple statements.

Simple statements like equality or inequalities, i.e. =, !=, >, etc. are more tried-and-true. They've been more extensively tested.

It's easy to see that _not x=1_ is the same as _x=1_. Similarly, _not x>5_ is just _x<=5_.

By contrast, negating a generic statement like _p(x)_ or _q(x,y)_ is more complicated. It could mean delaying a whole call, which may seem counter-intuitive to some programmers.

Nonetheless, it's valid logic code.

```javascript
rel p(x)
        x=2
    
not p(x) //x is not 2
x=1

Negating p(x) means keeping the call to p in hold until x is instantiated, i.e. x=1.

Procedural interpretation

The negation of A is obtained by delaying the evaluation of A until its parameters are instantiated. When it's done, the result will be false when A holds, and true when A doesn't.

Limitations --

rel p(x)
        not q(x)
    p(x)

p(x)

As stated previously, LP is not above infinite loops.

When running this example, q(x) is skipped entirely, leading straight to p(x). This will create an infinite loop.

Alternatives --

Our current implementation of negation plays it safe and may delay a statement until it can be sure it can be evaluated; if a statement is too complex for our not or if-statement, it’ll be delayed.

We fully expected someone very acquainted with logic programming may prefer to implement their own negation. If a more specific implementation of conditional/negation is required, this may be done using the operator when.

when(x=2)
        true
else
        x!=2

Finally, many examples of logic programming (such as the one given previously in the tutorial) do not need negation at all.

rel likes(x,y)
        when(x='alice')
                y='cake'
        else
                x='abbe'
                y='pastry'

when is simply a shortcut to 'and/or'. You are thus free to fine-tune your own negation or not at all.

Finally, you may use a function.

A function is constrained in many ways; it only needs to run once. Programmers writing deterministic programs may only need functions, and a function is more straightforward in those situations.

fun p(x)
        if(not q(x))//error
                p(x)

p(x)

A function will simply display an error if it can't execute the condition. This is much like what a procedural or functional language would do.

When using a relation, our main concern is that the operators behave in a purely relational way. As we're using a function, a way befit of a pure function is enough.

Pure functions are way simpler than pure relations. They don't return multiple answers, and don't work without non-instantiated arguments.

Logic-Procedural (II) --

Our main requirements if (and not) are that,

• It's a logically sound conditional.
• If it can't do that, an error is given. The reason for this is that we want to allow for different optimizations and ways to implement a logical conditional/negation to be tried.

  For us, being able to use a sound conditional is already an improvement. Prolog famously implemented negation in an unsound way (that also gave no warnings whatsoever).This

  This has highly limited experimentation with logic programming, as two operators were rendered mostly unusable.

  when --

  The when operator is simply syntax sugar for and/or,

  when(A) B else C <==> (A and B) or (C)

  It's similar to case.

  As you see, this is a rather barebones conditional.

• It's still pure code.
• However, the else is redundant, as the condition is not used anywhere. It can be replaced by case.
• It can be easily switched with other conditionals as they share similar keywords. By changing the keyword when to if and vice versa, you get different semantics. Functions --

  A relation can be turned into a function by changing the rel to a fun keyword.

  fun q(x,y)
          if(p(x))
                  y=2
          else
                  y=1

  As functions have simple requirements when compared to full relations, an if or when will accept any conditions when inside a function[1].

  Because a function is only meant to run once, there is no need to support backtracking or any other logical mechanisms.

  This may mean we're abandoning relational programming (LP) for a moment, though this is restricted to q.

  We'll explain this further in a later section.

  [1] The semantics of if or when will be turned into that of a commited-choice conditional. This is also the choose operator in Cosmos.

  List of Conditionals (Summary) --

  if: guaranteed to be logically sound, gives error if the condition is too complex. when: simple conversion to and/or. Even if it fulfills the condition and goes to if-code, it may still backtrack to else-code. Works for any condition. choose: simply checks if the condition is true, then chooses if or else-code. It'll not backtrack, but then this may not be logically sound.

  We recommend sticking to if when possible.

  if works best with simple conditions, i.e. simple inequalities such as != or >=.

  <!--If this is unreasonable, a simple solution is to fall back into functions, which we'll explain in a later section.-->

  A Conditional in Formal Logic --

  This is just trivia, but it may be of interest to some that the actual conditional A->B in formal logic does not have an else.

  It's simply something adopted from procedural programming where we often specify what to do if a condition is not true. This has often been adopted by LP aswell.[1]

  `If it's raining, then I will fly.`

  An odd thing about the formal conditional is that it holds true when the condition is false. As we said before, `if(A) B; <==> (not A) or B`.

  This is actually rather arbitrary! The implication is that if we confirm that it did not rain, then this odd implication is true.

  Perhaps we'd rather say: "you would not fly anyway!"

  That would maybe render the implication false.

  What about an implication that is always false when the condition is false? This would be arbitrary aswell.

  Though this may be a moot point, we may conclude the formal implication is not the same as the implication we use in natural language.

  There is no justification given for any of this.[2] It was simply deemed useful to take a conditional as true when we can't say otherwise.

  It's useful in our programs to say,

  `If the switch is set, our program will beep.`

  And, if the condition doesn't hold, simply go on with our program. We're going by the assumption that our machinery is correct.

  [1] See reif or `(->;)`. [2] One may consult the Principia Mathematica for a definition and explanation of the logical implication and see that, once again, it's an arbitrary definition.

Whitespace

It's about time we explained our use of whitespace and indendation.

Any major "structure" such as rel and if are delimited by an increase in whitespace (generally, comprised of four spaces or a single tab character).

if(true)
        false
else
        false

The code that comprises the if- and else-parts are evident through this method.

An error is issued if there's any inconsistency. For example, if you chose the first indent to be four spaces but the second to be three.

As long as you make a consistent rule, i.e. four spaces or one tab-only, there should be no issues.

Syntax-wise, whitespace only adds ands to the end of lines (unless a keyword like else or case pops up) and ends any unindent with a semicolon.

As such, the semantics of whitespace are very easy to understand.

Coming from a procedural language, it may seem odd that a semicolon is used to end a structure.

Still, this is in line with the syntax of most LP language.

2D. Data

Functors

We start with a declaration.

functor(F, Functor)

By using the special relation functor, we declare that F is a functor, or rather, it's what will be used to make functors. We can now make functors with the name/label F.

F(1, 2) = F(1, a)
print(a) //2

Functors are a kind of composite data. They are what some languages call a tuple.

If 1 or 'hello' is a single value, F(1,'hello') is a value made from combining both.

We made two functors. Since they are equal to each other, we can also conclude that its values are. Hence, a=2.

A Practical Example

Let's say we want to represent a person in our program. We could just make a functor Person.

functor(Person, Functor)

We can now make our first person.

bob = Person('bob', 23)

We have made the convention that the first two fields of Person stand for name and age, though if needed we could have included more fields. This allows us to make programs about one or more persons!

This is not the only way we could represent a person, of course. Cosmos™ has objects aswell, which are very similar in that sense! We'll cover them later.

Tuple

If we simply to group things together, any functor is enough. We may use a F or Tuple functor for this.

> Tuple(x,1) = Tuple('x', y)
| x='x' and y=1

Lists

A list is another kind of composite data. Let's say we want a list of things. This can easily be done with the syntax,

l = [1, 2, 3]

We've made a list with the values 1, 2 and 3.

There are operations we can make on lists. We can add, subtract or search for elements within it. These can be found in module list.

l2 = list.push(l, 55)
io.writeln(l)  //[1, 2, 3]
io.writeln(l2) //[1, 2, 3, 55]

If we push value 55 into l, we'll get a new list l2 with elements.

l = [1,2,3]
list.first(l, head) //head is 1
list.rest(l, tail) //tail is [2, 3]
l=[head|tail]

A common operation is to get the first element in a list, or the _rest_- the remaining, which itself is a new list. These are sometimes called the head and tail of a list.

This is a common pattern in declarative programming, and as such there's even a special syntax for it.

l=[head|tail]

The term [x|y] is very different from [x,y]- the latter is a list with two elements. The former is any list.

Lists as Functors

Lists are often implemented in terms of the functor Cons. These lists are identical:

l = [1, 2]
l = Cons(1, Cons(2, Cons))

2E. Modes

Functions [Incomplete] --

_"A function is a less general kind of relation." - The Cosmos Programming Language_

While a relation may give multiple answers-- or no answer at all (in which case the statement is taken to be false), a function is guaranteed to only give one.[1]

<!--Consider double.-->

[1] A function here is meant to mimic a typical programming function that you often see in procedural or functional languages. It's not necessarily a mathematical function. For example, it may accept more than one input.

A Factorial Function --

There's no clear rule on what should or shouldn't be a function.

We'll take factorial as a good case.

• It's often a function in mathematics.
• It may run multiple times. As such, it often benefits from whatever optimizations we may give it.
• You don't usually need anything more than simply getting its result.

  fun fact(x,y)
          if(x=0)
                  y=1
          else
                  y=x*fact(num(x-1))

  Mixing functions with relations --

  The language tries its best to accomodate both types of programming. When using a relation, if is a logical conditional and will behave soundly to the best of the language's ability. When using a function, if is allowed to use lower-level predicates, as there are way less requirements that a function needs to meet.

  fact(2,y)

  Even if "low-level LP" operators are present, they're subsumed by the use of functions. Although this may perhaps be considered leaving the paradigm, it's much cleaner than using the aforementioned operators. For what it's worth, functions themselves frequently appear in formal logic. This also opens up the possibility of using Cosmos as a functional language.

  At least, the language is not pretending something is a logical predicate when it isn't. When you do use a logic predicate--that is, a rel, the language will take it that you're programming in the logic paradigm, so as to make pure logic programming possible and easy.

  It's purposefully easy to switch between those and compare their behaviour, as one only has to change the appropriate keyword.

  A Main Function --

3. A "Prolog" tutorial

A common way to introduce logical languages is by emphasizing its relation to logic. Although it's not very practical, it's as good as any tutorial or introduction nonetheless.

Let's make it clear that,

• A program is a set of relations.
• These relations are then queried.

  rel human(x)
          x='socrates' or
          x='plato'
          
  rel mortal(x)
          human(x)

  This is easily translated to formal or informal logic,

  Socrates is human.
  Plato is human.
          
  If x is human, then x is mortal. (aka All humans are mortals).

  Or...

  human('socrates')
  human('plato')
  mortal(x) <- human(x)

  Although we converted our definition of mortal to a conditional, it's a conditional going the opposite way. The body of our relation, human(x), implies mortal(x).

Querying

We can then make a query to the interpreter,

> human(x)
| x='socrates'
| x='plato'
> human('socrates')
| true
> mortal('socrates')
| true
> human('bill')
| false

Naturally, no 'bill' exists in our program. We have not included Bill in our definition, so that either Bill is irrelevant to our program, not human, or our definition is flawed.

The query human(x) is akin to asking "Who is human?" Because x is a free variable in human(x), Cosmos™ will try to find valid matches-- what is a possible value for x--in other words, who is human in our program.

A peculiarity

When we define a relation, it's a complete definition.

When we defined human(x), we stated that a human really is either socrates or plato.

As far as our program is concerned, these are the only humans out there.

We chose to use String to represent our humans.

The string 'socrates' is of course not the same as the living Socrates.

Possible worlds

Logic is a tool for reasoning about both real, hypothetical or fictional worlds. Being real is not necessary. We can reason about possibilities aswell.

We dare say Cosmos™ is a good way to try reasoning with possible worlds, a relatively new formalism in logic.

t={
        rel human(x)
                x='socrates'
                
        rel mortal(x)
                human(x)
}

t2={
        rel human(x)
                x='alice'
                
        rel immortal(x)
                human(x)
}

We've made a different possible world t2 where Alice resides as an immortal. Such high-level theorizing is in the realm of possibility for the Cosmos™ programming language.

> t.human(x)
| x='socrates'
> t2.human(x)
| x='alice'

Using case or when --

As explained before, using case and when is a good way to avoid writing a lot of and/or statements and make our definitions concise.

This was not necessary for human(x), but it surely is for likes(x).

rel likes(x,y)
        when(x='alice')
                y='cake'
        else
                x='abbe'
                y='pastry'

Pure Logic Programming --

Most important for our language is the notion of a pure logic program.

Logic programs allow us to effectively write logical statements and have the language work them out correctly.

Writing pure relations

• Any relation made of pure relations and operators is pure.
• =, and, or are pure.
• not, if and regular arithmetic are pure. Knowing you have written a pure relation is quite simple. Both human and likes are pure because we have only used pure operators-- mainly, and/or and =.

  mortal is pure because it calls human and nothing else. human is pure, therefore mortal is pure.

3B. Pure Functional Programming

Functions [Incomplete] --

_"A function is a less general kind of relation." - The Cosmos Programming Language_

In Cosmos, we consider a function to be a more constrained type of relation.

It's for this reason that we have functions. A relation has a much broader behavior. A programmer who is not ready to tackle all the possible behavior of a relation, may prefer to start off with a function.

Using functions then is a matter of simple restraint. A programmer simply has to keep in mind what it is.

(1) A function receives an input and produces output.

• A function has one or more input parameters and a single output parameter, conventionally the last, typically. In factorial(2,y) and double(4,x), the input given is 2 and 4 while the output is y and x.

  (2) A function generates one output.

• The function mode (or fun keyword) ensures this. While a relation may give multiple answers-- or no answer at all (in which case the statement is taken to be false), a function is guaranteed to only give one.

  A Factorial Function --

  fun fact(x,y)
          if(x=0)
                  y=1
          else
                  y=x*fact(num(x-1))

  A programmer making a function should use it as a function. When making a fact function, we then decide that x should be the input and y the output. This is where most non-logical languages would have a return statement.

  When calling fact(2,y), a function will correctly receive 2 as input and work out y--the output or "return" value. It's even possible to try the reverse.

  As it's still a relational language, fact(x,2) will also work-- that is, computing the "reverse factorial". Two functions by the price of one.

  However, fact(x,y) is nonsense. This can only be interpreted as a statement, one that may yield multiple answers. The function mode is only made to support functional behavior, and will not handle this.[1]

  At this point, one should switch to relations. This can be done by changing fun to rel. Fortunately, fact still works well as a relation.

  [1] This is not completely enforced by the Cosmos™ compiler, as of yet. As such, we ask to have some caution when using it.

  Conclusion --

  One can work a functional program by simply determining which variable is the input and which is the output. In x=fact(2), 2 is an input and the answer returned is x.

  Knowing this, one might retrace the steps.

• 2 is compared to 0. That fails, hence we go to the else-part.
• By substitution, the output is y=2*fact(num(2-1)).
• fact is called again, etc.

Styles

A lot of this may be better off understood as styles.

• Use function syntax, despite them being relations. Pure Functional Programming --

  A function may not be pure in the logical sense-- however, it may be pure in the functional sense.

  You'll often hear about purity in functional languages, which have no relations. That refers to functional purity.

  A function is not pure when it does anything un-function-like, i.e. anything that is not receiving and returning values.

  fun write(x,y)
          if(x=0)
                  y=1
                  print(x)
          else
                  y=2

  Multiple inputs and outputs --

  A simple function like fact has two parameters--one input and one output. This is pretty straightforward and is generally how a function behaves in mathematics.

  Modern and even non-modern languages have a different understanding. A programing function may have multiple inputs, or even outputs.

  fun fn(x,y,s,s2)
          if(x=0)
                  y=1
          else
                  y=2
          if(s='')
                  s2='a'
          else
                  s2='b'

  Here, fn will take x and y as inputs and--depending on their values, make a value for y and s2.

3A. Paradigms

Typically, programming is divided into four main paradigms,

• Procedural (imperative)
• Functional
• Logic -- Logic-procedural (?!)
• Object-oriented Although this classification is increasingly dated, it's still used (and somewhat useful) to this day.

Logic

x=double(1)
y=x
print(y)

In the logic paradigm, or logic programming (LP), which we follow to the letter, as our language is in fact a logic programming language, this is simply a set of statements.

• The double of 1 is x.
• The variable of y equals x.
• The value of y is written on the screen. It's easy to see by looking at the program that this applies. We may simply look at the program and see that x equals the result of a function double, y is x, and the value is written (which we know the statement print does).

  We only have to take a moment to conclude that 2 will be written if we run the program, and it is.

  2

Procedural

If a paradigm is a way to look at a given program, the procedural paradigm looks at it imperatively. A program is a recipe, or list of instructions to be executed by the computer.

This paradigm follows the computer program closely. It's the closest to how the computer actually acts.

The following instructions would be executed.

1. A procedure or command double computes the double of 1.
2. This value is given to x. A variable in the procedural paradigm is like a slot. In fact, it's a slot of computer memory and occupies a space in the computer.
3. The value in x is then assigned to the slot y.
4. The command to write the statement on the screen, print, is given. The difference so far is subtle, and you could take it as a procedural program as-is. In fact, our language highly resembles procedural languages in syntax.

   This is on purpose, the language is made so that it can be reasoned or written in this style to some extent.

Logic-Procedural

Though you may often simply see the program as a series of statements in the logic paradigm, it's sometimes needed to know how the computer executes them.

After all, the program runs in a computer.

We'll call this the "Logic-Procedural" paradigm (though it's a made-up term we invented). The Logic-Procedural interpretation of a program is given by how our procedural logic engine executes the logic program.

Often, the only reason we need to know this is so our program has good performance. An efficient program will run as quickly as it can and not consume many memory slots.

Limitations

3B. Pure Logic Programming

Most important for our language is the notion of a pure logic program.

Logic programs allow us to effectively write logical statements and have the language work them out correctly.

Writing pure relations

• Any relation made of pure relations and operators is pure.
• =, and, or, not, if and regular arithmetic are pure. Knowing you have written a pure relation is quite simple.

  All you have to do is use pure relations! Then, your relation is also pure.

4B. Optimizations

if

Because it's the language's main conditional, any number of optimizations may be applied to it as long as it keeps its properties as a conditional (while the operator when is kept for fine-tuning; if you want a minimal conditional to which you can apply your own optimizations).

Currently,

• Any condition is negated.

when and choose

when is a sound but more barebones conditional. Implementing a pure factorial relation in when makes for a good case study.

when is a pure conditional. Still, since it requires you to manually negate the condition (which is redundant otherwise) and may not provide future optimizations, unless you implement them yourself, it's less preferred and not the default go-to for conditionals.

Remember that relations may backtrack or yield more than one solution. Even if a factorial relation succeeds the first time, an ill-defined program may cause on the second solution.

Let's compare this to,

rel fact(x,y) when(x=0) y=1 else y=x*fact(num(x-1))

... or ...

(x=0 and y=1) or (y=x*fact(num(x-1)))

While fact(1,x) would still work, it would for the first solution alone. Remember that relations may backtrack or yield more than one solution.

This would suffice were it a regular imperative or functional definition of factorial, since those are meant to run only once and never backtrack.

However, this is not so for a logical language and different reasoning is needed. If it ever tries to find a second solution, it will procceed to the second clause. The reader is invited to keep this in mind and accompany the execution of fact(1,x).

rel fact2(x,y) when(x=0) y=1 else x>0 y=x*fact2(num(x-1))

As you see, we have,

• Manually negated x=0, by typing x>0. The program now works soundly, even if it's not perfect! It still creates places to backtrack or choice points. This consumes memory, moreso if we call when a lot of times. But at least these are eliminated upon hitting x>0.

  ==

  What then if we simply removed backtracking?

  choose is a non-logical conditional that doesn't backtrack.

  As such, it's fit for code with side-effects. A simple prompt, for example.

  rel prompt() print('type a number') io.write('> ') io.read(x) choose(x='5') print('You typed 5!') else print('You didn't type 5!?') prompt()

  Such code is non-logical in the first place- and has no need for backtracking.

  The safest option is still to rely on if. It should never be unsafe to use if. For cases it can't be used, you'll invariably be given a warning or error at least. However, choose and when are available if you're aware of the limitations.

TCO (Tail-Call Optimization)