Conditionals allow a program to react to the data it is presented with. This can be data from the outside world, or data from some other module internal to a program composed out of modules (i.e. subroutines, for example). Without conditionals - or their moral equivalent - programs would not be able to react to their inputs or to changes in their internal state.

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There are Turing-complete languages without conditionals, though - even without data, as such! An example is the lambda calculus. If you want to learn more about this, read up on "Church encoding" and "Scott encoding".

The basic idea is to use functions and the visitor pattern to encode a kind of conditional behaviour that varies with the input supplied to it.

  True = (lambda (t f) (t))
  False = (lambda (t f) (f))
  If = (lambda (b tr fa) (b tr fa))

  (lambda (x)
    (If x
        (lambda () (display "It was true!"))
        (lambda () (display "It was false!"))))

Desk check the reductions of applying that function to True and to False. The "t" and "f" arguments to the boolean values act as a kind of "menu" (the visitor pattern), from which the datum selects in order to communicate what variant of data it is. The pattern extends readily to natural numbers, lists, and so on, giving a straightforward encoding of not only conditionals, but also (one-layer-deep) pattern matching.

> There are Turing-complete languages without conditionals, though...

> ... giving a straightforward encoding of not only conditionals, but also (one-layer-deep) pattern matching.

So you wrote a conditional out of something that didn't have one? But for that to work, something in the "If" has to decide whether to evaluate the first one or the second one. That is, there is something that is equivalent to a conditional somewhere in the parts that you use to build the "If".

This may come down to what the definition of "conditional" is, though...

Also, say I implement the visitor pattern in some language on a classical machine - aren't there branch instructions being used under the covers anyway?

(the code will probably be more optimized / easier to optimize for the compiler but you get my point)

Yes, because the machine language of ordinary CPUs uses a specific model of computation where branch instructions are the only way to effect conditional control transfer. But the idea of conditional control transfer is there in every interesting programming language to allow a program to react to its inputs.

this is interesting - thanks for your response.

I'm having a hard time understanding the program you supplied though, the subsequent description did not really help (probs more my fault than your own). I certainly understand the visitor pattern (and have had a sneaky suspicion it is important) but the lambda calculus, not so much.

What type of syntax is the program you supplied in - is it more of a pseudocode or an actual formal language?

thanks for the search keywords - definitely helped - if you have a favorite textbook/arxiv reference/author on this stuff, those would be greatly appreciated as well.

The syntax is concise definition of named values where the values are all anonymous functions.

True = (lambda (t f) (t))

-- Let True be the function that takes two parameters t and f and returns the first, t. --

False = (lambda (t f) (f))

-- Let False be the function that takes two parameters t and f and returns the second, f. --

If = (lambda (b tr fa) (b tr fa))

-- Let If be the function that takes three parameters, and applies the first parameter b to the second and third parameters. --

Notice that if the parameter b is the True value above, 'b tr fa' will return 'tr'. If 'b' is False, 'b tr fa' will return 'fa'

  (lambda (x)
    (If x
        (lambda () (display "It was true!"))
        (lambda () (display "It was false!"))))

So it follows that this will pass either "It was true!" or "It was false!" to 'display', waving aside all the stuff you have to define that makes 'display' and STRINGS work as you'd expect.

This is where it becomes clear that you need a lazy system for this to make sense. In an eager system all three of the parameters to 'If' are evaluated and then their values are substituted into the body of 'If'. In a lazy system the expressions for the values are substituted for the parameters in the body, so only the parameter 'b' and one of the other parameters will be evaluated.

When I play around with functional languages I would just write

  True t f = t
  False t f = f
  If b tr fa = b tr fa
  ShowIt x = If x (display "It was true!") (display "It was false!")

You could also write this in (I think this is) Scheme:

  (define True (t f) (t))
  (define False(t f) (f))
  (define If (b tr fa) (b tr fa))
  (define ShowIt (x) (If (x (display "It was true!") (display "It was false!"))))

Except that Scheme isn't usually lazy.

Yes, this is right except that the language as written is eager Scheme. You don't need a lazy system. The second and third arguments to If are nullary procedures. Their effects are delayed until invoked by whichever of True or False is supplied. Scheme's notation for invoking a nullary procedure -- `(t)`, vs `t` which just refers to the procedure -- doesn't help matters here :-)

The syntax is close to Scheme and with small changes will run in a Scheme interpreter. For example, here's a session at the REPL of Racket:

  ~$ racket
  Welcome to Racket v7.2.0.6.
  > (define True (lambda (t f) (t)))
  > (define False (lambda (t f) (f)))
  > (define If (lambda (b tr fa) (b tr fa)))
  > (define P (lambda (x)
                (If x
                    (lambda () (display "It was true!"))
                    (lambda () (display "It was false!")))))
  > (P True)
  It was true!
  > (P False)
  It was false!
  > 

Seen differently, it's also close to the mathematical notation of the lambda calculus.

To learn about the lambda calculus, check out chapter 5 of "Types and Programming Languages", Benjamin C. Pierce, MIT Press 2002. You might also get something out of "Semantics Engineering with PLT Redex", Felleisen, Findler & Flatt, MIT Press 2009.

Oh, see also https://en.wikipedia.org/wiki/Mogensen%E2%80%93Scott_encodin... for a quick overview of Church and Scott encodings. (Ignore the stuff about Mogensen-Scott. Also, you'll need to have played around with lambda calculus a bit for this stuff to make sense. This kind of thing might be helpful: https://jacksongl.github.io/files/demo/lambda/index.htm)

There are styles of programming with no conditionals, especially in purely functional languages.

A common technique inside compilers is to separate out which code gets run, and which result gets used. If everything is side-effect-free, then there's no harm in running the code whose result won't get used. In highly parallel systems, you can just run both versions and select at the end. So you can rewrite:

  if foo < 2:
    a = foo
  else
    a = 2

to:

  a = phi(foo<2, 2, foo)

(the phi operator is taken from SSA notation. See https://llvm.org/docs/LangRef.html#phi-instruction if you're unfamiliar)

I don't need to understand that code to tell you there is a control flow mechanism somewhere using a condition.

Some operations are basic enough that theoretically you can build any other operation out of them. “NAND” is commonly used as an example but in practice “nand” doesn’t really map to how people think. Conditionals are another one. They are a bit more complicated in a sense (three inputs rather than two) but they map better to how people think so people actually use them in practice.

So Turing completeness is basically two things. 1: the ability to compose an arbitrary number of conditional statements. 2: the ability to keep computing recursively, until some output is true.

Without conditionals, every program either terminates after a pre-specified amount of computation or never terminates. That leaves you extremely limited in what you can accomplish (less powerful than a DFA).

A DFA or maybe a 2DFA (which are equally powerful) is basically what you get when all you have is conditionals.

Without that, what you have is a box that can read finite input and compute a value. Equivalent to a finite lookup table.

It's not as much of a limitation as people think. One pattern is for the program to update a state every (fixed-length) iteration. You can build any arbitrary computation on top of that.

Can you provide an example? I have a hard time believing any arbitrary computation can be build without conditionals.

I suspect they are thinking about things like cellular automata.

If you squint just right, you may think there are no conditionals. Just straight-line data flow to compute cell values based on a fixed neighborhood of the preceding state. A big fluid dynamics simulation has a similar characteristic.

However, this squinty interpretation can just as simply show you that the whole thing is just a DFA, as mentioned a few posts up! The finite mutable register just encodes the name of each state in the state machine, and each simulation cycle is just computing the next state transition within that large but finite state space.

But cellular automata have conditionals. That is the entire point of having a state in each cell and then doing something based on that state.

I suspect the confusion in this thread really boils down to a lack of agreement about what "conditional" means. I think some commenters above think it means conditionally branched instruction streams. They don't consider arithmetic or logical operators, lookup tables, and demuxers to be conditionals.

The earlier description of a cycle of functional updates to state sounds like the classic basis for latched digital circuit design. But, assuming the commenters were interested in software examples, I brought up analogous programs for cellular automata and grid simulations.

Also, since the state transition is purely functional with a finite input and output, it could theoretically be implemented with any functionally equivalent method, including flat lookup tables. No branching or conditional execution is required, even though it might be convenient.

Yeah true but I wouldn't call storing every possible result of a system in for instance a lookup table a computation. That is just precomputing all the results and then running a lookup.

If I have a lookup table of all the values of the sine and then use that lookup table to get the result of sine for specific value I haven't computed sine, I just looked it up.

I'm not sure if we're actually debating anything at this point...

A cellular automaton or other grid simulation has modularity which means that the state space for one cell is relatively small. So, you might admit that the whole thing is doing computation without caring how the cell transition function is implemented?

But, lookup tables are at one extreme, and there are plenty of other design techniques for arbitrary functional computation that avoid conditional branching. At the digital design level, you can think of a continuum of logic gates where ROMs, mux/demux, or even ALU operations all have elements that can be seen as lookup-table or computation. The difference is in the observer's mental model, more so than in actual gate structures.

At the software level, many instructions can be used for demultiplexing. So, you can compute two different potential values, e.g. in separate registers, and select between results without branching. You could use a conditional-store instruction, or some kind of swizzle/pack instruction, but without any of that you could compute a binary test result, extend it to a whole register width, and use bitwise logical operations to combine both registers while effectively selecting one or the other. The essence of whether it is computation or conditional selection is in the eye of the beholder.

This is a common way to transform some code paths for vectorization, e.g. to generate one common SIMD instruction stream that has the effect of performing different optional computations in each lane, while actually running the same instructions on all lanes.

Conditionals actually destroy information. The way they are written does not even matter; Consider the evaluation of this functional pseudocode:

   min(a,b) := if(a < b, a, b)

Obviously, the evaluation of this function looses information about either a or b. This is of course by intention. But this is incompatible to a closed (quantum) system where information must be preserved.

Interestingly (but I think there is no deeper connection), vectorization uses masks to avoid branching. Programmatically, one implements then both branches at the same time, and on a per-data level different operations are performed. A simple pseudo-numpy example would be

   B[where(A < B)] = A
   which should actually read component by component, for
   instance, having A and B being matrices,

      B_{ij} = A_{ij}  if A_{ij} < B_{ij}
               B_{ij}  else  ("no-op")

One can write long programs following this paradigm which frequently excel at performance at GPUs and modern day CPUs (which are actually vector machines).

thanks for tying it back into quantum - also for the clarification that the way you write them doesn't matter - whether that be in terms of if/else, switch statements or actual CPU-level branch instructions.

Are you not approaching it a bit too much from a high level perspective?

From assembly(x86), most conditionals and loops break down to cmp/test and jmp,jx,jnx type instructions.

Computers need to make decisions based on comparisons of data or program state. You can abstract decision making however you want (e.g.: switch statements in place of if/else or usage of goto's) but the computer needs to test or compare and decide on what instruction to execute based on the result.

Let me invent a new condtional 'alot':

for i in range(0,100): alot i: exit(1)

instead of testing for boolean true/false. My conditional "alot" tests the result of the evaluation against the value range of the type,if the result is over half of the value range it executes the conditional statement (exit())

Of course I am being a bit silly but my point is my silly conditional much like if/else breaks down to comparing and jumping based on evaluation (i.e: instead of a jz/jnz it will break down to jge)

To simplify, software (generally) does this:

    input >>> output

By definition the output is conditional on the input, at least if the program works correctly. That's why conditional statements are important - they express the relationship between input and output.

Isn't this really basic stuff? Or am I missing something?

You can have output that’s conditional on the input without having conditional statements.

How is the output of the following not conditional upon the input? Are there any conditional statements in it?

F(x, y) = x + y

> You can have output that’s conditional on the input without having conditional statements.

Yes, of course you can.

mind bottling stuff here man thanks

In a trivial way, logical gates (AND, OR, etc) are conditionals. So, we need these gates to build any logical expressions. And many problems can be reduced to satisfiability, etc.

Taking in input, and making decisions based on that input, are what computers are for. And conditionals are the simplest version of "making a decision".

Most computer programs are, in practice, difference amplifiers. Conditionals are how they achieve that.

("Most" because you have programs like image classifiers that are trying to be similarity amplifiers. That turns out to be hard to do with the standard building blocks, which want to be difference amplifiers. Perhaps different building blocks would be more useful.)

thought you were going to tie it back into transistors for a second there... difference amplifiers are, after all, one of the simplest circuits out there. Not so sure about similarity amplifiers

It is possible to approximate logical expressions without conditionals. This is an old problem in neural networks. See more here: https://towardsdatascience.com/emulating-logical-gates-with-...

I apologize, but I really am having a hard time understanding your question at all. Every non-trivial computation requires some kind of conditional.

The first computers were literally people, they computed. There wasn't really a lot of "conditional" work to what they did, they did math (principally numerical work) like generating tables of data and compiling the results. They still needed some conditionals, like if a value was over a certain threshold, use a particular formula or change some parameter.

The early mechanical computers were largely tabulators of the same sort, but machines and not people. But as you want to express more complicated computations, you need the ability to branch. Conditionals are just a means for executing such a branch. Even the simplest of computational machines (in the theory sense) like regular expressions and deterministic finite automata include a branch or conditional element: Given input C while in state S go to state S'. A simple way to visualize this decision matrix is with a table:

  Input | S_0 | S_1
  -----------------
    0   | S_0 | S_0
    1   | S_1 | S_1
  Accepting states = {S_0}

Is a simple machine that will recognize any binary string ending in '0'. In each state, a decision is made on what the next state should be based on the input. This isn't even a Turing complete system, DFAs can only recognize regular languages. This cannot be done without some sort of conditional branching capability. You may be able to encode this in a hardware description language such that the conditional is obscured or obfuscated, but it's still there.

Even trying to compute taxes or payroll depended on conditionals (income level determines which tax brackets you pay into, over a certain amount of income you've maxed out Social Security). And that was one of the earliest (commercial) uses for computational devices.

Now, the way we express conditionals may change from one language or system to another. In Erlang, you can remove a lot of `if` expressions using pattern matching, consider the simple computation of the Fibonacci numbers as an example:

  fib(0) -> 1;
  fib(1) -> 1;
  fib(N) -> fib(N-1) + fib(N-2).

A conditional is right there: If N = 0, return 1; if N = 1, return 1; else return fib(N-1)+fib(N-2). Haskell and other functional languages use similar constructs, and you can include other guards:

  fib(0) -> 1;
  fib(1) -> 1;
  fib(N) when N > 1, is_integer(N) -> fib(N-1) + fib(N-2).

This removes the potential for infinite recursion by adding an additional guard, now this function is only valid for non-negative, integral inputs. It has been transformed now to: If N = 0, return 1; if N = 1, return 1; if N > 1 and N is an integer, return fib(N-1)+fib(N-2); else signal an error.