Haskell – How does non-strict and lazy differ

I often read that lazy is not the same as non-strict but I find it hard to understand the difference. They seem to be used interchangeably but I understand that they have different meanings. I would appreciate some help understanding the difference.

I have a few questions which are scattered about this post. I will summarize those questions at the end of this post. I have a few example snippets, I did not test them, I only presented them as concepts. I have added quotes to save you from looking them up. Maybe it will help someone later on with the same question.

Non-Strict Def:

A function f is said to be strict if, when applied to a nonterminating
expression, it also fails to terminate. In other words, f is strict
iff the value of f bot is |. For most programming languages, all
functions are strict. But this is not so in Haskell. As a simple
example, consider const1, the constant 1 function, defined by:

const1 x = 1

The value of const1 bot in Haskell is 1. Operationally speaking, since
const1 does not "need" the value of its argument, it never attempts to
evaluate it, and thus never gets caught in a nonterminating
computation. For this reason, non-strict functions are also called
"lazy functions", and are said to evaluate their arguments "lazily",
or "by need".

–A Gentle Introduction To Haskell: Functions

I really like this definition. It seems the best one I could find for understanding strict. Is const1 x = 1 lazy as well?

Non-strictness means that reduction (the mathematical term for
evaluation) proceeds from the outside in,

so if you have (a+(bc)) then first you reduce the +, then you reduce
the inner (bc).

–Haskell Wiki: Lazy vs non-strict

The Haskell Wiki really confuses me. I understand what they are saying about order but I fail to see how (a+(b*c)) would evaluate non-strictly if it was pass _|_?

In non-strict evaluation, arguments to a function are not evaluated
unless they are actually used in the evaluation of the function body.

Under Church encoding, lazy evaluation of operators maps to non-strict
evaluation of functions; for this reason, non-strict evaluation is
often referred to as "lazy". Boolean expressions in many languages use
a form of non-strict evaluation called short-circuit evaluation, where
evaluation returns as soon as it can be determined that an unambiguous
Boolean will result — for example, in a disjunctive expression where
true is encountered, or in a conjunctive expression where false is
encountered, and so forth. Conditional expressions also usually use
lazy evaluation, where evaluation returns as soon as an unambiguous
branch will result.

–Wikipedia: Evaluation Strategy

Lazy Def:

Lazy evaluation, on the other hand, means only evaluating an
expression when its results are needed (note the shift from
"reduction" to "evaluation"). So when the evaluation engine sees an
expression it builds a thunk data structure containing whatever values
are needed to evaluate the expression, plus a pointer to the
expression itself. When the result is actually needed the evaluation
engine calls the expression and then replaces the thunk with the
result for future reference.
…

Obviously there is a strong correspondence between a thunk and a
partly-evaluated expression. Hence in most cases the terms "lazy" and
"non-strict" are synonyms. But not quite.

–Haskell Wiki: Lazy vs non-strict

This seems like a Haskell specific answer. I take that lazy means thunks and non-strict means partial evaluation. Is that comparison too simplified? Does lazy always mean thunks and non-strict always mean partial evaluation.

In programming language theory, lazy evaluation or call-by-need1 is
an evaluation strategy which delays the evaluation of an expression
until its value is actually required (non-strict evaluation) and also
avoid repeated evaluations (sharing).

–Wikipedia: Lazy Evaluation

Imperative Examples

I know most people say forget imperative programming when learning a functional language. However, I would like to know if these qualify as non-strict, lazy, both or neither? At the very least it would provide something familiar.

Short Circuiting

f1() || f2()

C#, Python and other languages with "yield"

public static IEnumerable Power(int number, int exponent)
{
    int counter = 0;
    int result = 1;
    while (counter++ < exponent)
    {
        result = result * number;
        yield return result;
    }
}

–MSDN: yield (c#)

Callbacks

int f1() { return 1;}
int f2() { return 2;}

int lazy(int (*cb1)(), int (*cb2)() , int x) {
    if (x == 0)
        return cb1();
    else
        return cb2();
}

int eager(int e1, int e2, int x) {
    if (x == 0)
         return e1;
    else
         return e2;
}

lazy(f1, f2, x);
eager(f1(), f2(), x);

Questions

I know the answer is right in front of me with all those resources, but I can't grasp it. It all seems like the definition is too easily dismissed as implied or obvious.

I know I have a lot of questions. Feel free to answer whatever questions you feel are relevant. I added those questions for discussion.

• Is const1 x = 1 also lazy?
• How is evaluating from "inward" non-strict? Is it because inward allows reductions of unnecessary expressions, like in const1 x = 1? Reductions seem to fit the definition of lazy.
• Does lazy always mean thunks and non-strict always mean partial evaluation? Is this just a generalization?
• Are the following imperative concepts Lazy, Non-Strict, Both or Neither?
  • Short Circuiting
  • Using yield
  • Passing Callbacks to delay or avoid execution
• Is lazy a subset of non-strict or vice versa, or are they mutually exclusive. For example is it possible to be non-strict without being lazy, or lazy without being non-strict?
• Is Haskell's non-strictness achieved by laziness?

Thank you SO!