A Turing recognizable language is decidable or not? - turing-machines

Can a Turing recognizable language be decidable if it is possible to enumerate its strings in non-decreasing length?
I think it's not because you can go to infinity, and this will make it undecidable right?

The question is about searching for an element in an infinite stream S of ordered elements: a very natural algorithmic question.
This problem is indeed decidable, although in a somewhat sneaky way. You need to reason by cases. If elements in S have an upper bound, then S is finite, and hence it is decidable since every finite set is decidable. On the other side, if S has no bound, it contains elements larger than any number. So, if you are looking for w, it is enough to enumerate until you find an element larger or equal to w (that must exists), and then compare it with w.
However, the proof is not constructive, since you cannot decide if a r.e. set is finite or not. This means that you know (in classical sense) that there must exist some program deciding S, but you have no way to derive it from the code enumerating S. A quite frustrating situation :)

Related

Coq QArith division by zero is zero, why?

I noticed that in Coq's definition of rationals the inverse of zero is defined to zero. (Usually, division by zero is not well-defined/legal/allowed.)
Require Import QArith.
Lemma inv_zero_is_zero: (/ 0) == 0.
Proof. unfold Qeq. reflexivity. Qed.
Why is it so?
Could it cause problems in calculations with rationals, or is it safe?
The short answer is: yes, it is absolutely safe.
When we say that division by zero is not well-defined, what we actually mean is that zero doesn't have a multiplicative inverse. In particular, we can't have a function that computes a multiplicative inverse for zero. However, it is possible to write a function that computes the multiplicative inverse for all other elements, and returns some arbitrary value when such an inverse doesn't exists (e.g. for zero). This is exactly what this function is doing.
Having this inverse operator be defined everywhere means that we'll be able to define other functions that compute with it without having to argue explicitly that its argument is different from zero, making it more convenient to use. Indeed, imagine what a pain it would be if we made this function return an option instead, failing when we pass it zero: we would have to make our entire code monadic, making it harder to understand and reason about. We would have a similar problem if writing a function that requires a proof that its argument is non-zero.
So, what's the catch? Well, when trying to prove anything about a function that uses the inverse operator, we will have to add explicit hypotheses saying that we're passing it an argument that is different from zero, or argue that its argument can never be zero. The lemmas about this function then get additional preconditions, e.g.
forall q, q <> 0 -> q * (/ q) = 1
Many other libraries are structured like that, cf. for instance the definition of the field axioms in the algebra library of MathComp.
There are some cases where we want to internalize the additional preconditions required by certain functions as type-level constraints. This is what we do for instance when we use length-indexed vectors and a safe get function that can only be called on numbers that are in bounds. So how do we decide which one to go for when designing a library, i.e. whether to use a rich type with a lot of extra information and prevent bogus calls to certain functions (as in the length-indexed case) or to leave this information out and require it as explicit lemmas (as in the multiplicative inverse case)? Well, there's no definite answer here, and one really needs to analyze each case individually and decide which alternative will be better for that particular case.

What is the Pumping Lemma in Layman's terms?

I saw this question, and was curious as to what the pumping lemma was (Wikipedia didn't help much).
I understand that it's basically a theoretical proof that must be true in order for a language to be in a certain class, but beyond that I don't really get it.
Anyone care to try to explain it at a fairly granular level in a way understandable by non mathematicians/comp sci doctorates?
The pumping lemma is a simple proof to show that a language is not regular, meaning that a Finite State Machine cannot be built for it. The canonical example is the language (a^n)(b^n). This is the simple language which is just any number of as, followed by the same number of bs. So the strings
ab
aabb
aaabbb
aaaabbbb
etc. are in the language, but
aab
bab
aaabbbbbb
etc. are not.
It's simple enough to build a FSM for these examples:
This one will work all the way up to n=4. The problem is that our language didn't put any constraint on n, and Finite State Machines have to be, well, finite. No matter how many states I add to this machine, someone can give me an input where n equals the number of states plus one and my machine will fail. So if there can be a machine built to read this language, there must be a loop somewhere in there to keep the number of states finite. With these loops added:
all of the strings in our language will be accepted, but there is a problem. After the first four as, the machine loses count of how many as have been input because it stays in the same state. That means that after four, I can add as many as as I want to the string, without adding any bs, and still get the same return value. This means that the strings:
aaaa(a*)bbbb
with (a*) representing any number of as, will all be accepted by the machine even though they obviously aren't all in the language. In this context, we would say that the part of the string (a*) can be pumped. The fact that the Finite State Machine is finite and n is not bounded, guarantees that any machine which accepts all strings in the language MUST have this property. The machine must loop at some point, and at the point that it loops the language can be pumped. Therefore no Finite State Machine can be built for this language, and the language is not regular.
Remember that Regular Expressions and Finite State Machines are equivalent, then replace a and b with opening and closing Html tags which can be embedded within each other, and you can see why it is not possible to use regular expressions to parse Html
It's a device intended to prove that a given language cannot be of a certain class.
Let's consider the language of balanced parentheses (meaning symbols '(' and ')', and including all strings that are balanced in the usual meaning, and none that aren't). We can use the pumping lemma to show this isn't regular.
(A language is a set of possible strings. A parser is some sort of mechanism we can use to see if a string is in the language, so it has to be able to tell the difference between a string in the language or a string outside the language. A language is "regular" (or "context-free" or "context-sensitive" or whatever) if there is a regular (or whatever) parser that can recognize it, distinguishing between strings in the language and strings not in the language.)
LFSR Consulting has provided a good description. We can draw a parser for a regular language as a finite collection of boxes and arrows, with the arrows representing characters and the boxes connecting them (acting as "states"). (If it's more complicated than that, it isn't a regular language.) If we can get a string longer than the number of boxes, it means we went through one box more than once. That means we had a loop, and we can go through the loop as many times as we want.
Therefore, for a regular language, if we can create an arbitrarily long string, we can divide it into xyz, where x is the characters we need to get to the start of the loop, y is the actual loop, and z is whatever we need to make the string valid after the loop. The important thing is that the total lengths of x and y are limited. After all, if the length is greater than the number of boxes, we've obviously gone through another box while doing this, and so there's a loop.
So, in our balanced language, we can start by writing any number of left parentheses. In particular, for any given parser, we can write more left parens than there are boxes, and so the parser can't tell how many left parens there are. Therefore, x is some amount of left parens, and this is fixed. y is also some number of left parens, and this can increase indefinitely. We can say that z is some number of right parens.
This means that we might have a string of 43 left parens and 43 right parens recognized by our parser, but the parser can't tell that from a string of 44 left parens and 43 right parens, which isn't in our language, so the parser can't parse our language.
Since any possible regular parser has a fixed number of boxes, we can always write more left parens than that, and by the pumping lemma we can then add more left parens in a way that the parser can't tell. Therefore, the balanced parenthesis language can't be parsed by a regular parser, and therefore isn't a regular expression.
Its a difficult thing to get in layman's terms, but basically regular expressions should have a non-empty substring within it that can be repeated as many times as you wish while the entire new word remains valid for the language.
In practice, pumping lemmas are not sufficient to PROVE a language correct, but rather as a way to do a proof by contradiction and show a language does not fit in the class of languages (Regular or Context-Free) by showing the pumping lemma does not work for it.
Basically, you have a definition of a language (like XML), which is a way to tell whether a given string of characters (a "word") is a member of that language or not.
The pumping lemma establishes a method by which you can pick a "word" from the language, and then apply some changes to it. The theorem states that if the language is regular, these changes should yield a "word" that is still from the same language. If the word you come up with isn't in the language, then the language could not have been regular in the first place.
The simple pumping lemma is the one for regular languages, which are the sets of strings described by finite automata, among other things. The main characteristic of a finite automation is that it only has a finite amount of memory, described by its states.
Now suppose you have a string, which is recognized by a finite automaton, and which is long enough to "exceed" the memory of the automation, i.e. in which states must repeat. Then there is a substring where the state of the automaton at the beginning of the substring is the same as the state at the end of the substring. Since reading the substring doesn't change the state it may be removed or duplicated an arbitrary number of times, without the automaton being the wiser. So these modified strings must also be accepted.
There is also a somewhat more complicated pumping lemma for context-free languages, where you can remove/insert what may intuitively be viewed as matching parentheses at two places in the string.
By definition regular languages are those recognized by a finite state automaton. Think of it as a labyrinth : states are rooms, transitions are one-way corridors between rooms, there's an initial room, and an exit (final) room. As the name 'finite state automaton' says, there is a finite number of rooms. Each time you travel along a corridor, you jot down the letter written on its wall. A word can be recognized if you can find a path from the initial to the final room, going through corridors labelled with its letters, in the correct order.
The pumping lemma says that there is a maximum length (the pumping length) for which you can wander through the labyrinth without ever going back to a room through which you have gone before. The idea is that since there are only so many distinct rooms you can walk in, past a certain point, you have to either exit the labyrinth or cross over your tracks. If you manage to walk a longer path than this pumping length in the labyrinth, then you are taking a detour : you are inserting a(t least one) cycle in your path that could be removed (if you want your crossing of the labyrinth to recognize a smaller word) or repeated (pumped) indefinitely (allowing to recognize a super-long word).
There is a similar lemma for context-free languages. Those languages can be represented as word accepted by pushdown automata, which are finite state automata that can make use of a stack to decide which transitions to perform. Nonetheless, since there is stilla finite number of states, the intuition explained above carries over, even through the formal expression of the property may be slightly more complex.
In laymans terms, I think you have it almost right. It's a proof technique (two actually) for proving that a language is NOT in a certain class.
Fer example, consider a regular language (regexp, automata, etc) with an infinite number of strings in it. At a certain point, as starblue said, you run out of memory because the string is too long for the automaton. This means that there has to be a chunk of the string that the automaton can't tell how many copies of it you have (you're in a loop). So, any number of copies of that substring in the middle of the string, and you still are in the language.
This means that if you have a language that does NOT have this property, ie, there is a sufficiently long string with NO substring that you can repeat any number of times and still be in the language, then the language isn't regular.
For example, take this language L = anbn.
Now try to visualize finite automaton for the above language for some n's.
if n = 1, the string w = ab. Here we can make a finite automaton with out looping
if n = 2, the string w = a2b2. Here we can make a finite automaton with out looping
if n = p, the string w = apbp. Essentially a finite automaton can be assumed with 3 stages.
First stage, it takes a series of inputs and enter second stage. Similarly from stage 2 to stage 3. Let us call these stages as x, y and z.
There are some observations
Definitely x will contain 'a' and z will contain 'b'.
Now we have to be clear about y:
case a: y may contain 'a' only
case b: y may contain 'b' only
case c: y may contain a combination of 'a' and 'b'
So the finite automaton states for stage y should be able to take inputs 'a' and 'b' and also it should not take more a's and b's which cannot be countable.
If stage y is taking only one 'a' and one 'b', then there are two states required
If it is taking two 'a' and one 'b', three states are required with out loops
and so on....
So the design of stage y is purely infinite. We can only make it finite by putting some loops and if we put loops, the finite automaton can accept languages beyond L = anbn. So for this language we can't construct a finite automaton. Hence it is not regular.
This is not an explanation as such but it is simple.
For a^n b^n our FSM should be built in such a way that b must know the number of a's already parsed and will accept the same n number of b's. A FSM can not simply do stuff like that.

For 2 numbers only, is floating point addition and multiplication associative?

According to Is Floating point addition and multiplication associative?, I know A+B+C would have different results with A+C+B because the result of (A+B)+C and (A+C)+B is different.
But how about if there are 2 numbers only? Does A+B have the same result as B+A?
First, the question if A+B gives the same result as B+A is not about associativity, but commutativity. To answer your question, we need to know if the order of the operands for floating point addition and multiplication influences the result (as opposed to different groupings for the associativity question).
Now, IEEE-754 floating point addition and multiplication are commutative (but not associative, as you correctly pointed out in your question).
I should probably put an asterisk here, as different NaN's (actually, their so-called payloads) which are a part of IEEE-754 can violate commutativity, but I don't think you are interested in this particular case.
There is also another question on StackOverflow that actually answers the question of commutativity for floating-point operations – but I think that the confusion between the commutativity and associativity might be addressed for this question better.

Turing machine decidability ambiguous cases

1) Is a Turing machine M that accepts the language L = {ε}, accepting no entry?
In one hand, I think it can be false because the empty word could be an entry, but in another i think this could possibly be an indecidable problem.
2) Is every Turing machine whose language is decidable stops on any input ?
Same idea, intuitively I would have say yes, due to the definition of decidable, but I don't know, something trouble me.
3) Is the language of the palindromes decidable whatever the aphabet ?
For this one, I have almost no doubt that it's False, because with Rice's Theorem we can prove that, this probleme is indecidable.
1) I am not sure how to parse this but if a TM accepts the language consisting only of the empty set, it will eventually halt-accept on a blank tape. Whether that counts as an entry or not depends on your definition of "entry". I would count it as an entry, so I would answer "no".
2) The language consisting of only the empty string is decidable. However, we can write a TM that halt-accepts the empty string only and goes into an infinite loop for all other inputs. What is meant by "whose language" is debatable but for TMs that encode partial functions I would call the language of that TM the set of strings which it halt-accepts on, so I would answer "no".
3) It seems to me that, given an alphabet with n symbols, you can always construct a single-tape deterministic TM with O(n) states which halt-accepts on palindromes over that alphabet and halt-rejects other strings, thus deciding the language of palindromes over the alphabet. I would answer "yes", as long as the terms have their usual meanings. Note that Rice's theorem does not apply; it would apply to the problem of deciding whether a TM accepts the language of palindromes over an alphabet, but actually deciding whether something is a palindrome is of course possible (PDAs do it).

VDM-SL notation for a single, finite subset

Not sure if this is within the realm of SO but:
Using VDM-SL, I have been looking around for the 'best' way of describing a single, finite subset of ℕ. In my travels I have found several ways that people are conveying this but I wonder which is the most accepted.
I initially thought that F(ℕ) would do but I believe that this is the set of finite subsets of ℕ, rather than a single subset.
Would it be enough to say, "Let S be finite: S ⊂ ℕ?"
Or does such a notation exist?
All sets in VDM language are finite by definition, so I believe there is no need to explicitly specify that part. As defined here http://wiki.overturetool.org/images/c/cb/VDM10_lang_manV2.pdf section 3.2.1
Now, to model a type which is a subset of a set s2 , one of the ways is to use an invariant on that type. such as "inv t == s1 subset s2".

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