## Definition of liveness property in model checking

A property $$P$$ is simply a set of sequences of states and a certain program is characterized by its set of sequences of states, let’s call it $$T$$. A program is compliant to a specific property P if $$T\subseteq P$$.

Intuitively, a specific liveness property LP1 is the set of traces (sequences of states ) in each of which “something good” will happen, based on the specific property LP1 (for example “termination”). This means that if a program it’s compliant with LP1 in each of its traces the “good thing” will happen, that is the program will terminate.

However, I don’t understand how the formal definition of liveness property contains what was said, can you explain me? This definition to me makes no sense (i know all the terminology)…

$$\forall t\in S^{*}\,(\exists t’\in S^{\omega }:t\leq t^{‘}\wedge t^{‘}\in LP1)$$

## Is the definition of an Associative Array on Wikipedia correct?

I’m authoring the Automaton Standard Code for Information Interchange Data Types, which defines contiguous map and associative array types and I was checking my definitions of Associative Array on Wikipedia to be precise in my language and I noticed that the definition was murdered. An Abstract data type by definition is a data type that is defined by set theory. The Wiki read/currently reads after a revision undo volley that an Associative Array is a Collection of Key-value tuples such that each key appears only once, and the article refers to the mappings between the keys and values as “bindings”; which is no good because a Collection isn’t an Abstract data type nor is it part of Set theory. The Wiki article also states that an associative array is a map, which is like saying that all shapes are circles; all Dictnoaries are maps but not all maps are dictionaries; this confusion comes from C++ naming their dictionary map. An ASCII map is used to for instance to map a TCP port number to a socket and also to create hash table. Either my definition is wrong or the Wiki is completely wrong. This is problematic for the Computer Science community because the article makes no mention of a set, an operation on a set, and the result of the operations, which are requirements of abstract data types defined by set theory. “It’s defined as set theory, only we don’t mention a set” sounds REALLY bad!

The proposed definition I’m asking the community if it’s correct is that an Associative Array is a Surjective Abstract Data Type that maps a set of Unique strings in the domain, commonly called Keys and sometimes called Symbols, to instances of objects or the empty set in the codomain. The association between the key and value is called a Key-value Tuple. The association between a Key and a Value is defined by set theory as a “mapping”, and where a mapping may also refer to the action of modifying an existing mapping called remapping, and where the action of remapping a Key to the empty set is called Unmapping. The mappings are Surjective because each member of the codomain maps to at least one member of the domain.

The above definition defines the Abstract data type with set theory. I left out 3 words though: should have “The mappings are Surjective because each member of the codomain maps to at least one member of the domain read such that X->Y”, which is the Surjective definition; I implied it when I shouldn’t have. I can’t just engage in a revision undo war else I’ll be banned. We need a Computer Scientist with high credentials on Wikipedia to update the Wiki and fill in the gaps. Is my definition correct? Do you know of a definition with a Wiki-approved citation that is easier for those unknowledgeable of set theory to digest?

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## Definition of extensional and propositional equality in Martin-Lof extensional type theory

Martin Hofmann states in Extensional Concepts in Intensional Type Theory (§1.1 p.[4-5]) that:

A similar situation occurs in extensional Martin-Lof type theory where propositional and definitional equality are forcefully identified by the equality reflection rule

$$\frac{\Gamma \vdash P:Id_\sigma(M,N)}{\Gamma \vdash M=N: \sigma}\text{(Id-DefEq)}$$

Does the above mean that we purposefully drop the proof that M and N are equal and just consider them to be equal definitionally (like a presumption)?

Then it goes on and says:

This rule makes definitional equality extensional and undecidable.

How does it become extensional and what does it mean by becoming extensional in the first place?

And then it states:

Moreover, type checking becomes undecidable because $$Refl(M):Id_\sigma (M,N)$$ holds iff $$M$$ and $$N$$ are definitionally equal.

Why would $$Refl(M)$$ hold only if $$M$$ and $$N$$ are definitionally equal? And why would it make it undecidable?

## Is there a formal definition for iteration?

I wanted know if there is a formal definition of iteration. In the sense that, this definition will say that after the 4th iteration is termination or the 5th iteration and not the 2nd iteration. Basically a rigorous definition of iteration.

## Who is the author of AggregatedRating in schema.org definition?

Who is the author of AggregatedRating if the ratings are added by me, but they actually come from Google and Apple and maybe some other places?

The documentation does not give any explanation.

So if I added the JSON-LD there, am I the author? It comes from the App store so is the author App store(s)? Or is the author all the individuals who gave their rating?

Because Author is actually a required field or at least the testing tool gives an error “A value for the author field is required.” because I left it empty for now.

Example of the overall reviews (UserReview -> AggregateReview) .

## Struggling to understand the symbolism around the big oh formal definition

I’m struggling to understand what exactly T(n), and f(n) is in the above text:

When we compute the time complexity T(n) of an algorithm we rarely get an exact result, just an estimate. That’s fine, in computer science we are typically only interested in how fast T(n) is growing as a function of the input size n.

For example, if an algorithm increments each number in a list of length n, we might say: “This algorithm runs in O(n) time and performs O(1) work for each element”.

Here is the formal mathematical definition of Big O:

Let T(n) and f(n) be two positive functions. We write T(n) ∊ O(f(n)), and say that T(n) has order of f(n), if there are positive constants M and n₀ such that T(n) ≤ M·f(n) for all n ≥ n₀.

image

This graph shows a situation where all of the conditions in the definition are met.

I’m taking Tim Coughgarden course and reading his books. I’m still in the beginning and I can understand the explanations, but sometimes I struggle to understand the mathematical meaning of things… there are some implicitly about what is T(n) or f(n) (in this case) that is not explained at all.

## Can DFA with output (definition?) match expressiveness of NFA with unique output?

For a deterministic finite automaton (DFA), some output tasks are easy when done in one direction, but difficult (or impossible?) when done in the reverse direction. Let’s take a simple example of outputting a word lower-case, or upper-case, depending on a control sequence included in the input. So the task

• l:aNNa should become anna
• u:aNNa should become ANNA

(where aNNa could be an arbitrary long word) seems to be quite easy for a DFA with output. The task in the reverse direction

• aNNa:l should become anna
• aNNa:u should become ANNA

however seems to be impossible for a DFA with output, at least for the commonly encountered definitions of DFA with output. On the other hand, a nondeterministic finite automaton (NFA) with unique output has no problems doing a task in the reverse direction, if it can do it in one direction.

1. Is the observation correct that a DFA with output can’t do the task in the reverse direction, if it is forced to consume its input one symbol at a time?

2. Would it be possible to relax the restrictions for DFA and NFA, such that a NFA (with unique output) would not gain any additional expressiveness from the relaxed restrictions, but the DFA with output (for the relaxed restrictions) would be able to match the expressiveness of a NFA with unique output?

## Export a list definition to text file or script?

Is there any reasonable way to export a list definition to a text file or script so it can be used to create a new list after some tweaks to the column names, and the addition of some additional columns? I specifically am not interested in “Save list as template”, and am more interested in something which can be used via Powershell. I would love it if there was something similar to a SQL “CREATE TABLE” statement at our disposal.

My use case is, I have an existing list with about 40 columns in it. This list will be used as a pre-populator in a form where a user will fill in desired changes, along with additional columns, which will be submitted to a new list based on the source list. This is in a SP2010 Server environment.

For the dual reasons of accuracy and laziness, I wish to avoid having to hand-create those 40 columns.