Formal Languages - Science topic

I just encountered a notification about this article on Mathematical Proofs and the value of Proof Assistants, https://www.ams.org/journals/notices/202401/rnoti-p79.pdf

I resonate most with the positions by Lamport (author of LaTex and a Turing Award Laureate) and Laurence Paulson (author of ML for the Working Programmer and working very much in proof assistants). I think it will be clear that the situation about the presentation of proofs and incorporation of such proofs in mathematical publication is still very much up in the air.

I also did a ResearchGate search on "Proof Assistant" and the fire-hose of articles confirmed my view that this is yet stabilizing, although there are extensive favorite approaches.

Here is the above paper's abstract:

“A proof is one of the most important concepts of mathematics. However, there is a striking difference between how a proof is defined in theory and how it is used in practice. This puts the unique status of mathematics as exact science into peril. Now may be the time to reconcile theory and practice, i.e., precision and intuition, through the advent of computer proof assistants. This used to be a topic for experts in specialized communities. However, mathematical proofs have become increasingly sophisticated, stretching the boundaries of what is humanly comprehensible, so that leading mathematicians have asked for formal verification of their proofs. At the same time, major theorems in mathematics have recently been computer-verified by people from outside of these communities, even by beginning students. This article investigates the different definitions of a proof, the gap between them, and possibilities to build bridges. It is written as a polemic or a collage by different members of the communities in mathematics and computer science at different stages of their careers, challenging well-known preconceptions and exploring new perspectives.”

There is already an objection to this material on the list where I found it. The objection is to this statement: "This puts the unique status of mathematics as exact science into peril.” That statement disturbs me too, but maybe not for the same reason.

Peter Breuer Thank you for your comment, he clarified, he pointed out the precise idea that I had in mind when I asked the question; This general theoretical question arose from a practical problem: the completion of (knowledge) graphs and the need to create an ontology, based on a set of axioms, to admit or reject new links (arcs) and objects (nodes). In fact, we are talking about a monotonic extension of the given original set.

In fact, the original set of statements is implicitly assumed to be consistent (nothing to say otherwise). But here's a small detail: speaking of "consistency" could be interpreted as assuming the existence of some kind of formal system. Probably the problem should be posed as emerging from a situation in which, there being some relation (binary, for simplicity), written (of course) in some formalized language (hence w.f.f.), representing a set of non-contradictory statements (here, not contradictory means the absence of a statement and its -semantically understood- negation).

And yes, the set of all statements is a (trivial) system of axioms.

As for the assumption of the presence of some kind of logic, it is not assumed. Here, significantly, it is relevant to take into account what is meant by logic; As far as I know, despite their historical and genetic relationships (in the sense of genesis), they are not synonymous, logic (syntactic approach) is (mainly) a kind of formal system.

Well, here I end my ramblings. Thanks for the comment.

For me some basic knowledge of formal systems is essential for undergraduate CS students, as this is a necessary part of the scientific education.

If not, you might still learn a lot about software construction, but you wouldn't know nor understand its bedrock.

Thank you for your kind reply. I was curious to know if there was any specific article about that subject.

Thanks for the cooperation,

Giorgia

Kindly check also the following very good RG link:

Mathematics helps AI scientists to solve challenging deep abstract problems using traditional methods and techniques known for hundreds of years. Math is needed for AI because computers see the world differently from humans. Where humans see an image, a computer will see a 2D- or 3D-matrix. With the help of mathematics, we can input these dimensions into a computer, and linear algebra is about processing new data sets.

Here you can find good sources for this:

How about this one?

Sorry I put twice the same link: Here is the second link : https://cla.tcs.uj.edu.pl/

The answer is P=NP

Hola,

Para el cálculo de la frecuencia léxica de unidades simples o complejas, se suele utilizar WordSmith o AntCon.

Saludos

It will help if you use a suitable and powerful qualitative research software as Atlas.ti (https://atlasti.com/) or equivalent. This software allows you to introduce and research large amounts of text, written or oral, images, videos, etc. Then, you can select diverse research techniques, including frequencies, correlations, modulations, structures, and several other tools.

Define clearly the definition of Model Checking and its related definitions. Because mode checking is suite of concepts including type of models e.g. automata models or petri-nets, etc. The verification logic languages of a model-checker. First, take some model checkers on list and then see what are the different parts: model, engine, verification language, set of verification properties. Study each model checker and its each part in detail. See what each has edge over other, what types of application each can model, compare their performance and applicability. Case studies are helpful to understand tools practically.

Hello,

You should describe the source model (state machine), the target model ( fault tree ) and the transformation itself using a theorem prover like Isabelle/HOL or Coq. After that, you need to define some properties of this transformation and prove their correctness using the theorem prover.

Dear Prof. Ritamaria Bucciarelli:

Congratulations and thank you so much for asking this most important question in the context of both Applied and Computational Linguistics! In that regard, as far as the descriptive method is concerned, Chomsky (1956) undertakes highly focused research on several conceptions of linguistic structure to determine whether or not they can provide simple and “revealing” grammars that generate all of the sentences of English and only these. He finds that no finite-state Markov process that produces symbols with transition from state to state can serve as an English grammar. Furthermore, Chomsky (1956) draws attention to the particular subclass of such processes that produce n-order statistical approximations to English do not come closer, with increasing n, to matching the output of an English grammar. Like this manner, he formalizes the notions of phrase structures and shows that this gives us a method for describing language, which is essentially more powerful, though still representable as a rather elementary type of finite-state process. Nevertheless, Chomsky (1956) points out that it is successful only when limited to a small subset of simple sentences. In view of that, he studies the formal properties of a set of grammatical transformations that carry sentences with phrase structure into new sentences with derived phrase structure, showing that transformational grammars are processes of the same elementary type as phrase-structure grammars. Correspondingly, he underlines that the grammar of English is materially simplified if phrase structure description is limited to a kernel of simple sentences from which all other sentences are constructed by repeated transformations, and that this view of linguistic structure gives a certain insight into the use and understanding of language.

Subsequently, Chomsky (1956) states that general linguistic theory can be viewed as a metatheory, which is concerned with the problem of how to choose such a grammar in the case of each particular language on the basis of a finite corpus of sentences. In particular, as indicated by Chomsky (1956), it will consider and attempt to explicate the relation between the set of grammatical sentences and the set of observed sentences. In other words, this author emphasizes that linguistic theory attempts to explain the ability of a speaker to produce and understand- new sentences, and to reject as ungrammatical other new sequences, based on his limited linguistic experience. To this end, Chomsky (1956) gives the following example: suppose that for many languages there are certain clear cases of grammatical sentences and certain clear cases of ungrammatical sequences, e-e., (1) and (2). Respectively, in English.

(1) John ate a sandwich

(2) Sandwich a ate John.

In this case, we can test the adequacy of a proposed linguistic theory by determining, for each language, whether or not the clear cases are handled properly by the grammars constructed in accordance with this theory. For example, if a large corpus of English does not happen to contain either (1) or (2), we ask whether the grammar that is determined for this corpus will project the corpus to include (1) and exclude (2). In this sense, Chomsky (1956) points out that even though such clear cases may provide only a weak test of adequacy for the grammar of a given language taken in isolation. They provide a very strong test for any general linguistic theory and for the set of grammars to which it leads, since, as Chomsky (1956) claims, in the case of each language the clear cases be handled properly in a fixed and predetermined manner. Like so, Chomsky (1956) does foreground that we can take certain steps towards the construction of an operational characterization of “grammatical sentence” that will provide us with the clear cases required to set the task of linguistics significantly.

More to the point, Odlin (1994: 45) asserts that, nevertheless, at the practical level, the Universal Grammar Model by Chomsky suggests that more attention must be paid by teachers to the teaching of specifically syntactic aspects of vocabulary acquisition.

Best wishes,

Javier.

Yes, we can make science more clear and powerful with new language, but we can't neglect English because English currently plainly settled as the principle language of universal logical correspondence, specialists keep on distributing their work in different dialects than English too.We encourage mainstream researchers to attempt to handle this issue and propose potential methodologies both for incorporating non-English scientific knowledge viably and for upgrading the multilingualism of new and existing information accessible just in English for the clients of such learning.

Thank you dear Mr. Dennis Hamilton!

According to my studies, Zeno's great creation of “Achilles--Tortoise paradox” is not only a simple mistake but is a huge paradox family and its typical modern family member is the newly discovered Harmonic Series Paradox.

Let’s see following divergent proof of Harmonic Series which can still be found in many current higher mathematical books written in all kinds of languages:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．． （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．． （２） >1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．． （3） =1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity （4）

Because of not knowing what infinitesimals are, the unavoidable practical problem has been troubling us ever since is how many items (including infinitesimals of cause) in infinite decreasing Harmonic Series can be added up by “brackets-placing rule" to produce infinite numbers each bigger than 1/2?

This kind of “infinite-infinitesimals paradox” tells us:

1, in Harmonic Series, we can produce infinite numbers each bigger than 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite infinitesimals in Harmonic Series by “brackets-placing rule" to change an infinitely decreasing Harmonic Series with the property of Un--->0 into any infinite constant series with the property of Un--->constant or any infinitely increasing series with the property of Un--->infinity;

2, the “brackets-placing rule" to get 1/2 or 1 or 100 or 100000 or 10000000000 or… from infinite items in Harmonic Series corresponds to different runners with different speed in Zeno’s Paradox while the items in Harmonic Series corresponds to those steps of the tortoise in Zeno’s Paradox. So, not matter what kind of runner (even a runner with the speed of modern jet plane) held the race with the tortoise he will never catch up with it.

By the way, Robinson's non-standard analysis can do nothing to solve any of those suspended “infinite-infinitesimals paradoxes” either.

Sincerely yours,

Geng

Just as Peter, I can't make anything out of

" can we treat those “1/n--->0 standard real numbers in Harmonic Series” in the way of “letting be 0（1/n = 0）, taking the limit（lim 1/n =0）, taking the standard number（1/n =0）” as what we have been doing in the computing process of differentiation (derivation)? "

Dear André,

do you remember some of the 2/3 of your thoughts which you thought in internal language? How do you do that?

Can you rethink them and discuss them internally? How do you do that?

Are among those thoughts some which cannot come into mind like a picture (because they are in some way more abstract)? How do you remember them, rethink them, and discuss them internally?

Yes! my question was regarding the "exponential" with respect to the number of variables. (The question "arose" when I was trying a problem reduction from QBF. )

I understood that the question was irrelevant in any sense as the number of disjunctive clauses in the  CNF  depends on the atomics. I was about to delete the question and then saw your answer. Thank you for considering my question and taking time to answer it.

Good grief. It's a wonder anyone one wants to do formal specification anymore! High handed treatment of a young student aspiring to do some work in formal methods is very depressing indeed - we all have to start somewhere. Ashish - it's great that you're asking the questions in this forum. I started on a very similar vain many years ago by developing a simple translation from Z to Prolog for simulation. It can be done, and can provide insight into the meaning (semantics) of a formal language, and also make it "real". Keep at it!

It would appear to me that others may also have thought about using DSM for forming an autonomous organization. I would like to hear from anyone who may have implemented this concept and learn of their experience. Has anyone published anything on it?

And I would like to know of anyone who may be interested in the Explainer.I have used the Explainer to show how Congress can solve some of the problems that cause them to be in gridlock. But I have not seen any evidence that these problems are being solved. If anyone else has come up with such a method, I would like to hear about it. Maybe someone has come up with such a method and they have also had trouble getting people to look below the surface to see how it works and can be used.

Don

Dear Ahmed

the problem with left recursion, from a computational linguistics point of view, is that it leads to infinite recursion, as mentioned in other posts. And, sadly, linguists do tend to write an awful lot of such rules, as the example below shows (a very naive DCG grammar for English relative clauses). If you 'consult' this grammar with swi-prolog, all will apparently run smooth because swi-prolog can deal with such recursive rules appropriately. If you submit the following  goal "s([the,man,that,he,knows,sleeps],[]).", you'll get "true" as an answer. But, if you ask swi-prolog to search for more results (";"), then you'll get an "Out of local stack" error because of the left-recursion. 

The general strategy is "transform your left-recursive rules into right-recursive ones". It means you must tweak your grammar to eliminate such left-recursive rules and transform them into right-recursive ones, with the help of an intermediate non-terminal (cf. for eg. http://web.cs.wpi.edu/~kal/PLT/PLT4.1.2.html). 

From an algorithmic point of view, different approaches have been published, in order to deal with such left-recursive rules (as said earlier, this is how linguists spontaneously write formal grammars). If you're looking for algorithms, you can have a look at Bob Moore's paper http://research.microsoft.com/pubs/68869/naacl2k-proc-rev.pdf.

s --> np, vp.

np --> det, n.

np --> np, relc.%this is a left-recursive rule

relc --> pror, pro, vt.

vp --> vi.

vp --> vt, np.

det --> [the].

n --> [man].

pro --> [he].

pror --> [that].

vt --> [knows].

vi --> [sleeps].

Dear Muhammad ,

please find  the file in the attachment.

hope it helps .

----Available VDM Tools----

SpecBox-Syntax Checking,Document Printing,Semantic Analysis

VDM through Pictures(VtP)-for Editing,Visual Specification

mural-Proof Support

VDM Domain Compiler-Code Generation

I would like to know the steps that are to be followed in rewriting grammar, so that ANTLR generates appropriate parse trees.

Peter, you are right.

From your proposition, we can define exactly the primitive words as follow.

Let us call CP(x) the set of all cyclic permutations of x.

Then the primitive words over an alphabet E, PW(E), will be the set of words x built from E such that the cardinal of CP(x) is equal to the length of x.

E.g

"abc" gives CP(abc) = { bca, cab, abc } (thus |CP(abc)| = 3 = |abc| ) and therefore abc is primitive

whereas

"abab" gives CP(abab)={ baba, abab } and by definition is not primitive

More formally (we exclude the zero length words):

PW( E) = { x in E* such that |x| > |CP(x)| > 0 }

Hope this could help at some point !

Scade provides safe state machines to describe state-based behavoir. But you're right everything is finally mapped to an (at least locally) synchronous execution model,

However, when using UPPAAL or another model checker you usually do the design in another framework and build a verification model to prove your properties. That's is often non-trivial since the design model has to be abstracted effectively (e.g. UPPAAL comes to its limits easily if the system contains a lot of variables and clocks). Second, the abstraction-refinement relation has to be proven property preserving for the relevant properties, otherwise verifying something for the verification model may not tell anything about the design model.

So my first question would be whether you already have some design models for your systems, or do you constsruct your own verification models only?

It's a matter of value. In Scrum, the PO has to be convinced that formal modelling (or other formal techniques) have value above cost. If so convinced, the PO can make the corresponding engineering practices part of the Definition of Done. The Product Owner may herself use formal specification techniques if they are made transparent and if it can be shown that the benefit outweighs the cost. Both of these, of course, depend on doing true Scrum where the PO really is the boss and really has final say over the product vision, final say over the PBIs on the Product Backlog, and the wherewithal to come together with the team on the corresponding components of the Definition of Done.

To disagree slightly with Nick: Keep in mind that working code is not the first priority — it's self-organisation and feedback in Agile, and it's people and Kaizen mind in Lean and in Scrum. Few formal methods cater well to that agenda.

To disagree slightly with Yasar: Agile says nothing against tedious work.

To disagree with Maria: A system is as weak as its weakest link. Using formal methods on one part to robustify it does not make the system robust; there are many formal proofs around this concept. What she describes takes quite advanced risk analysis techniques; these are the purview of the PO, supported by input from the development team.

http://cpntools.org/

Download this tool bro....

Very interesting.

I looked at a converted postscript version the first time.

A quick first scan indicated the standard tools of parsing and execution of tokens. Toward the back of the document, the text appeared to apply these techniques to groups of unknown symbols (I could not read the symbols).

I looked at another copy of the work , this evening, and I can read all of the pages (all known symbols.)

So, there must have been a mixup in the language and symbol set in the first document that I scanned.

Take care and have fun,

Joe