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Cactus Language • Preliminaries 1
❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill's direction, and suddenly our entire reality would change.
❝Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photograph. Of the two images, the only part which coincides is the body. The two different sets of quills stick out into very different regions of space. The objective reality we see from within the first position, seemingly so full and spherical, actually agrees with the shifted reality only in the body of common knowledge. In every direction in which we look at all deeply, the realm of discovered scientific truth could be quite different. Yet in each of those two different situations, we would have thought the world complete, firmly known, and rather round in its penetration of the space of possible knowledge.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The task before us is to describe the syntax of a family of formal languages intended for use as a sentential calculus, and thus interpreted for the purpose of reasoning about propositions and their logical relations.
To carry out our discussion we need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things which can be named, indicated, described, discussed, and renamed if necessary, which can be placed, arranged, and rearranged within a suitable medium of expression — or else manipulated in the mind — which can be articulated and decomposed into their elementary signs, and which can be strung together in sequences to form complex signs.
Signs having signs as their objects are known as “higher order signs”, a topic which demands an adequate level of formalization, but in due time. The present discussion needs a quicker way to get into the subject, even if it settles for informal means which cannot be rendered absolutely precise.
Resources —
Cactus Language • Preliminaries
Survey of Animated Logical Graphs
Survey of Theme One Program
Cactus Language • Discussion 1
Answering a question from a reader …
Re: Cactus Language • Preliminaries 9
Re: Cybernetics • Joe Bury
JB:
❝What does subcatenation and surcatenation mean? Their definitions are not found in a dictionary. I get the formulas you wrote but I don't understand the meaning.❞
Thanks for the question, Joe,
The current presentation of Cactus Language is rather abstract and formal because that's what we need for a fully computational parsing algorithm, and there's quite a bit more to do on that score as we go, but I have written more intuitive introductions to the same material various times before — You might try one of the following for starters.
Logical Graphs • First Impressions
Logical Graphs • Formal Development
Keeping it short and simple as possible —
Under the Existential Interpretation —
• The syntactic connective of Concatenation is interpreted as the Logical Conjunction, which says all of its operands are true.
• The syntactic connective of Surcatenation is interpreted as the Minimal Negation Operation, which says exactly one of its operands is false.
Under the Entitative Interpretation —
• The syntactic connective of Concatenation is interpreted as the Logical Disjunction, which says some of its operands are true.
• The syntactic connective of Surcatenation is interpreted as the Dual of Minimal Negation, which says not just one of its operands is true.
Resources —
Cactus Language • Preliminaries
Survey of Animated Logical Graphs
Survey of Theme One Program
Cactus Language • Overview 1
❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.❞
— Herbert J. Bernstein • “Idols of Modern Science”
The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.
Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.
Resource —
For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.
Part 1
Cactus Language • Syntax
Part 2
Generalities About Formal Grammars
Part 3
Cactus Language • Stylistics
Cactus Language • Mechanics
Cactus Language • Semantics
Stretching Exercises
References
Document History
Automata theory provides a rigorous framework for modeling, analyzing, and verifying systems in both network protocols and bioinformatics. By abstracting complex systems into states and transitions, automata allow for the systematic exploration of system behavior, making it easier to detect errors, optimize performance, and understand underlying biological processes. This cross-disciplinary application of automata theory highlights its versatility and power in tackling real-world challenges.
Please recommend recent papers on the applications of fuzzy languages
As we know that, MC/DC require at-least a predicate which consists of at-least two atomic conditions in a program. Then only we can able to compute MC/DC score if we have a set of test data. Now, when compiler tries to compile a program then it decomposes a predicate into simplified form and this simplified form is in the syntax of Low level language code or Intermediate code. In this code we may not have boolean operators or any predicate, every thing is in atomic guard conditions. So, my point is that we can only compute MC/DC score for High language code not for intermediate code? But is it exactly what we expect from a test case generator? Because test case generator or constraints solver may not know about the actual program, but it tries to explore all the paths of intermediate code. But, can we say that the test cases generated by constraints solver is indirectly tends to high level language program.
Please share your views!! If anyone want more clarifications then do let me know, will explain through an example.
Thanks,
Sangha
Considering a grammar 'G' having certain semantic rules provided for the list of production 'P'. If intermediate Code needs to be generated and if I follow DAG method to represent it.
In that regard, What are the other variants of Syntax tree apart from DAG for the same?
I have implemented an algorithm for NFA by giving the adjacency matrix as an input, but I want to get it by structure.
One can certainly view the mechanics and behavior of the ribosome and conclude a correspondence with machines; to paraphrase musician and composer Frank Zappa, mechanism is not dead - it just smells funny.
So, might readers and especially biologists give council regarding the negatives to such a view?
How different type of algebra structure like lattice, integral lattice monoid and other algebra structures increase the power of formal language.
A Cellular Automata Transform as proposed by Olu Lafe is useful in image processing and other applications. Also one can suggest some good tutorial over it as free ebook is not available.
I read the following example in one of my professors notes.
1) we have a SLR(1) Grammar G as following. we use SLR(1) parser generator and generate a parse table S for G. we use LALR(1) parser generator and generate a parse table L for G.
S->AB
A->dAa
A-> lambda (lambda is a string with length=0)
B->aAb
Solution: the number of elements with R (reduce) in S is more than L.
but in one site I read:
2) Suppose T1, T2 is created with SLR(1) and LALR(1) for Grammar G. if G be a SLR(1) Grammar which of the following is TRUE?
a) T1 and T2 has not any difference.
b) total Number of non-error entries in T1 is lower than T2
c) total Number of error entries in T1 is lower than T2
Solution:
The LALR(1) algorithm generates exactly the same states as the SLR(1) algorithm, but it can generate different actions; it is capable of resolving more conflicts than the SLR(1) algorithm. However, if the grammar is SLR(1), both algorithms will produce exactly the same machine (a is right).
any one could describe for me which of them is true?
EDIT: infact my question is why for a given SLR(1) Grammar, the parse table of LALAR(1) and SLR(1) is exactly the same, (error and non-error entries are equal and number of reduce is equal) but for the above grammar, the number of Reduced in S is more than L.
Let A be a finite set. Suppose for each natural index i, there is a context free language Ci over alphabet A. Suppose further that for all indices I, we have Ci is contained in C{i+1}. The project is: to find conditions on {Ci} so that the ascending union of the Ci is still a context free language over A.
Note that at each stage i, a pumping lemma is satisfied, as will be Ogden's Lemma, and etc. So, one might need to work hard to find a good ``finiteness'' condition that would do the job.
I'm so glad that ask my third question on my favorite site.
Infact i ran into multiple choice question in recent exam on Compiler Course.
Suppose T1, T2 is created with SLR(1) and LALR(1) for Grammar G. if G be a SLR(1) Grammar which of the following is TRUE?
a) just T1 has meaning for G.
b) T1 and T2 has not any difference.
c) total Number of non-error element in T1 is lower than T2
d) total Number of error element in T1 is lower than T2
My solution:
we know table size and state of LALAR(1) and SLR(1) is the same. but someone say number of reduced state in LALAR(1) is lower than SLR(1) (free space in LALR(1) is more than SLR1(1) because using lookahead instead of follow) and so (d) is correct. but in answer sheet we see (b) is correct. anyone can describe it for us? which of these is true?
A word is primitive, if it is not the power (concatenation as multiplication) of another word. 0101 is not primitive while 01010 is.
For more than 20 years people have been trying to prove that the language consisting of all primitive words over two or more letters is not context-free. Without success. Do you have an idea?
Considering finite automata as a set of states with well defined transition function, how will one formally define the element 'state' in an automaton?
I am working on modelling the interaction between land-use changes and transport. I am using Metronamica which is a cellular automata based modelling package. One of the things I have come across from my reading, is that CA is not able to handle socio-economic variables. The problem is, in my case socioeconomic factors are very important drivers of urban change. Any suggestions on how I can overcome this?
Weighted transducers are finite-state transducers, in which each transition carries some weight in addition to the input and output labels. The weights are elements of a semiring
(S,⊕,⊗, 0, 1).
Is so what is its complexity?
So, now I propose the topic of computational efficiency, particularly with regard to shustring search.
A review of relevant literature mentions several concepts, each providing a portion to the published search algorithms. These concepts are:
LCP least common prefix array
Suffix array
Suffix tree
and variations on these examples.
It is quite possible to efficiently (say, with under 300Mb of memory and under 150 seconds of time for a sequence of 31Mbp) compute shustrings without, again, I say without the use of any of those crutches; the computation is instead direct. Further, it is a simple matter of sorting. Gross character of machine is also important, like speed of processor and processor environment overhead - dedicated processors solve one problem more quickly than does a multitasking processor.
My questions concern the run-time performance of algorithms that implement the above listed concepts. The gross measures are sufficient, amounts of time and memory versus volume of input but, order measures are useless to my particular need.
Has a reader any sense for such measures on algorithms for the above listed concepts?
Korshunov in 76 says: Almost all automata with $n$ states, $k$ input symbols and $m$ output symbols have the degree of distinguishability asymptotically (as n goes to \infty) equal to log_k(log_m(n)). Maybe there is an easy proof knowing that for almost all automata with $n$ states, $k$ input symbols the diameter is O(log_k(n))
I am looking to work on some interesting problem in Automata theory. I want to work on something from classical automata concepts such as FA's, TM, Grammars and RE. So far I am unable to narrow down some thing specific. Can any one guide or highlight any related problem set?
I am creating a project which "transforms the C code" to flow graph. Please suggest any tool or any materials. Should I change the compiler intermediate code to any specification of graph transformation system?
