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## Type theory for imperative programming languages?

The type theory that I have seen is all developed over lambda calculus, which is an inherently functional language.

Nevertheless, in practice imperative languages have type systems. Are there differences in the type theory for imperative vs functional languages?

## Is there a connection between representation theory and PDEs?

As a PhD student, if I want to do something algebraic / linear-algebraic such as representation theory as well as do PDEs, in both the theoretical and numerical aspects of PDEs, would this combination be compatible and / or useful? Is it feasible?

I’d be grateful for an online resource to look into.

Thanks,

## What is the difference between complexity theory and computability theory?

What is the difference between complexity theory and computability theory? Is there any overlap between the two or are they completely different? Or is one a subset of the other?

## Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.

• Cobordism Theory for DIFF/Differentiable/smooth manifolds

However, there are Topological Manifolds which are not Differentiable Manifolds.

So my question here for experts is that what do I need to beware and pay attention in order to master a cobordism theory of Topological Manifolds? What are the main differences of the computations of the bordism groups for the given following structures:

Say,

1. Cobordism Theory of TOP/topological manifolds

2. Cobordism Theory for PDIFF/piecewise differentiable manifolds

3. Cobordism Theory for PL/piecewise-linear manifolds

p.s. Are there Spin, Pin$$^+$$, and Pin$$^-$$ versions of these cobordism theories of Topological Manifolds computed in the literature explicitly?

## Which of the known alternative set theories is nearest in structure to this theory with a universal set and the complement of Russell set?

Before I’ll present the exposition of this theory, I’ll speak a little bit about the Mereological concept it is meant to catpure.

The idea is to work in Atomic General Extensional Mereology “AGEM”, one can think of it easily as a theory about collections of atoms, where atoms are indivisible objects, i.e. objects that do not have proper parts. The relation is an atomic part of is defined as:

$$\sf Definition:$$ $$x P^a y \iff atom(x) \land x P y$$.

where $$P$$ stands for “is a part of”, and atom(x) is defined as:

$$atom(x) \iff \not \exists y (y P x \land y \neq x)$$

This atomic part-hood relation can be regarded, conceptually speaking, as an instance of set membership relation.

Now the following theory is a try to define a set theory by a strategy of mimicking properties of this atomic part-hood relation with the background theory being AGEM.

Notation: let $$\phi^{P^a}$$ denote a formula that only use the binary relation $$P^a$$ or otherwise the equality relation, as predicate symbols. The notation $$\phi^{\in|P^a}$$ denotes the formula obtained by merely replacing each occurrence of the symbol $$“P^a”$$ in $$\phi^{P^a}$$ by the symbol $$“\in”$$.

Comprehension axiom schema: if $$\phi^{\in|P^a}(y)$$ doesn’t have the symbol $$x$$ occurring free, then all closures of:

$$\forall A [\exists x \forall y (y \ P^a \ x \leftrightarrow \phi^{P^a}(y)) \to \exists x \forall y (y \in x \leftrightarrow \phi^{\in|P^a}(y))]$$

are axioms.

In order to complete this theory we add axioms of Extensionality, Empty set and Singletons:

Extensionality: $$\forall xy [\forall z (z \in x \leftrightarrow z \in y) \to x=y]$$.

Empty set: $$\exists x \forall y (y \not \in x)$$

Singletons:: $$\forall A \exists x \forall y (y \in x \leftrightarrow y=A)$$

This theory has a universal set, also has a set of all sets that are in themselves, however it doesn’t have complements; axioms of Set union and Power are there. There are separation axioms for formulas $$\phi^{\in|P^a}$$ where $$\phi^{P^a}$$ holds of at least one object. Similarily replacement axioms are granted if $$\phi^{P^a}$$ formula replace atoms with atoms and of course is non empty.

The trick is that all formulas of the form $$x \not \in x$$, $$\exists x_1,..,x_n: \neg (x_1 \in x_2 \land…\land x_n \in x_1)$$; $$x \text { is well founded }$$, $$x \text{ is a von Neumann ordinal }$$, etc.. all of those won’t have their $$\phi^{P^a}$$ corresponding formulas hold of mereological atoms and so cannot be used in comprehension because there do not exist an object that has no atomic parts, since we are already working in AGEM.

Question: if one attempts to prove the consistency of this theory, which of the known alternative set theories have in some sense the nearest structure to this theory, other than positive set theory?

## Lambda Calculus as a branch of set theory

This answer to a question about whether C is the mother of all languages contained an interesting tidbit that I am curious about:

The functional paradigm, for example, was developed mathematically (by Alonzo Church) as a branch of set theory long before any programming language ever existed.

Is this true? What is the link between these topics that is so fundamental as to make lambda Calculus an outgrowth of set theory? The best I can come up with is that standard mathematical functions possess domains and codomains.

## NZEC error in a simple python code to predict the winner using combinatorial game theory

def calculateMex(Set):          Mex = 0      while Mex in Set:           Mex += 1       return Mex     def calculateGrundy(n,k):         if 0 <= n <= k:          return n        Set = set()      for i in range(1, k+1):           Set.add(calculateGrundy(n - i,k))         return calculateMex(Set)  sys.setrecursionlimit(10000) t=int(input()) while(t>0):     t=t-1     inp=input()     nk=inp.split(' ')     n=int(nk[0])     k=int(nk[1])     a=calculateGrundy(n,k)     if a==0:         print('Dishant')     else:         print('Arpa') 

Constraints: 1 < T < 10^5 and 1 < K < N < 10^18 where T is the no. of test cases; K is the atmost number of coins that can be taken away from a pile and N is the number of coins in the pile initially. Logically the code is working fine.But it gives NZEC error when I try to submit it at a competitive programming platform.I cant figure out the why and how to fix it.Any help would be appreciated.

## When should we use Theory of Reasoned Action (TRA)?

According to the Theory of Reasoned Action (TRA) [1, 2], the model aims to measure an individual’s behavior. The theory is applied in the social psychology literature defines relationships between beliefs, attitudes, norms, intentions, and behavior. However, it is not clear when and how we should use it.

For that reason, we ask the following question:

When and how should we use TRA?

During our work developments [3], we tried to understand when and how to use TRA among user behavior measurements. We followed both questions on ResearchGate and answers here on UX StackExchange. But still, we have no concrete solution.

[1] Paul, J., Modi, A. and Patel, J., 2016. Predicting green product consumption using theory of planned behavior and reasoned action. Journal of retailing and consumer services, 29, pp.123-134.

[2] Alryalat, M.A.A., Rana, N.P. and Dwivedi, Y.K., 2015. Citizen’s adoption of an E-Government system: Validating the extended theory of reasoned action (TRA). International Journal of Electronic Government Research (IJEGR), 11(4), pp.1-23.

[3] Calisto, F.M., Ferreira, A., Nascimento, J.C. and Gonçalves, D., 2017, October. Towards Touch-Based Medical Image Diagnosis Annotation. In Proceedings of the 2017 ACM International Conference on Interactive Surfaces and Spaces (pp. 390-395). ACM.

## Would this pure class theory about ordinals and their relations raise concerns about its arithmetic soundness?

The following theory is a class theory, where all classes are either classes of ordinals, or relations between classes of ordinals, i.e. classes of Kuratowski ordered pairs of ordinals, or otherwise classes of unordered pairs of ordinals. However, the size of its universe is weakly inaccessible. Ordinals are defined as von Neumann ordinals. The theory is formalized in first order logic with equality and membership.

Extensionality: $$\forall z (z \in x \leftrightarrow z \in y) \to x=y$$

Comprehension: if $$\phi$$ is a formula in which the symbol $$“x”$$ is not free, then all closures of: $$\exists x \forall y (y \in x \leftrightarrow \exists z(y \in z) \land \phi)$$; are axioms.

Ordinal pairing: $$\forall \text{ ordinals } \alpha \beta \ \exists x (\{\alpha,\beta\} \in x)$$

Define: $$\langle \alpha \beta \rangle = \{\{\alpha\},\{\alpha,\beta\}\}$$

Ordinal adjunction:: $$\forall \text { ordinal } \alpha \ \exists x (\alpha \cup \{\alpha\} \in x)$$

Relations: $$\forall \text{ ordinals } \alpha \beta \ \exists x (\langle \alpha, \beta \rangle \in x)$$

Elements: $$\exists y (x \in y) \to ordinal(x) \lor \exists \text{ ordinals } \alpha \beta \ (x=\langle \alpha,\beta \rangle \lor x=\{\alpha,\beta\})$$

Size: $$ORD \text { is weakly inaccessible}$$

Where $$ORD$$ is the class of all element ordinals.

/Theory definition finished.

Now this theory clearly can define various extended arithmetical operations on element ordinals. Also it proves transfinite induction over element ordinals. In some sense it can be regarded as stretching arithmetic to the infinite world. Of course $$PA$$ is interpretable in the finite segment of this theory.

In this posting Nik Weaver in his answer raised the concern of ZFC being arithmetically unsound.

My question: assuming this theory to be consistent, is the concern of it being arithmetically unsound is the same as that with ZFC?

The motive for this question is that it appears to me that the above theory is just a naive extension of numbers to the infinite world, it has no power set axiom nor the alike. One can say that this theory is in some sense purely mathematical in the sense that it’s only about numbers and their relations. Would this raise the same kind of suspicion about arithmetic unsoundness that is raised with ZFC.

My reasoning about that is that generally speaking when one raises the concern of arithmetic unsoundness of some theory, especially if that theory is well received by mathematicians working in set theory and foundations, then there must be some technical or intuitive argument behind that suspicion, otherwise that suspicion would be unfounded. The suspicion must not depend merely on the strength of the theory in question. Otherwise we’d not define any theory stronger than $$PA$$ based on such concerns.

From Nik Weaver’s answer it appears to me that his concern is based on ZFC not capturing a clear concept intuitively speaking. Now this theory is based on an intuitive concept that is generally similar to the one behind defining arithmetic for finite sets. It extends it in a very clear intuitive manner, higher ordinals are defined from prior ones in successive manner, and it doesn’t generally feel to be so different from the intuitive underpinnings of arithmetic in the finite world. So the question here is about if this theory still fall a prey to the arguments upon which the concerns about arithmetic unsoundness of ZFC are based.