# Tagged Questions

**4**

votes

**1**answer

300 views

### Application of Combinatorics, Logic and computability theory in physical science: Tiling of Wang Tile with proportionality

The original problem of Domino Tiling and Wang Tile has great theoretical interest on computability theory... However, the great emerging problem on application of Wang Tile in material science and ...

**3**

votes

**1**answer

141 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**6**

votes

**5**answers

520 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**0**

votes

**0**answers

157 views

### Shouldn't the problem size in CNF SAT be related to the number of clauses $m$ and number of literals $l$ rather than the number of variables $n$?

Note: This is a second (and hopefully better) attempt at asking the question that I utterly failed to communicate well yesterday at Approx. double exponential pattern in CNF SAT for low n and proving ...

**6**

votes

**0**answers

153 views

### When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of ...

**2**

votes

**0**answers

91 views

### What are natural examples of non-relativizable proofs? [duplicate]

As I understand it, a proof that P=NP or P≠NP would need to be non-relativizable (as in recursion theory oracles).
Virtually all proofs seem to be relativizable, though.
What are good examples of ...

**10**

votes

**1**answer

273 views

### Harvey Friedman's strict reverse mathematics vs. Cook-Nguyen's V$^0$

Harvey Friedman posted several manuscripts [1] proposing a program for "strict" reverse mathematics, in the sense that the base theory should be mathematically natural and coding-free.
In them he ...

**22**

votes

**10**answers

2k views

### Can We Decide Whether Small Computer Programs Halt?

The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not.
Are there some large $n$ ...

**4**

votes

**0**answers

198 views

### About “natural proof” of Razborov and Rudich

The famous "Natural Proof" paper ,http://www.cs.umd.edu/~gasarch/BLOGPAPERS/natural.pdf , of Razborov and Rudich gives a barrier for any proof that try to separate P and NP. It mainly shows that if ...

**17**

votes

**2**answers

760 views

### Deep theorems and long proofs

I ran across this discussion by Daniel Shanks,
"Is the quadratic reciprocity law a deep theorem?."
Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff.
which made me ...

**3**

votes

**2**answers

241 views

### Is There An Algorithmic Complexity Of A Random Distribution

Has anyone studied an equivalent to algorithmic complexity for probability distributions?
This would be a measure which was similar to Kolmogorov complexity but look at the complexity of a (discreet ...

**5**

votes

**0**answers

140 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**23**

votes

**1**answer

659 views

### Are sums of sequences decidable?

Suppose that $f,g$ are rational functions with integer coefficients such that $\sum_{n=0}^{\infty}f(n)$ and $\sum_{n=0}^{\infty}g(n)$ both converge. Is it decidable whether
...

**6**

votes

**1**answer

386 views

### Normality of Chaitin's constant

Can anyone provide an overview of the proof that Chaitin's constant is normal, or better yet, the guiding intuition?
Even if we replace the existential quantifiers in the assertion of non-normality ...

**1**

vote

**1**answer

145 views

### Can all programs reducible to ones with only arithmetic operations on inputs be simulated with polynomial overhead by arithmetic machine?

I failed to get an answer at http://math.stackexchange.com/questions/364061/can-all-programs-reducible-to-ones-with-only-arithmetic-operations-on-inputs-be, so I am asking here.
In ...

**16**

votes

**1**answer

504 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...

**9**

votes

**6**answers

1k views

### Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity.
The wikipedia article on constructive proof begins, "a constructive ...

**13**

votes

**1**answer

561 views

### Forcing over set theory versus forcing over arithmetic

I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...

**5**

votes

**3**answers

551 views

### computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That ...

**3**

votes

**1**answer

231 views

### Diagonalization and classes of computable functions

Fix a standard effective listing $(\phi_e)_{e\in\omega}$ of the partial computable functions from $\omega$ to $\omega$. Let $\mathcal{C}$ be a class of computable total functions $\omega\rightarrow ...

**14**

votes

**4**answers

3k views

### What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper:
Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computational complexity by ...

**4**

votes

**1**answer

438 views

### Infinite monkeys computing … triangle area?

I wonder if it is possible to specialize the question:
(a) What is the probability that a random Turing Machine program
will halt?, to: (b) What is the probability that a random Turing Machine
...

**10**

votes

**2**answers

456 views

### What is the computational-complexity-theoretic analogue of computable inseparability? For example, if P is not NP, are there disjoint NP sets with no separation in P?

Disjoint sets $A$ and $B$ are computably inseparable, if there
is no computable separating set, a computable set $C$ containing $A$ and disjoint from $B$. The
existence of c.e. computably inseparable ...

**12**

votes

**1**answer

533 views

### Does every feasible partial order relation on the natural numbers extend to a feasible linear order relation?

It is well known that every partial order on a set can be extended
to a linear order on that set. That is, for every partial order
$\lhd$ on a set $X$, there is a linear order $\prec$ on $X$ such
that ...

**3**

votes

**1**answer

216 views

### Is $MIN^P$ search problem (partial order) reducible to $MIN^L$ (linear order) search problem?

Search problem $MIN^P$ is, given a polynomial-time computable predicate that is a partial order, to find its minimum (any will do).
Search problem $MIN^L$ is, given a polynomial-time computable ...

**3**

votes

**2**answers

513 views

### Measure of progress towards a proof

Can one define some measure of progress towards a proof of a statement? I'm not sure if it's even possible for general first order logic statements so let's restrict ourselves to propositional ...

**7**

votes

**2**answers

834 views

### Distribution of the computable numbers on the real number line

If we order all the positive computable real numbers $r_1,r_2,r_3...$ by their Kolmogorov complexity in some language $L$, then make a histogram plot of the $r_i$ on the real line, and we scale it ...

**7**

votes

**3**answers

621 views

### Definition of relativization of complexity class

Is there any general definition, for a class $C$ of languages, what is the relativized class $C^A$ for an oracle $A$?
Usually, these classes and their relativizations seem to be defined in an ad-hoc ...

**2**

votes

**1**answer

443 views

### How much of P versus NP's difficulty stems from having to rule out the existence of Turing machines that “accidentally” solve, say, 3-SAT efficiently?

It seems like there is a sense in which a Turing machine that demonstrates P=NP could be said to "accidentally" exist. I'm wondering the extent to which the possibility of such machines is the main ...

**3**

votes

**1**answer

453 views

### Proof systems and their hierarchy

Why ZFC is placed in top of the proof system hierarchy? How it can p-simulate other systems?

**27**

votes

**2**answers

1k views

### The NP version of Matiyasevich's theorem

By Matiyasevich, for every recursively enumerable set $A$ of natural numbers there exists a polynomial $f(x_1,...,x_n)$ with integer coefficients such that for every $p\ge 0$, $f(x_1,...,x_n)=p$ has ...

**2**

votes

**0**answers

193 views

### Oracle separating FIP for bounded-depth Frege from FIP for Frege (and hardness conditions on DDH)

Is there an oracle such that in the relativized world, bd-Frege (bounded depth Frege propositional proof system) has FIP (feasible interpolation property) but Frege does not have FIP?
Such an ...

**2**

votes

**2**answers

528 views

### Natural numbers of great kolmogorov complexity

Before I ask my question, let me give you a mini-preamble: in 2006, during an animated discussion on feasibility, ultrafinitism, and what else on FOM, I introduced (informally, and to speak the tuth, ...

**1**

vote

**2**answers

1k views

### Do you believe P=NP? [closed]

Do you believe P=NP?
I've seen some mathematicians say that if P=NP their work would be worthless and restricted to enunciating theorems. They seem to believe that there exist an almost philosophical ...

**8**

votes

**5**answers

408 views

### Syntactically capturing complexity classes

Primitive recursive functions are syntactically constructible in the sense that from a set of "axioms" we can build every function in the set $PR$. This basicly means that we can build a machine that ...

**5**

votes

**4**answers

1k views

### Zero-knowledge proof that 0 = 1

Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent). Would it be possible to have a ...

**25**

votes

**8**answers

2k views

### Is P=NP relevant to finding proofs of everyday mathematical propositions?

Disclaimer: I don't know a whole lot about complexity theory beyond, say, a good undergrad class.
With increasing frequency I seem to be encountering claims by complexity theorists that, in the ...

**13**

votes

**2**answers

821 views

### Is there a name for sets for which it is easier to test membership than to find members---and vice versa?

This is a question my son Bob asked me. For some sets it is relatively easy
to test for membership but a lot more difficult to find members, and for others
the reverse is true. Here is an elementary ...

**11**

votes

**2**answers

703 views

### Which recursively-defined predicates can be expressed in Presburger Arithmetic?

In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for ...

**6**

votes

**2**answers

1k views

### Horn clauses and satisfiability

It is well known that satisfiability of Horn formulae can be checked in polynomial time using unit propagation.
But suppose we relax the condition for horn clauses from at most one un-negated ...

**7**

votes

**3**answers

1k views

### Decidable but nonrecursive sets

Until recently, I believed that recursive=decidable,
subscribing to this Wikipedia quote:
"In computability theory, a set is decidable, computable, or recursive if there
is an algorithm that ...

**14**

votes

**2**answers

1k views

### Structure theorems for Turing-decidable languages?

Languages decidable by weak models of computation often have certain necessary characteristics, e.g. the pumping lemma for regular languages or the pumping lemma for context-free languages. Such ...

**3**

votes

**0**answers

270 views

### To what extent MSO = WS1S, when adding relations?

Let me first clarify my definitions. For a word $w \in \Sigma^*$, with $\Sigma=\{a_1, \ldots, a_n\}$, I define two structures:
$${\mathbb{N}}(w) = \langle {\mathbb{N}}, <, Q_{a_1}, \ldots, Q_{a_n} ...

**3**

votes

**4**answers

443 views

### Deriving the complete set of “non-redundant” true statements in disjunctive form in propositional logic

Given a finite set of statements known to be true, I need to derive all the "non-redundant" statements in disjunctive form using only literals that can be derived from this set of statements, i e all ...

**23**

votes

**2**answers

1k views

### Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?

Question. Given a Turing-machine program $e$, which
is guaranteed to run in polynomial time, can we computably
find such a polynomial?
In other words, is there a
computable function $e\mapsto p_e$, ...

**8**

votes

**3**answers

950 views

### Is there a formal notion of what we do when we 'Let X be …'?

This is likely an elementary question to logicians or theoretical computer scientists, but I'm less than adequately informed on either topic and don't know where to find the answer. Please excuse the ...

**19**

votes

**3**answers

2k views

### Is the theory of categories decidable?

There are a lot of theorems in basic homological algebra, such as the five lemma or the snake lemma, that seem like they'd be more easily proven by computer than by hand. This led me to consider the ...

**0**

votes

**1**answer

391 views

### cardinal equivalence: for each boolean formula, |quantifications| = |assignments|. [closed]

Cardinal Equivalence Theorem
For each boolean formula, |quantifications| = |assignments|.
The set of valid quantifications has some cardinality, call that ...

**17**

votes

**3**answers

2k views

### Satisfiability of general Boolean formulas with at most two occurrences per variable

(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that ...