Questions tagged [proof-complexity]
propositional proof systems and corresponding bounded arithmetic theories
62
questions
-5
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1answer
114 views
Is mathematical proof itself NP-hard?
At the 8:00 mark of this video, he claims that proving things is itself an NP problem. I'm looking for more insight into this. Could someone help explain this concept to me and also provide a link to ...
2
votes
0answers
112 views
Proof systems induced by NP-complete problems
Satisfiability is the fundamental NP-complete problem. Cook-Reckhow theorem states that the existence of a propositional proof system that admits polynomial size proofs for all tautologies implies ...
asked Mar 24 '20 at 16:45
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
3
votes
0answers
122 views
Have the proofs in Buss's “On Goedel’s Theorems on Lengths of Proofs” been formalized (mechanically checked)?
The paper On Gödel's theorems on lengths of proofs I: Number of lines and speedup for arithmetics contains quite interesting results on proof length. But some of the details are still, to me, a bit ...
7
votes
1answer
186 views
Document references describing weaknesses for cutting planes and algebraic proof system?
Here, Fortnow says (section 4.3):
Since then complexity theorists have shown similar weaknesses in a number of other proof systems including cutting planes, algebraic proof systems based on ...
4
votes
0answers
106 views
Eliminating tautological axioms in tree-like $k$-DNF resolution
The propositional proof system $k$-DNF-resolution, a.k.a. $Res(k)$, is a generalization of propositional resolution, where the lines in a proof are $k$-DNF formulas, i.e., disjunctions of $k$-terms of ...
6
votes
1answer
97 views
Axioms of Minimum Size Resolution Refutations
Let $\phi$ be an unsatisfiable CNF formula and let $\Pi$ be a resolution refutation of $\phi$ of minimum size. Let $\psi$ be the subformula of $\phi$ containing the clauses that actually appear as ...
4
votes
0answers
246 views
Testing emptiness property complexity in Sum of Squares Proof systems
Take the set $$\mathcal T=\{f_1(x_1,\dots,x_n)=\dots=f_m(x_1,\dots,x_n)=0, h_1(x_1,\dots,x_n)\geq a_1,\dots,h_t(x_1,\dots,x_n)\geq a_t\}$$ where $$h_1(x_1,\dots,x_n),\dots,h_t(x_1,\dots,x_n)\in\mathbb ...
0
votes
1answer
58 views
Understanding the definition of a “restriction of a resolution derivation”
I am reading the paper An Introduction to Lower Bounds on Resolution Proof Systems. In its subsection 2.3 the author gives an inductive definition of the restriction of a resolution refutation $\pi$.
...
3
votes
0answers
59 views
Efficient transformation of clausal proof into resolution proof
Clausal proof is used to certify unsat results of SAT solvers. However the main theoretical results are on resolution proof (for instance, the non existence of a polynomial resolution proof for the ...
2
votes
0answers
106 views
Transforming a DAG-like resolution proof to a tree like resolution proof
How can a DAG-like resolution proof be transformed to a tree-like resolution proof?
Is such a transformation possible in polynomial time?
4
votes
1answer
83 views
Resolution augmented with the rule of symmetry or the rule of extension
Krishnamurthy (here is the abstract of his paper) showed that resolution augmented with the principle of symmetry forms a proof system (referred to as SR) in which certain formulas (which do not admit ...
3
votes
0answers
93 views
Primal/Dual of the Lasserre/ SOS SDP hierarchy
Sum of Squares proofs and the Lasserre hierarchy can both be stated as SDPs. It is often claimed without proof that these SDPs are dual to each other, although I do not see that this is obvious. I was ...
6
votes
1answer
206 views
Resolution vs Nondeterministic Search Problems
It is well known that each resolution refutation $\Pi$ for an unsatisfiable CNF formula $F = C_1\wedge C_2 \wedge ... \wedge C_m$ over variables $X$ can be translated in polynomial time (in the size ...
2
votes
2answers
109 views
IPS upper bound for subset sum axiom
I am reading the following paper
Michael A. Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson ,"Proof Complexity Lower Bounds from Algebraic Circuit Complexity", 2016.
IPS is defined as follows:
...
3
votes
2answers
133 views
Construct proof systems for common algorithmic task, like equivalence of regular expressions
A propositional proof system according to Cook and Reckhow for a language $L \subseteq \Sigma^{\ast}$ is a deterministic polynomial time function $f : \Sigma^{\ast} \to L$ that is onto.
For $y \in L$ ...
1
vote
0answers
124 views
Proof of Correctness of Bottleneck Dijkstra Algorithm [closed]
I am working on a bottleneck multicast tree for which I am using bottleneck Dijkstra algorithm.
My question is 1) bottleneck Dijkstra has the same correctness as that of (simple) Dijkstra or not ?
2)...
4
votes
0answers
70 views
Combinatorial characterization of hypergraph Tseitin satisfiability
The Tseitin formulas are as follows: Given a connected graph and a function $\alpha: V \rightarrow \{0,1\}$. Associate each edge $e$ with a variable $x_e$. The Tseitin formula $G(\alpha)$ is defined ...
4
votes
1answer
142 views
Is there any work relating type systems and Cook-Reckhow proof systems?
An important subfield of computational complexity is proof complexity, mostly due to Cook and Reckhow. E.g. one notable result is that a proof system that has efficient proofs (i.e., in length ...
0
votes
1answer
226 views
In regards to the tautologies of a polynomially-bounded propositional proof system
In the book 'Logical Foundations of Proof Complexity', co-authored by Stephen Cook, the following definition is given:
A proof-system $F$ is said to be polynomially-bounded if there is a polynomial p(...
9
votes
1answer
276 views
Lower Bounds for Frege and Extended Frege
Wikipedia [1] states that the best known lower bound for size of Frege proofs is quadratic, and that there is no known superlinear lower bounds for the number of lines of Frege proofs.
Questions:
1)...
-1
votes
1answer
105 views
On the difference between propositional proof system and polynomially-bounded proof system
For the definition of a propositional proof system we have:
An abstract proof system is a polynomial time function f whose range is equal to the set of tautologies. If τ is a tautology, then an f-...
-4
votes
1answer
267 views
NP-completeness of one generalized subset sum problem (target sum belongs to interval) [closed]
I need to prove that decision problem:
for a given set of positive integers $a_1, ..., a_n$, does it exist a subset that sums up to a value within interval $[\frac{1}{2}\sum a_i; \frac{1}{2}\sum a_i+...
6
votes
2answers
330 views
Are there any propositional proof systems which are not Cook-Reckhow proof systems?
An abstract proof system is a polynomial time function $f$ whose
range is equal to the set of tautologies. If $\tau$ is a tautology, then an $f$-proof of $\tau$ is any value $\pi$ such that $f(\pi) =...
-3
votes
1answer
81 views
On the (Cook) definition of a propositional proof system [closed]
Def: An abstract proof system is a polynomial time function f whose
range is equal to the set of tautologies. If T is a tautology, then an f-proof of T is any value w such that f(w)= T
I'm a bit ...
1
vote
0answers
104 views
Techniques for showing intermediate status between $\mathsf{NP \cap coNP}$ and coNP-completeness
Inspired by Suresh's post, for a new problem in $\mathsf{coNP}$, whose true proof complexity is intermediate between $\mathsf{NP \cap coNP}$ and being coNP-complete, I am interested in methods which ...
asked Oct 15 '15 at 17:32
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
1
vote
0answers
135 views
Do problems have to be statable in $\Pi_1$ to use Levin's universal search to find short proofs if P=NP
In If P=NP, could we obtain proofs of Goldbach's Conjecture etc.? it talks about the hypothetical world where P=NP and using the proof of it to prove a problem/theorem assuming that it has a short ...
10
votes
3answers
384 views
Graph theoretic restriction to Proofs in Proof Complexity Theory
Proof complexity is a most basic area of computational complexity theory. An ultimate purpose of this area is to prove $NP\neq coNP$, that is, any prover cannot give a proof of unsatisfiability of ...
28
votes
6answers
1k views
Natural NP-complete problems with “large” witnesses
The question on cstheory "What is NP restricted to linear size witnesses?" asks about the class NP restricted to linear size $O(n)$ witnesses, but
Are there natural NP-complete problems in which (...
7
votes
0answers
203 views
Implications of the impossibility of efficient sampling from random non-Hamiltonian graphs
Nisan's answer to this question shows the Impossiblity of efficient sampling from random non-Hamiltonian graphs (unless $NP=coNP$). I am interested in the implications of this conjecture.
Does the ...
8
votes
1answer
276 views
Proof complexity and lower bounds
One way to prove NP$\neq$ coNP is to show that for every propositional proof system $f$ computable in polynomial time, there exists a family of tautologies for which $f$ requires super polynomial ...
13
votes
4answers
1k views
Start learning proof complexity
I recently started to read a lot about proof complexity and have been really enjoying what I have been reading. I would really like to learn more about this, but I am having difficulty finding some ...
10
votes
4answers
592 views
An easy case of SAT that is not easy for tree resolution
Is there a natural class $C$ of CNF formulas - preferably one that has previously been studied in the literature - with the following properties:
$C$ is an easy case of SAT, like e.g. Horn or 2-CNF, ...
18
votes
3answers
552 views
Constructively efficient algorithms without efficient correctness and efficiency proof
I am looking for natural examples of efficient algorithms (i.e. in polynomial time) s.t.
their correctness and efficiency can be proven constructively (e.g. in $PRA$ or $HA$), but
no proof using only ...
8
votes
1answer
247 views
How do I use canonical ordering to reduce symmetry in the SAT encoding of the pigeonhole problem?
In the paper "Efficient CNF Encoding for Selecting 1 from N Objects", the authors introduce their "commander variable" technique for encoding the constraint, and then talk about ...
10
votes
1answer
429 views
Proofs in $S_{2}^{1}$
In a talk by Razborov, a curious little statement is posted.
If FACTORING is hard, then Fermat’s little theorem is not provable in $S_{2}^{1}$.
What is $S_{2}^{1}$ and why are current proofs not ...
13
votes
1answer
506 views
A 3-CNF formula that requires resolution width $5$
Recall that the width of a resolution refutation $R$ of a CNF formula $F$ is the maximal number of literals in any clause occurring in $R$. For every $w$, there are unsatisfiable formulas $F$ in 3-...
15
votes
2answers
3k views
Is propositional resolution a complete proof system?
This question is about propositional logic and all occurrences of "resolution" should be read as "propositional resolution".
This question is something extremely basic but it has been bothering me ...
24
votes
2answers
2k views
Sum-of-squares proof system
Recently I have seen several articles on arxiv that refer to a proof system called sum-of-squares.
Can someone explain what is a sum-of-squares proof and why such proofs are important/interesting?
...
4
votes
1answer
159 views
Consequences of a $p$-optimal proof system for $\operatorname{TAUT}$
I'm reading a paper which shows the result:
$(1)$ There is a $p$-optimal proof system for $\operatorname{TAUT}$. $\Leftrightarrow$
$(2)$ $L_{\leq}$ is a $P$-bounded logic for $P$.
Both $(1)$ and $(...
9
votes
2answers
294 views
Intuition behind proof systems
I'm trying to under stand the paper On p-Optimal Proof Systems and Logic for PTIME. There is a notion called proof systems in the paper and I do not get the intution:
$\Sigma = \{0,1\}$ ... We ...
9
votes
0answers
113 views
Reference for a propositional proof system being equivalent to its soundness?
I am looking for the original reference for the following statement:
Let $P$ be a propositional proof system containing $EF$. Then $P$ is equivalent to $EF+Sound(P)$.
Background
A propositional ...
15
votes
1answer
446 views
How efficient are DPLL-based SAT-solvers on satisfiable instances of PHP?
We know that DPLL based SAT-solvers fail to answer correctly on unsatisfiable instances of $\mathrm{PHP}$ (pigeon hole principle), e.g. on "there is a injective mapping from $n+1$ to $n$":
$$\mathrm{...
7
votes
2answers
1k views
Automated theorem proving via unsupervised approaches
This question Where and how did computers help prove a theorem? considers some automated theorem proving successes.
However they seem to be mostly supervised approaches, such as with the 4 color graph ...
0
votes
1answer
357 views
Proof Complexity and Circuit Lower bound for coNP
I have two questions
(1)Circuit lower bound for coNP
TAUT is a set of formulae such that any formula in TAUT is satisfied for all boolean assignments.
UnSAT is the complement problem of SAT.
It is ...
5
votes
1answer
238 views
Resolution vs Extended Resolution
Let $R(f)$ and $ER(f)$ be the minimum-size for unsat proofs of $f$ in Resolution and Extended Resolution respectively. What's the best bound we have on $D=\min_f (R(f)-ER(f))$ where $f$ belongs to a ...
8
votes
3answers
553 views
Graph problems with good characterization but not known to be in $P$
A decision problem has good characterization if it is in $NP \cap coNP$. Many natural graph problems have good characterizations. For instance, Kuratuwski's Theorem gives good characterization of ...
asked Apr 18 '12 at 17:48
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
14
votes
1answer
636 views
Consequences of sub-exponential proofs/algorithms for SAT
Would there be any major consequences if SAT had at most subexponential unsat proofs or even more strongly, SAT had subexponential-time algorithms?
18
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2answers
481 views
Uses of XORification
XORification is the technique to make a Boolean function or formula harder by replacing every variable $x$ by the XOR of $k\geq 2$ distinct variables $x_1 \oplus \ldots \oplus x_k$.
I am aware of ...
6
votes
1answer
325 views
Best lower bound for proof complexity of graph non-automorphism problem
Graph automorphism problem ( GA) of determining whether a graph has a nontrivial automorphism is a good candidate for a problem in NP-intermediate. I'm looking for references that study the ...
asked Oct 29 '11 at 22:02
Mohammad Al-Turkistany
20.3k4 gold badges56 silver badges142 bronze badges
8
votes
2answers
424 views
Postselection in geometric complexity theory
Context: As I understand, in geometric complexity theory, the existence of obstructions serves as a proof-certificate, so to speak, for the nonexistence of an efficient computational circuit for the ...
