Questions tagged [proof-complexity]
propositional proof systems and corresponding bounded arithmetic theories
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1answer
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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 $\pi$.
...
2
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0answers
47 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
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0answers
77 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
72 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
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0answers
44 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
186 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 ...
3
votes
2answers
94 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
130 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 $y \in L$ ...
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0answers
99 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
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0answers
65 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 ...
3
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1answer
125 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
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1answer
168 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(...
10
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1answer
217 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
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1answer
96 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
189 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
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2answers
318 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) =...
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1answer
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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 ...
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0answers
100 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 ...
1
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0answers
130 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
338 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
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6answers
935 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
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0answers
199 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
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1answer
243 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 ...
12
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4answers
853 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
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4answers
517 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, ...
17
votes
3answers
497 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
218 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 the pigeonhole problem....
10
votes
1answer
388 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
416 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
2k 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 ...
23
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2answers
1k 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?
...
2
votes
1answer
142 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
256 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
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0answers
107 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
420 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
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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
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1answer
339 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
218 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
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3answers
524 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 ...
11
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1answer
489 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?
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2answers
433 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
303 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 ...
8
votes
2answers
403 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 ...
7
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0answers
179 views
Oracle sparating 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 oracle ...
13
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2answers
462 views
Does coNP-complete problem have subexponential size certificate?
Assuming NP != coNP, then there is no polynomial size certificate for coNP-complete problem. But what about subexponential size certificate? Particularly for coSAT, is there subexponential size proof ...
37
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2answers
3k views
Axioms necessary for theoretical computer science
This question is inspired by a similar question about applied mathematics on mathoverflow, and that nagging thought that important questions of TCS such as P vs. NP might be independent of ZFC (or ...
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3answers
568 views
Theories which characterize classes of computational complexity
When reading the paper "An applicative theory for FPH" you can encounter the following passage:
Considering theories which characterize classes of computational complexity, there are three ...
11
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1answer
651 views
NP vs co-NP and second-order logic
Assume that NP=co-NP and polynomial $p(x)$ bounds the length of the proof of unsatisfiability for a 3-CNF instance $x$. Then are there any results on what form any proof of unsatisfiability for $x$ of ...
10
votes
1answer
212 views
A direct sum theorem for Resolution clause space complexity?
Resolution is a scheme to prove unsatisfiability of CNFs. A proof in resolution is a logical deduction of the empty clause for the initial clauses in of the CNF. In particular any initial clause can ...
35
votes
2answers
2k views
If P=NP, could we obtain proofs of Goldbach's Conjecture etc.?
This is a naive question, out of my expertise; apologies in advance.
Goldbach's Conjecture and many other unsolved questions in mathematics can be written
as short formulas in predicate calculus.
For ...
