Questions tagged [proof-complexity]

propositional proof systems and corresponding bounded arithmetic theories

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38
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2answers
4k 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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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 ...
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6answers
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Well known classes of boolean formulas that require exponentially long resolution proofs

You might often find cutting plane methods, variable propagation, branch and bound, clause learning, intelligent backtracking or even handwoven human heuristics in SAT solvers. Yet for decades the ...
28
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6answers
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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 (...
24
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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? ...
19
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3answers
924 views

What algorithms are known for computing Craig interpolants?

Is there any survey of algorithms for computing interpolants? What about papers on only one algorithm? The case I'm most interested in is $A=\lnot p\land q$ and $C=q$, plus the constraint that the ...
18
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3answers
558 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 ...
18
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2answers
483 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 ...
15
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3answers
596 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 ...
15
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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 ...
15
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1answer
447 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{...
14
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1answer
647 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?
14
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0answers
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Any example of an unsatisfiable integer program with non constant Rank Lower bounds for LS+ cuts but with short LS+ refutations?

Assume we want to refute an unsatisfiable CNF. We can interpret it as an integer program, thus a refutation can be done by applying Lovasz-Schrijver semidefinite cuts ($LS_{+}$ cuts) to its linear ...
13
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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 ...
13
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2answers
519 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 ...
13
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1answer
511 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-...
11
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1answer
707 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 ...
11
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1answer
644 views

Using Kolmogorov complexity to establish proof complexity lower bounds?

The motivation for this question is the fact that most n-bit strings are incompressible. Intuitively, we can propose by analogy that most proofs for Tautologies are incompressible to polynomial size. ...
10
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3answers
386 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 ...
10
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1answer
430 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 ...
10
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4answers
595 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, ...
10
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1answer
243 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 ...
9
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1answer
278 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)...
9
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2answers
295 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
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 ...
8
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3answers
554 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 ...
8
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2answers
880 views

PCP theorem and proof complexity?

It is known that if $P=NP$ then $CoNP= PCP[O(log(n)),O(1)]$. Also, it is known that $NEXP=PCP[poly(n),poly(n)]$. It appears that PCP can't tell us which natural problems are not in $NP$. I wonder if ...
8
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2answers
427 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 ...
8
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1answer
248 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 ...
8
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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 ...
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 ...
7
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1answer
188 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 ...
7
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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 ...
7
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0answers
183 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 ...
6
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2answers
331 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) =...
6
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1answer
98 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 ...
6
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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 ...
6
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1answer
326 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 ...
5
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1answer
242 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 ...
5
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1answer
85 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 ...
4
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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 ...
4
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1answer
162 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 $(...
4
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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 ...
4
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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 ...
4
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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 ...
3
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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 $y \in L$ ...
3
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0answers
123 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 ...
3
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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 ...
3
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0answers
98 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 ...
2
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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: ...