Questions tagged [proof-complexity]

propositional proof systems and corresponding bounded arithmetic theories

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Is there a relation between the techniques used by Dan Willard, versus those of Brown and Palsberg, to exclude diagonalization?

This question extends my inquiry from a previous post [0]. Dan Willard's Self-Justifying Axiom Systems/Self-Verifying Theories [1] and Brown and Palsberg's self-interpreter for F-Omega [2] both employ ...
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Boolean logic: What is the name of this trick to replace explicit negations by implications?

Consider a Boolean circuit $C$ composed of some finite set of input variables $A_1,\ldots, A_n$ and the connectives $\lor\land\neg\rightarrow$ (with $X\rightarrow Y=\neg X\lor Y$) (update: assume that ...
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Algorithmically determining proof complexity for Frege systems?

I apologize if this falls wildly short of research level - I am just learning the very basics of proof complexity and lack any real logic background. Let $F$ be a Frege proof system (a finite complete ...
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Proof of SPFA's worst-case complexity?

I am trying to prove the worst-case asymptotic time complexity of the Shortest Path Faster Algorithm (SPFA). I know the complexity is the same as the "original" Bellman-Ford (BF) algorithm, ...
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Examples of simulations in proof complexity that are not p-simulations

I am writing a paper on the complexity of some unorthodox proof systems, where I have two systems $P$ and $Q$ such that $P$ simulates $Q$ in the sense of it being possible to translate a $Q$-proof ...
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2 votes
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Asymptotic complexity lower bounds of proof checking

This paper on universal search mentions (on pp. 6-7) that proof checking can be done in $O(n^2)$ where $n$ is the length of the proof. Is this optimal? I don't want to specify the problem too ...
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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 ...
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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 ...
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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 ...
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7 votes
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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 ...
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4 votes
1 answer
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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 ...
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6 votes
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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 ...
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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 ...
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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 $\pi$. ...
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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 ...
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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?
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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 ...
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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 ...
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2 votes
2 answers
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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: ...
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3 votes
2 answers
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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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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)...
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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 ...
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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(...
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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)...
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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-...
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1 answer
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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+...
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2 answers
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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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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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1 vote
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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 ...
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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 ...
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3 answers
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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 ...
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28 votes
6 answers
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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 (...
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7 votes
0 answers
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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 ...
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9 votes
1 answer
291 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 ...
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4 answers
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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 ...
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10 votes
4 answers
626 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, ...
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18 votes
3 answers
575 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 ...
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8 votes
1 answer
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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 ...
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10 votes
1 answer
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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 ...
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13 votes
1 answer
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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-...
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16 votes
2 answers
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 ...
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25 votes
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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? ...
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4 votes
1 answer
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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 $(...
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9 votes
2 answers
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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 ...
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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 ...
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15 votes
1 answer
462 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{...
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8 votes
2 answers
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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 ...
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1 answer
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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 ...
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5 votes
1 answer
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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 ...
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8 votes
3 answers
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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 ...
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