propositional proof systems and corresponding bounded arithmetic theories

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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 ...
2
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1answer
84 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
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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(...
10
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1answer
103 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)...
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1answer
69 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-...
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1answer
93 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
284 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
50 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 ...
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0answers
85 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 ...
0
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0answers
112 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
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3answers
296 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 ...
26
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6answers
802 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
190 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
212 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
702 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
460 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
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3answers
438 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 ...
8
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1answer
180 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
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1answer
320 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
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1answer
346 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-...
13
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2answers
1k 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 ...
21
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2answers
801 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
120 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 $(...
8
votes
2answers
231 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
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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 ...
15
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1answer
383 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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2answers
731 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
282 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 ...
4
votes
1answer
188 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
506 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 ...
9
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1answer
329 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
399 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
276 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
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2answers
382 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
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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
424 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 ...
31
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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
530 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
603 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
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1answer
190 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 ...
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2answers
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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 ...
14
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0answers
259 views

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 ...
9
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1answer
483 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. ...
7
votes
2answers
738 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 ...
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3answers
697 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 ...
26
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6answers
809 views

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 ...