# Tag Info

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### Algebraic equivalent of SAT?

This is standard and widely used in computer science theory. There are many references that use boolean polynomials with False -> 0 and True -> 1, or in other words, a polynomial over GF(2) used ...
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### Lower Bounds for Frege and Extended Frege

1, 2, 4) The best known lower bounds on extended Frege are the same as for Frege: linear number of lines, and quadratic size. This applies e.g. to the tautologies $\neg^{2n}\top$ (basically, any ...

### Algebraic equivalent of SAT?

I think what you are asking about is also known as "polynomial calculus" in proof complexity and SAT solving. It was introduced in [1, 2] to investigate whether coNP can be separated from NP ...

### Are there any propositional proof systems which are not Cook-Reckhow proof systems?

Natural examples of propositional proof systems that do not fall under this definition are algebraic proof systems where the lines in the proof are arbitrary polynomials (not necessarily fully ...

### Graph theoretic restriction to Proofs in Proof Complexity Theory

Müller and Szeider study Resolution proofs where the proof DAG has bounded tree-width or bounded path-width (for suitable extensions of these graph complexity measures to directed graphs.) They show ...

### Do we currently know a polynomial-size Frege proof for Tseitin formulas?

Tseitin tautologies are unsatisfiable systems of linear equations over $\mathbb F_2$, and as such they can be refuted just by summing all the equations together (possibly after reconstructing the ...
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### Do we currently know a polynomial-size Frege proof for Tseitin formulas?

Section 6 of the following paper has a sketch: Alasdair Urquhart. Hard examples for resolution. Journal of the ACM, 34(1):209–219, 1987. DOI: https://doi.org/10.1145/7531.8928
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### Document references describing weaknesses for cutting planes and algebraic proof system?

For each of these proof systems we know that there are some formulas where the shortest proof needs to have exponential length. Some of the earliest examples are an exponential lower bound for the ...
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### Resolution augmented with the rule of symmetry or the rule of extension

First, ER p-simulates SR: for example, ER is p-equivalent to the extended Frege proof system (EF) which is p-equivalent to the substitution Frege proof system (SF), and it is easy to see that SF p-...
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### Axioms of Minimum Size Resolution Refutations

With the caveat that I am posting this quickly in a sleep-deprived state, I think the answer is "no" to all three questions. Take the pigeonhole principle formulas PHP^m_n for m pigeons and n holes. ...

### On the (Cook) definition of a propositional proof system

$f$ is not a prover, it's a proof-checker. $w$ is the proof. And the polynomial is a polynomial of the length of the proof, which could be much larger than the length of the thing being proved. If you ...

### Graph theoretic restriction to Proofs in Proof Complexity Theory

For strong enough proof systems the graph representation of a proof in the system seems less consequential, since (as Joshua Grochow already commented), DAG-like and tree-like Frege proofs are ...
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### Is there any work relating type systems and Cook-Reckhow proof systems?

Cook-Reckhow propositional proof systems are nonunifrom. E.g. the computational complexity counterpart to the class of polynomial-size $\mathsf{Extended Frege}$ proofs is the nonuniform complexity ...
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### Can CDCL Algorithm Derived Conflict Clauses Always Be Obtained Through Resolution from an Unsatisfiable CNF Formula?

It is indeed and here is the reference for it if needed: Pipatsrisawat, K., & Darwiche, A. (2011). On the power of clause-learning SAT solvers as resolution engines. Artificial intelligence, 175(2)...

### On the difference between propositional proof system and polynomially-bounded proof system

In Cook-Reckhow propositional proof systems proof checkers have to run in polynomial time w.r.t. the size of their input. The size of the input is the size of the proof. This is generally not a ...
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### Is there a relation between the techniques used by Dan Willard, versus those of Brown and Palsberg, to exclude diagonalization?

I'll defer if someone more knowledgeable on the subject arrives, but I think the answer is that they are not the same. Willard's work is about building theories that avoid the typical formulation of ...
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### Why isn't the proof obtained using Buss's proof of the derivational completeness of LK anchored?

The answer occurred to me right after I finished typing up the post. Rather than delete it, I figured I would post it anyway in case anyone else has the same question. Answer The mistake in the ...

### Construct proof systems for common algorithmic task, like equivalence of regular expressions

You have a non-deterministic algorithm deciding the problem. If you want to think of it as a proof system for $EQUIV$, then the proof of $(u,v) \in EQUIVE$ is just the string representing the ...

### Natural NP-complete problems with "large" witnesses

As for your first question, Allender states (in Amplifying Lower Bounds by Means of Self-Reducibility) that no natural NP-complete problem is known to lie outside of NTIME(n). This means that all ...

### IPS upper bound for subset sum axiom

First, Kaveh is correct that the verification for IPS is randomized, so all it would show is $\mathsf{NP} \subseteq \mathsf{coAM}$ (not $\mathsf{NP} = \mathsf{coNP}$). However, this alone would still ...
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### In regards to the tautologies of a polynomially-bounded propositional proof system

Your question is like asking what is the class of formulas for the problem SAT? In the definition of SAT it is fixed to some fixed class, say those based on $\{\lnot, \land, \lor\}$ but it doesn't ...

### Are there any propositional proof systems which are not Cook-Reckhow proof systems?

Assume that you have an algorithm $A$ which satisfies soundness and completeness. You can define a new proof checker which is sound, complete, and runs in polynomial time: it checks if a given $\pi$ ...

### IPS upper bound for subset sum axiom

I think what you are missing is probably the complexity of the proof verification algorithm for IPS. It is generally true that if we have a Cook-Reckhow proof system and have short proofs for a coNP-...

### Construct proof systems for common algorithmic task, like equivalence of regular expressions

Kaveh's response exemplifies well the Cook-Reckhow notion of an abstract proof system. Nonetheless, for comparison, I point to a recent preprint of mine and Damien Pous: A cut-free cyclic proof ...
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### Power of non-implicationally-complete Frege systems and Boolean equational calculus

$\let\eq\leftrightarrow\def\ru{\mathrel/}\let\ET\bigwedge$Frege systems are required to be implicationally complete to make all such systems p-equivalent, yielding a robust definition of the Frege ...

### Methods for Determining the minimal Width of Resolution Refutations for CNF Formulas

Partially answering question (2), the Prover-Delayer game of Atserias and Dalmau can be interpreted as a more general "dag-like query complexity" specialized to CNFs. See e.g. GGKS'18. And ...
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### Proof of SPFA's worst-case complexity?

Here's the algorithm (from the wikipedia page) then a proof of the time bound: ...
1 vote
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### Eliminating tautological axioms in tree-like $k$-DNF resolution

Since I could not find a proof in the literature, I wrote a short note containing the proof.
1 vote
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### Is mathematical proof itself NP-hard?

Here is a link to some lecture notes that answer my question - http://www.cs.cornell.edu/~sabhar/publications/iaspcmi-proofcomplexity00.pdf Proof complexity is a vast field of mathematics.

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