Trade-Offs Between Size and Degree in Polynomial Calculus

Building on [Clegg et al. \u2796], [Impagliazzo et al. \u2799] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(?(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen \u2716], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary

Kant’s (Non-Question-Begging) Refutation of Cartesian Scepticism

Interpreters of Kant’s Refutation of Idealism face a dilemma: it seems to either beg the question against the Cartesian sceptic or else offer a disappointingly Berkeleyan conclusion. In this article I offer an interpretation of the Refutation on which it does not beg the question against the Cartesian sceptic. After defending a principle about question-begging, I identify four premises concerning our representations that there are textual reasons to think Kant might be implicitly assuming. Using those assumptions, I offer a reconstruction of Kant’s Refutation that avoids the interpretative dilemma, though difficult questions about the argument remain

The Enhancement of Mathematical Communication and Self Regulated Learning of Senior High School Students Through PQ4R Strategy Accompanied by Refutation Text Reading

This study is experiment research with control group pretest-posttest design and aimed to examine the influence of PQ4R strategy and Refutation Text, school level, and student’s mathematical early knowledge toward achievement and enhancement of student’s mathematical communication ability and Self Regulated Learning. Subject of study as much as 241 students of class X from three Public Senior High School from high, medium, and low school level. Research instrument consist of one set of student’s mathematical communication, and one set of student’s Self Regulated Learning scale. Data analysis use Kosmogorov-Smirnov Test (Test-Z), Level Test, Test-t, one-way and two-way ANOVA, Post Hoc Test (Scheffe) and also Chi-Square Test. Study found that learning with PR4R strategy accompanied by Refutation Text Reading give consistent influence compared with conventional learning as viewed as a whole, based on school level and also mathematical early knowledge. In addition, study also found: (1) there is no interaction between learning (PQ4R) accompanied by Refutation Text reading and conventional and school level toward (a) student’s mathematical communication and (b) student’s Self Regulated Learning; (2) there is no significant interaction between learning and student’s mathematical early knowledge toward (a) student’s mathematical communication ability and (b) student’s Self Regulated Learning; and (3) there is association between student’s mathematical communication ability and student’s Self Regulated Learning. Keywords: PQ4R, Refutation Text, Mathematical Communication, and Self Regulated Learning

Width and size of regular resolution proofs

This paper discusses the topic of the minimum width of a regular resolution refutation of a set of clauses. The main result shows that there are examples having small regular resolution refutations, for which any regular refutation must contain a large clause. This forms a contrast with corresponding results for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods in Computer Scienc

Implicit Resolution

Let \Omega be a set of unsatisfiable clauses, an implicit resolution refutation of \Omega is a circuit \beta with a resolution proof {\alpha} of the statement "\beta describes a correct tree-like resolution refutation of \Omega". We show that such system is p-equivalent to Extended Frege. More generally, let {\tau} be a tautology, a [P, Q]-proof of {\tau} is a pair (\alpha,\beta) s.t. \alpha is a P-proof of the statement "\beta is a circuit describing a correct Q-proof of \tau". We prove that [EF,P] \leq p [R,P] for arbitrary Cook-Reckhow proof system P

The Completeness of Propositional Resolution: A Simple and Constructive<br> Proof

It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a resolution refutation. As a consequence, none of these proofs actually gives an algorithm for producing a resolution refutation from an unsatisfiable set of clauses. In this note, we give a simple and constructive proof of the completeness of propositional resolution which consists of an algorithm together with a proof of its correctness.Comment: 7 pages, submitted to LMC

How to refute a random CSP

Let $P$ be a $k$-ary predicate over a finite alphabet. Consider a random CSP$(P)$ instance $I$ over $n$ variables with $m$ constraints. When $m \gg n$ the instance $I$ will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When $P$ is the $3$-ary OR predicate, this is the well studied problem of refuting random $3$-SAT formulas, and an efficient algorithm is known only when $m \gg n^{3/2}$. Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a $k$-ary predicate, having $m \gg n^{\lceil k/2 \rceil}$ constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if $P$ fails to support a $t$-wise uniform distribution, then there is an efficient algorithm that refutes random CSP$(P)$ instances $I$ whp when $m \gg n^{t/2}$. Indeed, our algorithm will "somewhat strongly" refute $I$, certifying $\mathrm{Opt}(I) \leq 1-\Omega_k(1)$, if $t = k$ then we get the strongest possible refutation, certifying $\mathrm{Opt}(I) \leq \mathrm{E}[P] + o(1)$. This last result is new even in the context of random $k$-SAT. Regarding the optimality of our $m \gg n^{t/2}$ requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP$(P)$ may be impossible when $m \ll n^{t/2}$. Thus there is an indication our algorithm's dependence on $m$ is optimal for every $P$, at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the $O(1)$-round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz

Falsification and refutation

A scientific theory, according to Popper, can be legitimately saved from falsification by introducing an auxiliary hypothesis to generate new, falsifiable predictions. Also, if there are suspicions of bias or error, the researchers might introduce an auxiliary falsifiable hypothesis that would allow testing. But this technique can not solve the problem in general, because any auxiliary hypothesis can be challenged in the same way, ad infinitum. To solve this regression, Popper introduces the idea of ​​a basic statement, an empirical statement that can be used both to determine whether a given theory is falsifiable and, if necessary, to corroborate falsification assumptions. DOI: 10.13140/RG.2.2.22162.0992