We call a pseudorandom generator Gn : {0, 1}n → {0, 1}m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement Gn(x1, . . . , xn) 6= b for any string b ∈ {0, 1}m. We consider a variety of “combinatorial” pseudorandom generators inspired by the Nisan-Wigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. ...

The existence of a (p-)optimal propositional proof system is a major open question in (proof) complexity; many people conjecture that such systems do not exist. Kraj́ıček and Pudlák [KP89] show that this question is equivalent to the existence of an algorithm that is optimal1 on all propositional tautologies. Monroe [Mon09] recently gave a conjecture implying that such algorithm does not exist. ...

This report gives an account of a successful formalization of the proof of the Four Colour Theorem, which was fully checked by the Coq v7.3.1 proof assistant [13]. This proof is largely based on the mixed mathematics/computer proof [26] of Robertson et al, but contains original contributions as well. This document is organized as follows: section 1 gives a historical introduction to the problem...

Formal proofs generated by mechanised theorem proving systems may consist of a large number of inferences. As these theorem proving systems are usually very complex, it is extremely diicult if not impossible to formally verify them. This calls for an independent means of ensuring the consistency of mechanically generated proofs. This paper describes a method of recording HOL proofs in terms of ...

We present a link between the interactive proof assistant Isabelle/HOL and the Spark/Ada tool suite for the verification of high-integrity software. Using this link, we can tackle verification problems that are beyond reach of the proof tools currently available for Spark. To demonstrate that our methodology is suitable for real-world applications, we show how it can be used to verify an effici...

Interactive formal proof and automated veriication based on decision graphs are two contrasting formal hardware veriication techniques. In this paper, we compare these two approaches. In particular, we consider HOL and MDG. The former is an interactive theorem-proving system based on higher-order logic, while the latter is an automatic system based on Multiway Decision Graphs. As the basis for ...

Interactive formal proof and automated veri cation based on decision graphs are two contrasting formal hardware veri cation techniques. In this paper, we compare these two approaches. In particular we consider HOL and MDG. The former is an interactive theorem proving system based on higher-order logic, while the latter is an automatic system based on Multiway Decision Graphs. As the basis for o...

Failure to deduce false suppositions in proof by contradiction is still considered “more difficult” than proving the conditional contraposition. This study aims identify types of construction failures based on action steps contradiction, then offer a framework failure hypothesis specifically used contradiction. The research data were collected and analyzed from work students who have agreed be ...