Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems).

A propositional version of the construction shows that at least one of the following three statements is true:

1. 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm).

2. 2. $E \not \subseteq P/poly$.

3. 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${\text {NP}}$ sets.

In 2004 Atserias, Kolaitis, and Vardi proposed $\text {OBDD}$ -based propositional proof systems that prove unsatisfiability of a CNF formula by deduction of an identically false $\text {OBDD}$ from $\text {OBDD}$ s representing clauses of the initial formula. All $\text {OBDD}$ s in such proofs have the same order of variables. We initiate the study of $\text {OBDD}$ based proof systems that additionally contain a rule that allows changing the order in $\text {OBDD}$ s. At first we consider a proof system $\text {OBDD}(\land , \text{reordering})$ that uses the conjunction (join) rule and the rule that allows changing the order. We exponentially separate this proof system from $\text {OBDD}(\land )$ proof system that uses only the conjunction rule. We prove exponential lower bounds on the size of $\text {OBDD}(\land , \text{reordering})$ refutations of Tseitin formulas and the pigeonhole principle. The first lower bound was previously unknown even for $\text {OBDD}(\land )$ proofs and the second one extends the result of Tveretina et al. from $\text {OBDD}(\land )$ to $\text {OBDD}(\land , \text{reordering})$ .

In 2001 Aguirre and Vardi proposed an approach to the propositional satisfiability problem based on $\text {OBDD}$ s and symbolic quantifier elimination (we denote algorithms based on this approach as $\text {OBDD}(\land , \exists )$ algorithms). We augment these algorithms with the operation of reordering of variables and call the new scheme $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. We notice that there exists an $\text {OBDD}(\land , \exists )$ algorithm that solves satisfiable and unsatisfiable Tseitin formulas in polynomial time (a standard example of a hard system of linear equations over $\mathbb {F}_2$ ), but we show that there are formulas representing systems of linear equations over $\mathbb {F}_2$ that are hard for $\text {OBDD}(\land , \exists , \text{reordering})$ algorithms. Our hard instances are satisfiable formulas representing systems of linear equations over $\mathbb {F}_2$ that correspond to checksum matrices of error correcting codes.

The relativized weak pigeonhole principle states that if at least 2n out of n2 pigeons fly into n holes, then some hole must be doubly occupied. We prove that every DNF-refutation of the CNF encoding of this principle requires size $2^{\left( {{\rm{log\ }}n} \right)^{3/2 - \varepsilon } } $ for every ε﹥0 and every sufficiently large n. By reducing it to the standard weak pigeonhole principle with 2n pigeons and n holes, we also show that this lower bound is essentially tight in that there exist DNF-refutations of size $2^{\left( {{\rm{log\ }}n} \right)^{O\left( 1 \right)} } $ even in R(log). For the lower bound proof we need to discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition.

We introduce an extended tableau calculus for answer set programming (ASP). Theproof system is based on the ASP tableaux defined in the work by Gebser andSchaub (Tableau calculi for answer set programming. In Proceedings ofthe 22nd International Conference on Logic Programming (ICLP 2006),S. Etalle and M. Truszczynski, Eds. Lecture Notes in Computer Science, vol.4079. Springer, 11–25) with an added extension rule. We investigatethe power of Extended ASP Tableaux both theoretically and empirically. We studythe relationship of Extended ASP Tableaux with the Extended Resolution proofsystem defined by Tseitin for sets of clauses, and separate Extended ASPTableaux from ASP Tableaux by giving a polynomial-length proof for a family ofnormal logic programs {Φn} for which ASP Tableaux has exponential-length minimal proofs withrespect to n. Additionally, Extended ASP Tableaux implyinteresting insight into the effect of program simplification on the lengths ofproofs in ASP. Closely related to Extended ASP Tableaux, we empiricallyinvestigate the effect of redundant rules on the efficiency of ASP solving.

We prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the hard formulas are in CNF. We utilize (somewhat indirectly) feasible interpolation.

We define a proof system combining resolution and the OBDD proof system.

We give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1, ψ2, … s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel–Löb's logic, S andS4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.