Consistency of circuit lower bounds with bounded theories

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only show that the corresponding problems are hard on infinitely many input lengths. For instance, proving almost-everywhere circuit lower bounds is open even for problems in $\mathsf{MAEXP}$. Giving the notorious difficulty of proving lower bounds that hold for all large input lengths, we ask the following question: Can we show that a large set of techniques cannot prove that $\mathsf{NP}$ is easy infinitely often? Motivated by this and related questions about the interaction between mathematical proofs and computations, we investigate circuit complexity from the perspective of logic. Among other results, we prove that for any parameter $k \geq 1$ it is consistent with theory $T$ that computational class ${\mathcal C} \not \subseteq \textit{i.o.}\mathrm{SIZE}(n^k)$, where $(T, \mathcal{C})$ is one of the pairs: $T = \mathsf{T}^1_2$ and ${\mathcal C} = \mathsf{P}^\mathsf{NP}$, $T = \mathsf{S}^1_2$ and ${\mathcal C} = \mathsf{NP}$, $T = \mathsf{PV}$ and ${\mathcal C} = \mathsf{P}$. In other words, these theories cannot establish infinitely often circuit upper bounds for the corresponding problems. This is of interest because the weaker theory $\mathsf{PV}$ already formalizes sophisticated arguments, such as a proof of the PCP Theorem. These consistency statements are unconditional and improve on earlier theorems of [KO17] and [BM18] on the consistency of lower bounds with $\mathsf{PV}$

Separating Bounded Arithmetics by Herbrand Consistency

The problem of $\Pi_1-$separating the hierarchy of bounded arithmetic has been studied in the paper. It is shown that the notion of Herbrand Consistency, in its full generality, cannot $\Pi_1-$separate the theory ${\rm I\Delta_0+\bigwedge_j\Omega_j}$ from ${\rm I\Delta_0}$; though it can $\Pi_1-$separate ${\rm I\Delta_0+Exp}$ from ${\rm I\Delta_0}$. This extends a result of L. A. Ko{\l}odziejczyk (2006), by showing the unprovability of the Herbrand Consistency of ${\rm I\Delta_0}$ in the theory ${\rm I\Delta_0+\bigwedge_j\Omega_j}$.Comment: Published by Oxford University Press. arXiv admin note: text overlap with arXiv:1005.265

Herbrand Consistency of Some Arithmetical Theories

G\"odel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, \textit{Fundamenta Mathematicae} 171 (2002) 279--292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories ${\rm I\Delta_0+\Omega_m}$ with $m\geqslant 2$, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory $T\supseteq {\rm I\Delta_0+\Omega_2}$ in $T$ itself. In this paper, the above results are generalized for ${\rm I\Delta_0+\Omega_1}$. Also after tailoring the definition of Herbrand consistency for ${\rm I\Delta_0}$ we prove the corresponding theorems for ${\rm I\Delta_0}$. Thus the Herbrand version of G\"odel's second incompleteness theorem follows for the theories ${\rm I\Delta_0+\Omega_1}$ and ${\rm I\Delta_0}$

Resource Bounded Unprovability of Computational Lower Bounds

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P not= NP (more generally, any super-polynomial-time lower bound in PSPACE) is unprovable in a PTM-omega-consistent theory T, where T is a consistent PT-extension of PA. This result gives a unified view to the existing two major negative results on proving P not= NP, Natural Proofs and relativizable proofs, through the two manners of characterization of PTM-omega-consistency. We also show that the PTM-omega-consistency of T cannot be proven in any PTM-omega-consistent theory S, where S is a consistent PT-extension of T.Comment: 78 page

Indeterminism and Undecidability

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Goedel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e.\ almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories.Comment: 24 pages, v2 was major revision (doubled in size), v3 adds crucial clarifying footnote 24 and corrects a few typo

Connecting the provable with the unprovable: phase transitions for unprovability

Why are some theorems not provable in certain theories of mathematics? Why are most theorems from existing mathematics provable in very weak systems? Unprovability theory seeks answers for those questions. Logicians have obtained unprovable statements which resemble provable statements. These statements often contain some condition which seems to cause unprovability, as this condition can be modified, using a function parameter, in such a manner as to make the theorem provable. It turns out that in many cases there is a phase transition: By modifying the parameter slightly one changes the theorem from provable to unprovable. We study these transitions with the goal of gaining more insights into unprovability