Theories for TC0 and Other Small Complexity Classes

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc

2-D Tucker is PPA complete

The 2-D Tucker search problem is shown to be PPA-hard under many-one reductions; therefore it is complete for PPA. The same holds for k-D Tucker for all k≥2. This corrects a claim in the literature that the Tucker search problem is in PPAD.Peer ReviewedPostprint (author's final draft

Polynomial-size Frege and Resolution Proofs of st-Connectivity and Hex Tautologies

Abstract A grid graph has rectangularly arranged vertices with edges permit-ted only between orthogonally adjacent vertices. The st-connectivityprinciple states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of thegrid graph unless the paths cross somewhere. We prove that the propositional tautologies which encode the st-connectivity principle have polynomial size Frege proofs and poly-nomial size T C0-Frege proofs. For bounded width grid graphs, the st-connectivity tautologies have polynomial size resolution proofs. Akey part of the proof is to show that the group with two generators, both of order two, has word problem in alternating logtime (Alogtime)and even in T C0.Conversely, we show that constant depth Frege proofs of the st-connectivity tautologies require near-exponential size. The proofuses a reduction from the pigeonhole principle, via tautologies that express a "directed single source " principle SINK, which is related toPapadimitriou's search classes PPAD and PPADS (or, PSK). The st-connectivity principle is related to Urquhart's propositionalHex tautologies, and we establish the same upper and lower bounds on proof complexity for the Hex tautologies. In addition, the Hextautology is shown to be equivalent to the SINK tautologies and to the one-to-one onto pigeonhole principle. *Supported in part by NSF grants DMS-0100589 and DMS-0400848

Polynomial-size Frege and Resolution Proofs of st-Connectivity and Hex Tautologies

A grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices. The st-connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere