215 research outputs found
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
Is Quantum Mechanics An Island In Theoryspace?
This recreational paper investigates what happens if we change quantum
mechanics in several ways. The main results are as follows. First, if we
replace the 2-norm by some other p-norm, then there are no nontrivial
norm-preserving linear maps. Second, if we relax the demand that norm be
preserved, we end up with a theory that allows rapid solution of PP-complete
problems (as well as superluminal signalling). And third, if we restrict
amplitudes to be real, we run into a difficulty much simpler than the usual one
based on parameter-counting of mixed states.Comment: 9 pages, minor correction
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
Some subsystems of constant-depth Frege with parity
We consider three relatively strong families of subsystems of AC0[2]-Frege proof systems, i.e., propositional proof systems using constant-depth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. In order of increasing strength, the subsystems are (i) constant-depth proof systems with parity axioms and the (ii) treelike and (iii) daglike versions of systems introduced by Krajíček which we call PKcd(⊕). In a PKcd(⊕)-proof, lines are disjunctions (cedents) in which all disjuncts have depth at most d, parities can only appear as the outermost connectives of disjuncts, and all but c disjuncts contain no parity connective at all.
We prove that treelike PKO(1)O(1)(⊕) is quasipolynomially but not polynomially equivalent to constant-depth systems with parity axioms. We also verify that the technique for separating parity axioms from parity connectives due to Impagliazzo and Segerlind can be adapted to give a superpolynomial separation between daglike PKO(1)O(1)(⊕) and AC0[2]-Frege; the technique is inherently unable to prove superquasipolynomial separations.
We also study proof systems related to the system Res-Lin introduced by Itsykson and Sokolov. We prove that an extension of treelike Res-Lin is polynomially simulated by a system related to daglike PKO(1)O(1)(⊕), and obtain an exponential lower bound for this system.Peer ReviewedPostprint (author's final draft
Finite Model Theory and Proof Complexity Revisited: Distinguishing Graphs in Choiceless Polynomial Time and the Extended Polynomial Calculus
This paper extends prior work on the connections between logics from finite model theory and propositional/algebraic proof systems. We show that if all non-isomorphic graphs in a given graph class can be distinguished in the logic Choiceless Polynomial Time with counting (CPT), then they can also be distinguished in the bounded-degree extended polynomial calculus (EPC), and the refutations have roughly the same size as the resource consumption of the CPT-sentence. This allows to transfer lower bounds for EPC to CPT and thus constitutes a new potential approach towards better understanding the limits of CPT. A super-polynomial EPC lower bound for a Ptime-instance of the graph isomorphism problem would separate CPT from Ptime and thus solve a major open question in finite model theory. Further, using our result, we provide a model theoretic proof for the separation of bounded-degree polynomial calculus and bounded-degree extended polynomial calculus
Expander Construction in VNC1
We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen\u27s sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002)
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