6,769 research outputs found
Transformation of Refutation Graphs into Natural Deduction Proofs
Most computer generated proofs are stated in abstract representations not normally used by mathematicians. We describe a procedure to transform proofs represented as abstract refutation graphss into natural deduction proofs. The emphasis of this investigation is more on stylistic aspects rather than theoretical issues. In particular the topological properties of refutation graphs can be successfully exploited in order to obtain structured proofs
Limitations of Algebraic Approaches to Graph Isomorphism Testing
We investigate the power of graph isomorphism algorithms based on algebraic
reasoning techniques like Gr\"obner basis computation. The idea of these
algorithms is to encode two graphs into a system of equations that are
satisfiable if and only if if the graphs are isomorphic, and then to (try to)
decide satisfiability of the system using, for example, the Gr\"obner basis
algorithm. In some cases this can be done in polynomial time, in particular, if
the equations admit a bounded degree refutation in an algebraic proof systems
such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on
the polynomial calculus degree over all fields of characteristic different from
2 and also linear lower bounds for the degree of Positivstellensatz calculus
derivations.
We compare this approach to recently studied linear and semidefinite
programming approaches to isomorphism testing, which are known to be related to
the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the
power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system
that lies between degree-k Nullstellensatz and degree-k polynomial calculus
Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions
For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are
the amounts of time and memory used. In proof complexity, these resources
correspond to the length and space of resolution proofs. There has been a long
line of research investigating these proof complexity measures, but while
strong results have been established for length, our understanding of space and
how it relates to length has remained quite poor. In particular, the question
whether resolution proofs can be optimized for length and space simultaneously,
or whether there are trade-offs between these two measures, has remained
essentially open.
In this paper, we remedy this situation by proving a host of length-space
trade-off results for resolution. Our collection of trade-offs cover almost the
whole range of values for the space complexity of formulas, and most of the
trade-offs are superpolynomial or even exponential and essentially tight. Using
similar techniques, we show that these trade-offs in fact extend to the
exponentially stronger k-DNF resolution proof systems, which operate with
formulas in disjunctive normal form with terms of bounded arity k. We also
answer the open question whether the k-DNF resolution systems form a strict
hierarchy with respect to space in the affirmative.
Our key technical contribution is the following, somewhat surprising,
theorem: Any CNF formula F can be transformed by simple variable substitution
into a new formula F' such that if F has the right properties, F' can be proven
in essentially the same length as F, whereas on the other hand the minimal
number of lines one needs to keep in memory simultaneously in any proof of F'
is lower-bounded by the minimal number of variables needed simultaneously in
any proof of F. Applying this theorem to so-called pebbling formulas defined in
terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical
reports TR09-034 and TR09-047, and it hence subsumes these two report
On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and
of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound
in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a
(very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We
show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial
in the number of propositional variables. The proof technique for the lower bound extends the
idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle
beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal
Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. Ëť
We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles
On Graph Refutation for Relational Inclusions
We introduce a graphical refutation calculus for relational inclusions: it
reduces establishing a relational inclusion to establishing that a graph
constructed from it has empty extension. This sound and complete calculus is
conceptually simpler and easier to use than the usual ones.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning
Resolution refinements called w-resolution trees with lemmas (WRTL) and with
input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both
WRTL and WRTI when there is no regularity condition. For regular proofs, an
exponential separation between regular dag-like resolution and both regular
WRTL and regular WRTI is given.
It is proved that DLL proof search algorithms that use clause learning based
on unit propagation can be polynomially simulated by regular WRTI. More
generally, non-greedy DLL algorithms with learning by unit propagation are
equivalent to regular WRTI. A general form of clause learning, called
DLL-Learn, is defined that is equivalent to regular WRTL.
A variable extension method is used to give simulations of resolution by
regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and
non-greedy DLL algorithms with learning by unit propagation can use variable
extensions to simulate general resolution without doing restarts.
Finally, an exponential lower bound for WRTL where the lemmas are restricted
to short clauses is shown
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