84 research outputs found
Proof Complexity of Systems of (Non-Deterministic) Decision Trees and Branching Programs
This paper studies propositional proof systems in which lines are sequents of decision trees or branching programs, deterministic or non-deterministic. Decision trees (DTs) are represented by a natural term syntax, inducing the system LDT, and non-determinism is modelled by including disjunction, ?, as primitive (system LNDT). Branching programs generalise DTs to dag-like structures and are duly handled by extension variables in our setting, as is common in proof complexity (systems eLDT and eLNDT).
Deterministic and non-deterministic branching programs are natural nonuniform analogues of log-space (L) and nondeterministic log-space (NL), respectively. Thus eLDT and eLNDT serve as natural systems of reasoning corresponding to L and NL, respectively.
The main results of the paper are simulation and non-simulation results for tree-like and dag-like proofs in LDT, LNDT, eLDT and eLNDT. We also compare them with Frege systems, constant-depth Frege systems and extended Frege systems
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
An Improved Separation of Regular Resolution from Pool Resolution and Clause Learning
We prove that the graph tautology principles of Alekhnovich, Johannsen,
Pitassi and Urquhart have polynomial size pool resolution refutations that use
only input lemmas as learned clauses and without degenerate resolution
inferences. We also prove that these graph tautology principles can be refuted
by polynomial size DPLL proofs with clause learning, even when restricted to
greedy, unit-propagating DPLL search
Quasipolynomial size frege proofs of Frankl's Theorem on the trace of sets
We extend results of Bonet, Buss and Pitassi on Bondy's Theorem and of Nozaki, Arai and Arai on Bollobas' Theorem by proving that Frankl's Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl's Theorem has polynomial size AC(0)-Frege proofs from instances of the pigeonhole principle.Peer ReviewedPostprint (author's final draft
DRAT and Propagation Redundancy Proofs Without New Variables
We study the complexity of a range of propositional proof systems which allow
inference rules of the form: from a set of clauses derive the set of
clauses where, due to some syntactic condition, is satisfiable if is, but where does not
necessarily imply . These inference rules include BC, RAT, SPR and PR
(respectively short for blocked clauses, resolution asymmetric tautologies,
subset propagation redundancy and propagation redundancy), which arose from
work in satisfiability (SAT) solving. We introduce a new, more general rule SR
(substitution redundancy).
If the new clause is allowed to include new variables then the systems
based on these rules are all equivalent to extended resolution. We focus on
restricted systems that do not allow new variables. The systems with deletion,
where we can delete a clause from our set at any time, are denoted DBC,
DRAT, DSPR, DPR and DSR. The systems without deletion
are BC, RAT, SPR, PR and SR.
With deletion, we show that DRAT, DSPR and DPR are
equivalent. By earlier work of Kiesl, Rebola-Pardo and Heule, they are also
equivalent to DBC. Without deletion, we show that SPR can simulate
PR provided only short clauses are inferred by SPR inferences. We also
show that many of the well-known "hard" principles have small SPR
refutations. These include the pigeonhole principle, bit pigeonhole principle,
parity principle, Tseitin tautologies and clique-coloring tautologies.
SPR can also handle or-fication and xor-ification, and lifting with an
index gadget. Our final result is an exponential size lower bound for RAT
refutations, giving exponential separations between RAT and both
DRAT and SPR
Reordering Rule Makes OBDD Proof Systems Stronger
Atserias, Kolaitis, and Vardi showed that the proof system of Ordered Binary Decision Diagrams with conjunction and weakening, OBDD(^, weakening), simulates CP^* (Cutting Planes with unary coefficients). We show that OBDD(^, weakening) can give exponentially shorter proofs than dag-like cutting planes. This is proved by showing that the Clique-Coloring tautologies have polynomial size proofs in the OBDD(^, weakening) system.
The reordering rule allows changing the variable order for OBDDs. We show that OBDD(^, weakening, reordering) is strictly stronger than OBDD(^, weakening). This is proved using the Clique-Coloring tautologies, and by transforming tautologies using coded permutations and orification. We also give CNF formulas which have polynomial size OBDD(^) proofs but require superpolynomial (actually, quasipolynomial size) resolution proofs, and thus we partially resolve an open question proposed by Groote and Zantema.
Applying dag-like and tree-like lifting techniques to the mentioned results, we completely analyze which of the systems among CP^*, OBDD(^), OBDD(^, reordering), OBDD(^, weakening) and OBDD(^, weakening, reordering) polynomially simulate each other. For dag-like proof systems, some of our separations are quasipolynomial and some are exponential; for tree-like systems, all of our separations are exponential
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