16,283 research outputs found
Space complexity in polynomial calculus
During the last decade, an active line of research in proof complexity has been to study space
complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of
intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused
on weak systems that are used by SAT solvers.
There has been a relatively long sequence of papers on space in resolution, which is now reasonably
well understood from this point of view. For other natural candidates to study, however, such as
polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial
space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been
for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is
smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent
with current knowledge that polynomial calculus could be able to refute any k-CNF formula in
constant space.
In this paper, we prove several new results on space in polynomial calculus (PC), and in the
extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]:
1. We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole
principle formulas PHPm
n with m pigeons and n holes, and show that this is tight.
2. For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole
principle. These formulas have width O(log n), and hence this is an exponential
improvement over [Alekhnovich et al. ’02] measured in the width of the formulas.
3. We then present another encoding of the pigeonhole principle that has constant width, and
prove an Ω(n) space lower bound in PCR for these formulas as well.
4. Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential
size and linear space (which holds for resolution and thus for PCR, but was not obviously
the case for PC). We also characterize a natural class of CNF formulas for which the space
complexity in resolution and PCR does not change when the formula is transformed into 3-CNF
in the canonical way, something that we believe can be useful when proving PCR space lower
bounds for other well-studied formula families in proof complexity
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
Improved bounds for the rate loss of multiresolution source codes
We present new bounds for the rate loss of multiresolution source codes (MRSCs). Considering an M-resolution code, the rate loss at the ith resolution with distortion D/sub i/ is defined as L/sub i/=R/sub i/-R(D/sub i/), where R/sub i/ is the rate achievable by the MRSC at stage i. This rate loss describes the performance degradation of the MRSC compared to the best single-resolution code with the same distortion. For two-resolution source codes, there are three scenarios of particular interest: (i) when both resolutions are equally important; (ii) when the rate loss at the first resolution is 0 (L/sub 1/=0); (iii) when the rate loss at the second resolution is 0 (L/sub 2/=0). The work of Lastras and Berger (see ibid., vol.47, p.918-26, Mar. 2001) gives constant upper bounds for the rate loss of an arbitrary memoryless source in scenarios (i) and (ii) and an asymptotic bound for scenario (iii) as D/sub 2/ approaches 0. We focus on the squared error distortion measure and (a) prove that for scenario (iii) L/sub 1/<1.1610 for all D/sub 2/<0.7250; (c) tighten the Lastras-Berger bound for scenario (i) from L/sub i//spl les/1/2 to L/sub i/<0.3802, i/spl isin/{1,2}; and (d) generalize the bounds for scenarios (ii) and (iii) to M-resolution codes with M/spl ges/2. We also present upper bounds for the rate losses of additive MRSCs (AMRSCs). An AMRSC is a special MRSC where each resolution describes an incremental reproduction and the kth-resolution reconstruction equals the sum of the first k incremental reproductions. We obtain two bounds on the rate loss of AMRSCs: one primarily good for low-rate coding and another which depends on the source entropy
Decidability of the Monadic Shallow Linear First-Order Fragment with Straight Dismatching Constraints
The monadic shallow linear Horn fragment is well-known to be decidable and
has many application, e.g., in security protocol analysis, tree automata, or
abstraction refinement. It was a long standing open problem how to extend the
fragment to the non-Horn case, preserving decidability, that would, e.g.,
enable to express non-determinism in protocols. We prove decidability of the
non-Horn monadic shallow linear fragment via ordered resolution further
extended with dismatching constraints and discuss some applications of the new
decidable fragment.Comment: 29 pages, long version of CADE-26 pape
Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in<br> SAT-Based Planning
In Verification and in (optimal) AI Planning, a successful method is to
formulate the application as boolean satisfiability (SAT), and solve it with
state-of-the-art DPLL-based procedures. There is a lack of understanding of why
this works so well. Focussing on the Planning context, we identify a form of
problem structure concerned with the symmetrical or asymmetrical nature of the
cost of achieving the individual planning goals. We quantify this sort of
structure with a simple numeric parameter called AsymRatio, ranging between 0
and 1. We run experiments in 10 benchmark domains from the International
Planning Competitions since 2000; we show that AsymRatio is a good indicator of
SAT solver performance in 8 of these domains. We then examine carefully crafted
synthetic planning domains that allow control of the amount of structure, and
that are clean enough for a rigorous analysis of the combinatorial search
space. The domains are parameterized by size, and by the amount of structure.
The CNFs we examine are unsatisfiable, encoding one planning step less than the
length of the optimal plan. We prove upper and lower bounds on the size of the
best possible DPLL refutations, under different settings of the amount of
structure, as a function of size. We also identify the best possible sets of
branching variables (backdoors). With minimum AsymRatio, we prove exponential
lower bounds, and identify minimal backdoors of size linear in the number of
variables. With maximum AsymRatio, we identify logarithmic DPLL refutations
(and backdoors), showing a doubly exponential gap between the two structural
extreme cases. The reasons for this behavior -- the proof arguments --
illuminate the prototypical patterns of structure causing the empirical
behavior observed in the competition benchmarks
Variations on a Theme: A Bibliography on Approaches to Theorem Proving Inspired From Satchmo
This articles is a structured bibliography on theorem provers,
approaches to theorem proving, and theorem proving applications inspired
from Satchmo, the model generation theorem prover developed
in the mid 80es of the 20th century at ECRC, the European Computer-
Industry Research Centre. Note that the bibliography given in this article
is not exhaustive
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