2,547 research outputs found
Proving Finite Satisfiability of Deductive Databases
It is shown how certain refutation methods can be extended into semi-decision
procedures that are complete for both unsatisfiability and finite satisfiability. The proposed extension
is justified by a new characterization of finite satisfiability. This research was motivated
by a database design problem: Deduction rules and integrity constraints in definite databases
have to be finitely satisfiabl
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
From singularities to graphs
In this paper I analyze the problems which led to the introduction of graphs
as tools for studying surface singularities. I explain how such graphs were
initially only described using words, but that several questions made it
necessary to draw them, leading to the elaboration of a special calculus with
graphs. This is a non-technical paper intended to be readable both by
mathematicians and philosophers or historians of mathematics.Comment: 23 pages, 27 figures. Expanded version of the talk given at the
conference "Quand la forme devient substance : puissance des gestes,
intuition diagrammatique et ph\'enom\'enologie de l'espace", which took place
at Lyc\'ee Henri IV in Paris from 25 to 27 January 201
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
Why Philosophers Should Care About Computational Complexity
One might think that, once we know something is computable, how efficiently
it can be computed is a practical question with little further philosophical
importance. In this essay, I offer a detailed case that one would be wrong. In
particular, I argue that computational complexity theory---the field that
studies the resources (such as time, space, and randomness) needed to solve
computational problems---leads to new perspectives on the nature of
mathematical knowledge, the strong AI debate, computationalism, the problem of
logical omniscience, Hume's problem of induction, Goodman's grue riddle, the
foundations of quantum mechanics, economic rationality, closed timelike curves,
and several other topics of philosophical interest. I end by discussing aspects
of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and
beyond," MIT Press, 2012. Some minor clarifications and corrections; new
references adde
NP-complete Problems and Physical Reality
Can NP-complete problems be solved efficiently in the physical universe? I
survey proposals including soap bubbles, protein folding, quantum computing,
quantum advice, quantum adiabatic algorithms, quantum-mechanical
nonlinearities, hidden variables, relativistic time dilation, analog computing,
Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and
"anthropic computing." The section on soap bubbles even includes some
"experimental" results. While I do not believe that any of the proposals will
let us solve NP-complete problems efficiently, I argue that by studying them,
we can learn something not only about computation but also about physics.Comment: 23 pages, minor correction
Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models
AbstractA new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model
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