731 research outputs found
Efficient Solving of Quantified Inequality Constraints over the Real Numbers
Let a quantified inequality constraint over the reals be a formula in the
first-order predicate language over the structure of the real numbers, where
the allowed predicate symbols are and . Solving such constraints is
an undecidable problem when allowing function symbols such or . In
the paper we give an algorithm that terminates with a solution for all, except
for very special, pathological inputs. We ensure the practical efficiency of
this algorithm by employing constraint programming techniques
Collection analysis for Horn clause programs
We consider approximating data structures with collections of the items that
they contain. For examples, lists, binary trees, tuples, etc, can be
approximated by sets or multisets of the items within them. Such approximations
can be used to provide partial correctness properties of logic programs. For
example, one might wish to specify than whenever the atom is proved
then the two lists and contain the same multiset of items (that is,
is a permutation of ). If sorting removes duplicates, then one would like to
infer that the sets of items underlying and are the same. Such results
could be useful to have if they can be determined statically and automatically.
We present a scheme by which such collection analysis can be structured and
automated. Central to this scheme is the use of linear logic as a omputational
logic underlying the logic of Horn clauses
Constraint programming in computational linguistics
Constraint programming is a programming paradigm that was originally invented in computer science to deal with hard combinatorial problems. Recently, constraint programming has evolved into a technology which permits to solve hard industrial scheduling and optimization problems. We argue that existing constraint programming technology can be useful for applications in natural language processing. Some problems whose treatment with traditional methods requires great care to avoid combinatorial explosion of (potential) readings seem to be solvable in an efficient and elegant manner using constraint programming. We illustrate our claim by two recent examples, one from the area of underspecified semantics and one from parsing
Deep Inference and Symmetry in Classical Proofs
In this thesis we see deductive systems for classical propositional and predicate logic which use deep inference, i.e. inference rules apply arbitrarily deep inside formulas, and a certain symmetry, which provides an involution on derivations. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they enjoy various new interesting properties. Not only the identity axiom, but also cut, weakening and even contraction are reducible to atomic form. This leads to inference rules that are local, meaning that the effort of applying them is bounded, and finitary, meaning that, given a conclusion, there is only a finite number of premises to choose from. The systems also enjoy new normal forms for derivations and, in the propositional case, a cut elimination procedure that is drastically simpler than the ones for sequent systems
Presburger arithmetic, rational generating functions, and quasi-polynomials
Presburger arithmetic is the first-order theory of the natural numbers with
addition (but no multiplication). We characterize sets that can be defined by a
Presburger formula as exactly the sets whose characteristic functions can be
represented by rational generating functions; a geometric characterization of
such sets is also given. In addition, if p=(p_1,...,p_n) are a subset of the
free variables in a Presburger formula, we can define a counting function g(p)
to be the number of solutions to the formula, for a given p. We show that every
counting function obtained in this way may be represented as, equivalently,
either a piecewise quasi-polynomial or a rational generating function. Finally,
we translate known computational complexity results into this setting and
discuss open directions.Comment: revised, including significant additions explaining computational
complexity results. To appear in Journal of Symbolic Logic. Extended abstract
in ICALP 2013. 17 page
A Survey of Satisfiability Modulo Theory
Satisfiability modulo theory (SMT) consists in testing the satisfiability of
first-order formulas over linear integer or real arithmetic, or other theories.
In this survey, we explain the combination of propositional satisfiability and
decision procedures for conjunctions known as DPLL(T), and the alternative
"natural domain" approaches. We also cover quantifiers, Craig interpolants,
polynomial arithmetic, and how SMT solvers are used in automated software
analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest,
Romania. 201
Automatic modular abstractions for template numerical constraints
We propose a method for automatically generating abstract transformers for
static analysis by abstract interpretation. The method focuses on linear
constraints on programs operating on rational, real or floating-point variables
and containing linear assignments and tests. In addition to loop-free code, the
same method also applies for obtaining least fixed points as functions of the
precondition, which permits the analysis of loops and recursive functions. Our
algorithms are based on new quantifier elimination and symbolic manipulation
techniques. Given the specification of an abstract domain, and a program block,
our method automatically outputs an implementation of the corresponding
abstract transformer. It is thus a form of program transformation. The
motivation of our work is data-flow synchronous programming languages, used for
building control-command embedded systems, but it also applies to imperative
and functional programming
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