3,904 research outputs found
The First-Order Theory of Sets with Cardinality Constraints is Decidable
We show that the decidability of the first-order theory of the language that
combines Boolean algebras of sets of uninterpreted elements with Presburger
arithmetic operations. We thereby disprove a recent conjecture that this theory
is undecidable. Our language allows relating the cardinalities of sets to the
values of integer variables, and can distinguish finite and infinite sets. We
use quantifier elimination to show the decidability and obtain an elementary
upper bound on the complexity.
Precise program analyses can use our decidability result to verify
representation invariants of data structures that use an integer field to
represent the number of stored elements.Comment: 18 page
Quantifier elimination in quasianalytic structures via non-standard analysis
The paper is a continuation of our earlier article where we developed a
theory of active and non-active infinitesimals and intended to establish
quantifier elimination in quasianalytic structures. That article, however, did
not attain full generality, which refers to one of its results, namely the
theorem on an active infinitesimal, playing an essential role in our
non-standard analysis. The general case was covered in our subsequent preprint,
which constitutes a basis for the approach presented here. We also provide a
quasianalytic exposition of the results concerning rectilinearization of terms
and of definable functions from our earlier research. It will be used to
demonstrate a quasianalytic structure corresponding to a Denjoy-Carleman class
which, unlike the classical analytic structure, does not admit quantifier
elimination in the language of restricted quasianalytic functions augmented
merely by the reciprocal function. More precisely, we construct a plane
definable curve, which indicates both that the classical theorem by J. Denef
and L. van den Dries as well as \L{}ojasiewicz's theorem that every subanalytic
curve is semianalytic are no longer true for quasianalytic structures. Besides
rectilinearization of terms, our construction makes use of some theorems on
power substitution for Denjoy-Carleman classes and on non-extendability of
quasianalytic function germs. The last result relies on Grothendieck's
factorization and open mapping theorems for (LF)-spaces. Note finally that this
paper comprises our earlier preprints on the subject from May 2012.Comment: Final version, 36 pages. arXiv admin note: substantial text overlap
with arXiv:1310.130
Proving theorems by program transformation
In this paper we present an overview of the unfold/fold proof method, a method for proving theorems about programs, based on program transformation. As a metalanguage for specifying programs and program properties we adopt constraint logic programming (CLP), and we present a set of transformation rules (including the familiar unfolding and folding rules) which preserve the semantics of CLP programs. Then, we show how program transformation strategies can be used, similarly to theorem proving tactics, for guiding the application of the transformation rules and inferring the properties to be proved. We work out three examples: (i) the proof of predicate equivalences, applied to the verification of equality between CCS processes, (ii) the proof of first order formulas via an extension of the quantifier elimination method, and (iii) the proof of temporal properties of infinite state concurrent systems, by using a transformation strategy that performs program specialization
Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition
Cylindrical algebraic decomposition(CAD) is a key tool in computational
algebraic geometry, particularly for quantifier elimination over real-closed
fields. When using CAD, there is often a choice for the ordering placed on the
variables. This can be important, with some problems infeasible with one
variable ordering but easy with another. Machine learning is the process of
fitting a computer model to a complex function based on properties learned from
measured data. In this paper we use machine learning (specifically a support
vector machine) to select between heuristics for choosing a variable ordering,
outperforming each of the separate heuristics.Comment: 16 page
SMT Solving for Functional Programming over Infinite Structures
We develop a simple functional programming language aimed at manipulating
infinite, but first-order definable structures, such as the countably infinite
clique graph or the set of all intervals with rational endpoints. Internally,
such sets are represented by logical formulas that define them, and an external
satisfiability modulo theories (SMT) solver is regularly run by the interpreter
to check their basic properties.
The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038
Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
We provide a scheme for inferring causal relations from uncontrolled
statistical data based on tools from computational algebraic geometry, in
particular, the computation of Groebner bases. We focus on causal structures
containing just two observed variables, each of which is binary. We consider
the consequences of imposing different restrictions on the number and
cardinality of latent variables and of assuming different functional
dependences of the observed variables on the latent ones (in particular, the
noise need not be additive). We provide an inductive scheme for classifying
functional causal structures into distinct observational equivalence classes.
For each observational equivalence class, we provide a procedure for deriving
constraints on the joint distribution that are necessary and sufficient
conditions for it to arise from a model in that class. We also demonstrate how
this sort of approach provides a means of determining which causal parameters
are identifiable and how to solve for these. Prospects for expanding the scope
of our scheme, in particular to the problem of quantum causal inference, are
also discussed.Comment: Accepted for publication in Journal of Causal Inference. Revised and
updated in response to referee feedback. 16+5 pages, 26+2 figures. Comments
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