scholarly journals Synchronizing Constrained Horn Clauses

10.29007/gr5c ◽  
2018 ◽  
Author(s):  
Dmitry Mordvinov ◽  
Grigory Fedyukovich
Keyword(s):  
Iterative Algorithm ◽  
Recurrence Relations ◽  
Multiple Recurrence ◽  
Integer Arithmetic ◽  
Horn Clauses ◽  
Constraint Language ◽  
Non Linear

Simultaneous occurrences of multiple recurrence relations in a system of non-linear constrained Horn clauses are crucial for proving its satis ability. A solution of such system is often inexpressible in the constraint language. We propose to synchronize recurrent computations, thus increasing the chances for a solution to be found. We introduce a notion of CHC product allowing to formulate a lightweight iterative algorithm of merging recurrent computations into groups and prove its soundness. The evaluation over a set of systems handling lists and linear integer arithmetic confirms that the transformed systems are drastically more simple to solve than the original ones.

scholarly journals A linear algorithm for solving non-linear isothermal ice-shelf equations

2014 ◽  
Vol 7 (2) ◽  
pp. 1829-1864
Author(s):  
A. Sargent ◽  
J. L. Fastook
Keyword(s):  
Differential Equations ◽  
Iterative Algorithm ◽  
Linear Equations ◽  
Direct Methods ◽  
Two Dimensional ◽  
Ice Shelf ◽  
Linear Algorithm ◽  
Manufactured Solutions ◽  
Non Linear ◽  
Non Linear Equations

Abstract. A linear non-iterative algorithm is suggested for solving nonlinear isothermal steady-state Morland–MacAyeal ice shelf equations. The idea of the algorithm is in replacing the problem of solving the non-linear second order differential equations for velocities with a system of linear first order differential equations for stresses. The resulting system of linear equations can be solved numerically with direct methods which are faster than iterative methods for solving corresponding non-linear equations. The suggested algorithm is applicable if the boundary conditions for stresses can be specified. The efficiency of the linear algorithm is demonstrated for one-dimensional and two-dimensional ice shelf equations by comparing the linear algorithm and the traditional iterative algorithm on derived manufactured solutions. The linear algorithm is shown to be as accurate as the traditional iterative algorithm but significantly faster. The method may be valuable as the way to increase the efficiency of complex ice sheet models a part of which requires solving the ice shelf model as well as to solve efficiently two-dimensional ice-shelf equations.

scholarly journals Tree dimension in verification of constrained Horn clauses

2018 ◽  
Vol 18 (2) ◽  
pp. 224-251 ◽  
Author(s):  
BISHOKSAN KAFLE ◽  
JOHN P. GALLAGHER ◽  
PIERRE GANTY
Keyword(s):  
Experimental Results ◽  
Horn Clause ◽  
Higher Dimension ◽  
Dimension Zero ◽  
Horn Clauses ◽  
Verification Problem ◽  
Non Linear

AbstractIn this paper, we show how the notion of tree dimension can be used in the verification of constrained Horn clauses (CHCs). The dimension of a tree is a numerical measure of its branching complexity and the concept here applies to Horn clause derivation trees. Derivation trees of dimension zero correspond to derivations using linear CHCs, while trees of higher dimension arise from derivations using non-linear CHCs. We show how to instrument CHCs predicates with an extra argument for the dimension, allowing a CHC verifier to reason about bounds on the dimension of derivations. Given a set of CHCsP, we define a transformation ofPyielding adimension-boundedset of CHCsP≤k. The set of derivations forP≤kconsists of the derivations forPthat have dimension at mostk. We also show how to construct a set of clauses denotedP>kwhose derivations have dimension exceedingk. We then present algorithms using these constructions to decompose a CHC verification problem. One variation of this decomposition considers derivations of successively increasing dimension. The paper includes descriptions of implementations and experimental results.

scholarly journals Property-Directed Inference of Relational Invariants

2019 ◽  
Vol 26 (4) ◽  
pp. 550-571
Author(s):  
Dmitry A. Mordvinov
Keyword(s):  
Experimental Evaluation ◽  
Horn Clauses ◽  
Non Linear ◽  
Symbolic Constraints ◽  
Novel Algorithm

Property Directed Reachability (PDR) is an efficient and scalable approach to solving systems of symbolic constraints also known as Constrained Horn Clauses (CHC). In the case of non-linear CHCs, which may arise, e.g., from relational verification tasks, PDR aims to infer an inductive invariant for each uninterpreted predicate. However, in many practical cases this reasoning is not successful, as invariants should be derived for groups of predicates instead of individual predicates. The article describes a novel algorithm that identifies these groups automatically and complements the existing PDR technique. The key feature of the algorithm is that it does not require a possibly expensive synchronization transformation over the system of CHCs. We have implemented the algorithm on top of a up-to-date CHC solver Spacer. Our experimental evaluation shows that for some CHC systems, on which existing solvers diverge, our tool is able to discover relational invariants.

scholarly journals On a Quasi-Linear Equation

1956 ◽  
Vol 8 ◽  
pp. 198-202 ◽  
Author(s):  
Richard Bellman
Keyword(s):  
Linear Equation ◽  
Limit Theorems ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Initial Values ◽  
Non Linear ◽  
Image Position

1. Introduction. The purpose of this note is to establish some limit theorems for the non-linear recurrence relations1.1, i= 1,2, …, N; n ≥ 0under certain assumptions concerning the initial values ci = xi(0), and the coefficient matrices A (q) = (aij(q)).

scholarly journals Solving non-linear Horn clauses using a linear Horn clause solver

2016 ◽  
Vol 219 ◽  
pp. 33-48 ◽  
Author(s):  
Bishoksan Kafle ◽  
John P. Gallagher ◽  
Pierre Ganty
Keyword(s):  
Horn Clause ◽  
Horn Clauses ◽  
Non Linear
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