We develop algorithms for computing Craig interpolants for first-order formulas over real numbers with a wide range of nonlinear functions, including transcendental functions and differential equations. We transform proof traces from δ-complete decision procedures into interpolants that consist of Boolean combinations of linear constraints. The algorithms are guaranteed to find the interpolants...

Interpolation (together with completeness and decidability) has become one of the standard properties that logicians investigate when designing a logic. In this paper, we provide strong evidence that the presence of interpolants is not only cogent for scientific reasoning but has also important practical implications in computer science. We illustrate that interpolation in general, and uniform ...

We present ongoing work to extend proof tree preserving interpolation to inductive sequences and tree interpolants. We present an algorithm to compute tree interpolants inductively over a resolution proof. Correctness of the algorithm is justified by the concept of partial tree interpolants and the appropriate definition of a projection function for conjunctions of literals onto nodes of the tr...

We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform interpolants and their existence in terms of bisimulations, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of modeltheoretic a...

For the problem of constructing smooth functions over arbitrary surfaces from discrete data, we propose to use Loop’s subdivision functions as the interpolants. Results on the existence, uniqueness and error bound of the interpolants are established. An efficient progressive computation algorithm for the interpolants is also presented. Mathematics subject classification:

Approximation orders of shift-invariant subspaces of L p (IR d), 2 p 1, generated by the shifts of one compactly supported function are considered. In that course, explicit approximation maps are constructed. The approach avoids quasi-interpolation and applies to stationary and non-stationary reenements. The general results are specialized to box spline spaces, to obtain new results on their ap...

This paper describes the Vinter tool for extracting interpolants from proofs and minimising such interpolants using various measures. Vinter takes an input problem written in either SMT-LIB or TPTP syntax, generates so called local proofs and then uses a technique of playing in the grey areas of proofs to find interpolants minimal with respect to various measures. Proofs are found using either ...

Logical interpolants have found a wide array of applications in automated verification, including symbolic model checking and predicate abstraction. It is often critical to these applications that reported interpolants exhibit desired properties, correctness being first and foremost. In this paper, we introduce a method in which interpolants are computed by type inference within the trusted cor...

Interpolant-based model checking is an approximate method for computing invariants of transition systems. The performance of the model checker is contingent on the approximation computed, which in turn depends on the logical strength of the interpolants. A good approximation is coarse enough to enable rapid convergence but strong enough to be contained within the weakest inductive invariant. We...

One of the main challenges in software verification is efficient and precise compositional analysis of programs with procedures and loops. Interpolation methods remains one of the most promising techniques for such verification, and are closely related to solving Horn clause constraints. We introduce a new notion of interpolation, disjunctive interpolation, which solve a more general class of p...