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First-Order Interpolation and Interpolating Proof Systems

Laura Kovács, Andrei Voronkov
unpublished
We finally propose a new direction related to interpolation vialocal proofs in first-order theories.  ...  for alternatives for interpolating proof systems, we consider severalvariations on interpolation and local proofs.  ...  We thank Stefan Hetzl and Matthias Baaz for pointing out relevant works on relational interpolants and discussing proof-theoretic approaches on constructing such interpolants.  ... 
doi:10.29007/1qb8 fatcat:ea7b4s5xy5f5zmdpotltjlxd4q

Interpolant Strength Revisited [chapter]

Georg Weissenbacher
2012 Lecture Notes in Computer Science  
Finally, we show that, when applied to local (or split) proofs, our extension generalises two existing interpolation systems for first-order logic and relates them in logical strength.  ...  There is a variety of interpolation systems which derive interpolants from refutation proofs; these systems are ad-hoc and rigid in the sense that they provide exactly one interpolant for a given proof  ...  -When applied to local (or split) proofs [13] , the extended interpolation system generalises the existing interpolation systems for first-order logic presented in [15] and [25] and relates them in  ... 
doi:10.1007/978-3-642-31612-8_24 fatcat:oq2e2n4oo5garguppejho4lqca

Interpolation Systems for Ground Proofs in Automated Deduction: a Survey

Maria Paola Bonacina, Moa Johansson
2015 Journal of automated reasoning  
We analyze interpolation in the presence of equality, and we demonstrate the color-based approach by giving a complete interpolation system for ground proofs by superposition.  ...  An interpolation system takes a refutation and extracts an interpolant by building it inductively from partial interpolants.  ...  proof produced by the SAT-solver to a proof in the first-order theory, and interpolate the latter [58] .  ... 
doi:10.1007/s10817-015-9325-5 fatcat:tdwepokjfvcjxfx2ieykouexee

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Angelo Brillout, Daniel Kroening, Philipp Rümmer, Thomas Wahl
2011 Journal of automated reasoning  
We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks.  ...  We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants.  ...  Interpolants exist for any two first-order formulae A and C such that A ⇒ C.  ... 
doi:10.1007/s10817-011-9237-y fatcat:e53cvfqtmvgrlj6i6is6zaquhm

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic [chapter]

Angelo Brillout, Daniel Kroening, Philipp Rümmer, Thomas Wahl
2010 Lecture Notes in Computer Science  
We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks.  ...  We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants.  ...  Interpolants exist for any two first-order formulae A and C such that A ⇒ C.  ... 
doi:10.1007/978-3-642-14203-1_33 fatcat:evdehmnl45btlcqpwt5hf75f4i

On Interpolation in Automated Theorem Proving

Maria Paola Bonacina, Moa Johansson
2014 Journal of automated reasoning  
Known interpolation systems for ground proofs use colors to track symbols.  ...  An interpolation system takes a refutation of the inconsistent formulae and extracts an interpolant by building it inductively from partial interpolants.  ...  Inference Systems and their Proof Trees We consider resolution-based clausal refutational inference systems for first-order logic with equality, and use Γ as the name of such a system.  ... 
doi:10.1007/s10817-014-9314-0 fatcat:p4l7gjyafbbxrg3fbe6npjbwqm

Interpolation in practical formal development

J Bicarregui
2001 Logic Journal of the IGPL  
We illustrate that interpolation in general, and uniform splitting interpolants, in particular, play an important role in applications where formality and modularity are invoked.  ...  Interpolation (together with completeness and decidability) has become one of the standard properties that logicians investigate when designing a logic.  ...  The basic theory of the B-Method is the amalgamated union of a first order set theory, a first order theory of natural numbers and a first order theory of generalised substitutions. 8 .  ... 
doi:10.1093/jigpal/9.2.231 fatcat:v4tf6vnt6nb47m7qaefu43siny

Splitting Proofs for Interpolation [chapter]

Bernhard Gleiss, Laura Kovács, Martin Suda
2017 Lecture Notes in Computer Science  
We study interpolant extraction from local first-order refutations.  ...  We implemented the new algorithm in first-order theorem prover VAMPIRE and evaluated it on a large number of examples coming from the first-order proving community.  ...  -We present a new algorithm for first-order-interpolation using local proofs in arbitrary sound inference systems.  ... 
doi:10.1007/978-3-319-63046-5_18 fatcat:asu2e5ocpbgmddwk7pezfi2bdy

Splitting Proofs for Interpolation [article]

Bernhard Gleiss, Laura Kovacs, Martin Suda
2017 arXiv   pre-print
We study interpolant extraction from local first-order refutations.  ...  We implemented the new algorithm in first-order theorem prover VAMPIRE and evaluated it on a large number of examples coming from the first-order proving community.  ...  -We present a new algorithm for first-order-interpolation using local proofs in arbitrary sound inference systems.  ... 
arXiv:1711.02503v1 fatcat:ae4ltmfy55hhnfxhspjisurbsi

Interpolants in Nonlinear Theories Over the Reals [chapter]

Sicun Gao, Damien Zufferey
2016 Lecture Notes in Computer Science  
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 show applications of the methods in control and robotic design, and hybrid system verification.  ...  We thank Martin Fränzle for pointing out the work on Craig interpolation in iSat, and the anonymous reviewers for their helpful feedback.  ... 
doi:10.1007/978-3-662-49674-9_41 fatcat:u5akjwy3lng4vmm7woaslhkfme

Lemma Localization: A Practical Method for Downsizing SMT-Interpolants

Florian Pigorsch, Christoph Scholl
2013 Design, Automation & Test in Europe Conference & Exhibition (DATE), 2013  
The latter systems demand interpolation for fragments of first-order logic. When it comes to model checking, the structural compactness of interpolants is necessary for efficient algorithms.  ...  Craig interpolation has become a powerful and universal tool in the formal verification domain, where it is used not only for Boolean systems, but also for timed systems, hybrid systems, and software programs  ...  In this paper we consider interpolants for first-order formulas.  ... 
doi:10.7873/date.2013.287 dblp:conf/date/PigorschS13 fatcat:nges3ejbpvb4ba23xfi2adscku

Labelled Interpolation Systems for Hyper-Resolution, Clausal, and Local Proofs

Matthias Schlaipfer, Georg Weissenbacher
2016 Journal of automated reasoning  
Finally, we show that, when applied to local (or split) proofs, our extension generalises two existing interpolation systems for first-order logic and relates them in logical strength.  ...  There is a variety of interpolation systems which derive interpolants from refutation proofs; these systems are ad-hoc and rigid in the sense that they provide exactly one interpolant for a given proof  ...  -When applied to local (or split) proofs [30] , the extended interpolation system generalises the existing interpolation systems for first-order logic presented in [32, 55] and relates them in logical  ... 
doi:10.1007/s10817-016-9364-6 pmid:30174360 pmcid:PMC6109788 fatcat:if6rf4dpgbbdjfwawnkiwpwqli

Craig Interpolation with Clausal First-Order Tableaux [article]

Christoph Wernhard
2021 arXiv   pre-print
We present the first interpolation method for first-order proofs represented by closed tableaux that proceeds in two stages, similar to known interpolation methods for resolution proofs.  ...  In addition, we discuss various subtle aspects that are relevant for the investigation and practical realization of first-order interpolation based on clausal tableaux.  ...  The construction of the tableaux for interpolation is performed there with the first-order prover CMProver that proceeds in the goal-oriented top-down way.  ... 
arXiv:2008.03489v2 fatcat:f3phjnrmgnb6vhsdfzcffkpbbi

Applications of Craig Interpolants in Model Checking [chapter]

K. L. McMillan
2005 Lecture Notes in Computer Science  
An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and proof systems.  ...  A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B.  ...  Finally, it would be useful to extend the extraction of interpolants from proofs to other theories, for example, first-order arrays and bit vectors.  ... 
doi:10.1007/978-3-540-31980-1_1 fatcat:m7nfhuc7crbrzdbwpwp433644i

Applications of Craig Interpolation to Model Checking [chapter]

Kenneth McMillan
2005 Lecture Notes in Computer Science  
An interpolant can be efficiently derived from a refutation of A ∧ B, for certain theories and proof systems.  ...  A Craig interpolant for a mutually inconsistent pair of formulas (A, B) is a formula that is (1) implied by A, (2) inconsistent with B, and (3) expressed over the common variables of A and B.  ...  Finally, it would be useful to extend the extraction of interpolants from proofs to other theories, for example, first-order arrays and bit vectors.  ... 
doi:10.1007/11494744_2 fatcat:ujvuhwop65cutft52winn3vtdq
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