Implementation and Evaluation of Contextual Natural Deduction for Minimal Logic

The contextual natural deduction calculus (\({\mathbf{ND }^\mathbf{c }}\)) extends the usual natural deduction calculus (\(\mathbf{ND }\)) by allowing the implication introduction and elimination rules to operate on formulas that occur inside contexts. It has been shown that, asymptotically in the best case, \({\mathbf{ND }^\mathbf{c }}\)-proofs can be quadratically smaller than the smallest \(\mathbf{ND }\)-proofs of the same theorems. In this paper we describe the first implementation of a theorem prover for minimal logic based on \({\mathbf{ND }^\mathbf{c }}\). Furthermore, we empirically compare it to an equally simple \(\mathbf{ND }\) theorem prover on thousands of randomly generated conjectures.

[1]  Richard McKinley,et al.  An Algorithmic Interpretation of a Deep Inference System , 2008, LPAR.

[2]  Tobias Nipkow,et al.  Sledgehammer: Judgement Day , 2010, IJCAR.

[3]  Roberto Bruttomesso,et al.  Resolution proof transformation for compression and interpolation , 2014, Formal Methods Syst. Des..

[4]  Alessio Guglielmi,et al.  On the proof complexity of deep inference , 2009, TOCL.

[5]  Alexander Leitsch,et al.  Towards Algorithmic Cut-Introduction , 2012, LPAR.

[6]  Alexander Leitsch,et al.  Herbrand Sequent Extraction , 2008, AISC/MKM/Calculemus.

[7]  Bruno Woltzenlogel Paleo,et al.  Atomic Cut Introduction by Resolution: Proof Structuring and Compression , 2010, LPAR.

[8]  Scott Cotton.  Two Techniques for Minimizing Resolution Proofs , 2010, SAT.

[9]  Bruno Woltzenlogel Paleo,et al.  Towards the Compression of First-Order Resolution Proofs by Lowering Unit Clauses , 2015, CADE.

[10]  Ofer Strichman,et al.  Linear-Time Reductions of Resolution Proofs , 2008, Haifa Verification Conference.

[11]  Stephan Merz,et al.  Compression of Propositional Resolution Proofs via Partial Regularization , 2011, CADE.

[12]  Joseph Boudou,et al.  Compression of Propositional Resolution Proofs by Lowering Subproofs , 2013, TABLEAUX.

[13]  Alwen Tiu,et al.  A Local System for Intuitionistic Logic , 2006, LPAR.

[14]  Joseph Boudou,et al.  Skeptik: A Proof Compression System , 2014, IJCAR.

[15]  Nicolas Guenot.  Nested proof search as reduction in the Lambda-calculus , 2011, PPDP.

[16]  Alessio Guglielmi,et al.  A system of interaction and structure , 1999, TOCL.

[17]  Tom Gundersen,et al.  A Quasipolynomial Cut-Elimination Procedure in Deep Inference via Atomic Flows and Threshold Formulae , 2010, LPAR.

[18]  Kai Brünnler,et al.  Atomic Cut Elimination for classical Logic , 2003, CSL.

[19]  Bruno Woltzenlogel Paleo,et al.  Reducing redundancy in cut-elimination by resolution , 2017, J. Log. Comput..

[20]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[21]  Bruno Woltzenlogel Paleo,et al.  Contextual Natural Deduction , 2013, LFCS.