Combinatorial Flows and Proof Compression Lutz Straßburger
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Certification of First-Order Proofs in Classical and Intuitionistic Logics
Certification of First-order proofs in classical and intuitionistic logics 1 Zakaria Chihani August 21, 2015 1This work has been funded by the ERC Advanced Grant ProofCert. Abstract The field of automated reasoning contains a plethora of methods and tools, each with its own language and its community, often evolving separately. These tools express their proofs, or some proof evidence, in different formats such as resolution refutations, proof scripts, natural deductions, expansion trees, equational rewritings and many others. The disparity in formats reduces communication and trust be- tween the different communities. Related efforts were deployed to fill the gaps in communication including libraries and languages bridging two or more tools. This thesis proposes a novel approach at filling this gap for first-order classical and intuitionistic logics. Rather than translating proofs written in various languages to proofs written in one chosen language, this thesis introduces a framework for describing the semantics of a wide range of proof evidence languages through a rela- tional specification, called Foundational Proof Certification (FPC). The description of the semantics of a language can then be appended to any proof evidence written in that language, forming a proof certificate, which allows a small kernel checker to verify them independently from the tools that created them. The use of seman- tics description for one language rather than proof translation from one language to another relieves one from the need to radically change the notion of proof. Proof evidence, unlike complete proof, does not have to contain all details. Us- ing proof reconstruction, a kernel checker can rebuild missing parts of the proof, allowing for compression and gain in storage space. -
Combinatorial Proofs and Decomposition Theorems for First
Combinatorial Proofs and Decomposition Theorems for First-order Logic Dominic J. D. Hughes Lutz Straßburger Jui-Hsuan Wu Logic Group Inria, Equipe Partout Ecole Normale Sup´erieure U.C. Berkeley Ecole Polytechnique, LIX France USA France Abstract—We uncover a close relationship between combinato- rial and syntactic proofs for first-order logic (without equality). ax ⊢ pz, pz Whereas syntactic proofs are formalized in a deductive proof wk system based on inference rules, a combinatorial proof is a ax ⊢ pw, pz, pz ⊢ p,p wk syntax-free presentation of a proof that is independent from any wk ⊢ pw, pz, pz, ∀y.py set of inference rules. We show that the two proof representations ⊢ p,q,p ∨ ∨ ax ⊢ pw, pz, pz ∨ (∀y.py) are related via a deep inference decomposition theorem that ⊢ p ∨ q,p ⊢ p,p ∃ ∧ ⊢ pw, pz, ∃x.(px ∨ (∀y.py)) establishes a new kind of normal form for syntactic proofs. This ⊢ (p ∨ q) ∧ p,p,p ∀ yields (a) a simple proof of soundness and completeness for first- ctr ⊢ pw, ∀y.py, ∃x.(px ∨ (∀y.py)) ⊢ (p ∨ q) ∧ p,p ∨ order combinatorial proofs, and (b) a full completeness theorem: ∨ ⊢ pw ∨ (∀y.py), ∃x.(px ∨ (∀y.py)) ⊢ ((p ∨ q) ∧ p) ∨ p ∃ every combinatorial proof is the image of a syntactic proof. ⊢ ∃x.(px ∨ (∀y.py)), ∃x.(px ∨ (∀y.py)) ctr ⊢ ∃x.(px ∨ (∀y.py)) I. INTRODUCTION First-order predicate logic is a cornerstone of modern logic. ↓ ↓ Since its formalisation by Frege [1] it has seen a growing usage in many fields of mathematics and computer science. Upon the development of proof theory by Hilbert [2], proofs became first-class citizens as mathematical objects that could be studied on their own. -
Algorithmic Structuring and Compression of Proofs (ASCOP)
Project Presentation: Algorithmic Structuring and Compression of Proofs (ASCOP) Stefan Hetzl Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria [email protected] Abstract. Computer-generated proofs are typically analytic, i.e. they essentially consist only of formulas which are present in the theorem that is shown. In contrast, mathematical proofs written by humans almost never are: they are highly structured due to the use of lemmas. The ASCOP-project aims at developing algorithms and software which structure and abbreviate analytic proofs by computing useful lemmas. These algorithms will be based on recent groundbreaking results estab- lishing a new connection between proof theory and formal language the- ory. This connection allows the application of efficient algorithms based on formal grammars to structure and compress proofs. 1 Introduction Proofs are the most important carriers of mathematical knowledge. Logic has endowed us with formal languages for proofs which make them amenable to al- gorithmic treatment. From the early days of automated deduction to the current state of the art in automated and interactive theorem proving we have witnessed a huge increase in the ability of computers to search for, formalise and work with proofs. Due to the continuing formalisation of computer science (e.g. in areas such as hardware and software verification) the importance of formal proofs will grow further. Formal proofs which are generated automatically are usually difficult or even impossible to understand for a human reader. This is due to several reasons: one is a potentially extreme length as in the well-known cases of the four colour theorem or the Kepler conjecture. -
Applications of Foundational Proof Certificates in Theorem Proving
Applications of Foundational Proof Certificates in theorem proving par M. Roberto Blanco Martínez Thèse de doctorat de l’ préparée à Spécialité de doctorat: Informatique École doctorale no 580 Sciences et technologies de l’information et de la communication nnt: XXX Thèse présentée et soutenue à Palaiseau, le XX décembre 2017 Composition du Jury: Mme Laura Kovács Professeure des universités Rapporteuse Technische Universität Wien & Chalmers tekniska högskola M. Dale Miller Directeur de recherche Directeur de thèse Inria & LIX/École polytechnique M. Claudio Sacerdoti Coen Professeur des universités Rapporteur Alma Mater Studiorum - Università di Bologna D D E E F F Applications of Foundational Proof Certificates in theorem proving For Jason. Contents Contents v List of figures ix Preface xiii 1 Introduction 1 I Logical foundations 7 2 Structural proof theory 9 2.1 Concept of proof . .9 2.2 Evolution of proof theory . 10 2.3 Classical and intuitionistic logics . 12 2.4 Sequent calculus . 13 2.5 Focusing . 20 2.6 Soundness and completeness . 25 2.7 Notes . 26 3 Foundational Proof Certificates 29 3.1 Proof as trusted communication . 29 3.2 Augmented sequent calculus . 32 3.3 Running example: CNF decision procedure . 38 3.4 Running example: oracle strings . 39 3.5 Running example: decide depth . 41 3.6 Running example: binary resolution . 42 3.7 Checkers, kernels, clients and certificates . 45 3.8 Notes . 48 v vi contents II Logics without fixed points 53 4 Logic programming in intuitionistic logic 55 4.1 Logic and computation . 55 4.2 Logic programming . 56 4.3 λProlog . 59 4.4 FPC kernels . -
Craig Interpolation and Proof Manipulation
Craig Interpolation and Proof Manipulation Theory and Applications to Model Checking Doctoral Dissertation submitted to the Faculty of Informatics of the Università della Svizzera Italiana in partial fulfillment of the requirements for the degree of Doctor of Philosophy presented by Simone Fulvio Rollini under the supervision of Prof. Natasha Sharygina February 2014 Dissertation Committee Prof. Rolf Krause Università della Svizzera Italiana, Switzerland Prof. Evanthia Papadopoulou Università della Svizzera Italiana, Switzerland Prof. Roberto Sebastiani Università di Trento, Italy Prof. Ofer Strichman Technion, Israel Dissertation accepted on 06 February 2014 Prof. Natasha Sharygina Prof. Igor Pivkin Prof. Stefan Wolf i I certify that except where due acknowledgement has been given, the work pre- sented in this thesis is that of the author alone; the work has not been submitted previously, in whole or in part, to qualify for any other academic award; and the content of the thesis is the result of work which has been carried out since the offi- cial commencement date of the approved research program. ii Abstract Model checking is one of the most appreciated methods for automated formal veri- fication of software and hardware systems. The main challenge in model checking, i.e. scalability to complex systems of extremely large size, has been successfully ad- dressed by means of symbolic techniques, which rely on an efficient representation and manipulation of the systems based on first order logic. Symbolic model checking has been considerably enhanced with the introduction in the last years of Craig interpolation as a means of overapproximation. Interpolants can be efficiently computed from proofs of unsatisfiability based on their structure. -
Resolution Proof Transformation for Compression and Interpolation
Resolution Proof Transformation for Compression and Interpolation S.F. Rollini1, R. Bruttomesso2, N. Sharygina1, and A. Tsitovich3 1 University of Lugano, Switzerland fsimone.fulvio.rollini, [email protected] 2 Atrenta Advanced R&D of Grenoble, France [email protected] 3 Phonak, Switzerland [email protected] Abstract. Verification methods based on SAT, SMT, and Theorem Proving often rely on proofs of unsatisfiability as a powerful tool to extract information in order to reduce the overall effort. For example a proof may be traversed to identify a minimal reason that led to unsatisfi- ability, for computing abstractions, or for deriving Craig interpolants. In this paper we focus on two important aspects that concern efficient han- dling of proofs of unsatisfiability: compression and manipulation. First of all, since the proof size can be very large in general (exponential in the size of the input problem), it is indeed beneficial to adopt techniques to compress it for further processing. Secondly, proofs can be manipulated as a flexible preprocessing step in preparation for interpolant computa- tion. Both these techniques are implemented in a framework that makes use of local rewriting rules to transform the proofs. We show that a care- ful use of the rules, combined with existing algorithms, can result in an effective simplification of the original proofs. We have evaluated several heuristics on a wide range of unsatisfiable problems deriving from SAT and SMT test cases. 1 Introduction Symbolic verification methods rely on a representation of the state space as a set of formulae, which are manipulated by formal engines such as SAT- and SMT- arXiv:1307.2028v3 [cs.LO] 15 Apr 2014 solvers. -
Resolution Proof Transformation for Compression and Interpolation
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Form Methods Syst Des (2014) 45:1–41 DOI 10.1007/s10703-014-0208-x Resolution proof transformation for compression and interpolation Simone Fulvio Rollini · Roberto Bruttomesso · Natasha Sharygina · Aliaksei Tsitovich Published online: 27 April 2014 © Springer Science+Business Media New York 2014 Abstract Verification methods based on SAT, SMT, and theorem proving often rely on proofs of unsatisfiability as a powerful tool to extract information in order to reduce the overall effort. For example a proof may be traversed to identify a minimal reason that led to unsatisfiability, for computing abstractions, or for deriving Craig interpolants. In this paper we focus on two important aspects that concern efficient handling of proofs of unsatisfiability: compression and manipulation. First of all, since the proof size can be very large in general (exponential in the size of the input problem), it is indeed beneficial to adopt techniques to compress it for further processing. Secondly, proofs can be manipulated as a flexible preprocessing step in preparation for interpolant computation. Both these techniques are implemented in a framework that makes use of local rewriting rules to transform the proofs. We show that a careful use of the rules, combined with existing algorithms, can result in an effective simplification of the original proofs. We have evaluated several heuristics on a wide range of unsatisfiable problems deriving from SAT and SMT test cases. Keywords Resolution system · Craig interpolation · Proof compression · Formal verification S. F. Rollini (B) · N.