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Unformatted text preview: Interactive Theorem Proving with PVS N. Shankar Computer Science Laboratory SRI International Menlo Park, CA May 16, 2011 Background Basic Proof Construction Proof Obligations Applications Course Outline An Introduction to interactive theorem proving (ITP) using PVS 1 An Introduction to PVS 2 Advanced interactive proof techniques 3 Examples and Applications PVS combines an expressive language (like Coq) with interaction (like the LCF provers HOL, Coq, Isabelle) and automation (like ACL2). Sam Owre contributed significantly to the preparation of these slides. N. Shankar Interactive Theorem Proving with PVS Background Basic Proof Construction Proof Obligations Applications Logic PVS Overview Background Slides The next series of slides covers 1 Logic Background 2 Basic information about PVS These slides are not part of the main lectures N. Shankar Interactive Theorem Proving with PVS Background Basic Proof Construction Proof Obligations Applications Logic PVS Overview What is Logic? Logic is the art and science of effective reasoning. How can we draw general and reliable conclusions from a collection of facts? Formal logic: Precise, syntactic characterizations of wellformed expressions and valid deductions. Formal logic makes it possible to calculate consequences so that each step is verifiable by means of proof. Computers can be used to automate such symbolic calculations. N. Shankar Interactive Theorem Proving with PVS Background Basic Proof Construction Proof Obligations Applications Logic PVS Overview Logic Basics Logic studies the trinity between language , interpretation , and proof . Language circumscribes the syntax that is used to construct sensible assertions. Interpretation ascribes an intended sense to these assertions by fixing the meaning of certain symbols, e.g., the logical connectives, equality , and delimiting the variation in the meanings of other symbols, e.g., variables, functions, and predicates . An assertion is valid if it holds in all interpretations. Checking validity through interpretations is not always possible, so proofs in the form axioms and inference rules are used to demonstrate the validity of assertions. N. Shankar Interactive Theorem Proving with PVS Background Basic Proof Construction Proof Obligations Applications Logic PVS Overview Language Signature Σ[ X ] contains functions and predicate symbols with associated arities, and X is a set of variables. The signature can be used to construct Terms τ := x  f ( τ 1 , .. . ,τ n ) Atoms α := p ( τ 1 , .. . τ n ), Literals λ := α  ¬ α Constraints λ 1 ∧ . .. ∧ λ n , Clauses λ 1 ∨ . .. ∨ λ n , Formulas ψ := p ( τ 1 , .. . ,τ n )  τ = τ 1  ¬ ψ  ψ ∨ ψ 1  ψ ∧ ψ 1  ( ∃ x : ψ )  ( ∀ x : ψ ) N. Shankar Interactive Theorem Proving with PVS Background Basic Proof Construction Proof Obligations Applications Logic PVS Overview Structure A Σstructure M consists of A domain  M  A map M ( f ) from  M  n → M for each nary function...
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This note was uploaded on 02/07/2012 for the course CS 4322 taught by Professor Martinrinard during the Spring '11 term at MIT.
 Spring '11
 MartinRinard
 Computer Science

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