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Hoare Logic and VDM: MachineChecked Soundness and Completeness Proofs
, 1998
"... Investigating soundness and completeness of verification calculi for imperative programming languages is a challenging task. Many incorrect results have been published in the past. We take advantage of the computeraided proof tool LEGO to interactively establish soundness and completeness of both H ..."
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Cited by 31 (1 self)
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Investigating soundness and completeness of verification calculi for imperative programming languages is a challenging task. Many incorrect results have been published in the past. We take advantage of the computeraided proof tool LEGO to interactively establish soundness and completeness of both Hoare Logic and the operation decomposition rules of the Vienna Development Method (VDM) with respect to operational semantics. We deal with parameterless recursive procedures and local variables in the context of total correctness. As a case study, we use LEGO to verify the correctness of Quicksort in Hoare Logic. As our main contribution, we illuminate the rle of auxiliary variables in Hoare Logic. They are required to relate the value of program variables in the final state with the value of program variables in the initial state. In our formalisation, we reflect their purpose by interpreting assertions as relations on states and a domain of auxiliary variables. Furthermore, we propose a new structural rule for adjusting auxiliary variables when strengthening preconditions and weakening postconditions. This rule is stronger than all previously suggested structural rules, including rules of adaptation. With the new treatment, we are able to show that, contrary to common belief, Hoare Logic subsumes VDM in that every derivation in VDM can be naturally embedded in Hoare Logic. Moreover, we establish completeness results uniformly as corollaries of Most General Formula theorems which remove the need to reason about arbitrary assertions.
Hoare Logics for Recursive Procedures and Unbounded Nondeterminism
 COMPUTER SCIENCE LOGIC (CSL 2002), VOLUME 2471 OF LNCS
, 2002
"... This paper presents sound and complete Hoare logics for partial and total correctness of recursive parameterless procedures in the context of unbounded nondeterminism. For total correctness, the literature so far has either restricted recursive procedures to be deterministic or has studied unbounde ..."
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Cited by 26 (3 self)
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This paper presents sound and complete Hoare logics for partial and total correctness of recursive parameterless procedures in the context of unbounded nondeterminism. For total correctness, the literature so far has either restricted recursive procedures to be deterministic or has studied unbounded nondeterminism only in conjunction with loops rather than procedures. We consider both single procedures and systems of mutually recursive procedures. All proofs have been checked with the theorem prover Isabelle/HOL.
Finding lexicographic orders for termination proofs in Isabelle/HOL
 Theorem Proving in Higher Order Logics: TPHOLs 2007, volume 4732 of Lecture Notes in Computer Science
, 2007
"... Abstract. We present a simple method to formally prove termination of recursive functions by searching for lexicographic combinations of size measures. Despite its simplicity, the method turns out to be powerful enough to solve a large majority of termination problems encountered in daily theorem pr ..."
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Cited by 15 (5 self)
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Abstract. We present a simple method to formally prove termination of recursive functions by searching for lexicographic combinations of size measures. Despite its simplicity, the method turns out to be powerful enough to solve a large majority of termination problems encountered in daily theorem proving practice. 1
PROSPECTUS  Sound Foundations for Effective Proofs of Programs
"... When considering the correctness of programs, the only absolute demonstration of quality is mathematical proof. Yet the complexity of these proofs makes them all but impossible both to construct and read, and the correctness of the proofs themselves come into question. We take an approach to the ..."
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When considering the correctness of programs, the only absolute demonstration of quality is mathematical proof. Yet the complexity of these proofs makes them all but impossible both to construct and read, and the correctness of the proofs themselves come into question. We take an approach to the creation of these proofs based on specifying an axiomatic semantics for the programming language, and using that semantics to automatically create a Verification Condition Generator, a program that takes a general program written in the language and creates the proof of that program, modulo a set of verification conditions, to be proven by hand. This automates much of the detailed work of creating the proof. Yet even this VCG technique depends on the soundness of the axiomatic semantics, and in fact, many proposed axiomatic semantics have suffered from unsoundness. We take the difficult but secure approach of foundationally defining an operational semantics of the programming language...
4. Verification Condition Generators..................................................5
, 1993
"... When considering the correctness of programs, the only absolute demonstration of quality is mathematical proof. Yet the complexity of these proofs makes them all but impossible both to construct and read, and the correctness of the proofs themselves come into question. We take an approach to the cre ..."
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When considering the correctness of programs, the only absolute demonstration of quality is mathematical proof. Yet the complexity of these proofs makes them all but impossible both to construct and read, and the correctness of the proofs themselves come into question. We take an approach to the creation of these proofs based on specifying an axiomatic semantics for the programming language, and using that semantics to automatically create a Verification Condition Generator, a program that takes a general program written in the language and creates the proof of that program, modulo a set of verification conditions, to be proven by hand. This automates much of the detailed work of creating the proof. Yet even this VCG technique depends on the soundness of the axiomatic semantics, and in fact, many proposed axiomatic semantics have suffered from unsoundness. We take the difficult but secure approach of foundationally defining an operational semantics of the programming language, including concurrency, and then proving the axioms and rules of inference of the axiomatic semantics from the operational semantics as theorems. Once this is done, the correctness of the VCG function itself can be proven, so the proofs of concurrent programs as constructed by the VCG in a way that is known to be sound, modulo the truth of the