Formal proof

Summary

In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the de

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Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follo

In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that th

A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are t

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Foundations for SCALA

Vincent Cremet

SCALA is an attractive programming language because it is both very expressive and statically strongly typed. This marriage against nature comes at the price of a certain complexity in the language constructs and the static analysis. This complexity makes type safety unclear. The goal of this thesis is to initiate a rigorous and formal process of validation of the SCALA type system. The correctness of a type system can only be established in relation to a formal description of a program execution. The first contribution of this thesis is the definition of a formal semantics for SCALA, by translation in a minimal class-based calculus. SCALA offers two means for expressing type abstraction: type parameters and type members, also called virtual types. Virtual types seem more primitive since they allow to encode type parameters whereas the existence of a reverse encoding is not clear. The soundness of virtual types has been the object of a long debate in the community; they are now commonly believed to be safe, but at this time, there exists no formal argument that would confirm this belief. The second contribution of this thesis is to provide a formal proof of type safety for virtual types.

EPFL2006

Symbolic Proofs of Temporal Properties

1994

An Equational Theory for Transactions

Vincent Cremet, Rachid Guerraoui, Martin Odersky

Transactions are commonly described as being ACID: All-or-nothing, Consistent, Isolated and Durable. However, although these words convey a powerful intuition, the ACID properties have never been given a precise semantics in a way that disentangles each property from the others. Among the benefits of such a semantics would be the ability to trade-off the value of a property against the cost of its implementation. This paper gives a sound equational semantics for the transaction properties. We define three categories of actions, A-actions, I-actions and D-actions, while we view Consistency as an induction rule that enables us to derive system-wide consistency from local consistency. The three kinds of action can be nested, leading to different forms of transactions, each with a well-defined semantics. Conventional transactions are obtained as ADI-actions. >From the equational semantics we develop a formal proof principle for transactional programs, from which we derive the induction rule for Consistency.

2003