220 research outputs found
A Universal Machine for Biform Theory Graphs
Broadly speaking, there are two kinds of semantics-aware assistant systems
for mathematics: proof assistants express the semantic in logic and emphasize
deduction, and computer algebra systems express the semantics in programming
languages and emphasize computation. Combining the complementary strengths of
both approaches while mending their complementary weaknesses has been an
important goal of the mechanized mathematics community for some time. We pick
up on the idea of biform theories and interpret it in the MMTt/OMDoc framework
which introduced the foundations-as-theories approach, and can thus represent
both logics and programming languages as theories. This yields a formal,
modular framework of biform theory graphs which mixes specifications and
implementations sharing the module system and typing information. We present
automated knowledge management work flows that interface to existing
specification/programming tools and enable an OpenMath Machine, that
operationalizes biform theories, evaluating expressions by exhaustively
applying the implementations of the respective operators. We evaluate the new
biform framework by adding implementations to the OpenMath standard content
dictionaries.Comment: Conferences on Intelligent Computer Mathematics, CICM 2013 The final
publication is available at http://link.springer.com
Reconstructing a logic for inductive proofs of properties of functional programs
A logical framework consisting of a polymorphic call-by-value functional language and a first-order logic on the values is presented, which is a reconstruction of the logic of the verification system VeriFun. The reconstruction uses contextual semantics to define the logical value of equations. It equates undefinedness and non-termination, which is a standard semantical approach. The main results of this paper are: Meta-theorems about the globality of several classes of theorems in the logic, and proofs of global correctness of transformations and deduction rules. The deduction rules of VeriFun are globally correct if rules depending on termination are appropriately formulated. The reconstruction also gives hints on generalizations of the VeriFun framework: reasoning on nonterminating expressions and functions, mutual recursive functions and abstractions in the data values, and formulas with arbitrary quantifier prefix could be allowed
Mathematical Proof Between Generations
A proof is one of the most important concepts of mathematics. However, there
is a striking difference between how a proof is defined in theory and how it is
used in practice. This puts the unique status of mathematics as exact science
into peril. Now may be the time to reconcile theory and practice, i.e.
precision and intuition, through the advent of computer proof assistants. For
the most time this has been a topic for experts in specialized communities.
However, mathematical proofs have become increasingly sophisticated, stretching
the boundaries of what is humanly comprehensible, so that leading
mathematicians have asked for formal verification of their proofs. At the same
time, major theorems in mathematics have recently been computer-verified by
people from outside of these communities, even by beginning students. This
article investigates the gap between the different definitions of a proof and
possibilities to build bridges. It is written as a polemic or a collage by
different members of the communities in mathematics and computer science at
different stages of their careers, challenging well-known preconceptions and
exploring new perspectives.Comment: 17 pages, 1 figur
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