512 lines
15 KiB
TeX
512 lines
15 KiB
TeX
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\part{Good Definitions}
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\frame[plain]{\partpage}
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\section{General Discussion}
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\begin{frame}
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\frametitle{Importance of Good Definitions}
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\begin{itemize}
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\item using \textit{good} definitions is very important
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\begin{itemize}
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\item good definitions are vital for \emph{clarity}
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\item \emph{proofs} depend a lot on the form of definitions
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\end{itemize}
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\item unluckily, it is hard to state what a good definition is
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\item even harder to come up with good definitions
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\item let's look at it a bit closer anyhow
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Importance of Good Definitions --- Clarity I}
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\begin{itemize}
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\item HOL guarantees that theorems do indeed hold
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\item However, does the theorem mean what you think it does?
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\item you can separate your development in
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\begin{itemize}
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\item main theorems you care for
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\item auxiliary stuff used to derive your main theorems
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\end{itemize}
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\item it is essential to understand your main theorems
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Importance of Good Definitions --- Clarity II}
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\begin{minipage}[t]{.45\textwidth}
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\begin{block}{Guarded by HOL}
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\begin{itemize}
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\item proofs checked
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\item internal, technical definitions
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\item technical lemmata
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\item proof tools
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\end{itemize}
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\end{block}
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\end{minipage}
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\qquad
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\begin{minipage}[t]{.45\textwidth}
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\begin{block}{Manual review needed for}
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\begin{itemize}
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\item meaning of main theorems
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\item meaning of definitions used by main theorems
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\item meaning of types used by main theorems
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\end{itemize}
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\end{block}
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\end{minipage}
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\end{frame}
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\begin{frame}
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\frametitle{Importance of Good Definitions --- Clarity III}
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\begin{itemize}
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\item it is essential to understand your main theorems
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\begin{itemize}
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\item you need to understand all the definitions directly used
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\item you need to understand the indirectly used ones as well
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\item you need to convince others that you express the intended statement
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\item therefore, it is vital to \alert{use very simple, clear definitions}
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\end{itemize}
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\item defining concepts is often the main development task
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\item checking resulting model against real artefact is vital
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\begin{itemize}
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\item testing via \eg \hol{EVAL}
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\item formal sanity
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\item conformance testing
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\end{itemize}
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\item wrong models are main source of error when using HOL
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\item proofs, auxiliary lemmata and auxiliary definitions
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\begin{itemize}
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\item can be as technical and complicated as you like
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\item correctness is guaranteed by HOL
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\item reviewers don't need to care
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Importance of Good Definitions --- Proofs}
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\begin{itemize}
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\item good definitions can shorten proofs significantly
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\item they improve maintainability
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\item they can improve automation drastically
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\item unluckily for proofs definitions often need to be technical
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\item this contradicts clarity aims
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{How to come up with good definitions}
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\begin{itemize}
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\item unluckily, it is hard to state what a good definition is
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\item it is even harder to come up with them
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\begin{itemize}
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\item there are often many competing interests
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\item a lot of experience and detailed tool knowledge is needed
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\item much depends on personal style and taste
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\end{itemize}
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\item general advice: use more than one definition
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\begin{itemize}
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\item in HOL you can derive equivalent definitions as theorems
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\item define a concept as clearly and easily as possible
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\item derive equivalent definitions for various purposes
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\begin{itemize}
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\item one very close to your favourite textbook
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\item one nice for certain types of proofs
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\item another one good for evaluation
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\item \ldots
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\end{itemize}
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\end{itemize}
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\item lessons from functional programming apply
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\end{itemize}
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\end{frame}
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\section{Functional Programming}
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\begin{frame}
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\frametitle{Good Definitions in Functional Programming}
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\begin{block}{Objectives}
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\begin{itemize}
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\item clarity (readability, maintainability)
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\item performance (runtime speed, memory usage, ...)
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\end{itemize}
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\end{block}
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\begin{block}{General Advice}
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\begin{itemize}
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\item use the powerful type-system
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\item use many small function definitions
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\item encode invariants in types and function signatures
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions -- no number encodings}
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\begin{itemize}
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\item many programmers familiar with C encode everything as a number
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\item enumeration types are very cheap in SML and HOL
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\item use them instead
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\end{itemize}
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\begin{exampleblock}{Example Enumeration Types}\scriptsize
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In C the result of an order comparison is an integer with 3 equivalence classes:
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0, negative and positive integers. In SML and HOL, it is better to use a variant type.
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\begin{semiverbatim}
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val _ = Datatype `ordering = LESS | EQUAL | GREATER`;
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val compare_def = Define `
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(compare LESS lt eq gt = lt)
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\holAnd{} (compare EQUAL lt eq gt = eq)
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\holAnd{} (compare GREATER lt eq gt = gt) `;
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val list_compare_def = Define `
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(list_compare cmp [] [] = EQUAL) \holAnd{} (list_compare cmp [] l2 = LESS)
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\holAnd{} (list_compare cmp l1 [] = GREATER)
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\holAnd{} (list_compare cmp (x::l1) (y::l2) = compare (cmp (x:'a) y)
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(* x<y *) LESS
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(* x=y *) (list_compare cmp l1 l2)
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(* x>y *) GREATER) `;
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\end{semiverbatim}
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\end{exampleblock}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions --- Isomorphic Types}
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\begin{itemize}
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\item the type-checker is your friend
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\begin{itemize}
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\item it helps you find errors
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\item code becomes more robust
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\item using good types is a great way of writing self-documenting code
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\end{itemize}
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\item therefore, use many types
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\item even use types isomorphic to existing ones
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\end{itemize}
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\begin{exampleblock}{Virtual and Physical Memory Addresses}\scriptsize
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Virtual and physical addresses might in a development both be numbers. It is
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still nice to use separate types to avoid mixing them up.
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\begin{semiverbatim}
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val _ = Datatype `vaddr = VAddr num`;
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val _ = Datatype `paddr = PAddr num`;
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val virt_to_phys_addr_def = Define `
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virt_to_phys_addr (VAddr a) = PAddr( \textrm{\textit{translation of}} a )`;
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\end{semiverbatim}
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\end{exampleblock}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions --- Record Types I}
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\begin{itemize}
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\item often people use tuples where records would be more appropriate
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\item using large tuples quickly becomes awkward
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\begin{itemize}
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\item it is easy to mix up order of tuple entries
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\begin{itemize}
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\item often types coincide, so type-checker does not help
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\end{itemize}
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\item no good error messages for tuples
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\begin{itemize}
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\item hard to decipher type mismatch messages for long product types
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\item hard to figure out which entry is missing at which position
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\item non-local error messages
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\item variable in last entry can hide missing entries
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\end{itemize}
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\end{itemize}
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\item records sometimes require slightly more proof effort
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\item however, records have many benefits
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions --- Record Types II}
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\begin{itemize}
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\item using records
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\begin{itemize}
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\item introduces field names
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\item provides automatically defined accessor and update functions
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\item leads to better type-checking error messages
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\end{itemize}
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\item records improve readability
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\begin{itemize}
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\item accessors and update functions lead to shorter code
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\item field names act as documentation
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\end{itemize}
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\item records improve maintainability
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\begin{itemize}
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\item improved error messages
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\item much easier to add extra fields
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions --- Encoding Invariants}
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\begin{itemize}
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\item try to encode as many invariants as possible in the types
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\item this allows the type-checker to ensure them for you
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\item you don't have to check them manually any more
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\item your code becomes more robust and clearer
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\end{itemize}
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\begin{exampleblock}{Network Connections (Example by Yaron Minsky from Jane Street)}\scriptsize
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Consider the following datatype for network connections. It has many implicit invariants.
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\begin{semiverbatim}
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datatype connection_state = Connected | Disconnected | Connecting;
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type connection_info = \{
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state : connection_state,
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server : inet_address,
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last_ping_time : time option,
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last_ping_id : int option,
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session_id : string option,
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when_initiated : time option,
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when_disconnected : time option
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\}
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\end{semiverbatim}
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\end{exampleblock}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions --- Encoding Invariants II}
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\begin{exampleblock}{Network Connections (Example by Yaron Minsky from Jane Street) II}\scriptsize
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The following definition of \texttt{connection\_info} makes the invariants explicit:
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\begin{semiverbatim}
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type connected = \{ last_ping : (time * int) option,
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session_id : string \};
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type disconnected = \{ when_disconnected : time \};
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type connecting = \{ when_initiated : time \};
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datatype connection_state =
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Connected of connected
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| Disconnected of disconneted
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| Connecting of connecting;
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type connection_info = \{
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state : connection_state,
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server : inet_address
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\}
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\end{semiverbatim}
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\end{exampleblock}
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\end{frame}
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\section{HOL}
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\begin{frame}
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\frametitle{Good Definitions in HOL}
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\begin{block}{Objectives}
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\begin{itemize}
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\item clarity (readability)
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\item good for proofs
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\item performance (good for automation, easily evaluatable, ...)
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\end{itemize}
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\end{block}
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\begin{block}{General Advice}
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\begin{itemize}
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\item same advice as for functional programming applies
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\item use even smaller definitions
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\begin{itemize}
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\item introduce auxiliary definitions for important function parts
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\item use extra definitions for important constants
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\item ...
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\end{itemize}
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\item tiny definitions
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\begin{itemize}
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\item allow keeping proof state small by unfolding only needed ones
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\item allow many small lemmata
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\item improve maintainability
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\end{itemize}
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\end{itemize}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions in HOL II}
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\begin{block}{Technical Issues}
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\begin{itemize}
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\item write definitions such that they work well with HOL's tools
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\item this requires you to know HOL well
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\item a lot of experience is required
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\item general advice
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\begin{itemize}
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\item avoid explicit case-expressions
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\item prefer curried functions
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\end{itemize}
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\end{itemize}
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\end{block}
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\begin{block}{Example}
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\begin{semiverbatim}\scriptsize
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val ZIP_GOOD_def = Define `(ZIP (x::xs) (y::ys) = (x,y)::(ZIP xs ys)) \holAnd{}
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(ZIP _ _ = [])`
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val ZIP_BAD1_def = Define `ZIP xs ys = case (xs, ys) of
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(x::xs, y::ys) => (x,y)::(ZIP xs ys)
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| (_, _) => []`
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val ZIP_BAD2_def = Define `(ZIP (x::xs, y::ys) = (x,y)::(ZIP (xs, ys))) \holAnd{}
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(ZIP _ = [])`
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\end{semiverbatim}
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\end{block}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Good Definitions in HOL III}
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\begin{block}{Multiple Equivalent Definitions}
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\begin{itemize}
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\item satisfy competing requirements by having multiple equivalent definitions
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\item derive them as theorems
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\item initial definition should be as clear as possible
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\begin{itemize}
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\item clarity allows simpler reviews
|
||
|
|
\item simplicity reduces the likelihood of errors
|
||
|
|
\end{itemize}
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
|
||
|
|
\begin{block}{Example - \texttt{ALL\_DISTINCT}}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
|- (ALL_DISTINCT [] <=> T) \holAnd{}
|
||
|
|
(!h t. ALL_DISTINCT (h::t) <=> \holNeg{}MEM h t \holAnd{} ALL_DISTINCT t)
|
||
|
|
|
||
|
|
|- !l. ALL_DISTINCT l <=>
|
||
|
|
(!x. MEM x l ==> (FILTER (\$= x) l = [x]))
|
||
|
|
|
||
|
|
|- !ls. ALL_DISTINCT ls <=> (CARD (set ls) = LENGTH ls):
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\section{Formal Sanity}
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Formal Sanity}
|
||
|
|
|
||
|
|
\begin{block}{Formal Sanity}
|
||
|
|
\begin{itemize}
|
||
|
|
\item to ensure correctness test your definitions via \eg \hol{EVAL}
|
||
|
|
\item in HOL testing means symbolic evaluation, \ie proving lemmata
|
||
|
|
\item \emph{formally proving sanity check lemmata} is very beneficial
|
||
|
|
\begin{itemize}
|
||
|
|
\item they should express core properties of your definition
|
||
|
|
\item thereby they check your intuition against your actual definitions
|
||
|
|
\item these lemmata are often useful for following proofs
|
||
|
|
\item using them improves robustness and maintainability of your development
|
||
|
|
\end{itemize}
|
||
|
|
\item I highly recommend using formal sanity checks
|
||
|
|
\end{itemize}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Formal Sanity Example I}
|
||
|
|
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
> \hol{val ALL_DISTINCT = Define `
|
||
|
|
(ALL_DISTINCT [] = T) \holAnd{}
|
||
|
|
(ALL_DISTINCT (h::t) = \holNeg{}MEM h t \holAnd{} ALL_DISTINCT t)`};
|
||
|
|
\end{semiverbatim}
|
||
|
|
|
||
|
|
\begin{block}{Example Sanity Check Lemmata}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
|- ALL_DISTINCT []\smallskip
|
||
|
|
|- !x xs. ALL_DISTINCT (x::xs) <=> \holNeg{}MEM x xs \holAnd{} ALL_DISTINCT xs\smallskip
|
||
|
|
|- !x. ALL_DISTINCT [x]\smallskip
|
||
|
|
|- !x xs. \holNeg{}(ALL_DISTINCT (x::x::xs))\smallskip
|
||
|
|
|- !l. ALL_DISTINCT (REVERSE l) <=> ALL_DISTINCT l\smallskip
|
||
|
|
|- !x l. ALL_DISTINCT (SNOC x l) <=> \holNeg{}MEM x l \holAnd{} ALL_DISTINCT l\smallskip
|
||
|
|
|- !l1 l2. ALL_DISTINCT (l1 ++ l2) <=>
|
||
|
|
ALL_DISTINCT l1 \holAnd{} ALL_DISTINCT l2 \holAnd{} !e. MEM e l1 ==> \holNeg{}MEM e l2
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Formal Sanity Example II 1}
|
||
|
|
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
> \hol{val ZIP_def = Define `
|
||
|
|
(ZIP [] ys = []) \holAnd{} (ZIP xs [] = []) \holAnd{}
|
||
|
|
(ZIP (x::xs) (y::ys) = (x, y)::(ZIP xs ys))`}
|
||
|
|
|
||
|
|
val ZIP_def =
|
||
|
|
|- (!ys. ZIP [] ys = []) \holAnd{} (!v3 v2. ZIP (v2::v3) [] = []) \holAnd{}
|
||
|
|
(!ys y xs x. ZIP (x::xs) (y::ys) = (x,y)::ZIP xs ys)
|
||
|
|
\end{semiverbatim}\vspace{-1em}
|
||
|
|
\begin{itemize}
|
||
|
|
\item above definition of \hol{ZIP} looks straightforward
|
||
|
|
\item small changes cause heuristics to produce different theorems
|
||
|
|
\item use formal sanity lemmata to compensate
|
||
|
|
\end{itemize}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
> \hol{val ZIP_def = Define `
|
||
|
|
(ZIP xs [] = []) \holAnd{} (ZIP [] ys = []) \holAnd{}
|
||
|
|
(ZIP (x::xs) (y::ys) = (x, y)::(ZIP xs ys))`}
|
||
|
|
|
||
|
|
val ZIP_def =
|
||
|
|
|- (!xs. ZIP xs [] = []) \holAnd{} (!v3 v2. ZIP [] (v2::v3) = []) \holAnd{}
|
||
|
|
(!ys y xs x. ZIP (x::xs) (y::ys) = (x,y)::ZIP xs ys0
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
\begin{frame}[fragile]
|
||
|
|
\frametitle{Formal Sanity Example II 2}
|
||
|
|
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
val ZIP_def =
|
||
|
|
|- (!ys. ZIP [] ys = []) \holAnd{} (!v3 v2. ZIP (v2::v3) [] = []) \holAnd{}
|
||
|
|
(!ys y xs x. ZIP (x::xs) (y::ys) = (x,y)::ZIP xs ys)
|
||
|
|
\end{semiverbatim}
|
||
|
|
\begin{block}{Example Formal Sanity Lemmata}
|
||
|
|
\begin{semiverbatim}\scriptsize
|
||
|
|
|- (!xs. ZIP xs [] = []) \holAnd{} (!ys. ZIP [] ys = []) \holAnd{}
|
||
|
|
(!y ys x xs. ZIP (x::xs) (y::ys) = (x,y)::ZIP xs ys)\smallskip
|
||
|
|
|- !xs ys. LENGTH (ZIP xs ys) = MIN (LENGTH xs) (LENGTH ys)\smallskip
|
||
|
|
|- !x y xs ys. MEM (x, y) (ZIP xs ys) ==> (MEM x xs \holAnd{} MEM y ys)\smallskip
|
||
|
|
|- !xs1 xs2 ys1 ys2. LENGTH xs1 = LENGTH ys1 ==>
|
||
|
|
(ZIP (xs1++xs2) (ys1++ys2) = (ZIP xs1 ys1 ++ ZIP xs2 ys2))\smallskip
|
||
|
|
...
|
||
|
|
\end{semiverbatim}
|
||
|
|
\end{block}
|
||
|
|
\begin{itemize}
|
||
|
|
\item in your proofs use sanity lemmata, not original definition
|
||
|
|
\item this makes your development robust against
|
||
|
|
\begin{itemize}
|
||
|
|
\item small changes to the definition required later
|
||
|
|
\item changes to \hol{Define} and its heuristics
|
||
|
|
\item bugs in function definition package
|
||
|
|
\end{itemize}
|
||
|
|
\end{itemize}
|
||
|
|
\end{frame}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
%%% Local Variables:
|
||
|
|
%%% mode: latex
|
||
|
|
%%% TeX-master: "current"
|
||
|
|
%%% End:
|