\documentclass{article} \usepackage{open-axiom} \begin{document} \title{src/algebra aggcat.spad} \author{Michael Monagan, Manuel Bronstein, Richard Jenks, Stephen Watt} \maketitle \begin{abstract} \end{abstract} \tableofcontents \eject \section{category AGG Aggregate} <>= import Type import Boolean import NonNegativeInteger )abbrev category AGG Aggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ The notion of aggregate serves to model any data structure aggregate, ++ designating any collection of objects, ++ with heterogenous or homogeneous members, ++ with a finite or infinite number ++ of members, explicitly or implicitly represented. ++ An aggregate can in principle ++ represent everything from a string of characters to abstract sets such ++ as "the set of x satisfying relation {\em r(x)}" ++ An attribute \spadatt{finiteAggregate} is used to assert that a domain ++ has a finite number of elements. Aggregate: Category == Type with eq?: (%,%) -> Boolean ++ eq?(u,v) tests if u and v are same objects. copy: % -> % ++ copy(u) returns a top-level (non-recursive) copy of u. ++ Note: for collections, \axiom{copy(u) == [x for x in u]}. empty: () -> % ++ empty()$D creates an aggregate of type D with 0 elements. ++ Note: The {\em $D} can be dropped if understood by context, ++ e.g. \axiom{u: D := empty()}. empty?: % -> Boolean ++ empty?(u) tests if u has 0 elements. sample: constant -> % ++ sample yields a value of type % if % has finiteAggregate then #: % -> NonNegativeInteger ++ # u returns the number of items in u. add eq?(a,b) == %peq(a,b)$Foreign(Builtin) sample() == empty() @ \section{} \section{category HOAGG HomogeneousAggregate} <>= import Boolean import OutputForm import SetCategory import Aggregate import CoercibleTo OutputForm import Evalable )abbrev category HOAGG HomogeneousAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991, May 1995 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A homogeneous aggregate is an aggregate of elements all of the ++ same type. ++ In the current system, all aggregates are homogeneous. ++ Two attributes characterize classes of aggregates. ++ Aggregates from domains with attribute \spadatt{finiteAggregate} ++ have a finite number of members. ++ Those with attribute \spadatt{shallowlyMutable} allow an element ++ to be modified or updated without changing its overall value. HomogeneousAggregate(S:Type): Category == Aggregate with if S has CoercibleTo(OutputForm) then CoercibleTo(OutputForm) if S has BasicType then BasicType if S has SetCategory then SetCategory if S has SetCategory then if S has Evalable S then Evalable S map : (S->S,%) -> % ++ map(f,u) returns a copy of u with each element x replaced by f(x). ++ For collections, \axiom{map(f,u) = [f(x) for x in u]}. if % has shallowlyMutable then map!: (S->S,%) -> % ++ map!(f,u) destructively replaces each element x of u by \axiom{f(x)}. if % has finiteAggregate then any?: (S->Boolean,%) -> Boolean ++ any?(p,u) tests if \axiom{p(x)} is true for any element x of u. ++ Note: for collections, ++ \axiom{any?(p,u) = reduce(or,map(f,u),false,true)}. every?: (S->Boolean,%) -> Boolean ++ every?(f,u) tests if p(x) is true for all elements x of u. ++ Note: for collections, ++ \axiom{every?(p,u) = reduce(and,map(f,u),true,false)}. count: (S->Boolean,%) -> NonNegativeInteger ++ count(p,u) returns the number of elements x in u ++ such that \axiom{p(x)} is true. For collections, ++ \axiom{count(p,u) = reduce(+,[1 for x in u | p(x)],0)}. parts: % -> List S ++ parts(u) returns a list of the consecutive elements of u. ++ For collections, \axiom{parts([x,y,...,z]) = (x,y,...,z)}. members: % -> List S ++ members(u) returns a list of the consecutive elements of u. ++ For collections, \axiom{parts([x,y,...,z]) = (x,y,...,z)}. if S has BasicType then count: (S,%) -> NonNegativeInteger ++ count(x,u) returns the number of occurrences of x in u. ++ For collections, \axiom{count(x,u) = reduce(+,[x=y for y in u],0)}. member?: (S,%) -> Boolean ++ member?(x,u) tests if x is a member of u. ++ For collections, ++ \axiom{member?(x,u) = reduce(or,[x=y for y in u],false)}. add if S has Evalable S then eval(u:%,l:List Equation S):% == map(eval(#1,l),u) if % has finiteAggregate then #c == # parts c any?(f, c) == or/[f x for x in parts c] every?(f, c) == and/[f x for x in parts c] count(f:S -> Boolean, c:%) == +/[1 for x in parts c | f x] members x == parts x if S has BasicType then x = y == #x = #y and (and/[a = b for a in parts x for b in parts y]) if S has BasicType then count(s:S, x:%) == count(s = #1, x) member?(e, c) == any?(e = #1,c) if S has CoercibleTo(OutputForm) then coerce(x:%):OutputForm == bracket commaSeparate [a::OutputForm for a in parts x]$List(OutputForm) @ \section{Aggregates of finite extent} <>= )abbrev category FINAGG FiniteAggregate ++ Author: Gabriel Dos Reis ++ Date Created: May 15, 2013 ++ Date Last Modified: May 15, 2013 ++ Description: ++ A finite aggregate is a homogeneous aggregate with a finite ++ number of elements. FiniteAggregate(S: Type): Category == Exports where Exports == HomogeneousAggregate S with finiteAggregate -- FIXME: TEMPORARY FOR TRANSITIONAL PURPOSE ONLY #: % -> NonNegativeInteger ++ \spad{#u} returns the number of items in u. any?: (S->Boolean,%) -> Boolean ++ \spad{any?(p,u)} tests if \spad{p(x)} is true for ++ any element \spad{x} of \spad{u}. ++ Note: for collections, ++ \axiom{any?(p,u) = reduce(or,map(f,u),false,true)}. every?: (S->Boolean,%) -> Boolean ++ \spad{every?(f,u)} tests if p(x) holds for all ++ elements \spad{x} of \spad{u}. ++ Note: for collections, ++ \axiom{every?(p,u) = reduce(and,map(f,u),true,false)}. count: (S->Boolean,%) -> NonNegativeInteger ++ \spad{count(p,u)} returns the number of elements \spad{x} ++ in \spad{u} such that \axiom{p(x)} holds. For collections, ++ \axiom{count(p,u) = reduce(+,[1 for x in u | p(x)],0)}. members: % -> List S ++ \spad{members(u)} returns a list of the consecutive elements of u. ++ For collections, \axiom{parts([x,y,...,z]) = (x,y,...,z)}. if S has BasicType then count: (S,%) -> NonNegativeInteger ++ \spad{count(x,u)} returns the number of occurrences ++ of \spad{x} in \spad{u}. ++ For collections, \axiom{count(x,u) = reduce(+,[x=y for y in u],0)}. member?: (S,%) -> Boolean ++ \spad{member?(x,u)} tests if \spad{x} is a member of \spad{u}. ++ For collections, ++ \axiom{member?(x,u) = reduce(or,[x=y for y in u],false)}. add empty? x == #x = 0 #x == # members x any?(f, x) == or/[f e for e in members x] every?(f, x) == and/[f e for e in members x] count(f:S -> Boolean, x:%) == +/[1 for e in parts x | f e] if S has BasicType then count(s:S, x:%) == count(s = #1, x) member?(e, x) == any?(e = #1,x) @ \section{} <>= )abbrev category SMAGG ShallowlyMutableAggregate ++ Author: Gabriel Dos Reis ++ Date Created: May 17, 2013 ++ Date Last Created: May 17, 2013 ++ Description: ++ This category describes the class of homogeneous aggregates ++ that support in place mutation that do not change their general ++ shapes. ShallowlyMutableAggregate(S: Type): Category == Exports where Exports == HomogeneousAggregate S with shallowlyMutable -- FIXME: TEMPORARY. map!: (S->S,%) -> % ++ \spad{map!(f,u)} destructively replaces each element ++ \spad{x} of \spad{u} by \spad{f(x)} @ \section{category CLAGG Collection} <>= import Boolean import HomogeneousAggregate )abbrev category CLAGG Collection ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A collection is a homogeneous aggregate which can built from ++ list of members. The operation used to build the aggregate is ++ generically named \spadfun{construct}. However, each collection ++ provides its own special function with the same name as the ++ data type, except with an initial lower case letter, e.g. ++ \spadfun{list} for \spadtype{List}, ++ \spadfun{flexibleArray} for \spadtype{FlexibleArray}, and so on. Collection(S:Type): Category == HomogeneousAggregate(S) with construct: List S -> % ++ \axiom{construct(x,y,...,z)} returns the collection of elements \axiom{x,y,...,z} ++ ordered as given. Equivalently written as \axiom{[x,y,...,z]$D}, where ++ D is the domain. D may be omitted for those of type List. find: (S->Boolean, %) -> Union(S, "failed") ++ find(p,u) returns the first x in u such that \axiom{p(x)} is true, and ++ "failed" otherwise. if % has finiteAggregate then reduce: ((S,S)->S,%) -> S ++ reduce(f,u) reduces the binary operation f across u. For example, ++ if u is \axiom{[x,y,...,z]} then \axiom{reduce(f,u)} returns \axiom{f(..f(f(x,y),...),z)}. ++ Note: if u has one element x, \axiom{reduce(f,u)} returns x. ++ Error: if u is empty. reduce: ((S,S)->S,%,S) -> S ++ reduce(f,u,x) reduces the binary operation f across u, where x is ++ the identity operation of f. ++ Same as \axiom{reduce(f,u)} if u has 2 or more elements. ++ Returns \axiom{f(x,y)} if u has one element y, ++ x if u is empty. ++ For example, \axiom{reduce(+,u,0)} returns the ++ sum of the elements of u. remove: (S->Boolean,%) -> % ++ remove(p,u) returns a copy of u removing all elements x such that ++ \axiom{p(x)} is true. ++ Note: \axiom{remove(p,u) == [x for x in u | not p(x)]}. select: (S->Boolean,%) -> % ++ select(p,u) returns a copy of u containing only those elements such ++ \axiom{p(x)} is true. ++ Note: \axiom{select(p,u) == [x for x in u | p(x)]}. if S has SetCategory then reduce: ((S,S)->S,%,S,S) -> S ++ reduce(f,u,x,z) reduces the binary operation f across u, stopping ++ when an "absorbing element" z is encountered. ++ As for \axiom{reduce(f,u,x)}, x is the identity operation of f. ++ Same as \axiom{reduce(f,u,x)} when u contains no element z. ++ Thus the third argument x is returned when u is empty. remove: (S,%) -> % ++ remove(x,u) returns a copy of u with all ++ elements \axiom{y = x} removed. ++ Note: \axiom{remove(y,c) == [x for x in c | x ~= y]}. removeDuplicates: % -> % ++ removeDuplicates(u) returns a copy of u with all duplicates removed. if S has ConvertibleTo InputForm then ConvertibleTo InputForm add if % has finiteAggregate then #c == # parts c count(f:S -> Boolean, c:%) == +/[1 for x in parts c | f x] any?(f, c) == or/[f x for x in parts c] every?(f, c) == and/[f x for x in parts c] find(f:S -> Boolean, c:%) == find(f, parts c) reduce(f:(S,S)->S, x:%) == reduce(f, parts x) reduce(f:(S,S)->S, x:%, s:S) == reduce(f, parts x, s) remove(f:S->Boolean, x:%) == construct remove(f, parts x) select(f:S->Boolean, x:%) == construct select(f, parts x) if S has SetCategory then remove(s:S, x:%) == remove(#1 = s, x) reduce(f:(S,S)->S, x:%, s1:S, s2:S) == reduce(f, parts x, s1, s2) removeDuplicates(x) == construct removeDuplicates parts x @ \section{category BGAGG BagAggregate} <>= import HomogeneousAggregate import List )abbrev category BGAGG BagAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A bag aggregate is an aggregate for which one can insert and extract objects, ++ and where the order in which objects are inserted determines the order ++ of extraction. ++ Examples of bags are stacks, queues, and dequeues. BagAggregate(S:Type): Category == HomogeneousAggregate S with shallowlyMutable ++ shallowlyMutable means that elements of bags may be destructively changed. bag: List S -> % ++ bag([x,y,...,z]) creates a bag with elements x,y,...,z. extract!: % -> S ++ extract!(u) destructively removes a (random) item from bag u. insert!: (S,%) -> % ++ insert!(x,u) inserts item x into bag u. inspect: % -> S ++ inspect(u) returns an (random) element from a bag. add bag(l) == x:=empty() for s in l repeat x:=insert!(s,x) x @ \section{category SKAGG StackAggregate} <>= import NonNegativeInteger import BagAggregate )abbrev category SKAGG StackAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A stack is a bag where the last item inserted is the first item extracted. StackAggregate(S:Type): Category == Join(BagAggregate S,FiniteAggregate S) with push!: (S,%) -> S ++ push!(x,s) pushes x onto stack s, i.e. destructively changing s ++ so as to have a new first (top) element x. ++ Afterwards, pop!(s) produces x and pop!(s) produces the original s. pop!: % -> S ++ pop!(s) returns the top element x, destructively removing x from s. ++ Note: Use \axiom{top(s)} to obtain x without removing it from s. ++ Error: if s is empty. top: % -> S ++ top(s) returns the top element x from s; s remains unchanged. ++ Note: Use \axiom{pop!(s)} to obtain x and remove it from s. depth: % -> NonNegativeInteger ++ depth(s) returns the number of elements of stack s. ++ Note: \axiom{depth(s) = #s}. @ \section{category QUAGG QueueAggregate} <>= import NonNegativeInteger import BagAggregate )abbrev category QUAGG QueueAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A queue is a bag where the first item inserted is the first item extracted. QueueAggregate(S:Type): Category == Join(BagAggregate S,FiniteAggregate S) with enqueue!: (S, %) -> S ++ enqueue!(x,q) inserts x into the queue q at the back end. dequeue!: % -> S ++ dequeue! s destructively extracts the first (top) element from queue q. ++ The element previously second in the queue becomes the first element. ++ Error: if q is empty. rotate!: % -> % ++ rotate! q rotates queue q so that the element at the front of ++ the queue goes to the back of the queue. ++ Note: rotate! q is equivalent to enqueue!(dequeue!(q)). length: % -> NonNegativeInteger ++ length(q) returns the number of elements in the queue. ++ Note: \axiom{length(q) = #q}. front: % -> S ++ front(q) returns the element at the front of the queue. ++ The queue q is unchanged by this operation. ++ Error: if q is empty. back: % -> S ++ back(q) returns the element at the back of the queue. ++ The queue q is unchanged by this operation. ++ Error: if q is empty. @ \section{category DQAGG DequeueAggregate} <>= import StackAggregate import QueueAggregate )abbrev category DQAGG DequeueAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A dequeue is a doubly ended stack, that is, a bag where first items ++ inserted are the first items extracted, at either the front or the back end ++ of the data structure. DequeueAggregate(S:Type): Category == Join(StackAggregate S,QueueAggregate S) with dequeue: () -> % ++ dequeue()$D creates an empty dequeue of type D. dequeue: List S -> % ++ dequeue([x,y,...,z]) creates a dequeue with first (top or front) ++ element x, second element y,...,and last (bottom or back) element z. height: % -> NonNegativeInteger ++ height(d) returns the number of elements in dequeue d. ++ Note: \axiom{height(d) = # d}. top!: % -> S ++ top!(d) returns the element at the top (front) of the dequeue. bottom!: % -> S ++ bottom!(d) returns the element at the bottom (back) of the dequeue. insertTop!: (S,%) -> S ++ insertTop!(x,d) destructively inserts x into the dequeue d, that is, ++ at the top (front) of the dequeue. ++ The element previously at the top of the dequeue becomes the ++ second in the dequeue, and so on. insertBottom!: (S,%) -> S ++ insertBottom!(x,d) destructively inserts x into the dequeue d ++ at the bottom (back) of the dequeue. extractTop!: % -> S ++ extractTop!(d) destructively extracts the top (front) element ++ from the dequeue d. ++ Error: if d is empty. extractBottom!: % -> S ++ extractBottom!(d) destructively extracts the bottom (back) element ++ from the dequeue d. ++ Error: if d is empty. reverse!: % -> % ++ reverse!(d) destructively replaces d by its reverse dequeue, i.e. ++ the top (front) element is now the bottom (back) element, and so on. @ \section{category PRQAGG PriorityQueueAggregate} <>= import BagAggregate )abbrev category PRQAGG PriorityQueueAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A priority queue is a bag of items from an ordered set where the item ++ extracted is always the maximum element. PriorityQueueAggregate(S:OrderedSet): Category == Join(BagAggregate S,FiniteAggregate S) with max: % -> S ++ max(q) returns the maximum element of priority queue q. merge: (%,%) -> % ++ merge(q1,q2) returns combines priority queues q1 and q2 to return ++ a single priority queue q. merge!: (%,%) -> % ++ merge!(q,q1) destructively changes priority queue q to include the ++ values from priority queue q1. @ \section{category DIOPS DictionaryOperations} <>= import Boolean import Collection import BagAggregate )abbrev category DIOPS DictionaryOperations ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ This category is a collection of operations common to both ++ categories \spadtype{Dictionary} and \spadtype{MultiDictionary} DictionaryOperations(S:SetCategory): Category == Join(BagAggregate S, Collection(S)) with dictionary: () -> % ++ dictionary()$D creates an empty dictionary of type D. dictionary: List S -> % ++ dictionary([x,y,...,z]) creates a dictionary consisting of ++ entries \axiom{x,y,...,z}. -- insert: (S,%) -> S ++ insert an entry -- member?: (S,%) -> Boolean ++ search for an entry -- remove!: (S,%,NonNegativeInteger) -> % -- ++ remove!(x,d,n) destructively changes dictionary d by removing -- ++ up to n entries y such that \axiom{y = x}. -- remove!: (S->Boolean,%,NonNegativeInteger) -> % -- ++ remove!(p,d,n) destructively changes dictionary d by removing -- ++ up to n entries x such that \axiom{p(x)} is true. if % has finiteAggregate then remove!: (S,%) -> % ++ remove!(x,d) destructively changes dictionary d by removing ++ all entries y such that \axiom{y = x}. remove!: (S->Boolean,%) -> % ++ remove!(p,d) destructively changes dictionary d by removeing ++ all entries x such that \axiom{p(x)} is true. select!: (S->Boolean,%) -> % ++ select!(p,d) destructively changes dictionary d by removing ++ all entries x such that \axiom{p(x)} is not true. add construct l == dictionary l dictionary() == empty() if % has finiteAggregate then copy d == dictionary parts d coerce(s:%):OutputForm == prefix("dictionary"@String :: OutputForm, [x::OutputForm for x in parts s]) @ \section{category DIAGG Dictionary} <>= import Boolean import DictionaryOperations )abbrev category DIAGG Dictionary ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A dictionary is an aggregate in which entries can be inserted, ++ searched for and removed. Duplicates are thrown away on insertion. ++ This category models the usual notion of dictionary which involves ++ large amounts of data where copying is impractical. ++ Principal operations are thus destructive (non-copying) ones. Dictionary(S:SetCategory): Category == DictionaryOperations S add dictionary l == d := dictionary() for x in l repeat insert!(x, d) d if % has finiteAggregate then -- remove(f:S->Boolean,t:%) == remove!(f, copy t) -- select(f, t) == select!(f, copy t) select!(f, t) == remove!(not f #1, t) --extract! d == -- empty? d => error "empty dictionary" -- remove!(x := first parts d, d, 1) -- x s = t == eq?(s,t) => true #s ~= #t => false and/[member?(x, t) for x in parts s] remove!(f:S->Boolean, t:%) == for m in parts t repeat if f m then remove!(m, t) t @ \section{category MDAGG MultiDictionary} <>= import NonNegativeInteger import DictionaryOperations )abbrev category MDAGG MultiDictionary ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A multi-dictionary is a dictionary which may contain duplicates. ++ As for any dictionary, its size is assumed large so that ++ copying (non-destructive) operations are generally to be avoided. MultiDictionary(S:SetCategory): Category == DictionaryOperations S with -- count: (S,%) -> NonNegativeInteger ++ multiplicity count insert!: (S,%,NonNegativeInteger) -> % ++ insert!(x,d,n) destructively inserts n copies of x into dictionary d. -- remove!: (S,%,NonNegativeInteger) -> % -- ++ remove!(x,d,n) destructively removes (up to) n copies of x from -- ++ dictionary d. removeDuplicates!: % -> % ++ removeDuplicates!(d) destructively removes any duplicate values ++ in dictionary d. duplicates: % -> List Record(entry:S,count:NonNegativeInteger) ++ duplicates(d) returns a list of values which have duplicates in d -- ++ duplicates(d) returns a list of ++ duplicates iterator -- to become duplicates: % -> Iterator(D,D) @ \section{category SETAGG SetAggregate} <>= import SetCategory import Collection )abbrev category SETAGG SetAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: 14 Oct, 1993 by RSS ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A set category lists a collection of set-theoretic operations ++ useful for both finite sets and multisets. ++ Note however that finite sets are distinct from multisets. ++ Although the operations defined for set categories are ++ common to both, the relationship between the two cannot ++ be described by inclusion or inheritance. SetAggregate(S:SetCategory): Category == Join(SetCategory, Collection(S)) with partiallyOrderedSet part? : (%, %) -> Boolean ++ s < t returns true if all elements of set aggregate s are also ++ elements of set aggregate t. brace : () -> % ++ brace()$D (otherwise written {}$D) ++ creates an empty set aggregate of type D. ++ This form is considered obsolete. Use \axiomFun{set} instead. brace : List S -> % ++ brace([x,y,...,z]) ++ creates a set aggregate containing items x,y,...,z. ++ This form is considered obsolete. Use \axiomFun{set} instead. set : () -> % ++ set()$D creates an empty set aggregate of type D. set : List S -> % ++ set([x,y,...,z]) creates a set aggregate containing items x,y,...,z. intersect: (%, %) -> % ++ intersect(u,v) returns the set aggregate w consisting of ++ elements common to both set aggregates u and v. ++ Note: equivalent to the notation (not currently supported) ++ {x for x in u | member?(x,v)}. difference : (%, %) -> % ++ difference(u,v) returns the set aggregate w consisting of ++ elements in set aggregate u but not in set aggregate v. ++ If u and v have no elements in common, \axiom{difference(u,v)} ++ returns a copy of u. ++ Note: equivalent to the notation (not currently supported) ++ \axiom{{x for x in u | not member?(x,v)}}. difference : (%, S) -> % ++ difference(u,x) returns the set aggregate u with element x removed. ++ If u does not contain x, a copy of u is returned. ++ Note: \axiom{difference(s, x) = difference(s, {x})}. symmetricDifference : (%, %) -> % ++ symmetricDifference(u,v) returns the set aggregate of elements x which ++ are members of set aggregate u or set aggregate v but not both. ++ If u and v have no elements in common, \axiom{symmetricDifference(u,v)} ++ returns a copy of u. ++ Note: \axiom{symmetricDifference(u,v) = union(difference(u,v),difference(v,u))} subset? : (%, %) -> Boolean ++ subset?(u,v) tests if u is a subset of v. ++ Note: equivalent to ++ \axiom{reduce(and,{member?(x,v) for x in u},true,false)}. union : (%, %) -> % ++ union(u,v) returns the set aggregate of elements which are members ++ of either set aggregate u or v. union : (%, S) -> % ++ union(u,x) returns the set aggregate u with the element x added. ++ If u already contains x, \axiom{union(u,x)} returns a copy of u. union : (S, %) -> % ++ union(x,u) returns the set aggregate u with the element x added. ++ If u already contains x, \axiom{union(x,u)} returns a copy of u. add symmetricDifference(x, y) == union(difference(x, y), difference(y, x)) union(s:%, x:S) == union(s, {x}) union(x:S, s:%) == union(s, {x}) difference(s:%, x:S) == difference(s, {x}) @ \section{category FSAGG FiniteSetAggregate} <>= import Dictionary import SetAggregate )abbrev category FSAGG FiniteSetAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: 14 Oct, 1993 by RSS ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A finite-set aggregate models the notion of a finite set, that is, ++ a collection of elements characterized by membership, but not ++ by order or multiplicity. ++ See \spadtype{Set} for an example. FiniteSetAggregate(S:SetCategory): Category == Join(Dictionary S, SetAggregate S,FiniteAggregate S) with cardinality: % -> NonNegativeInteger ++ cardinality(u) returns the number of elements of u. ++ Note: \axiom{cardinality(u) = #u}. if S has Finite then Finite complement: % -> % ++ complement(u) returns the complement of the set u, ++ i.e. the set of all values not in u. universe: () -> % ++ universe()$D returns the universal set for finite set aggregate D. if S has OrderedSet then max: % -> S ++ max(u) returns the largest element of aggregate u. min: % -> S ++ min(u) returns the smallest element of aggregate u. add part?(s,t) == #s < #t and s = intersect(s,t) s = t == #s = #t and empty? difference(s,t) brace l == construct l set l == construct l cardinality s == #s construct l == (s := set(); for x in l repeat insert!(x,s); s) count(x:S, s:%) == (member?(x, s) => 1; 0) subset?(s, t) == #s < #t and (and/[member?(x, t) for x in parts s]) coerce(s:%):OutputForm == brace [x::OutputForm for x in parts s]$List(OutputForm) intersect(s, t) == i := {} for x in parts s | member?(x, t) repeat insert!(x, i) i difference(s:%, t:%) == m := copy s for x in parts t repeat remove!(x, m) m symmetricDifference(s, t) == d := copy s for x in parts t repeat if member?(x, s) then remove!(x, d) else insert!(x, d) d union(s:%, t:%) == u := copy s for x in parts t repeat insert!(x, u) u if S has Finite then universe() == {index(i::PositiveInteger) for i in 1..size()$S} complement s == difference(universe(), s ) size() == 2 ** size()$S index i == {index(j::PositiveInteger)$S for j in 1..size()$S | bit?(i-1,j-1)} random() == index((random()$Integer rem (size()$% + 1))::PositiveInteger) lookup s == n:PositiveInteger := 1 for x in parts s repeat n := n + 2 ** ((lookup(x) - 1)::NonNegativeInteger) n if S has OrderedSet then max s == empty?(l := parts s) => error "Empty set" reduce("max", l) min s == empty?(l := parts s) => error "Empty set" reduce("min", l) @ \section{category MSETAGG MultisetAggregate} <>= import MultiDictionary import SetAggregate )abbrev category MSETAGG MultisetAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A multi-set aggregate is a set which keeps track of the multiplicity ++ of its elements. MultisetAggregate(S:SetCategory): Category == Join(MultiDictionary S, SetAggregate S) @ \section{category OMSAGG OrderedMultisetAggregate} <>= import MultisetAggregate import PriorityQueueAggregate import List )abbrev category OMSAGG OrderedMultisetAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An ordered-multiset aggregate is a multiset built over an ordered set S ++ so that the relative sizes of its entries can be assessed. ++ These aggregates serve as models for priority queues. OrderedMultisetAggregate(S:OrderedSet): Category == Join(MultisetAggregate S,PriorityQueueAggregate S) with -- max: % -> S ++ smallest entry in the set -- duplicates: % -> List Record(entry:S,count:NonNegativeInteger) ++ to become an in order iterator -- parts: % -> List S ++ in order iterator min: % -> S ++ min(u) returns the smallest entry in the multiset aggregate u. @ \section{category KDAGG KeyedDictionary} <>= import Dictionary import List )abbrev category KDAGG KeyedDictionary ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A keyed dictionary is a dictionary of key-entry pairs for which there is ++ a unique entry for each key. KeyedDictionary(Key:SetCategory, Entry:SetCategory): Category == Join(Dictionary Record(key:Key,entry:Entry),IndexedAggregate(Key,Entry)) with key?: (Key, %) -> Boolean ++ key?(k,t) tests if k is a key in table t. keys: % -> List Key ++ keys(t) returns the list the keys in table t. -- to become keys: % -> Key* and keys: % -> Iterator(Entry,Entry) remove!: (Key, %) -> Union(Entry,"failed") ++ remove!(k,t) searches the table t for the key k removing ++ (and return) the entry if there. ++ If t has no such key, \axiom{remove!(k,t)} returns "failed". search: (Key, %) -> Union(Entry,"failed") ++ search(k,t) searches the table t for the key k, ++ returning the entry stored in t for key k. ++ If t has no such key, \axiom{search(k,t)} returns "failed". add key?(k, t) == search(k, t) case Entry member?(p: Record(key: Key,entry: Entry), t: %): Boolean == r := search(p.key, t) r case Entry and r::Entry = p.entry if % has finiteAggregate then keys t == [x.key for x in parts t] elt(t, k) == (r := search(k, t)) case Entry => r::Entry error "key not in table" elt(t, k, e) == (r := search(k, t)) case Entry => r::Entry e @ \section{category ELTAB Eltable} <>= import Type import SetCategory )abbrev category ELTAB Eltable ++ Author: Michael Monagan; revised by Manuel Bronstein ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 25, 2010 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An eltable over domains \spad{S} and \spad{T} is a structure which ++ can be viewed as a function from \spad{S} to \spad{T}. ++ Examples of eltable structures range from data structures, e.g. those ++ of type \spadtype{List}, to algebraic structures, e.g. \spadtype{Polynomial}. Eltable(S: Type, T: Type): Category == Type with elt : (%, S) -> T ++ \spad{elt(u,s)} (also written: \spad{u.s}) returns the value ++ of \spad{u} at \spad{s}. ++ Error: if \spad{u} is not defined at \spad{s}. @ \section{category ELTAGG EltableAggregate} <>= import Type import SetCategory )abbrev category ELTAGG EltableAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An eltable aggregate is one which can be viewed as a function. ++ For example, the list \axiom{[1,7,4]} can applied to 0,1, and 2 respectively ++ will return the integers 1,7, and 4; thus this list may be viewed ++ as mapping 0 to 1, 1 to 7 and 2 to 4. In general, an aggregate ++ can map members of a domain {\em Dom} to an image domain {\em Im}. EltableAggregate(Dom:SetCategory, Im:Type): Category == Eltable(Dom, Im) with elt : (%, Dom, Im) -> Im ++ elt(u, x, y) applies u to x if x is in the domain of u, ++ and returns y otherwise. ++ For example, if u is a polynomial in \axiom{x} over the rationals, ++ \axiom{elt(u,n,0)} may define the coefficient of \axiom{x} ++ to the power n, returning 0 when n is out of range. qelt: (%, Dom) -> Im ++ qelt(u, x) applies \axiom{u} to \axiom{x} without checking whether ++ \axiom{x} is in the domain of \axiom{u}. If \axiom{x} is not in the ++ domain of \axiom{u} a memory-access violation may occur. If a check ++ on whether \axiom{x} is in the domain of \axiom{u} is required, use ++ the function \axiom{elt}. if % has shallowlyMutable then setelt : (%, Dom, Im) -> Im ++ setelt(u,x,y) sets the image of x to be y under u, ++ assuming x is in the domain of u. ++ Error: if x is not in the domain of u. -- this function will soon be renamed as setelt!. qsetelt!: (%, Dom, Im) -> Im ++ qsetelt!(u,x,y) sets the image of \axiom{x} to be \axiom{y} under ++ \axiom{u}, without checking that \axiom{x} is in the domain of ++ \axiom{u}. ++ If such a check is required use the function \axiom{setelt}. add qelt(a, x) == elt(a, x) if % has shallowlyMutable then qsetelt!(a, x, y) == (a.x := y) @ \section{category IXAGG IndexedAggregate} <>= import Type import SetCategory import HomogeneousAggregate import EltableAggregate import List )abbrev category IXAGG IndexedAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An indexed aggregate is a many-to-one mapping of indices to entries. ++ For example, a one-dimensional-array is an indexed aggregate where ++ the index is an integer. Also, a table is an indexed aggregate ++ where the indices and entries may have any type. IndexedAggregate(Index: SetCategory, Entry: Type): Category == Join(HomogeneousAggregate(Entry), EltableAggregate(Index, Entry)) with entries: % -> List Entry ++ entries(u) returns a list of all the entries of aggregate u ++ in no assumed order. -- to become entries: % -> Entry* and entries: % -> Iterator(Entry,Entry) index?: (Index,%) -> Boolean ++ index?(i,u) tests if i is an index of aggregate u. indices: % -> List Index ++ indices(u) returns a list of indices of aggregate u in no ++ particular order. -- to become indices: % -> Index* and indices: % -> Iterator(Index,Index). -- map: ((Entry,Entry)->Entry,%,%,Entry) -> % -- ++ exists c = map(f,a,b,x), i:Index where -- ++ c.i = f(a(i,x),b(i,x)) | index?(i,a) or index?(i,b) if Entry has SetCategory and % has finiteAggregate then entry?: (Entry,%) -> Boolean ++ entry?(x,u) tests if x equals \axiom{u . i} for some index i. if Index has OrderedSet then maxIndex: % -> Index ++ maxIndex(u) returns the maximum index i of aggregate u. ++ Note: in general, ++ \axiom{maxIndex(u) = reduce(max,[i for i in indices u])}; ++ if u is a list, \axiom{maxIndex(u) = #u}. minIndex: % -> Index ++ minIndex(u) returns the minimum index i of aggregate u. ++ Note: in general, ++ \axiom{minIndex(a) = reduce(min,[i for i in indices a])}; ++ for lists, \axiom{minIndex(a) = 1}. first : % -> Entry ++ first(u) returns the first element x of u. ++ Note: for collections, \axiom{first([x,y,...,z]) = x}. ++ Error: if u is empty. if % has shallowlyMutable then fill!: (%,Entry) -> % ++ fill!(u,x) replaces each entry in aggregate u by x. ++ The modified u is returned as value. swap!: (%,Index,Index) -> Void ++ swap!(u,i,j) interchanges elements i and j of aggregate u. ++ No meaningful value is returned. add elt(a, i, x) == (index?(i, a) => qelt(a, i); x) if % has finiteAggregate then entries x == parts x if Entry has SetCategory then entry?(x, a) == member?(x, a) if Index has OrderedSet then maxIndex a == "max"/indices(a) minIndex a == "min"/indices(a) first a == a minIndex a if % has shallowlyMutable then map(f, a) == map!(f, copy a) map!(f, a) == for i in indices a repeat qsetelt!(a, i, f qelt(a, i)) a fill!(a, x) == for i in indices a repeat qsetelt!(a, i, x) a swap!(a, i, j) == t := a.i qsetelt!(a, i, a.j) qsetelt!(a, j, t) @ \section{category TBAGG TableAggregate} <>= import SetCategory import KeyedDictionary import IndexedAggregate import Boolean import OutputForm import List )abbrev category TBAGG TableAggregate ++ Author: Michael Monagan, Stephen Watt; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: May 17, 2013. ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A table aggregate is a model of a table, i.e. a discrete many-to-one ++ mapping from keys to entries. TableAggregate(Key:SetCategory, Entry:SetCategory): Category == Join(KeyedDictionary(Key,Entry), FiniteAggregate Record(key:Key,entry:Entry)) with table: () -> % ++ table()$T creates an empty table of type T. table: List Record(key:Key,entry:Entry) -> % ++ table([x,y,...,z]) creates a table consisting of entries ++ \axiom{x,y,...,z}. -- to become table: Record(key:Key,entry:Entry)* -> % map: ((Entry, Entry) -> Entry, %, %) -> % ++ map(fn,t1,t2) creates a new table t from given tables t1 and t2 with ++ elements fn(x,y) where x and y are corresponding elements from t1 ++ and t2 respectively. add table() == empty() table l == dictionary l -- empty() == dictionary() insert!(p, t) == (t(p.key) := p.entry; t) indices t == keys t coerce(t:%):OutputForm == prefix("table"::OutputForm, [k::OutputForm = (t.k)::OutputForm for k in keys t]) map!(f: Entry->Entry, t: %) == for k in keys t repeat t.k := f t.k t map(f:(Entry, Entry) -> Entry, s:%, t:%) == z := table() for k in keys s | key?(k, t) repeat z.k := f(s.k, t.k) z -- map(f, s, t, x) == -- z := table() -- for k in keys s repeat z.k := f(s.k, t(k, x)) -- for k in keys t | not key?(k, s) repeat z.k := f(t.k, x) -- z parts(t:%):List Record(key:Key,entry:Entry) == [[k, t.k] for k in keys t] parts(t:%):List Entry == [t.k for k in keys t] entries(t:%):List Entry == parts(t) s:% = t:% == eq?(s,t) => true #s ~= #t => false for k in keys s repeat (e := search(k, t)) case "failed" or (e::Entry) ~= s.k => return false true map(f: Record(key:Key,entry:Entry)->Record(key:Key,entry:Entry), t: %): % == z := table() for k in keys t repeat ke: Record(key:Key,entry:Entry) := f [k, t.k] z ke.key := ke.entry z map!(f: Record(key:Key,entry:Entry)->Record(key:Key,entry:Entry), t: %): % == lke: List Record(key:Key,entry:Entry) := nil() for k in keys t repeat lke := cons(f [k, remove!(k,t)::Entry], lke) for ke in lke repeat t ke.key := ke.entry t inspect(t: %): Record(key:Key,entry:Entry) == ks := keys t empty? ks => error "Cannot extract from an empty aggregate" [first ks, t first ks] find(f: Record(key:Key,entry:Entry)->Boolean, t:%): Union(Record(key:Key,entry:Entry), "failed") == for ke in parts(t)@List(Record(key:Key,entry:Entry)) repeat if f ke then return ke "failed" index?(k: Key, t: %): Boolean == search(k,t) case Entry remove!(x:Record(key:Key,entry:Entry), t:%) == if member?(x, t) then remove!(x.key, t) t extract!(t: %): Record(key:Key,entry:Entry) == k: Record(key:Key,entry:Entry) := inspect t remove!(k.key, t) k any?(f: Entry->Boolean, t: %): Boolean == for k in keys t | f t k repeat return true false every?(f: Entry->Boolean, t: %): Boolean == for k in keys t | not f t k repeat return false true count(f: Entry->Boolean, t: %): NonNegativeInteger == tally: NonNegativeInteger := 0 for k in keys t | f t k repeat tally := tally + 1 tally @ \section{category RCAGG RecursiveAggregate} <>= import Type import SetCategory import List import Boolean )abbrev category RCAGG RecursiveAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A recursive aggregate over a type S is a model for a ++ a directed graph containing values of type S. ++ Recursively, a recursive aggregate is a {\em node} ++ consisting of a \spadfun{value} from S and 0 or more \spadfun{children} ++ which are recursive aggregates. ++ A node with no children is called a \spadfun{leaf} node. ++ A recursive aggregate may be cyclic for which some operations as noted ++ may go into an infinite loop. RecursiveAggregate(S:Type): Category == HomogeneousAggregate(S) with children: % -> List % ++ children(u) returns a list of the children of aggregate u. -- should be % -> %* and also needs children: % -> Iterator(S,S) nodes: % -> List % ++ nodes(u) returns a list of all of the nodes of aggregate u. -- to become % -> %* and also nodes: % -> Iterator(S,S) leaf?: % -> Boolean ++ leaf?(u) tests if u is a terminal node. value: % -> S ++ value(u) returns the value of the node u. elt: (%,"value") -> S ++ elt(u,"value") (also written: \axiom{a. value}) is ++ equivalent to \axiom{value(a)}. cyclic?: % -> Boolean ++ cyclic?(u) tests if u has a cycle. leaves: % -> List S ++ leaves(t) returns the list of values in obtained by visiting the ++ nodes of tree \axiom{t} in left-to-right order. distance: (%,%) -> Integer ++ distance(u,v) returns the path length (an integer) from node u to v. if S has SetCategory then child?: (%,%) -> Boolean ++ child?(u,v) tests if node u is a child of node v. node?: (%,%) -> Boolean ++ node?(u,v) tests if node u is contained in node v ++ (either as a child, a child of a child, etc.). if % has shallowlyMutable then setchildren!: (%,List %)->% ++ setchildren!(u,v) replaces the current children of node u ++ with the members of v in left-to-right order. setelt: (%,"value",S) -> S ++ setelt(a,"value",x) (also written \axiom{a . value := x}) ++ is equivalent to \axiom{setvalue!(a,x)} setvalue!: (%,S) -> S ++ setvalue!(u,x) sets the value of node u to x. add elt(x,"value") == value x if % has shallowlyMutable then setelt(x,"value",y) == setvalue!(x,y) if S has SetCategory then child?(x,l) == member?(x,children(l)) @ \section{category BRAGG BinaryRecursiveAggregate} <>= import Type import RecursiveAggregate )abbrev category BRAGG BinaryRecursiveAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A binary-recursive aggregate has 0, 1 or 2 children and ++ serves as a model for a binary tree or a doubly-linked aggregate structure BinaryRecursiveAggregate(S:Type):Category == RecursiveAggregate S with -- needs preorder, inorder and postorder iterators left: % -> % ++ left(u) returns the left child. elt: (%,"left") -> % ++ elt(u,"left") (also written: \axiom{a . left}) is ++ equivalent to \axiom{left(a)}. right: % -> % ++ right(a) returns the right child. elt: (%,"right") -> % ++ elt(a,"right") (also written: \axiom{a . right}) ++ is equivalent to \axiom{right(a)}. if % has shallowlyMutable then setelt: (%,"left",%) -> % ++ setelt(a,"left",b) (also written \axiom{a . left := b}) is equivalent ++ to \axiom{setleft!(a,b)}. setleft!: (%,%) -> % ++ setleft!(a,b) sets the left child of \axiom{a} to be b. setelt: (%,"right",%) -> % ++ setelt(a,"right",b) (also written \axiom{b . right := b}) ++ is equivalent to \axiom{setright!(a,b)}. setright!: (%,%) -> % ++ setright!(a,x) sets the right child of t to be x. add cycleMax ==> 1000 elt(x,"left") == left x elt(x,"right") == right x leaf? x == empty? x or empty? left x and empty? right x leaves t == empty? t => empty()$List(S) leaf? t => [value t] concat(leaves left t,leaves right t) nodes x == l := empty()$List(%) empty? x => l concat(nodes left x,concat([x],nodes right x)) children x == l := empty()$List(%) empty? x => l empty? left x => [right x] empty? right x => [left x] [left x, right x] if % has SetAggregate(S) and S has SetCategory then node?(u,v) == empty? v => false u = v => true for y in children v repeat node?(u,y) => return true false x = y == empty?(x) => empty?(y) empty?(y) => false value x = value y and left x = left y and right x = right y if % has finiteAggregate then member?(x,u) == empty? u => false x = value u => true member?(x,left u) or member?(x,right u) if S has CoercibleTo(OutputForm) then coerce(t:%): OutputForm == empty? t => bracket(empty()$OutputForm) v := value(t):: OutputForm empty? left t => empty? right t => v r := (right t)::OutputForm bracket ["."::OutputForm, v, r] l := (left t)::OutputForm r := empty? right t => "."::OutputForm (right t)::OutputForm bracket [l, v, r] if % has finiteAggregate then aggCount: (%,NonNegativeInteger) -> NonNegativeInteger #x == aggCount(x,0) aggCount(x,k) == empty? x => 0 k := k + 1 k = cycleMax and cyclic? x => error "cyclic tree" for y in children x repeat k := aggCount(y,k) k isCycle?: (%, List %) -> Boolean eqMember?: (%, List %) -> Boolean cyclic? x == not empty? x and isCycle?(x,empty()$(List %)) isCycle?(x,acc) == empty? x => false eqMember?(x,acc) => true for y in children x | not empty? y repeat isCycle?(y,acc) => return true false eqMember?(y,l) == for x in l repeat eq?(x,y) => return true false if % has shallowlyMutable then setelt(x,"left",b) == setleft!(x,b) setelt(x,"right",b) == setright!(x,b) @ \section{category DLAGG DoublyLinkedAggregate} <>= import Type import RecursiveAggregate )abbrev category DLAGG DoublyLinkedAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A doubly-linked aggregate serves as a model for a doubly-linked ++ list, that is, a list which can has links to both next and previous ++ nodes and thus can be efficiently traversed in both directions. DoublyLinkedAggregate(S:Type): Category == RecursiveAggregate S with last: % -> S ++ last(l) returns the last element of a doubly-linked aggregate l. ++ Error: if l is empty. head: % -> % ++ head(l) returns the first element of a doubly-linked aggregate l. ++ Error: if l is empty. tail: % -> % ++ tail(l) returns the doubly-linked aggregate l starting at ++ its second element. ++ Error: if l is empty. previous: % -> % ++ previous(l) returns the doubly-link list beginning with its previous ++ element. ++ Error: if l has no previous element. ++ Note: \axiom{next(previous(l)) = l}. next: % -> % ++ next(l) returns the doubly-linked aggregate beginning with its next ++ element. ++ Error: if l has no next element. ++ Note: \axiom{next(l) = rest(l)} and \axiom{previous(next(l)) = l}. if % has shallowlyMutable then concat!: (%,%) -> % ++ concat!(u,v) destructively concatenates doubly-linked aggregate v to the end of doubly-linked aggregate u. setprevious!: (%,%) -> % ++ setprevious!(u,v) destructively sets the previous node of doubly-linked aggregate u to v, returning v. setnext!: (%,%) -> % ++ setnext!(u,v) destructively sets the next node of doubly-linked aggregate u to v, returning v. @ \section{category URAGG UnaryRecursiveAggregate} <>= import Type import RecursiveAggregate import NonNegativeInteger )abbrev category URAGG UnaryRecursiveAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A unary-recursive aggregate is a one where nodes may have either ++ 0 or 1 children. ++ This aggregate models, though not precisely, a linked ++ list possibly with a single cycle. ++ A node with one children models a non-empty list, with the ++ \spadfun{value} of the list designating the head, or \spadfun{first}, of the ++ list, and the child designating the tail, or \spadfun{rest}, of the list. ++ A node with no child then designates the empty list. ++ Since these aggregates are recursive aggregates, they may be cyclic. UnaryRecursiveAggregate(S:Type): Category == RecursiveAggregate S with concat: (%,%) -> % ++ concat(u,v) returns an aggregate w consisting of the elements of u ++ followed by the elements of v. ++ Note: \axiom{v = rest(w,#a)}. concat: (S,%) -> % ++ concat(x,u) returns aggregate consisting of x followed by ++ the elements of u. ++ Note: if \axiom{v = concat(x,u)} then \axiom{x = first v} ++ and \axiom{u = rest v}. first: % -> S ++ first(u) returns the first element of u ++ (equivalently, the value at the current node). elt: (%,"first") -> S ++ elt(u,"first") (also written: \axiom{u . first}) is equivalent to first u. first: (%,NonNegativeInteger) -> % ++ first(u,n) returns a copy of the first n (\axiom{n >= 0}) elements of u. rest: % -> % ++ rest(u) returns an aggregate consisting of all but the first ++ element of u ++ (equivalently, the next node of u). elt: (%,"rest") -> % ++ elt(%,"rest") (also written: \axiom{u.rest}) is ++ equivalent to \axiom{rest u}. rest: (%,NonNegativeInteger) -> % ++ rest(u,n) returns the \axiom{n}th (n >= 0) node of u. ++ Note: \axiom{rest(u,0) = u}. last: % -> S ++ last(u) resturn the last element of u. ++ Note: for lists, \axiom{last(u) = u . (maxIndex u) = u . (# u - 1)}. elt: (%,"last") -> S ++ elt(u,"last") (also written: \axiom{u . last}) is equivalent to last u. last: (%,NonNegativeInteger) -> % ++ last(u,n) returns a copy of the last n (\axiom{n >= 0}) nodes of u. ++ Note: \axiom{last(u,n)} is a list of n elements. tail: % -> % ++ tail(u) returns the last node of u. ++ Note: if u is \axiom{shallowlyMutable}, ++ \axiom{setrest(tail(u),v) = concat(u,v)}. second: % -> S ++ second(u) returns the second element of u. ++ Note: \axiom{second(u) = first(rest(u))}. third: % -> S ++ third(u) returns the third element of u. ++ Note: \axiom{third(u) = first(rest(rest(u)))}. cycleEntry: % -> % ++ cycleEntry(u) returns the head of a top-level cycle contained in ++ aggregate u, or \axiom{empty()} if none exists. cycleLength: % -> NonNegativeInteger ++ cycleLength(u) returns the length of a top-level cycle ++ contained in aggregate u, or 0 is u has no such cycle. cycleTail: % -> % ++ cycleTail(u) returns the last node in the cycle, or ++ empty if none exists. if % has shallowlyMutable then concat!: (%,%) -> % ++ concat!(u,v) destructively concatenates v to the end of u. ++ Note: \axiom{concat!(u,v) = setlast!(u,v)}. concat!: (%,S) -> % ++ concat!(u,x) destructively adds element x to the end of u. ++ Note: \axiom{concat!(a,x) = setlast!(a,[x])}. cycleSplit!: % -> % ++ cycleSplit!(u) splits the aggregate by dropping off the cycle. ++ The value returned is the cycle entry, or nil if none exists. ++ For example, if \axiom{w = concat(u,v)} is the cyclic list where v is ++ the head of the cycle, \axiom{cycleSplit!(w)} will drop v off w thus ++ destructively changing w to u, and returning v. setfirst!: (%,S) -> S ++ setfirst!(u,x) destructively changes the first element of a to x. setelt: (%,"first",S) -> S ++ setelt(u,"first",x) (also written: \axiom{u.first := x}) is ++ equivalent to \axiom{setfirst!(u,x)}. setrest!: (%,%) -> % ++ setrest!(u,v) destructively changes the rest of u to v. setelt: (%,"rest",%) -> % ++ setelt(u,"rest",v) (also written: \axiom{u.rest := v}) is equivalent to ++ \axiom{setrest!(u,v)}. setlast!: (%,S) -> S ++ setlast!(u,x) destructively changes the last element of u to x. setelt: (%,"last",S) -> S ++ setelt(u,"last",x) (also written: \axiom{u.last := b}) ++ is equivalent to \axiom{setlast!(u,v)}. split!: (%,Integer) -> % ++ split!(u,n) splits u into two aggregates: \axiom{v = rest(u,n)} ++ and \axiom{w = first(u,n)}, returning \axiom{v}. ++ Note: afterwards \axiom{rest(u,n)} returns \axiom{empty()}. add cycleMax ==> 1000 findCycle: % -> % elt(x, "first") == first x elt(x, "last") == last x elt(x, "rest") == rest x second x == first rest x third x == first rest rest x cyclic? x == not empty? x and not empty? findCycle x last x == first tail x nodes x == l := empty()$List(%) while not empty? x repeat l := concat(x, l) x := rest x reverse! l children x == l := empty()$List(%) empty? x => l concat(rest x,l) leaf? x == empty? x value x == empty? x => error "value of empty object" first x #x == k: NonNegativeInteger := 0 while not empty? x repeat k = cycleMax and cyclic? x => error "cyclic list" x := rest x k := k + 1 k tail x == empty? x => error "empty list" y := rest x for k in 0.. while not empty? y repeat k = cycleMax and cyclic? x => error "cyclic list" y := rest(x := y) x findCycle x == y := rest x while not empty? y repeat if eq?(x, y) then return x x := rest x y := rest y if empty? y then return y if eq?(x, y) then return y y := rest y y cycleTail x == empty?(y := x := cycleEntry x) => x z := rest x while not eq?(x,z) repeat (y := z; z := rest z) y cycleEntry x == empty? x => x empty?(y := findCycle x) => y z := rest y l: NonNegativeInteger := 1 while not eq?(y,z) repeat z := rest z l := l + 1 y := x for k in 1..l repeat y := rest y while not eq?(x,y) repeat (x := rest x; y := rest y) x cycleLength x == empty? x => 0 empty?(x := findCycle x) => 0 y := rest x k: NonNegativeInteger := 1 while not eq?(x,y) repeat y := rest y k := k + 1 k rest(x, n) == for i in 1..n repeat empty? x => error "Index out of range" x := rest x x if % has finiteAggregate then last(x, n) == n > (m := #x) => error "index out of range" copy rest(x, (m - n)::NonNegativeInteger) if S has SetCategory then x = y == eq?(x, y) => true for k in 0.. while not empty? x and not empty? y repeat k = cycleMax and cyclic? x => error "cyclic list" first x ~= first y => return false x := rest x y := rest y empty? x and empty? y node?(u, v) == for k in 0.. while not empty? v repeat u = v => return true k = cycleMax and cyclic? v => error "cyclic list" v := rest v u=v if % has shallowlyMutable then setelt(x, "first", a) == setfirst!(x, a) setelt(x, "last", a) == setlast!(x, a) setelt(x, "rest", a) == setrest!(x, a) concat(x:%, y:%) == concat!(copy x, y) setlast!(x, s) == empty? x => error "setlast: empty list" setfirst!(tail x, s) s setchildren!(u,lv) == #lv=1 => setrest!(u, first lv) error "wrong number of children specified" setvalue!(u,s) == setfirst!(u,s) split!(p, n) == n < 1 => error "index out of range" p := rest(p, (n - 1)::NonNegativeInteger) q := rest p setrest!(p, empty()) q cycleSplit! x == empty?(y := cycleEntry x) or eq?(x, y) => y z := rest x while not eq?(z, y) repeat (x := z; z := rest z) setrest!(x, empty()) y @ \section{category STAGG StreamAggregate} <>= import Type import UnaryRecursiveAggregate import LinearAggregate import Boolean )abbrev category STAGG StreamAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A stream aggregate is a linear aggregate which possibly has an infinite ++ number of elements. A basic domain constructor which builds stream ++ aggregates is \spadtype{Stream}. From streams, a number of infinite ++ structures such power series can be built. A stream aggregate may ++ also be infinite since it may be cyclic. ++ For example, see \spadtype{DecimalExpansion}. StreamAggregate(S:Type): Category == Join(UnaryRecursiveAggregate S, LinearAggregate S) with explicitlyFinite?: % -> Boolean ++ explicitlyFinite?(s) tests if the stream has a finite ++ number of elements, and false otherwise. ++ Note: for many datatypes, \axiom{explicitlyFinite?(s) = not possiblyInfinite?(s)}. possiblyInfinite?: % -> Boolean ++ possiblyInfinite?(s) tests if the stream s could possibly ++ have an infinite number of elements. ++ Note: for many datatypes, \axiom{possiblyInfinite?(s) = not explictlyFinite?(s)}. less?: (%,NonNegativeInteger) -> Boolean ++ less?(u,n) tests if u has less than n elements. more?: (%,NonNegativeInteger) -> Boolean ++ more?(u,n) tests if u has greater than n elements. size?: (%,NonNegativeInteger) -> Boolean ++ size?(u,n) tests if u has exactly n elements. add c2: (%, %) -> S explicitlyFinite? x == not cyclic? x possiblyInfinite? x == cyclic? x first(x, n) == construct [c2(x, x := rest x) for i in 1..n] c2(x, r) == empty? x => error "Index out of range" first x elt(x:%, i:Integer) == i := i - minIndex x negative? i or empty?(x := rest(x, i::NonNegativeInteger)) => error "index out of range" first x elt(x:%, i:UniversalSegment(Integer)) == l := lo(i) - minIndex x negative? l => error "index out of range" not hasHi i => copy(rest(x, l::NonNegativeInteger)) (h := hi(i) - minIndex x) < l => empty() first(rest(x, l::NonNegativeInteger), (h - l + 1)::NonNegativeInteger) if % has shallowlyMutable then concat(x:%, y:%) == concat!(copy x, y) concat l == empty? l => empty() concat!(copy first l, concat rest l) map!(f, l) == y := l while not empty? l repeat setfirst!(l, f first l) l := rest l y fill!(x, s) == y := x while not empty? y repeat (setfirst!(y, s); y := rest y) x setelt(x:%, i:Integer, s:S) == i := i - minIndex x negative? i or empty?(x := rest(x,i::NonNegativeInteger)) => error "index out of range" setfirst!(x, s) setelt(x:%, i:UniversalSegment(Integer), s:S) == negative?(l := lo(i) - minIndex x) => error "index out of range" h := if hasHi i then hi(i) - minIndex x else maxIndex x h < l => s y := rest(x, l::NonNegativeInteger) z := rest(y, (h - l + 1)::NonNegativeInteger) while not eq?(y, z) repeat (setfirst!(y, s); y := rest y) s concat!(x:%, y:%) == empty? x => y setrest!(tail x, y) x @ \section{category LNAGG LinearAggregate} <>= import Type import Collection import IndexedAggregate import NonNegativeInteger import Integer )abbrev category LNAGG LinearAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A linear aggregate is an aggregate whose elements are indexed by integers. ++ Examples of linear aggregates are strings, lists, and ++ arrays. ++ Most of the exported operations for linear aggregates are non-destructive ++ but are not always efficient for a particular aggregate. ++ For example, \spadfun{concat} of two lists needs only to copy its first ++ argument, whereas \spadfun{concat} of two arrays needs to copy both arguments. ++ Most of the operations exported here apply to infinite objects (e.g. streams) ++ as well to finite ones. ++ For finite linear aggregates, see \spadtype{FiniteLinearAggregate}. LinearAggregate(S:Type): Category == Join(IndexedAggregate(Integer, S), Collection(S),_ Eltable(UniversalSegment Integer, %)) with new : (NonNegativeInteger,S) -> % ++ new(n,x) returns \axiom{fill!(new n,x)}. concat: (%,S) -> % ++ concat(u,x) returns aggregate u with additional element x at the end. ++ Note: for lists, \axiom{concat(u,x) == concat(u,[x])} concat: (S,%) -> % ++ concat(x,u) returns aggregate u with additional element at the front. ++ Note: for lists: \axiom{concat(x,u) == concat([x],u)}. concat: (%,%) -> % ++ concat(u,v) returns an aggregate consisting of the elements of u ++ followed by the elements of v. ++ Note: if \axiom{w = concat(u,v)} then \axiom{w.i = u.i for i in indices u} ++ and \axiom{w.(j + maxIndex u) = v.j for j in indices v}. concat: List % -> % ++ concat(u), where u is a lists of aggregates \axiom{[a,b,...,c]}, returns ++ a single aggregate consisting of the elements of \axiom{a} ++ followed by those ++ of b followed ... by the elements of c. ++ Note: \axiom{concat(a,b,...,c) = concat(a,concat(b,...,c))}. map: ((S,S)->S,%,%) -> % ++ map(f,u,v) returns a new collection w with elements \axiom{z = f(x,y)} ++ for corresponding elements x and y from u and v. ++ Note: for linear aggregates, \axiom{w.i = f(u.i,v.i)}. delete: (%,Integer) -> % ++ delete(u,i) returns a copy of u with the \axiom{i}th element deleted. ++ Note: for lists, \axiom{delete(a,i) == concat(a(0..i - 1),a(i + 1,..))}. delete: (%,UniversalSegment(Integer)) -> % ++ delete(u,i..j) returns a copy of u with the \axiom{i}th through ++ \axiom{j}th element deleted. ++ Note: \axiom{delete(a,i..j) = concat(a(0..i-1),a(j+1..))}. insert: (S,%,Integer) -> % ++ insert(x,u,i) returns a copy of u having x as its \axiom{i}th element. ++ Note: \axiom{insert(x,a,k) = concat(concat(a(0..k-1),x),a(k..))}. insert: (%,%,Integer) -> % ++ insert(v,u,k) returns a copy of u having v inserted beginning at the ++ \axiom{i}th element. ++ Note: \axiom{insert(v,u,k) = concat( u(0..k-1), v, u(k..) )}. if % has shallowlyMutable then setelt: (%,UniversalSegment(Integer),S) -> S ++ setelt(u,i..j,x) (also written: \axiom{u(i..j) := x}) destructively ++ replaces each element in the segment \axiom{u(i..j)} by x. ++ The value x is returned. ++ Note: u is destructively change so ++ that \axiom{u.k := x for k in i..j}; ++ its length remains unchanged. add indices a == [i for i in minIndex a .. maxIndex a] index?(i, a) == i >= minIndex a and i <= maxIndex a concat(a:%, x:S) == concat(a, new(1, x)) concat(x:S, y:%) == concat(new(1, x), y) insert(x:S, a:%, i:Integer) == insert(new(1, x), a, i) if % has finiteAggregate then maxIndex l == #l - 1 + minIndex l --if % has shallowlyMutable then new(n, s) == fill!(new n, s) @ \section{category FLAGG FiniteLinearAggregate} <>= import Type import SetCategory import OrderedSet import LinearAggregate import Boolean import Integer )abbrev category FLAGG FiniteLinearAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A finite linear aggregate is a linear aggregate of finite length. ++ The finite property of the aggregate adds several exports to the ++ list of exports from \spadtype{LinearAggregate} such as ++ \spadfun{reverse}, \spadfun{sort}, and so on. FiniteLinearAggregate(S:Type): Category == Join(LinearAggregate S,FiniteAggregate S) with merge: ((S,S)->Boolean,%,%) -> % ++ merge(p,a,b) returns an aggregate c which merges \axiom{a} and b. ++ The result is produced by examining each element x of \axiom{a} and y ++ of b successively. If \axiom{p(x,y)} is true, then x is inserted into ++ the result; otherwise y is inserted. If x is chosen, the next element ++ of \axiom{a} is examined, and so on. When all the elements of one ++ aggregate are examined, the remaining elements of the other ++ are appended. ++ For example, \axiom{merge(<,[1,3],[2,7,5])} returns \axiom{[1,2,3,7,5]}. reverse: % -> % ++ reverse(a) returns a copy of \axiom{a} with elements in reverse order. sort: ((S,S)->Boolean,%) -> % ++ sort(p,a) returns a copy of \axiom{a} sorted using total ordering predicate p. sorted?: ((S,S)->Boolean,%) -> Boolean ++ sorted?(p,a) tests if \axiom{a} is sorted according to predicate p. position: (S->Boolean, %) -> Integer ++ position(p,a) returns the index i of the first x in \axiom{a} such that ++ \axiom{p(x)} is true, and \axiom{minIndex(a) - 1} if there is no such x. if S has SetCategory then position: (S, %) -> Integer ++ position(x,a) returns the index i of the first occurrence of x in a, ++ and \axiom{minIndex(a) - 1} if there is no such x. position: (S,%,Integer) -> Integer ++ position(x,a,n) returns the index i of the first occurrence of x in ++ \axiom{a} where \axiom{i >= n}, and \axiom{minIndex(a) - 1} if no such x is found. if S has OrderedSet then OrderedSet merge: (%,%) -> % ++ merge(u,v) merges u and v in ascending order. ++ Note: \axiom{merge(u,v) = merge(<=,u,v)}. sort: % -> % ++ sort(u) returns an u with elements in ascending order. ++ Note: \axiom{sort(u) = sort(<=,u)}. sorted?: % -> Boolean ++ sorted?(u) tests if the elements of u are in ascending order. if % has shallowlyMutable then copyInto!: (%,%,Integer) -> % ++ copyInto!(u,v,i) returns aggregate u containing a copy of ++ v inserted at element i. reverse!: % -> % ++ reverse!(u) returns u with its elements in reverse order. sort!: ((S,S)->Boolean,%) -> % ++ sort!(p,u) returns u with its elements ordered by p. if S has OrderedSet then sort!: % -> % ++ sort!(u) returns u with its elements in ascending order. add if S has SetCategory then position(x:S, t:%) == position(x, t, minIndex t) if S has OrderedSet then -- sorted? l == sorted?(_<$S, l) sorted? l == sorted?(#1 < #2 or #1 = #2, l) merge(x, y) == merge(_<$S, x, y) sort l == sort(_<$S, l) if % has shallowlyMutable then reverse x == reverse! copy x sort(f, l) == sort!(f, copy l) if S has OrderedSet then sort! l == sort!(_<$S, l) @ \section{category A1AGG OneDimensionalArrayAggregate} <>= import Type import FiniteLinearAggregate )abbrev category A1AGG OneDimensionalArrayAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ One-dimensional-array aggregates serves as models for one-dimensional arrays. ++ Categorically, these aggregates are finite linear aggregates ++ with the \spadatt{shallowlyMutable} property, that is, any component of ++ the array may be changed without affecting the ++ identity of the overall array. ++ Array data structures are typically represented by a fixed area in storage and ++ therefore cannot efficiently grow or shrink on demand as can list structures ++ (see however \spadtype{FlexibleArray} for a data structure which ++ is a cross between a list and an array). ++ Iteration over, and access to, elements of arrays is extremely fast ++ (and often can be optimized to open-code). ++ Insertion and deletion however is generally slow since an entirely new ++ data structure must be created for the result. OneDimensionalArrayAggregate(S:Type): Category == FiniteLinearAggregate S with shallowlyMutable add parts x == [qelt(x, i) for i in minIndex x .. maxIndex x] sort!(f, a) == quickSort(f, a)$FiniteLinearAggregateSort(S, %) any?(f, a) == for i in minIndex a .. maxIndex a repeat f qelt(a, i) => return true false every?(f, a) == for i in minIndex a .. maxIndex a repeat not(f qelt(a, i)) => return false true position(f:S -> Boolean, a:%) == for i in minIndex a .. maxIndex a repeat f qelt(a, i) => return i minIndex(a) - 1 find(f, a) == for i in minIndex a .. maxIndex a repeat f qelt(a, i) => return qelt(a, i) "failed" count(f:S->Boolean, a:%) == n:NonNegativeInteger := 0 for i in minIndex a .. maxIndex a repeat if f(qelt(a, i)) then n := n+1 n map!(f, a) == for i in minIndex a .. maxIndex a repeat qsetelt!(a, i, f qelt(a, i)) a setelt(a:%, s:UniversalSegment(Integer), x:S) == l := lo s; h := if hasHi s then hi s else maxIndex a l < minIndex a or h > maxIndex a => error "index out of range" for k in l..h repeat qsetelt!(a, k, x) x reduce(f, a) == empty? a => error "cannot reduce an empty aggregate" r := qelt(a, m := minIndex a) for k in m+1 .. maxIndex a repeat r := f(r, qelt(a, k)) r reduce(f, a, identity) == for k in minIndex a .. maxIndex a repeat identity := f(identity, qelt(a, k)) identity if S has SetCategory then reduce(f, a, identity,absorber) == for k in minIndex a .. maxIndex a while identity ~= absorber repeat identity := f(identity, qelt(a, k)) identity -- this is necessary since new has disappeared. stupidnew: (NonNegativeInteger, %, %) -> % stupidget: List % -> S -- a and b are not both empty if n > 0 stupidnew(n, a, b) == zero? n => empty() new(n, (empty? a => qelt(b, minIndex b); qelt(a, minIndex a))) -- at least one element of l must be non-empty stupidget l == for a in l repeat not empty? a => return first a error "Should not happen" map(f, a, b) == m := max(minIndex a, minIndex b) n := min(maxIndex a, maxIndex b) l := max(0, n - m + 1)::NonNegativeInteger c := stupidnew(l, a, b) for i in minIndex(c).. for j in m..n repeat qsetelt!(c, i, f(qelt(a, j), qelt(b, j))) c -- map(f, a, b, x) == -- m := min(minIndex a, minIndex b) -- n := max(maxIndex a, maxIndex b) -- l := (n - m + 1)::NonNegativeInteger -- c := new l -- for i in minIndex(c).. for j in m..n repeat -- qsetelt!(c, i, f(a(j, x), b(j, x))) -- c merge(f, a, b) == r := stupidnew(#a + #b, a, b) i := minIndex a m := maxIndex a j := minIndex b n := maxIndex b k := minIndex(r) while i <= m and j <= n repeat if f(qelt(a, i), qelt(b, j)) then qsetelt!(r, k, qelt(a, i)) i := i+1 else qsetelt!(r, k, qelt(b, j)) j := j+1 k := k + 1 while i <= m repeat qsetelt!(r, k, elt(a, i)) k := k + 1 i := i + 1 while j <= n repeat qsetelt!(r, k, elt(b, j)) k := k + 1 j := j + 1 r elt(a:%, s:UniversalSegment(Integer)) == l := lo s h := if hasHi s then hi s else maxIndex a l < minIndex a or h > maxIndex a => error "index out of range" r := stupidnew(max(0, h - l + 1)::NonNegativeInteger, a, a) for k in minIndex r.. for i in l..h repeat qsetelt!(r, k, qelt(a, i)) r insert(a:%, b:%, i:Integer) == m := minIndex b n := maxIndex b i < m or i > n => error "index out of range" y := stupidnew(#a + #b, a, b) k := minIndex y for j in m..i-1 repeat qsetelt!(y, k, qelt(b, j)) k := k + 1 for j in minIndex a .. maxIndex a repeat qsetelt!(y, k, qelt(a, j)) k := k + 1 for j in i..n repeat qsetelt!(y, k, qelt(b, j)) k := k + 1 y copy x == y := stupidnew(#x, x, x) for i in minIndex x .. maxIndex x for j in minIndex y .. repeat qsetelt!(y, j, qelt(x, i)) y copyInto!(y, x, s) == s < minIndex y or s + #x > maxIndex y + 1 => error "index out of range" for i in minIndex x .. maxIndex x for j in s.. repeat qsetelt!(y, j, qelt(x, i)) y construct l == -- a := new(#l) empty? l => empty() a := new(#l, first l) for i in minIndex(a).. for x in l repeat qsetelt!(a, i, x) a delete(a:%, s:UniversalSegment(Integer)) == l := lo s; h := if hasHi s then hi s else maxIndex a l < minIndex a or h > maxIndex a => error "index out of range" h < l => copy a r := stupidnew((#a - h + l - 1)::NonNegativeInteger, a, a) k := minIndex(r) for i in minIndex a..l-1 repeat qsetelt!(r, k, qelt(a, i)) k := k + 1 for i in h+1 .. maxIndex a repeat qsetelt!(r, k, qelt(a, i)) k := k + 1 r delete(x:%, i:Integer) == i < minIndex x or i > maxIndex x => error "index out of range" y := stupidnew((#x - 1)::NonNegativeInteger, x, x) k := minIndex y for j in minIndex x..i-1 repeat qsetelt!(y, k, qelt(x, j)) k := k + 1 for j in i+1 .. maxIndex x repeat qsetelt!(y, k, qelt(x, j)) k := k + 1 y reverse! x == m := minIndex x n := maxIndex x for i in 0..((n-m) quo 2) repeat swap!(x, m+i, n-i) x concat l == empty? l => empty() n := +/[#a for a in l] i := minIndex(r := new(n, stupidget l)) for a in l repeat copyInto!(r, a, i) i := i + #a r sorted?(f, a) == for i in minIndex(a)..maxIndex(a)-1 repeat not f(qelt(a, i), qelt(a, i + 1)) => return false true concat(x:%, y:%) == z := stupidnew(#x + #y, x, y) copyInto!(z, x, i := minIndex z) copyInto!(z, y, i + #x) z if S has CoercibleTo(OutputForm) then coerce(r:%):OutputForm == bracket commaSeparate [qelt(r, k)::OutputForm for k in minIndex r .. maxIndex r] if S has SetCategory then x = y == #x ~= #y => false for i in minIndex x .. maxIndex x repeat not(qelt(x, i) = qelt(y, i)) => return false true position(x:S, t:%, s:Integer) == n := maxIndex t s < minIndex t or s > n => error "index out of range" for k in s..n repeat qelt(t, k) = x => return k minIndex(t) - 1 if S has OrderedSet then a < b == for i in minIndex a .. maxIndex a for j in minIndex b .. maxIndex b repeat qelt(a, i) ~= qelt(b, j) => return a.i < b.j #a < #b @ \section{category ELAGG ExtensibleLinearAggregate} <>= import Type import LinearAggregate import OrderedSet import Integer )abbrev category ELAGG ExtensibleLinearAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An extensible aggregate is one which allows insertion and deletion of entries. ++ These aggregates are models of lists and streams which are represented ++ by linked structures so as to make insertion, deletion, and ++ concatenation efficient. However, access to elements of these ++ extensible aggregates is generally slow since access is made from the end. ++ See \spadtype{FlexibleArray} for an exception. ExtensibleLinearAggregate(S:Type):Category == LinearAggregate S with shallowlyMutable concat!: (%,S) -> % ++ concat!(u,x) destructively adds element x to the end of u. concat!: (%,%) -> % ++ concat!(u,v) destructively appends v to the end of u. ++ v is unchanged delete!: (%,Integer) -> % ++ delete!(u,i) destructively deletes the \axiom{i}th element of u. delete!: (%,UniversalSegment(Integer)) -> % ++ delete!(u,i..j) destructively deletes elements u.i through u.j. remove!: (S->Boolean,%) -> % ++ remove!(p,u) destructively removes all elements x of ++ u such that \axiom{p(x)} is true. insert!: (S,%,Integer) -> % ++ insert!(x,u,i) destructively inserts x into u at position i. insert!: (%,%,Integer) -> % ++ insert!(v,u,i) destructively inserts aggregate v into u at position i. merge!: ((S,S)->Boolean,%,%) -> % ++ merge!(p,u,v) destructively merges u and v using predicate p. select!: (S->Boolean,%) -> % ++ select!(p,u) destructively changes u by keeping only values x such that ++ \axiom{p(x)}. if S has SetCategory then remove!: (S,%) -> % ++ remove!(x,u) destructively removes all values x from u. removeDuplicates!: % -> % ++ removeDuplicates!(u) destructively removes duplicates from u. if S has OrderedSet then merge!: (%,%) -> % ++ merge!(u,v) destructively merges u and v in ascending order. add delete(x:%, i:Integer) == delete!(copy x, i) delete(x:%, i:UniversalSegment(Integer)) == delete!(copy x, i) remove(f:S -> Boolean, x:%) == remove!(f, copy x) insert(s:S, x:%, i:Integer) == insert!(s, copy x, i) insert(w:%, x:%, i:Integer) == insert!(copy w, copy x, i) select(f, x) == select!(f, copy x) concat(x:%, y:%) == concat!(copy x, y) concat(x:%, y:S) == concat!(copy x, new(1, y)) concat!(x:%, y:S) == concat!(x, new(1, y)) if S has SetCategory then remove(s:S, x:%) == remove!(s, copy x) remove!(s:S, x:%) == remove!(#1 = s, x) removeDuplicates(x:%) == removeDuplicates!(copy x) if S has OrderedSet then merge!(x, y) == merge!(_<$S, x, y) @ \section{category LSAGG ListAggregate} <>= import Type import StreamAggregate import FiniteLinearAggregate import ExtensibleLinearAggregate )abbrev category LSAGG ListAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A list aggregate is a model for a linked list data structure. ++ A linked list is a versatile ++ data structure. Insertion and deletion are efficient and ++ searching is a linear operation. ListAggregate(S:Type): Category == Join(StreamAggregate S, FiniteLinearAggregate S, ExtensibleLinearAggregate S) with list: S -> % ++ list(x) returns the list of one element x. add cycleMax ==> 1000 mergeSort: ((S, S) -> Boolean, %, Integer) -> % sort!(f, l) == mergeSort(f, l, #l) list x == concat(x, empty()) reduce(f, x) == empty? x => error "reducing over an empty list needs the 3 argument form" reduce(f, rest x, first x) merge(f, p, q) == merge!(f, copy p, copy q) select!(f, x) == while not empty? x and not f first x repeat x := rest x empty? x => x y := x z := rest y while not empty? z repeat if f first z then (y := z; z := rest z) else (z := rest z; setrest!(y, z)) x merge!(f, p, q) == empty? p => q empty? q => p eq?(p, q) => error "cannot merge a list into itself" r: % t: % if f(first p, first q) then (r := t := p; p := rest p) else (r := t := q; q := rest q) while not empty? p and not empty? q repeat if f(first p, first q) then (setrest!(t, p); t := p; p := rest p) else (setrest!(t, q); t := q; q := rest q) setrest!(t, if empty? p then q else p) r insert!(s:S, x:%, i:Integer) == i < (m := minIndex x) => error "index out of range" i = m => concat(s, x) y := rest(x, (i - 1 - m)::NonNegativeInteger) z := rest y setrest!(y, concat(s, z)) x insert!(w:%, x:%, i:Integer) == i < (m := minIndex x) => error "index out of range" i = m => concat!(w, x) y := rest(x, (i - 1 - m)::NonNegativeInteger) z := rest y setrest!(y, w) concat!(y, z) x remove!(f:S -> Boolean, x:%) == while not empty? x and f first x repeat x := rest x empty? x => x p := x q := rest x while not empty? q repeat if f first q then q := setrest!(p, rest q) else (p := q; q := rest q) x delete!(x:%, i:Integer) == i < (m := minIndex x) => error "index out of range" i = m => rest x y := rest(x, (i - 1 - m)::NonNegativeInteger) setrest!(y, rest(y, 2)) x delete!(x:%, i:UniversalSegment(Integer)) == (l := lo i) < (m := minIndex x) => error "index out of range" h := if hasHi i then hi i else maxIndex x h < l => x l = m => rest(x, (h + 1 - m)::NonNegativeInteger) t := rest(x, (l - 1 - m)::NonNegativeInteger) setrest!(t, rest(t, (h - l + 2)::NonNegativeInteger)) x find(f, x) == while not empty? x and not f first x repeat x := rest x empty? x => "failed" first x position(f:S -> Boolean, x:%) == k := minIndex(x) while not empty? x and not f first x repeat x := rest x k := k + 1 empty? x => minIndex(x) - 1 k mergeSort(f, p, n) == if n = 2 and f(first rest p, first p) then p := reverse! p n < 3 => p l := (n quo 2)::NonNegativeInteger q := split!(p, l) p := mergeSort(f, p, l) q := mergeSort(f, q, n - l) merge!(f, p, q) sorted?(f, l) == empty? l => true p := rest l while not empty? p repeat not f(first l, first p) => return false p := rest(l := p) true reduce(f, x, i) == r := i while not empty? x repeat (r := f(r, first x); x := rest x) r if S has SetCategory then reduce(f, x, i,a) == r := i while not empty? x and r ~= a repeat r := f(r, first x) x := rest x r new(n, s) == l := empty() for k in 1..n repeat l := concat(s, l) l map(f, x, y) == z := empty() while not empty? x and not empty? y repeat z := concat(f(first x, first y), z) x := rest x y := rest y reverse! z -- map(f, x, y, d) == -- z := empty() -- while not empty? x and not empty? y repeat -- z := concat(f(first x, first y), z) -- x := rest x -- y := rest y -- z := reverseInPlace z -- if not empty? x then -- z := concat!(z, map(f(#1, d), x)) -- if not empty? y then -- z := concat!(z, map(f(d, #1), y)) -- z reverse! x == empty? x => x empty?(y := rest x) => x setrest!(x, empty()) while not empty? y repeat z := rest y setrest!(y, x) x := y y := z x copy x == y := empty() for k in 0.. while not empty? x repeat k = cycleMax and cyclic? x => error "cyclic list" y := concat(first x, y) x := rest x reverse! y copyInto!(y, x, s) == s < (m := minIndex y) => error "index out of range" z := rest(y, (s - m)::NonNegativeInteger) while not empty? z and not empty? x repeat setfirst!(z, first x) x := rest x z := rest z y if S has SetCategory then position(w, x, s) == s < (m := minIndex x) => error "index out of range" x := rest(x, (s - m)::NonNegativeInteger) k := s while not empty? x and w ~= first x repeat x := rest x k := k + 1 empty? x => minIndex x - 1 k removeDuplicates! l == p := l while not empty? p repeat p := setrest!(p, remove!(#1 = first p, rest p)) l if S has OrderedSet then x < y == while not empty? x and not empty? y repeat first x ~= first y => return(first x < first y) x := rest x y := rest y empty? x => not empty? y false @ \section{category ALAGG AssociationListAggregate} <>= import SetCategory import TableAggregate import ListAggregate )abbrev category ALAGG AssociationListAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ An association list is a list of key entry pairs which may be viewed ++ as a table. It is a poor mans version of a table: ++ searching for a key is a linear operation. AssociationListAggregate(Key:SetCategory,Entry:SetCategory): Category == Join(TableAggregate(Key, Entry), ListAggregate Record(key:Key,entry:Entry)) with assoc: (Key, %) -> Maybe Record(key:Key,entry:Entry) ++ assoc(k,u) returns the element x in association list u stored ++ with key k, or \spad{nothing} if u has no key k. @ \section{category SRAGG StringAggregate} <>= import OneDimensionalArrayAggregate Character import UniversalSegment import Boolean import Character import CharacterClass import Integer )abbrev category SRAGG StringAggregate ++ Author: Stephen Watt and Michael Monagan. revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ A string aggregate is a category for strings, that is, ++ one dimensional arrays of characters. StringAggregate: Category == OneDimensionalArrayAggregate Character with lowerCase : % -> % ++ lowerCase(s) returns the string with all characters in lower case. lowerCase!: % -> % ++ lowerCase!(s) destructively replaces the alphabetic characters ++ in s by lower case. upperCase : % -> % ++ upperCase(s) returns the string with all characters in upper case. upperCase!: % -> % ++ upperCase!(s) destructively replaces the alphabetic characters ++ in s by upper case characters. prefix? : (%, %) -> Boolean ++ prefix?(s,t) tests if the string s is the initial substring of t. ++ Note: \axiom{prefix?(s,t) == reduce(and,[s.i = t.i for i in 0..maxIndex s])}. suffix? : (%, %) -> Boolean ++ suffix?(s,t) tests if the string s is the final substring of t. ++ Note: \axiom{suffix?(s,t) == reduce(and,[s.i = t.(n - m + i) for i in 0..maxIndex s])} ++ where m and n denote the maxIndex of s and t respectively. substring?: (%, %, Integer) -> Boolean ++ substring?(s,t,i) tests if s is a substring of t beginning at ++ index i. ++ Note: \axiom{substring?(s,t,0) = prefix?(s,t)}. match: (%, %, Character) -> NonNegativeInteger ++ match(p,s,wc) tests if pattern \axiom{p} matches subject \axiom{s} ++ where \axiom{wc} is a wild card character. If no match occurs, ++ the index \axiom{0} is returned; otheriwse, the value returned ++ is the first index of the first character in the subject matching ++ the subject (excluding that matched by an initial wild-card). ++ For example, \axiom{match("*to*","yorktown","*")} returns \axiom{5} ++ indicating a successful match starting at index \axiom{5} of ++ \axiom{"yorktown"}. match?: (%, %, Character) -> Boolean ++ match?(s,t,c) tests if s matches t except perhaps for ++ multiple and consecutive occurrences of character c. ++ Typically c is the blank character. replace : (%, UniversalSegment(Integer), %) -> % ++ replace(s,i..j,t) replaces the substring \axiom{s(i..j)} of s by string t. position : (%, %, Integer) -> Integer ++ position(s,t,i) returns the position j of the substring s in string t, ++ where \axiom{j >= i} is required. position : (CharacterClass, %, Integer) -> Integer ++ position(cc,t,i) returns the position \axiom{j >= i} in t of ++ the first character belonging to cc. coerce : Character -> % ++ coerce(c) returns c as a string s with the character c. split: (%, Character) -> List % ++ split(s,c) returns a list of substrings delimited by character c. split: (%, CharacterClass) -> List % ++ split(s,cc) returns a list of substrings delimited by characters in cc. trim: (%, Character) -> % ++ trim(s,c) returns s with all characters c deleted from right ++ and left ends. ++ For example, \axiom{trim(" abc ", char " ")} returns \axiom{"abc"}. trim: (%, CharacterClass) -> % ++ trim(s,cc) returns s with all characters in cc deleted from right ++ and left ends. ++ For example, \axiom{trim("(abc)", charClass "()")} returns \axiom{"abc"}. leftTrim: (%, Character) -> % ++ leftTrim(s,c) returns s with all leading characters c deleted. ++ For example, \axiom{leftTrim(" abc ", char " ")} returns \axiom{"abc "}. leftTrim: (%, CharacterClass) -> % ++ leftTrim(s,cc) returns s with all leading characters in cc deleted. ++ For example, \axiom{leftTrim("(abc)", charClass "()")} returns \axiom{"abc)"}. rightTrim: (%, Character) -> % ++ rightTrim(s,c) returns s with all trailing occurrences of c deleted. ++ For example, \axiom{rightTrim(" abc ", char " ")} returns \axiom{" abc"}. rightTrim: (%, CharacterClass) -> % ++ rightTrim(s,cc) returns s with all trailing occurences of ++ characters in cc deleted. ++ For example, \axiom{rightTrim("(abc)", charClass "()")} returns \axiom{"(abc"}. elt: (%, %) -> % ++ elt(s,t) returns the concatenation of s and t. It is provided to ++ allow juxtaposition of strings to work as concatenation. ++ For example, \axiom{"smoo" "shed"} returns \axiom{"smooshed"}. add trim(s: %, c: Character) == leftTrim(rightTrim(s, c), c) trim(s: %, cc: CharacterClass) == leftTrim(rightTrim(s, cc), cc) lowerCase s == lowerCase! copy s upperCase s == upperCase! copy s prefix?(s, t) == substring?(s, t, minIndex t) coerce(c:Character):% == new(1, c) elt(s:%, t:%): % == concat(s,t)$% @ \section{category BTAGG BitAggregate} <>= import OrderedSet import Logic import OneDimensionalArrayAggregate Boolean )abbrev category BTAGG BitAggregate ++ Author: Michael Monagan; revised by Manuel Bronstein and Richard Jenks ++ Date Created: August 87 through August 88 ++ Date Last Updated: April 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ The bit aggregate category models aggregates representing large ++ quantities of Boolean data. BitAggregate(): Category == Join(OrderedSet, BooleanLogic, Logic, OneDimensionalArrayAggregate Boolean) with nand : (%, %) -> % ++ nand(a,b) returns the logical {\em nand} of bit aggregates \axiom{a} ++ and \axiom{b}. nor : (%, %) -> % ++ nor(a,b) returns the logical {\em nor} of bit aggregates \axiom{a} and ++ \axiom{b}. xor : (%, %) -> % ++ xor(a,b) returns the logical {\em exclusive-or} of bit aggregates ++ \axiom{a} and \axiom{b}. add not v == map(_not, v) ~ v == map(_~, v) v /\ u == map(_/_\, v, u) v \/ u == map(_\_/, v, u) nand(v, u) == map(nand, v, u) nor(v, u) == map(nor, v, u) @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2010, Gabriel Dos Reis. --All rights reserved. -- --Redistribution and use in source and binary forms, with or without --modification, are permitted provided that the following conditions are --met: -- -- - Redistributions of source code must retain the above copyright -- notice, this list of conditions and the following disclaimer. -- -- - Redistributions in binary form must reproduce the above copyright -- notice, this list of conditions and the following disclaimer in -- the documentation and/or other materials provided with the -- distribution. -- -- - Neither the name of The Numerical ALgorithms Group Ltd. nor the -- names of its contributors may be used to endorse or promote products -- derived from this software without specific prior written permission. -- --THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS --IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED --TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A --PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER --OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, --EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, --PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR --PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF --LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING --NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS --SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. @ <<*>>= <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}