\documentclass{article} \usepackage{open-axiom} \title{src/algebra boolean.spad} \author{Stephen M. Watt, Michael Monagan, Gabriel Dos~Reis} \begin{document} \maketitle \begin{abstract} \end{abstract} \tableofcontents \eject \section{Categories an domains for logic} <>= )abbrev category BOOLE BooleanLogic ++ Author: Gabriel Dos Reis ++ Date Created: April 04, 2010 ++ Date Last Modified: April 04, 2010 ++ Description: ++ This is the category of Boolean logic structures. BooleanLogic(): Category == Logic with not: % -> % ++ \spad{not x} returns the complement or negation of \spad{x}. and: (%,%) -> % ++ \spad{x and y} returns the conjunction of \spad{x} and \spad{y}. or: (%,%) -> % ++ \spad{x or y} returns the disjunction of \spad{x} and \spad{y}. add not x == ~ x x and y == x /\ y x or y == x \/ y @ <>= )abbrev category PROPLOG PropositionalLogic ++ Author: Gabriel Dos Reis ++ Date Created: Januray 14, 2008 ++ Date Last Modified: May 27, 2009 ++ Description: This category declares the connectives of ++ Propositional Logic. PropositionalLogic(): Category == Join(BooleanLogic,SetCategory) with true: % ++ true is a logical constant. false: % ++ false is a logical constant. implies: (%,%) -> % ++ implies(p,q) returns the logical implication of `q' by `p'. equiv: (%,%) -> % ++ equiv(p,q) returns the logical equivalence of `p', `q'. @ \section{domain PROPFRML PropositionalFormula} <>= )set mess autoload on )abbrev domain PROPFRML PropositionalFormula ++ Author: Gabriel Dos Reis ++ Date Created: Januray 14, 2008 ++ Date Last Modified: February, 2011 ++ Description: This domain implements propositional formula build ++ over a term domain, that itself belongs to PropositionalLogic PropositionalFormula(T: SetCategory): Public == Private where Public == Join(PropositionalLogic, CoercibleFrom T) with isAtom : % -> Maybe T ++ \spad{isAtom f} returns a value \spad{v} such that ++ \spad{v case T} holds if the formula \spad{f} is a term. isNot : % -> Maybe % ++ \spad{isNot f} returns a value \spad{v} such that ++ \spad{v case %} holds if the formula \spad{f} is a negation. isAnd : % -> Maybe Pair(%,%) ++ \spad{isAnd f} returns a value \spad{v} such that ++ \spad{v case Pair(%,%)} holds if the formula \spad{f} ++ is a conjunction formula. isOr : % -> Maybe Pair(%,%) ++ \spad{isOr f} returns a value \spad{v} such that ++ \spad{v case Pair(%,%)} holds if the formula \spad{f} ++ is a disjunction formula. isImplies : % -> Maybe Pair(%,%) ++ \spad{isImplies f} returns a value \spad{v} such that ++ \spad{v case Pair(%,%)} holds if the formula \spad{f} ++ is an implication formula. isEquiv : % -> Maybe Pair(%,%) ++ \spad{isEquiv f} returns a value \spad{v} such that ++ \spad{v case Pair(%,%)} holds if the formula \spad{f} ++ is an equivalence formula. conjunction: (%,%) -> % ++ \spad{conjunction(p,q)} returns a formula denoting the ++ conjunction of \spad{p} and \spad{q}. disjunction: (%,%) -> % ++ \spad{disjunction(p,q)} returns a formula denoting the ++ disjunction of \spad{p} and \spad{q}. Private == add Rep == Union(T, Kernel %) import Kernel % import BasicOperator import KernelFunctions2(Identifier,%) import List % -- Local names for proposition logical operators macro FALSE == '%false macro TRUE == '%true macro NOT == '%not macro AND == '%and macro OR == '%or macro IMP == '%implies macro EQV == '%equiv -- Return the nesting level of a formula level(f: %): NonNegativeInteger == f' := rep f f' case T => 0 height f' -- A term is a formula coerce(t: T): % == per t false == per constantKernel FALSE true == per constantKernel TRUE ~ p == per kernel(operator(NOT, 1::Arity), [p], 1 + level p) conjunction(p,q) == per kernel(operator(AND, 2), [p, q], 1 + max(level p, level q)) p /\ q == conjunction(p,q) disjunction(p,q) == per kernel(operator(OR, 2), [p, q], 1 + max(level p, level q)) p \/ q == disjunction(p,q) implies(p,q) == per kernel(operator(IMP, 2), [p, q], 1 + max(level p, level q)) equiv(p,q) == per kernel(operator(EQV, 2), [p, q], 1 + max(level p, level q)) isAtom f == f' := rep f f' case T => just(f'@T) nothing isNot f == f' := rep f f' case Kernel(%) and is?(f', NOT) => just(first argument f') nothing isBinaryOperator(f: Kernel %, op: Symbol): Maybe Pair(%, %) == not is?(f, op) => nothing args := argument f just pair(first args, second args) isAnd f == f' := rep f f' case Kernel % => isBinaryOperator(f', AND) nothing isOr f == f' := rep f f' case Kernel % => isBinaryOperator(f', OR) nothing isImplies f == f' := rep f f' case Kernel % => isBinaryOperator(f', IMP) nothing isEquiv f == f' := rep f f' case Kernel % => isBinaryOperator(f', EQV) nothing -- Unparsing grammar. -- -- Ideally, the following syntax would the external form -- Formula: -- EquivFormula -- -- EquivFormula: -- ImpliesFormula -- ImpliesFormula <=> EquivFormula -- -- ImpliesFormula: -- OrFormula -- OrFormula => ImpliesFormula -- -- OrFormula: -- AndFormula -- AndFormula or OrFormula -- -- AndFormula -- NotFormula -- NotFormula and AndFormula -- -- NotFormula: -- PrimaryFormula -- not NotFormula -- -- PrimaryFormula: -- Term -- ( Formula ) -- -- Note: Since the token '=>' already means a construct different -- from what we would like to have as a notation for -- propositional logic, we will output the formula `p => q' -- as implies(p,q), which looks like a function call. -- Similarly, we do not have the token `<=>' for logical -- equivalence; so we unparser `p <=> q' as equiv(p,q). -- -- So, we modify the nonterminal PrimaryFormula to read -- PrimaryFormula: -- Term -- implies(Formula, Formula) -- equiv(Formula, Formula) formula: % -> OutputForm coerce(p: %): OutputForm == formula p primaryFormula(p: %): OutputForm == p' := rep p p' case T => p'@T::OutputForm case constantIfCan p' is c@Identifier => c::OutputForm otherwise => is?(p', IMP) or is?(p', EQV) => args := argument p' elt(operator(p')::OutputForm, [formula first args, formula second args])$OutputForm paren(formula p)$OutputForm notFormula(p: %): OutputForm == case isNot p is f@% => elt(outputForm 'not, [notFormula f])$OutputForm otherwise => primaryFormula p andFormula(f: %): OutputForm == case isAnd f is p@Pair(%,%) => -- ??? idealy, we should be using `and$OutputForm' but -- ??? a bug in the compiler currently prevents that. infix(outputForm 'and, notFormula first p, andFormula second p)$OutputForm otherwise => notFormula f orFormula(f: %): OutputForm == case isOr f is p@Pair(%,%) => -- ??? idealy, we should be using `or$OutputForm' but -- ??? a bug in the compiler currently prevents that. infix(outputForm 'or, andFormula first p, orFormula second p)$OutputForm otherwise => andFormula f formula f == -- Note: this should be equivFormula, but see the explanation above. orFormula f @ <>= )abbrev package PROPFUN1 PropositionalFormulaFunctions1 ++ Author: Gabriel Dos Reis ++ Date Created: April 03, 2010 ++ Date Last Modified: April 03, 2010 ++ Description: ++ This package collects unary functions operating on propositional ++ formulae. PropositionalFormulaFunctions1(T): Public == Private where T: SetCategory Public == Type with dual: PropositionalFormula T -> PropositionalFormula T ++ \spad{dual f} returns the dual of the proposition \spad{f}. atoms: PropositionalFormula T -> Set T ++ \spad{atoms f} ++ returns the set of atoms appearing in ++ the formula \spad{f}. simplify: PropositionalFormula T -> PropositionalFormula T ++ \spad{simplify f} returns a formula logically equivalent ++ to \spad{f} where obvious tautologies have been removed. Private == add macro F == PropositionalFormula T inline Pair(F,F) dual f == f = true$F => false$F f = false$F => true$F isAtom f case T => f (f1 := isNot f) case F => not dual f1 (f2 := isAnd f) case Pair(F,F) => disjunction(dual first f2, dual second f2) (f2 := isOr f) case Pair(F,F) => conjunction(dual first f2, dual second f2) error "formula contains `equiv' or `implies'" atoms f == (t := isAtom f) case T => { t } (f1 := isNot f) case F => atoms f1 (f2 := isAnd f) case Pair(F,F) => union(atoms first f2, atoms second f2) (f2 := isOr f) case Pair(F,F) => union(atoms first f2, atoms second f2) empty()$Set(T) -- one-step simplification helper function simplifyOneStep(f: F): F == (f1 := isNot f) case F => f1 = true$F => false$F f1 = false$F => true$F (f1' := isNot f1) case F => f1' -- assume classical logic f (f2 := isAnd f) case Pair(F,F) => first f2 = false$F or second f2 = false$F => false$F first f2 = true$F => second f2 second f2 = true$F => first f2 f (f2 := isOr f) case Pair(F,F) => first f2 = false$F => second f2 second f2 = false$F => first f2 first f2 = true$F or second f2 = true$F => true$F f (f2 := isImplies f) case Pair(F,F) => first f2 = false$F or second f2 = true$F => true$F first f2 = true$F => second f2 second f2 = false$F => not first f2 f (f2 := isEquiv f) case Pair(F,F) => first f2 = true$F => second f2 second f2 = true$F => first f2 first f2 = false$F => not second f2 second f2 = false$F => not first f2 f f simplify f == (f1 := isNot f) case F => simplifyOneStep(not simplify f1) (f2 := isAnd f) case Pair(F,F) => simplifyOneStep(conjunction(simplify first f2, simplify second f2)) (f2 := isOr f) case Pair(F,F) => simplifyOneStep(disjunction(simplify first f2, simplify second f2)) (f2 := isImplies f) case Pair(F,F) => simplifyOneStep(implies(simplify first f2, simplify second f2)) (f2 := isEquiv f) case Pair(F,F) => simplifyOneStep(equiv(simplify first f2, simplify second f2)) f @ <>= )abbrev package PROPFUN2 PropositionalFormulaFunctions2 ++ Author: Gabriel Dos Reis ++ Date Created: April 03, 2010 ++ Date Last Modified: April 03, 2010 ++ Description: ++ This package collects binary functions operating on propositional ++ formulae. PropositionalFormulaFunctions2(S,T): Public == Private where S: SetCategory T: SetCategory Public == Type with map: (S -> T, PropositionalFormula S) -> PropositionalFormula T ++ \spad{map(f,x)} returns a propositional formula where ++ all atoms in \spad{x} have been replaced by the result ++ of applying the function \spad{f} to them. Private == add macro FS == PropositionalFormula S macro FT == PropositionalFormula T map(f,x) == x = true$FS => true$FT x = false$FS => false$FT (t := isAtom x) case S => f(t)::FT (f1 := isNot x) case FS => not map(f,f1) (f2 := isAnd x) case Pair(FS,FS) => conjunction(map(f,first f2), map(f,second f2)) (f2 := isOr x) case Pair(FS,FS) => disjunction(map(f,first f2), map(f,second f2)) (f2 := isImplies x) case Pair(FS,FS) => implies(map(f,first f2), map(f,second f2)) (f2 := isEquiv x) case Pair(FS,FS) => equiv(map(f,first f2), map(f,second f2)) error "invalid propositional formula" @ \section{domain REF Reference} <>= )abbrev domain REF Reference ++ Author: Stephen M. Watt ++ Date Created: ++ Date Last Changed: October 11, 2011 ++ Basic Operations: deref, ref, setref, = ++ Related Constructors: ++ Keywords: reference ++ Description: \spadtype{Reference} is for making a changeable instance ++ of something. Reference(S:Type): SetCategory with ref : S -> % ++ \spad{ref(s)} creates a reference to the object \spad{s}. deref : % -> S ++ \spad{deref(r)} returns the object referenced by \spad{r} setref: (%, S) -> S ++ setref(r,s) reset the reference \spad{r} to refer to \spad{s} = : (%, %) -> Boolean ++ \spad{a=b} tests if \spad{a} and \spad{b} are equal. == add Rep == Record(value: S) import %peq: (%,%) -> Boolean from Foreign Builtin p = q == %peq(p,q) ref v == per [v] deref p == rep(p).value setref(p, v) == rep(p).value := v coerce p == obj := S has CoercibleTo OutputForm => rep(p).value::OutputForm '?::OutputForm prefix('ref::OutputForm, [obj]) @ \section{category LOGIC Logic} <>= )abbrev category LOGIC Logic ++ Author: ++ Date Created: ++ Date Last Changed: May 27, 2009 ++ Basic Operations: ~, /\, \/ ++ Related Constructors: ++ Keywords: boolean ++ Description: ++ `Logic' provides the basic operations for lattices, ++ e.g., boolean algebra. Logic: Category == Type with ~: % -> % ++ ~(x) returns the logical complement of x. /\: (%, %) -> % ++ \spadignore { /\ }returns the logical `meet', e.g. `and'. \/: (%, %) -> % ++ \spadignore{ \/ } returns the logical `join', e.g. `or'. add x \/ y == ~(~x /\ ~y) @ \section{domain BOOLEAN Boolean} <>= )abbrev domain BOOLEAN Boolean ++ Author: Stephen M. Watt ++ Date Created: ++ Date Last Changed: May 27, 2009 ++ Basic Operations: true, false, not, and, or, xor, nand, nor, implies ++ Related Constructors: ++ Keywords: boolean ++ Description: \spadtype{Boolean} is the elementary logic with 2 values: ++ true and false Boolean(): Join(OrderedFinite, PropositionalLogic, ConvertibleTo InputForm) with xor : (%, %) -> % ++ xor(a,b) returns the logical exclusive {\em or} ++ of Boolean \spad{a} and b. nand : (%, %) -> % ++ nand(a,b) returns the logical negation of \spad{a} and b. nor : (%, %) -> % ++ nor(a,b) returns the logical negation of \spad{a} or b. == add import %false: % from Foreign Builtin import %true: % from Foreign Builtin import %peq: (%,%) -> Boolean from Foreign Builtin import %and: (%,%) -> % from Foreign Builtin import %or: (%,%) -> % from Foreign Builtin import %not: % -> % from Foreign Builtin true == %true false == %false not b == %not b ~b == %not b a and b == %and(a,b) a /\ b == %and(a,b) a or b == %or(a,b) a \/ b == %or(a,b) xor(a, b) == (a => %not b; b) nor(a, b) == (a => %false; %not b) nand(a, b) == (a => %not b; %true) a = b == %peq(a,b) implies(a, b) == (a => b; %true) equiv(a,b) == %peq(a, b) a < b == (b => %not a; %false) size() == 2 index i == even?(i::Integer) => %false %true lookup a == a => 1 2 random() == even?(random()$Integer) => %false %true convert(x:%):InputForm == x => 'true 'false coerce(x:%):OutputForm == x => 'true 'false @ \section{domain IBITS IndexedBits} <>= )abbrev domain IBITS IndexedBits ++ Author: Stephen Watt and Michael Monagan ++ Date Created: ++ July 86 ++ Change History: ++ Oct 87 ++ Basic Operations: range ++ Related Constructors: ++ Keywords: indexed bits ++ Description: \spadtype{IndexedBits} is a domain to compactly represent ++ large quantities of Boolean data. IndexedBits(mn:Integer): BitAggregate() == add import %2bool: NonNegativeInteger -> Boolean from Foreign Builtin import %2bit: Boolean -> NonNegativeInteger from Foreign Builtin import %bitveccopy: % -> % from Foreign Builtin import %bitveclength: % -> NonNegativeInteger from Foreign Builtin import %bitvecref: (%,Integer) -> NonNegativeInteger from Foreign Builtin import %bitveceq: (%,%) -> Boolean from Foreign Builtin import %bitveclt: (%,%) -> Boolean from Foreign Builtin import %bitvecnot: % -> % from Foreign Builtin import %bitvecand: (%,%) -> % from Foreign Builtin import %bitvecor: (%,%) -> % from Foreign Builtin import %bitvecxor: (%,%) -> % from Foreign Builtin import %bitvector: (NonNegativeInteger,NonNegativeInteger) -> % from Foreign Builtin minIndex u == mn -- range check index of `i' into `v'. range(v: %, i: Integer): Integer == i >= 0 and i < #v => i error "Index out of range" coerce(v):OutputForm == t:Character := char "1" f:Character := char "0" s := new(#v, space()$Character)$String for i in minIndex(s)..maxIndex(s) for j in mn.. repeat s.i := if v.j then t else f s::OutputForm new(n, b) == %bitvector(n, %2bit(b)$Foreign(Builtin)) empty() == %bitvector(0,0) copy v == %bitveccopy v #v == %bitveclength v v = u == %bitveceq(v,u) v < u == %bitveclt(v,u) u and v == (#v=#u => %bitvecand(v,u); map("and",v,u)) u or v == (#v=#u => %bitvecor(v,u); map("or", v,u)) xor(v,u) == (#v=#u => %bitvecxor(v,u); map("xor",v,u)) setelt(v:%, i:Integer, f:Boolean) == %2bool %store(%bitvecref(v,range(v,i-mn)),%2bit f)$Foreign(Builtin) elt(v:%, i:Integer) == %2bool %bitvecref(v,range(v,i-mn)) ~v == %bitvecnot v u /\ v == (#v=#u => %bitvecand(v,u); map("and",v,u)) u \/ v == (#v=#u => %bitvecor(v,u); map("or", v,u)) @ \section{domain BITS Bits} <>= )abbrev domain BITS Bits ++ Author: Stephen M. Watt ++ Date Created: ++ Change History: ++ Basic Operations: And, Not, Or ++ Related Constructors: ++ Keywords: bits ++ Description: \spadtype{Bits} provides logical functions for Indexed Bits. Bits(): Exports == Implementation where Exports == BitAggregate() with bits: (NonNegativeInteger, Boolean) -> % ++ bits(n,b) creates bits with n values of b Implementation == IndexedBits(1) add bits(n,b) == new(n,b) @ \section{Kleene's Three-Valued Logic} <>= )abbrev domain KTVLOGIC KleeneTrivalentLogic ++ Author: Gabriel Dos Reis ++ Date Created: September 20, 2008 ++ Date Last Modified: January 14, 2012 ++ Description: ++ This domain implements Kleene's 3-valued propositional logic. KleeneTrivalentLogic(): Public == Private where Public == Join(PropositionalLogic,Finite) with unknown: % ++ the indefinite `unknown' case: (%,[| false |]) -> Boolean ++ x case false holds if the value of `x' is `false' case: (%,[| unknown |]) -> Boolean ++ x case unknown holds if the value of `x' is `unknown' case: (%,[| true |]) -> Boolean ++ s case true holds if the value of `x' is `true'. Private == Maybe Boolean add false == per just(false@Boolean) unknown == per nothing true == per just(true@Boolean) x = y == rep x = rep y x case true == x = true@% x case false == x = false@% x case unknown == x = unknown not x == x case false => true x case unknown => unknown false x and y == x case false => false x case unknown => y case false => false unknown y x or y == x case false => y x case true => x y case true => y unknown implies(x,y) == x case false => true x case true => y y case true => true unknown equiv(x,y) == x case unknown => x x case true => y not y coerce(x: %): OutputForm == case rep x is y@Boolean => y::OutputForm otherwise => outputForm 'unknown size() == 3 index n == n > 3 => error "index: argument out of bound" n = 1 => false n = 2 => unknown true lookup x == x = false => 1 x = unknown => 2 3 @ \section{License} <>= --Copyright (c) 1991-2002, The Numerical Algorithms Group Ltd. --All rights reserved. --Copyright (C) 2007-2012, 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}