Difference between revisions of "User:Gwern/kenn"
(→Stage 2 – Move Negation Inwards: hm, the double slashes show up here as well) 
(updated tagsoup script) 

(3 intermediate revisions by the same user not shown)  
Line 1:  Line 1:  
−  <blockquote>Classical firstorder logic has the pleasant property that a formula can be reduced to an elegant ''implicative normal form'' through a series of syntactic simplifications. Using these transformations as a vehicle, this article demonstrates how to use 
+  <blockquote>Classical firstorder logic has the pleasant property that a formula can be reduced to an elegant ''implicative normal form'' through a series of syntactic simplifications. Using these transformations as a vehicle, this article demonstrates how to use Haskell�s type system, specifically a variation on Swierstra�s �Data Types à la Carte� method, to enforce the structural correctness property that these constructors are, in fact, eliminated by each stage of the transformation. 
</blockquote> 
</blockquote> 

−  <blockquote>< 
+  <blockquote><haskell class="haskell">{# LANGUAGE RankNTypes,TypeOperators,PatternSignatures #} 
{# LANGUAGE UndecidableInstances,IncoherentInstances #} 
{# LANGUAGE UndecidableInstances,IncoherentInstances #} 

{# LANGUAGE MultiParamTypeClasses,TypeSynonymInstances #} 
{# LANGUAGE MultiParamTypeClasses,TypeSynonymInstances #} 

Line 8:  Line 8:  
import Text.PrettyPrint.HughesPJ 
import Text.PrettyPrint.HughesPJ 

import Control.Monad.State 
import Control.Monad.State 

−  import Prelude hiding (or,and,not)</ 
+  import Prelude hiding (or,and,not)</haskell> 
{LANGUAGE pragma and module imports} 
{LANGUAGE pragma and module imports} 

</blockquote> 
</blockquote> 

== FirstOrder Logic == 
== FirstOrder Logic == 

−  Consider the optimistic statement 
+  Consider the optimistic statement �Every person has a heart.� If we were asked to write this formally in a logic or philosophy class, we might write the following formula of classical firstorder logic: 
<math>\forall p.\, Person(p) \Rightarrow \exists h.\, Heart(h) \wedge Has(p,h) \\ </math> 
<math>\forall p.\, Person(p) \Rightarrow \exists h.\, Heart(h) \wedge Has(p,h) \\ </math> 

Line 19:  Line 19:  
If asked to write the same property for testing by QuickCheck {quickcheck}, we might instead produce this code: 
If asked to write the same property for testing by QuickCheck {quickcheck}, we might instead produce this code: 

−  < 
+  <haskell class="haskell"> heartFact :: Person > Bool 
heartFact p = has p (heart p) 
heartFact p = has p (heart p) 

−  where heart :: Person  
+  where heart :: Person > Heart 
−  ...</ 
+  ...</haskell> 
These look rather different. Ignoring how some of the predicates moved into our types, there are two other transformations involved. First, the universally quantified <math>p</math> has been made a parameter, essentially making it a free variable of the formula. Second, the existentially quantified <math>h</math> has been replaced by a function heart that, for any person, returns their heart. How did we know to encode things this way? We have performed firstorder quantifier elimination in our heads! 
These look rather different. Ignoring how some of the predicates moved into our types, there are two other transformations involved. First, the universally quantified <math>p</math> has been made a parameter, essentially making it a free variable of the formula. Second, the existentially quantified <math>h</math> has been replaced by a function heart that, for any person, returns their heart. How did we know to encode things this way? We have performed firstorder quantifier elimination in our heads! 

−  This idea has an elegant instantiation for classical firstorder logic which (along with some other simple transformations) yields a useful normal form for any formula. This article is a tour of the normalization process, implemented in Haskell, using a number of Haskell programming tricks. We will begin with just a couple of formal definitions, but quickly move on to 
+  This idea has an elegant instantiation for classical firstorder logic which (along with some other simple transformations) yields a useful normal form for any formula. This article is a tour of the normalization process, implemented in Haskell, using a number of Haskell programming tricks. We will begin with just a couple of formal definitions, but quickly move on to �all code, all the time.� 
First, we need the primitive set of terms <math>t</math>, which are either variables <math>x</math> or function symbols <math>f</math> applied to a list of terms (constants are functions of zero arguments). <math>t ::= x ~\mid~ f(t_1, \cdots, t_n) 
First, we need the primitive set of terms <math>t</math>, which are either variables <math>x</math> or function symbols <math>f</math> applied to a list of terms (constants are functions of zero arguments). <math>t ::= x ~\mid~ f(t_1, \cdots, t_n) 

Line 40:  Line 40:  
<math>implicative~normal~f\!orm ~ ::= ~ \bigwedge \left[\bigwedge t^* \Rightarrow \bigvee t ^*\right]^* </math> 
<math>implicative~normal~f\!orm ~ ::= ~ \bigwedge \left[\bigwedge t^* \Rightarrow \bigvee t ^*\right]^* </math> 

−  The normal form may be very large compared to the input formula, but it is convenient for many purposes, such as using 
+  The normal form may be very large compared to the input formula, but it is convenient for many purposes, such as using Prolog�s resolution procedure or an SMT (Satisfiability Modulo Theories) solver. The following process for normalizing a formula is described by Russell and Norvig {Russell2003} in seven steps: 
# Eliminate implications. 
# Eliminate implications. 

Line 55:  Line 55:  
Experienced Haskellers may translate the above definitions into the following Haskell data types immediately upon reading them: 
Experienced Haskellers may translate the above definitions into the following Haskell data types immediately upon reading them: 

−  < 
+  <haskell class="haskell">data Term = Const String [Term] 
−   Var String</ 
+   Var String</haskell> 
We will reuse the constructor names from FOL later, though, so this is not part of the code for the demonstration. 
We will reuse the constructor names from FOL later, though, so this is not part of the code for the demonstration. 

−  < 
+  <haskell class="haskell">data FOL = Impl FOL FOL 
 Atom String [Term]  Not FOL 
 Atom String [Term]  Not FOL 

 TT  FF 
 TT  FF 

 Or FOL FOL  And FOL FOL 
 Or FOL FOL  And FOL FOL 

−   Exists String FOL  Forall String FOL</ 
+   Exists String FOL  Forall String FOL</haskell> 
−  To make things more interesting, let us work with the formula representing the statement 
+  To make things more interesting, let us work with the formula representing the statement �If there is a person that eats every food, then there is no food that noone eats.� 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

Line 72:  Line 72:  
</math> 
</math> 

−  < 
+  <haskell class="haskell">foodFact = 
(Impl 
(Impl 

−  (Exists 
+  (Exists "p" 
−  (And (Atom 
+  (And (Atom "Person" [Var "p"]) 
−  (Forall 
+  (Forall "f" 
−  (Impl (Atom 
+  (Impl (Atom "Food" [Var "f"]) 
−  (Atom 
+  (Atom "Eats" [Var "p", Var "f"]))))) 
−  (Not (Exists 
+  (Not (Exists "f" 
−  (And (Atom 
+  (And (Atom "Food" [Var "f"]) 
−  (Not (Exists 
+  (Not (Exists "p" 
−  (And (Atom 
+  (And (Atom "Person" [Var "p"]) 
−  (Atom 
+  (Atom "Eats" [Var "f"]))))))))</haskell> 
=== HigherOrder Abstract Syntax === 
=== HigherOrder Abstract Syntax === 

−  While the above encoding is natural to write down, it has drawbacks for actual work. The first thing to notice is that we are using the String type to represent variables, and may have to carefully manage scoping. But what do variables range over? Terms. And Haskell already has variables that range over the data type Term, so we can reuse 
+  While the above encoding is natural to write down, it has drawbacks for actual work. The first thing to notice is that we are using the String type to represent variables, and may have to carefully manage scoping. But what do variables range over? Terms. And Haskell already has variables that range over the data type Term, so we can reuse Haskell�s implementation; this technique is known as higherorder abstract syntax (HOAS). 
−  < 
+  <haskell class="haskell">data FOL = Impl FOL FOL 
 Atom String [Term]  Not FOL 
 Atom String [Term]  Not FOL 

 TT  FF 
 TT  FF 

 Or FOL FOL  And FOL FOL 
 Or FOL FOL  And FOL FOL 

−   Exists (Term  
+   Exists (Term > FOL)  Forall (Term > FOL)</haskell> 
−  In a HOAS encoding, the binder of the object language (the quantifiers of firstorder logic) are implemented using the binders of the metalanguage (Haskell). For example, where in the previous encoding we would represent <math>\exists x.\, P(x)</math> as Exists 
+  In a HOAS encoding, the binder of the object language (the quantifiers of firstorder logic) are implemented using the binders of the metalanguage (Haskell). For example, where in the previous encoding we would represent <math>\exists x.\, P(x)</math> as Exists "x" (Const "P" [Var "x"]) we now represent it with Exists ( > (Const "P" [x])). And our example becomes: 
−  < 
+  <haskell class="haskell">foodFact = 
(Impl 
(Impl 

−  (Exists $ \p  
+  (Exists $ \p > 
−  (And (Atom 
+  (And (Atom "Person" [p]) 
−  (Forall $ \f  
+  (Forall $ \f > 
−  (Impl (Atom 
+  (Impl (Atom "Food" [f]) 
−  (Atom 
+  (Atom "Eats" [p, f]))))) 
−  (Not (Exists $ \f  
+  (Not (Exists $ \f > 
−  (And (Atom 
+  (And (Atom "Food" [f]) 
−  (Not (Exists $ \p  
+  (Not (Exists $ \p > 
−  (And (Atom 
+  (And (Atom "Person" [p]) 
−  (Atom 
+  (Atom "Eats" [f]))))))))</haskell> 
−  Since the variables p and f have taken the place of the String variable names, 
+  Since the variables p and f have taken the place of the String variable names, Haskell�s binding structure now ensures that we cannot construct a firstorder logic formula with unbound variables, unless we use the Var constructor, which is still present because we will need it later. Another important benefit is that the type now expresses that the variables range over the Term data type, while before it was up to us to properly interpret the String variable names. 
<blockquote>Modify the code of this article so that the Var constructor is not introduced until it is required in stage 5. 
<blockquote>Modify the code of this article so that the Var constructor is not introduced until it is required in stage 5. 

Line 113:  Line 113:  
=== Data Types à la Carte === 
=== Data Types à la Carte === 

−  But even using this improved encoding, all our transformations will be of type FOL  
+  But even using this improved encoding, all our transformations will be of type FOL > FOL. Because this type does not express the structure of the computation very precisely, we must rely on human inspection to ensure that each stage is written correctly. More importantly, we are not making manifest the requirement of certain stages that the prior stages� transformations have been performed. For example, our elimination of universal quantification is only a correct transformation when existentials have already been eliminated. A good goal for refining our type structure is to describe our data with types that reflect which connectives may be present. 
−  Swierstra proposes a technique {dtalc} whereby a variant data type is built up by mixing and matching constructors of different functors using their ''coproduct'' (:+:), which is the 
+  Swierstra proposes a technique {dtalc} whereby a variant data type is built up by mixing and matching constructors of different functors using their ''coproduct'' (:+:), which is the �smallest� functor containing both of its arguments. 
−  < 
+  <haskell class="haskell">data (f :+: g) a = Inl (f a)  Inr (g a) 
infixr 6 :+: 
infixr 6 :+: 

−  instance (Functor f, Functor g) = 
+  instance (Functor f, Functor g) => Functor (f :+: g) where 
fmap f (Inl x) = Inl (fmap f x) 
fmap f (Inl x) = Inl (fmap f x) 

−  fmap f (Inr x) = Inr (fmap f x)</ 
+  fmap f (Inr x) = Inr (fmap f x)</haskell> 
−  The :+: constructor is like Either but it operates on functors. This difference is crucial 
+  The :+: constructor is like Either but it operates on functors. This difference is crucial � if f and g represent two constructors that we wish to combine into a larger ''recursive'' data type, then the type parameter a represents the type of their subformulae. 
−  To work conveniently with coproducts, we define a type class : 
+  To work conveniently with coproducts, we define a type class :<: that implements subtyping by explicitly providing an injection from one of the constructors to the larger coproduct data type. There are some technical aspects to making sure current Haskell implementations can figure out the needed instances of :<:, but in this example we need only Swierstra�s original subsumption instances, found in Figure fig:Subsumption. For your own use of the technique, discussion on Phil Wadler�s blog {wadlerdtalc} and the HaskellCafe mailing list {haskellcafedtalc} may be helpful. 
−  <blockquote>< 
+  <blockquote><haskell class="haskell">class (Functor sub, Functor sup) => sub :<: sup where 
−  inj :: sub a  
+  inj :: sub a > sup a 
−  instance Functor f = 
+  instance Functor f => (:<:) f f where 
inj = id 
inj = id 

−  instance (Functor f, Functor g) = 
+  instance (Functor f, Functor g) => (:<:) f (f :+: g) where 
inj = Inl 
inj = Inl 

−  instance (Functor f, Functor g, Functor h, (f : 
+  instance (Functor f, Functor g, Functor h, (f :<: g)) 
−  = 
+  => (:<:) f (h :+: g) where 
−  inj = Inr . inj</ 
+  inj = Inr . inj</haskell> 
{Subsumption instances} (fig:Subsumption) 
{Subsumption instances} (fig:Subsumption) 

</blockquote> 
</blockquote> 

If the above seems a bit abstract or confusing, it will become very clear when we put it into practice. Let us immediately do so by encoding the constructors of firstorder logic in this modular fashion. 
If the above seems a bit abstract or confusing, it will become very clear when we put it into practice. Let us immediately do so by encoding the constructors of firstorder logic in this modular fashion. 

−  < 
+  <haskell class="haskell">data TT a = TT 
data FF a = FF 
data FF a = FF 

data Atom a = Atom String [Term] 
data Atom a = Atom String [Term] 

Line 150:  Line 150:  
data And a = And a a 
data And a = And a a 

data Impl a = Impl a a 
data Impl a = Impl a a 

−  data Exists a = Exists (Term  
+  data Exists a = Exists (Term > a) 
−  data Forall a = Forall (Term  
+  data Forall a = Forall (Term > a)</haskell> 
Each constructor is parameterized by a type a of subformulae; TT, FF, and Atom do not have any subformulae so they ignore their parameter. Logical operations such as And have two subformulae. Correspondingly, the And constructor takes two arguments of type a. 
Each constructor is parameterized by a type a of subformulae; TT, FF, and Atom do not have any subformulae so they ignore their parameter. Logical operations such as And have two subformulae. Correspondingly, the And constructor takes two arguments of type a. 

The compound functor Input is now the specification of which constructors may appear in a firstorder logic formula. 
The compound functor Input is now the specification of which constructors may appear in a firstorder logic formula. 

−  < 
+  <haskell class="haskell">type Input = TT :+: FF :+: Atom 
:+: Not :+: Or :+: And :+: Impl 
:+: Not :+: Or :+: And :+: Impl 

−  :+: Exists :+: Forall</ 
+  :+: Exists :+: Forall</haskell> 
−  The final step is to 
+  The final step is to �tie the knot� with the following Formula data type, which generates a recursive formula over whatever constructors are present in its functor argument f. 
−  < 
+  <haskell class="haskell">data Formula f = In { out :: f (Formula f) }</haskell> 
If you have not seen this trick before, that definition may be hard to read and understand. Consider the types of In and out. 
If you have not seen this trick before, that definition may be hard to read and understand. Consider the types of In and out. 

−  < 
+  <haskell class="haskell">In :: f (Formula f) > Formula f 
−  out :: Formula f  
+  out :: Formula f > f (Formula f)</haskell> 
−  Observe that <math>In . out == out . In == id</math>. This pair of inverses allows us to 
+  Observe that <math>In . out == out . In == id</math>. This pair of inverses allows us to �roll� and �unroll� one layer of a formula in order to operate on the outermost constructor. Haskell does this same thing when you patternmatch against �normal� recursive data types. Like Haskell, we want to hide this rolling and unrolling. To hide the rolling, we define some helper constructors, found in Figure fig:FOLboilerplate, that inject a constructor into an arbitrary supertype, and then apply In to yield a Formula. 
−  To hide the unrolling, we use the fact that a fixpoint of a functor comes with a fold operation, or ''catamorphism'', which we will use to traverse a 
+  To hide the unrolling, we use the fact that a fixpoint of a functor comes with a fold operation, or ''catamorphism'', which we will use to traverse a formula�s syntax. The function foldFormula takes as a parameter an ''algebra'' of the functor f. Intuitively, algebra tells us how to fold �one layer� of a formula, assuming all subformulae have already been processed. The fixpoint then provides the recursive structure of the computation once and for all. 
−  < 
+  <haskell class="haskell">foldFormula :: Functor f => (f a > a) > Formula f > a 
−  foldFormula algebra = algebra . fmap (foldFormula algebra) . out</ 
+  foldFormula algebra = algebra . fmap (foldFormula algebra) . out</haskell> 
−  We are already reaping some of the benefit of our 
+  We are already reaping some of the benefit of our �à la carte� technique: The boilerplate Functor instances in Figure fig:FOLboilerplate are not much larger than the code of foldFormula would have been, and they are defined modularly! Unlike a monolithic foldFormula implementation, this one function will work no matter which constructors are present. If the definition of foldFormula is unfamiliar, it is worth imagining a Formula f flowing through the three stages: First, out unrolls the formula one layer, then fmap recursively folds over all the subformulae. Finally, the results of the recursion are combined by algebra. 
−  <blockquote>< 
+  <blockquote><haskell class="haskell">instance Functor TT where fmap _ _ = TT 
instance Functor FF where fmap _ _ = FF 
instance Functor FF where fmap _ _ = FF 

instance Functor Atom where fmap _ (Atom p ts) = Atom p ts 
instance Functor Atom where fmap _ (Atom p ts) = Atom p ts 

Line 184:  Line 184:  
instance Functor Exists where fmap f (Exists phi) = Exists (f . phi) 
instance Functor Exists where fmap f (Exists phi) = Exists (f . phi) 

−  inject :: (g : 
+  inject :: (g :<: f) => g (Formula f) > Formula f 
inject = In . inj 
inject = In . inj 

−  tt :: (TT : 
+  tt :: (TT :<: f) => Formula f 
tt = inject TT 
tt = inject TT 

−  ff :: (FF : 
+  ff :: (FF :<: f) => Formula f 
ff = inject FF 
ff = inject FF 

−  atom :: (Atom : 
+  atom :: (Atom :<: f) => String > [Term] > Formula f 
atom p ts = inject (Atom p ts) 
atom p ts = inject (Atom p ts) 

−  not :: (Not : 
+  not :: (Not :<: f) => Formula f > Formula f 
not = inject . Not 
not = inject . Not 

−  or :: (Or : 
+  or :: (Or :<: f) => Formula f > Formula f > Formula f 
or phi1 phi2 = inject (Or phi1 phi2) 
or phi1 phi2 = inject (Or phi1 phi2) 

−  and :: (And : 
+  and :: (And :<: f) => Formula f > Formula f > Formula f 
and phi1 phi2 = inject (And phi1 phi2) 
and phi1 phi2 = inject (And phi1 phi2) 

−  impl :: (Impl : 
+  impl :: (Impl :<: f) => Formula f > Formula f > Formula f 
impl phi1 phi2 = inject (Impl phi1 phi2) 
impl phi1 phi2 = inject (Impl phi1 phi2) 

−  forall :: (Forall : 
+  forall :: (Forall :<: f) => (Term > Formula f) > Formula f 
forall = inject . Forall 
forall = inject . Forall 

−  exists :: (Exists : 
+  exists :: (Exists :<: f) => (Term > Formula f) > Formula f 
−  exists = inject . Exists</ 
+  exists = inject . Exists</haskell> 
{Boilerplate for FirstOrder Logic Constructors} (fig:FOLboilerplate) 
{Boilerplate for FirstOrder Logic Constructors} (fig:FOLboilerplate) 

</blockquote> 
</blockquote> 

Here is what our running example looks like with this encoding: 
Here is what our running example looks like with this encoding: 

−  < 
+  <haskell class="haskell">foodFact :: Formula Input 
−  foodFact = (exists $ \p  
+  foodFact = (exists $ \p > atom "Person" [p] 
−  `and` (forall $ \f  
+  `and` (forall $ \f > atom "Food" [f] 
−  `impl` atom 
+  `impl` atom "Eats" [p,f])) 
`impl` 
`impl` 

−  (not (exists $ \f  
+  (not (exists $ \f > atom "Food" [f] 
−  `and` (not (exists $ \p  
+  `and` (not (exists $ \p > atom "Person" [p] 
−  `and` atom 
+  `and` atom "Eats" [p,f]))))</haskell> 
−  A TeX prettyprinter is included as an appendix to this article. To make things readable, though, 
+  A TeX prettyprinter is included as an appendix to this article. To make things readable, though, I�ll doctor its output into a nice table, and remove extraneous parentheses. But I won�t rewrite the variable names, since variables and binding are a key aspect of managing formulae. By convention, the printer uses <math>c</math> for existentially quantified variables and <math>x</math> for universally quantified variables. 
−  < 
+  <haskell>*Main> texprint foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& (\exists c_1.\, Person(c_1) \wedge \forall x_2.\, Food(x_2) \Rightarrow Eats(c_1, x_2)) \\ 
& (\exists c_1.\, Person(c_1) \wedge \forall x_2.\, Food(x_2) \Rightarrow Eats(c_1, x_2)) \\ 

Line 234:  Line 234:  
</math> 
</math> 

−  == Stage 1 
+  == Stage 1 � Eliminate Implications == 
−  The first transformation is an easy one, in which we 
+  The first transformation is an easy one, in which we �desugar� <math>\phi_1 \Rightarrow \phi_2</math> into <math>\neg \phi_1 \vee \phi_2</math>. The highlevel overview is given by the type and body of elimImp. 
−  < 
+  <haskell class="haskell">type Stage1 = TT :+: FF :+: Atom :+: Not :+: Or :+: And :+: Exists :+: Forall 
−  elimImp :: Formula Input  
+  elimImp :: Formula Input > Formula Stage1 
−  elimImp = foldFormula elimImpAlg</ 
+  elimImp = foldFormula elimImpAlg</haskell> 
We take a formula containing all the constructors of firstorder logic, and return a formula built without use of Impl. The way that elimImp does this is by folding the algebras elimImpAlg for each constructor over the recursive structure of a formula. 
We take a formula containing all the constructors of firstorder logic, and return a formula built without use of Impl. The way that elimImp does this is by folding the algebras elimImpAlg for each constructor over the recursive structure of a formula. 

−  The function elimImpAlg we provide by making each constructor an instance of the ElimImp type class. This class specifies for a given constructor how to eliminate implications 
+  The function elimImpAlg we provide by making each constructor an instance of the ElimImp type class. This class specifies for a given constructor how to eliminate implications � for most constructors this is just the identity function, though we must rebuild an identical term to alter its type. Perhaps there is a way to use generic programming to eliminate the uninteresting cases. 
−  < 
+  <haskell class="haskell">class Functor f => ElimImp f where 
−  elimImpAlg :: f (Formula Stage1)  
+  elimImpAlg :: f (Formula Stage1) > Formula Stage1 
instance ElimImp Impl where elimImpAlg (Impl phi1 phi2) = (not phi1) `or` phi2 
instance ElimImp Impl where elimImpAlg (Impl phi1 phi2) = (not phi1) `or` phi2 

Line 258:  Line 258:  
instance ElimImp And where elimImpAlg (And phi1 phi2) = phi1 `and` phi2 
instance ElimImp And where elimImpAlg (And phi1 phi2) = phi1 `and` phi2 

instance ElimImp Exists where elimImpAlg (Exists phi) = exists phi 
instance ElimImp Exists where elimImpAlg (Exists phi) = exists phi 

−  instance ElimImp Forall where elimImpAlg (Forall phi) = forall phi</ 
+  instance ElimImp Forall where elimImpAlg (Forall phi) = forall phi</haskell> 
We extend ElimImp in the natural way over coproducts, so that whenever all our constructors are members of the type class, then their coproduct is as well. 
We extend ElimImp in the natural way over coproducts, so that whenever all our constructors are members of the type class, then their coproduct is as well. 

−  < 
+  <haskell class="haskell">instance (ElimImp f, ElimImp g) => ElimImp (f :+: g) where 
elimImpAlg (Inr phi) = elimImpAlg phi 
elimImpAlg (Inr phi) = elimImpAlg phi 

−  elimImpAlg (Inl phi) = elimImpAlg phi</ 
+  elimImpAlg (Inl phi) = elimImpAlg phi</haskell> 
Our running example is now 
Our running example is now 

−  < 
+  <haskell>*Main> texprint . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& \neg (\exists c_1.\, Person(c_1) \wedge \forall x_2.\, \neg Food(x_2) \vee Eats(c_1, x_2)) \\ 
& \neg (\exists c_1.\, Person(c_1) \wedge \forall x_2.\, \neg Food(x_2) \vee Eats(c_1, x_2)) \\ 

Line 275:  Line 275:  
<blockquote>Design a solution where only the Impl case of elimImpAlg needs to be written. 
<blockquote>Design a solution where only the Impl case of elimImpAlg needs to be written. 

</blockquote> 
</blockquote> 

−  == Stage 2 
+  == Stage 2 � Move Negation Inwards == 
−  Now that implications are gone, we are left with entirely symmetrical constructions, and can always push negations in or out using duality: <math>\neg(\neg \phi) \Leftrightarrow \phi 
+  Now that implications are gone, we are left with entirely symmetrical constructions, and can always push negations in or out using duality: <math>\neg(\neg \phi) \Leftrightarrow \phi \\ 
−  \neg(\phi_1 \wedge \phi_2) \Leftrightarrow \neg\phi_1 \vee \neg\phi_2 
+  \neg(\phi_1 \wedge \phi_2) \Leftrightarrow \neg\phi_1 \vee \neg\phi_2 \\ 
−  \neg(\phi_1 \vee \phi_2) \Leftrightarrow \neg\phi_1 \wedge \neg\phi_2 
+  \neg(\phi_1 \vee \phi_2) \Leftrightarrow \neg\phi_1 \wedge \neg\phi_2 \\ 
−  \neg(\exists x.\, \phi) \Leftrightarrow \forall x.\, \neg\phi 
+  \neg(\exists x.\, \phi) \Leftrightarrow \forall x.\, \neg\phi \\ 
\neg(\forall x.\, \phi) \Leftrightarrow \exists x.\, \neg\phi 
\neg(\forall x.\, \phi) \Leftrightarrow \exists x.\, \neg\phi 

</math> 
</math> 

Line 286:  Line 286:  
Our eventual goal is to move negation all the way inward so it is only applied to atomic predicates. To express this structure in our types, we define a new constructor for negated atomic predicates as well as the type for the output of Stage 2: 
Our eventual goal is to move negation all the way inward so it is only applied to atomic predicates. To express this structure in our types, we define a new constructor for negated atomic predicates as well as the type for the output of Stage 2: 

−  < 
+  <haskell class="haskell">data NAtom a = NAtom String [Term] 
instance Functor NAtom where fmap f (NAtom p ts) = NAtom p ts 
instance Functor NAtom where fmap f (NAtom p ts) = NAtom p ts 

−  natom :: (NAtom : 
+  natom :: (NAtom :<: f) => String > [Term] > Formula f 
natom p ts = inject (NAtom p ts) 
natom p ts = inject (NAtom p ts) 

Line 296:  Line 296:  
:+: NAtom 
:+: NAtom 

:+: Or :+: And 
:+: Or :+: And 

−  :+: Exists :+: Forall</ 
+  :+: Exists :+: Forall</haskell> 
One could imagine implementing duality with a multiparameter type class that records exactly the dual of each constructor, as 
One could imagine implementing duality with a multiparameter type class that records exactly the dual of each constructor, as 

−  < 
+  <haskell class="haskell">class (Functor f, Functor g) => Dual f g where 
−  dual :: f a  
+  dual :: f a > g a</haskell> 
Unfortunately, this leads to a situation where our subtyping must use the commutativity of coproducts, which it is incapable of doing as written. For this article, let us just define an algebra to dualize a whole formula at a time. 
Unfortunately, this leads to a situation where our subtyping must use the commutativity of coproducts, which it is incapable of doing as written. For this article, let us just define an algebra to dualize a whole formula at a time. 

−  < 
+  <haskell class="haskell">dualize :: Formula Stage2 > Formula Stage2 
dualize = foldFormula dualAlg 
dualize = foldFormula dualAlg 

−  class Functor f = 
+  class Functor f => Dualize f where 
−  dualAlg :: f (Formula Stage2)  
+  dualAlg :: f (Formula Stage2) > Formula Stage2 
instance Dualize TT where dualAlg TT = ff 
instance Dualize TT where dualAlg TT = ff 

Line 318:  Line 318:  
instance Dualize Forall where dualAlg (Forall phi) = exists phi 
instance Dualize Forall where dualAlg (Forall phi) = exists phi 

−  instance (Dualize f, Dualize g) = 
+  instance (Dualize f, Dualize g) => Dualize (f :+: g) where 
dualAlg (Inl phi) = dualAlg phi 
dualAlg (Inl phi) = dualAlg phi 

−  dualAlg (Inr phi) = dualAlg phi</ 
+  dualAlg (Inr phi) = dualAlg phi</haskell> 
−  Now perhaps the pattern of these transformations is becoming clear. It is remarkably painless, involving just a little type class syntax as overhead, to write these functor algebras. The definition of pushNotInwards is another straightforward fold, with a helper type class PushNot. 
+  Now perhaps the pattern of these transformations is becoming clear. It is remarkably painless, involving just a little type class syntax as overhead, to write these functor algebras. The definition of pushNotInwards is another straightforward fold, with a helper type class PushNot. I�ve separated the instance for Not since it is the only one that does anything. 
−  < 
+  <haskell class="haskell">class Functor f => PushNot f where 
−  pushNotAlg :: f (Formula Stage2)  
+  pushNotAlg :: f (Formula Stage2) > Formula Stage2 
instance PushNot Not where pushNotAlg (Not phi) = dualize phi 
instance PushNot Not where pushNotAlg (Not phi) = dualize phi 

Line 336:  Line 336:  
instance PushNot Forall where pushNotAlg (Forall phi) = forall phi 
instance PushNot Forall where pushNotAlg (Forall phi) = forall phi 

−  instance (PushNot f, PushNot g) = 
+  instance (PushNot f, PushNot g) => PushNot (f :+: g) where 
pushNotAlg (Inr phi) = pushNotAlg phi 
pushNotAlg (Inr phi) = pushNotAlg phi 

−  pushNotAlg (Inl phi) = pushNotAlg phi</ 
+  pushNotAlg (Inl phi) = pushNotAlg phi</haskell> 
All we have to do now is define a fold that calls pushNotAlg. 
All we have to do now is define a fold that calls pushNotAlg. 

−  < 
+  <haskell class="haskell">pushNotInwards :: Formula Stage1 > Formula Stage2 
−  pushNotInwards = foldFormula pushNotAlg</ 
+  pushNotInwards = foldFormula pushNotAlg</haskell> 
Our running example now becomes: 
Our running example now becomes: 

−  < 
+  <haskell>*Main> texprint . pushNotInwards . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& (\forall x_1.\, \neg Person(x_1) \vee \exists c_2.\, Food(c_2) \wedge \neg Eats(x_1, c_2)) \\ 
& (\forall x_1.\, \neg Person(x_1) \vee \exists c_2.\, Food(c_2) \wedge \neg Eats(x_1, c_2)) \\ 

Line 356:  Line 356:  
<blockquote>Encode a form of subtyping that can reason using commutativity of coproducts, and rewrite the Dualize algebra using a twoparameter Dual type class as described above. 
<blockquote>Encode a form of subtyping that can reason using commutativity of coproducts, and rewrite the Dualize algebra using a twoparameter Dual type class as described above. 

</blockquote> 
</blockquote> 

⚫  
⚫  
⚫  
⚫  
⚫  
⚫  
⚫  It is interesting to arrive at the definition of Skolemization via the CurryHoward correspondence. You may be familiar with the idea that terms of type a > b are proofs of the proposition �<math>a</math> implies <math>b</math>�, assuming a and b are interpreted as propositions as well. This rests on a notion that a proof of a > b must be some process that can take a proof of a and generate a proof of b, a very computational notion. Rephrasing the above, a function of type a > b is a guarantee that ''for all'' elements of type a, ''there exists'' a corresponding element of type b. So a function type expresses an alternation of a universal quantifier with an existential. We will use this to replace all the existential quantifiers with freshlygenerated functions. We can of course, pass a unit type to a function, or a tuple of many arguments, to have as many universal quantifiers as we like. 

−  
⚫  It is interesting to arrive at the definition of Skolemization via the CurryHoward correspondence. You may be familiar with the idea that terms of type a  

Suppose we have <math>\forall x.\, \forall y.\, \exists z.\, P(x,y,z)</math>, then in general there may be many choices for <math>z</math>, given a particular <math>x</math> and <math>y</math>. By the axiom of choice, we can create a function <math>f</math> that associates each <math>\langle x,y \rangle</math> pair with a corresponding <math>z</math> arbitrarily, and then rewrite the above formula as <math>\forall x.\, 
Suppose we have <math>\forall x.\, \forall y.\, \exists z.\, P(x,y,z)</math>, then in general there may be many choices for <math>z</math>, given a particular <math>x</math> and <math>y</math>. By the axiom of choice, we can create a function <math>f</math> that associates each <math>\langle x,y \rangle</math> pair with a corresponding <math>z</math> arbitrarily, and then rewrite the above formula as <math>\forall x.\, 

−  P(x, y, f(x,y))</math>. Technically, this formula is only equisatisfiable, but by convention 
+  P(x, y, f(x,y))</math>. Technically, this formula is only equisatisfiable, but by convention I�m assuming constants to be existentially quantified. 
So we need to traverse the syntax tree gathering free variables and replacing existentially quantified variables with functions of a fresh name. Since we are eliminating a binding construct, we now need to reason about fresh unique names. 
So we need to traverse the syntax tree gathering free variables and replacing existentially quantified variables with functions of a fresh name. Since we are eliminating a binding construct, we now need to reason about fresh unique names. 

−  +  Today�s formulas are small, so let us use a naïve and wasteful splittable unique identifier supply. Our supply is an infinite binary tree, where moving left prepends a 0 to the bit representation of the current counter, while moving right prepends a 1. Hence, the left and right subtrees are both infinite, nonoverlapping supplies of identifiers. The code for our unique identifier supplies is in Figure fig:unq. 

Launchbury and PeytonJones {launchbury95state} have discussed how to use the ST monad to implement a much more sophisticated and spaceefficient version of the same idea. 
Launchbury and PeytonJones {launchbury95state} have discussed how to use the ST monad to implement a much more sophisticated and spaceefficient version of the same idea. 

<blockquote> 
<blockquote> 

−  < 
+  <haskell class="haskell">type Unique = Int 
data UniqueSupply = UniqueSupply Unique UniqueSupply UniqueSupply 
data UniqueSupply = UniqueSupply Unique UniqueSupply UniqueSupply 

Line 383:  Line 382:  
(genSupply (2*n+1)) 
(genSupply (2*n+1)) 

−  splitUniqueSupply :: UniqueSupply  
+  splitUniqueSupply :: UniqueSupply > (UniqueSupply, UniqueSupply) 
splitUniqueSupply (UniqueSupply _ l r) = (l,r) 
splitUniqueSupply (UniqueSupply _ l r) = (l,r) 

−  getUnique :: UniqueSupply  
+  getUnique :: UniqueSupply > (Unique, UniqueSupply) 
getUnique (UniqueSupply n l r) = (n,l) 
getUnique (UniqueSupply n l r) = (n,l) 

Line 392:  Line 391:  
fresh :: Supply Int 
fresh :: Supply Int 

−  fresh = do supply 
+  fresh = do supply < get 
let (uniq,rest) = getUnique supply 
let (uniq,rest) = getUnique supply 

put rest 
put rest 

Line 398:  Line 397:  
freshes :: Supply UniqueSupply 
freshes :: Supply UniqueSupply 

−  freshes = do supply 
+  freshes = do supply < get 
let (l,r) = splitUniqueSupply supply 
let (l,r) = splitUniqueSupply supply 

put r 
put r 

−  return l</ 
+  return l</haskell> 
{Unique supplies} (fig:unq) 
{Unique supplies} (fig:unq) 

</blockquote> 
</blockquote> 

−  The helper algebra for Skolemization is more complex than before because a Formula Stage2 is not directly transformed into Formula Stage4 but into a function from its free variables to a new formula. On top of that, the final computation takes place in the Supply monad because of the need to generate fresh names for Skolem functions. Correspondingly, we choose the return type of the algebra to be [Term]  
+  The helper algebra for Skolemization is more complex than before because a Formula Stage2 is not directly transformed into Formula Stage4 but into a function from its free variables to a new formula. On top of that, the final computation takes place in the Supply monad because of the need to generate fresh names for Skolem functions. Correspondingly, we choose the return type of the algebra to be [Term] > Supply (Formula Stage4). Thankfully, many instances are just boilerplate. 
−  < 
+  <haskell class="haskell">type Stage4 = TT :+: FF :+: Atom :+: NAtom :+: Or :+: And :+: Forall 
−  class Functor f = 
+  class Functor f => Skolem f where 
−  skolemAlg :: f ([Term]  
+  skolemAlg :: f ([Term] > Supply (Formula Stage4)) 
−   
+  > [Term] > Supply (Formula Stage4) 
instance Skolem TT where 
instance Skolem TT where 

Line 425:  Line 424:  
skolemAlg (And phi1 phi2) xs = liftM2 and (phi1 xs) (phi2 xs) 
skolemAlg (And phi1 phi2) xs = liftM2 and (phi1 xs) (phi2 xs) 

−  instance (Skolem f, Skolem g) = 
+  instance (Skolem f, Skolem g) => Skolem (f :+: g) where 
skolemAlg (Inr phi) = skolemAlg phi 
skolemAlg (Inr phi) = skolemAlg phi 

−  skolemAlg (Inl phi) = skolemAlg phi</ 
+  skolemAlg (Inl phi) = skolemAlg phi</haskell> 
−  In the case for a universal quantifier <math>\forall x.\, \phi</math>, any existentials contained within <math>\phi</math> are parameterized by the variable <math>x</math>, so we add <math>x</math> to the list of free variables and Skolemize the body <math>\phi</math>. Implementing this in Haskell, the algebra instance must be a function from Forall (Term  
+  In the case for a universal quantifier <math>\forall x.\, \phi</math>, any existentials contained within <math>\phi</math> are parameterized by the variable <math>x</math>, so we add <math>x</math> to the list of free variables and Skolemize the body <math>\phi</math>. Implementing this in Haskell, the algebra instance must be a function from Forall (Term > [Term] > Supply (Formula Stage4)) to [Term] > Supply (Forall (Term > Formula Stage4)), which involves some juggling of the unique supply. 
−  < 
+  <haskell class="haskell">instance Skolem Forall where 
skolemAlg (Forall phi) xs = 
skolemAlg (Forall phi) xs = 

−  do supply 
+  do supply < freshes 
−  return (forall $ \x  
+  return (forall $ \x > evalState (phi x (x:xs)) supply)</haskell> 
From the recursive result phi, we need to construct a new body for the forall constructor that has a ''pure'' body: It must not run in the Supply monad. Yet the body must contain only names that do not conflict with those used in the rest of this fold. This is why we need a moderately complex UniqueSupply data structure: To break off a disjointyetinfinite supply for use by evalState in the body of a forall, restoring purity to the body by running the Supply computation to completion. 
From the recursive result phi, we need to construct a new body for the forall constructor that has a ''pure'' body: It must not run in the Supply monad. Yet the body must contain only names that do not conflict with those used in the rest of this fold. This is why we need a moderately complex UniqueSupply data structure: To break off a disjointyetinfinite supply for use by evalState in the body of a forall, restoring purity to the body by running the Supply computation to completion. 

−  Finally, the key instance for existentials is actually quite simple 
+  Finally, the key instance for existentials is actually quite simple � just generate a fresh name and apply the Skolem function to all the arguments xs. The application phi (Const name xs) is how we express replacement of the existentially bound term with Const name xs with higherorder abstract syntax. 
−  < 
+  <haskell class="haskell">instance Skolem Exists where 
skolemAlg (Exists phi) xs = 
skolemAlg (Exists phi) xs = 

−  do uniq 
+  do uniq < fresh 
−  phi (Const ( 
+  phi (Const ("Skol" ++ show uniq) xs) xs</haskell> 
After folding the Skolemization algebra over a formula, Since we are assuming the formula is closed, we apply the result of folding skolemAlg to the empty list of free variables. Then the resulting Supply (Formula Stage4) computation is run to completion starting with the initialUniqueSupply. 
After folding the Skolemization algebra over a formula, Since we are assuming the formula is closed, we apply the result of folding skolemAlg to the empty list of free variables. Then the resulting Supply (Formula Stage4) computation is run to completion starting with the initialUniqueSupply. 

−  < 
+  <haskell class="haskell">skolemize :: Formula Stage2 > Formula Stage4 
skolemize formula = evalState (foldResult []) initialUniqueSupply 
skolemize formula = evalState (foldResult []) initialUniqueSupply 

−  where foldResult :: [Term]  
+  where foldResult :: [Term] > Supply (Formula Stage4) 
−  foldResult = foldFormula skolemAlg formula</ 
+  foldResult = foldFormula skolemAlg formula</haskell> 
And the output is starting to get interesting: 
And the output is starting to get interesting: 

−  < 
+  <haskell>*Main> texprint . skolemize . pushNotInwards . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& (\forall x_1.\, \neg Person(x_1) \vee Food(Skol_2(x_1)) \wedge \neg Eats(x_1, Skol_2(x_1))) \\ 
& (\forall x_1.\, \neg Person(x_1) \vee Food(Skol_2(x_1)) \wedge \neg Eats(x_1, Skol_2(x_1))) \\ 

Line 457:  Line 456:  
</math> 
</math> 

−  In the first line, <math>Skol_2</math> maps a person to a food they 
+  In the first line, <math>Skol_2</math> maps a person to a food they don�t eat. In the second line, <math>Skol_6</math> maps a food to a person who doesn�t eat it. 
−  <blockquote>In the definition of skolemAlg, we use liftM2 to thread the Supply monad through the boring cases, but the ( 
+  <blockquote>In the definition of skolemAlg, we use liftM2 to thread the Supply monad through the boring cases, but the (>) [Term] monad is managed manually. Augment the (>) [Term] monad to handle the Forall and Exists cases, and then combine this monad with Supply using either StateT or the monad coproduct {monadcoproduct}. 
</blockquote> 
</blockquote> 

−  == Stage 5 
+  == Stage 5 � Prenex Normal Form == 
−  Now that all the existentials have been eliminated, we can also eliminate the universally quantified variables. A formula is in ''prenex normal form'' when all the quantifiers have been pushed to the outside of other connectives. We have already removed existential quantifiers, so we are dealing only with universal quantifiers. As long as variable names 
+  Now that all the existentials have been eliminated, we can also eliminate the universally quantified variables. A formula is in ''prenex normal form'' when all the quantifiers have been pushed to the outside of other connectives. We have already removed existential quantifiers, so we are dealing only with universal quantifiers. As long as variable names don�t conflict, we can freely push them as far out as we like and commute all binding sites. By convention, free variables are universally quantifed, so a formula is valid if and only if the body of its prenex form is valid. Though this may sound technical, all we have to do to eliminate universal quantification is choose fresh names for all the variables and forget about their binding sites. 
−  < 
+  <haskell class="haskell">type Stage5 = TT :+: FF :+: Atom :+: NAtom :+: Or :+: And 
−  prenex :: Formula Stage4  
+  prenex :: Formula Stage4 > Formula Stage5 
prenex formula = evalState (foldFormula prenexAlg formula) 
prenex formula = evalState (foldFormula prenexAlg formula) 

initialUniqueSupply 
initialUniqueSupply 

−  class Functor f = 
+  class Functor f => Prenex f where 
−  prenexAlg :: f (Supply (Formula Stage5))  
+  prenexAlg :: f (Supply (Formula Stage5)) > Supply (Formula Stage5) 
instance Prenex Forall where 
instance Prenex Forall where 

−  prenexAlg (Forall phi) = do uniq 
+  prenexAlg (Forall phi) = do uniq < fresh 
−  phi (Var ( 
+  phi (Var ("x" ++ show uniq)) 
instance Prenex TT where 
instance Prenex TT where 

Line 491:  Line 490:  
prenexAlg (And phi1 phi2) = liftM2 and phi1 phi2 
prenexAlg (And phi1 phi2) = liftM2 and phi1 phi2 

−  instance (Prenex f, Prenex g) = 
+  instance (Prenex f, Prenex g) => Prenex (f :+: g) where 
prenexAlg (Inl phi) = prenexAlg phi 
prenexAlg (Inl phi) = prenexAlg phi 

−  prenexAlg (Inr phi) = prenexAlg phi</ 
+  prenexAlg (Inr phi) = prenexAlg phi</haskell> 
Since prenex is just forgetting the binders, our example is mostly unchanged. 
Since prenex is just forgetting the binders, our example is mostly unchanged. 

−  < 
+  <haskell>*Main> texprint . prenex . skolemize . pushNotInwards 
−  . elimImp $ foodFact</ 
+  . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& (\neg Person(x_1) \vee Food(Skol_2(x_1)) \wedge \neg Eats(x_1, Skol_2(x_1))) \\ 
& (\neg Person(x_1) \vee Food(Skol_2(x_1)) \wedge \neg Eats(x_1, Skol_2(x_1))) \\ 

Line 504:  Line 503:  
</math> 
</math> 

−  == Stage 6 
+  == Stage 6 � Conjunctive Normal Form == 
Now all we have left is possiblynegated atomic predicates connected by <math>\wedge</math> and <math>\vee</math>. This secondtolast stage distributes these over each other to reach a canonical form with all the conjunctions at the outer layer, and all the disjunctions in the inner layer. 
Now all we have left is possiblynegated atomic predicates connected by <math>\wedge</math> and <math>\vee</math>. This secondtolast stage distributes these over each other to reach a canonical form with all the conjunctions at the outer layer, and all the disjunctions in the inner layer. 

−  At this point, we no longer have a recursive type for formulas, so 
+  At this point, we no longer have a recursive type for formulas, so there�s not too much point to reusing the old constructors. 
−  < 
+  <haskell class="haskell">type Literal = (Atom :+: NAtom) () 
type Clause = [Literal]  implicit disjunction 
type Clause = [Literal]  implicit disjunction 

type CNF = [Clause]  implicit conjunction 
type CNF = [Clause]  implicit conjunction 

−  (\/) :: Clause  
+  (\/) :: Clause > Clause > Clause 
(\/) = (++) 
(\/) = (++) 

−  (/\) :: CNF  
+  (/\) :: CNF > CNF > CNF 
−  (/\) = (++)</ 
+  (/\) = (++)</haskell> 
−  < 
+  <haskell class="haskell">cnf :: Formula Stage5 > CNF 
cnf = foldFormula cnfAlg 
cnf = foldFormula cnfAlg 

−  class Functor f = 
+  class Functor f => ToCNF f where 
−  cnfAlg :: f CNF  
+  cnfAlg :: f CNF > CNF 
instance ToCNF TT where 
instance ToCNF TT where 

Line 536:  Line 535:  
cnfAlg (And phi1 phi2) = phi1 /\ phi2 
cnfAlg (And phi1 phi2) = phi1 /\ phi2 

instance ToCNF Or where 
instance ToCNF Or where 

−  cnfAlg (Or phi1 phi2) = [ a \/ b  a 
+  cnfAlg (Or phi1 phi2) = [ a \/ b  a < phi1, b < phi2 ] 
−  instance (ToCNF f, ToCNF g) = 
+  instance (ToCNF f, ToCNF g) => ToCNF (f :+: g) where 
cnfAlg (Inl phi) = cnfAlg phi 
cnfAlg (Inl phi) = cnfAlg phi 

−  cnfAlg (Inr phi) = cnfAlg phi</ 
+  cnfAlg (Inr phi) = cnfAlg phi</haskell> 
And we can now watch our formula get really large and ugly, as our running example illustrates: 
And we can now watch our formula get really large and ugly, as our running example illustrates: 

−  < 
+  <haskell>*Main> texprint . cnf . prenex . skolemize 
−  . pushNotInwards . elimImp $ foodFact</ 
+  . pushNotInwards . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& (\neg Person(x1) \vee Food(Skol2(x1))\vee \neg Food(x2)\vee Person(Skol6(x2))) \\ 
& (\neg Person(x1) \vee Food(Skol2(x1))\vee \neg Food(x2)\vee Person(Skol6(x2))) \\ 

Line 553:  Line 552:  
</math> 
</math> 

−  == Stage 7 
+  == Stage 7 � Implicative Normal Form == 
There is one more step we can take to remove all those aethetically displeasing negations in the CNF result above, reaching the particularly elegant ''implicative normal form''. We just gather all negated literals and push them to left of an implicit implication arrow, i.e. utilize this equivalence: 
There is one more step we can take to remove all those aethetically displeasing negations in the CNF result above, reaching the particularly elegant ''implicative normal form''. We just gather all negated literals and push them to left of an implicit implication arrow, i.e. utilize this equivalence: 

Line 562:  Line 561:  
</math> 
</math> 

−  < 
+  <haskell class="haskell">data IClause = IClause  implicit implication 
[Atom ()]  implicit conjunction 
[Atom ()]  implicit conjunction 

[Atom ()]  implicit disjunction 
[Atom ()]  implicit disjunction 

Line 568:  Line 567:  
type INF = [IClause]  implicit conjuction 
type INF = [IClause]  implicit conjuction 

−  inf :: CNF  
+  inf :: CNF > INF 
inf formula = map toImpl formula 
inf formula = map toImpl formula 

−  where toImpl disj = IClause [ Atom p ts  Inr (NAtom p ts) 
+  where toImpl disj = IClause [ Atom p ts  Inr (NAtom p ts) < disj ] 
−  [ t  Inl t@(Atom _ _ ) 
+  [ t  Inl t@(Atom _ _ ) < disj ]</haskell> 
This form is especially useful for a resolution procedure because one always resolves a term on the left of an IClause with a term on the right. 
This form is especially useful for a resolution procedure because one always resolves a term on the left of an IClause with a term on the right. 

−  < 
+  <haskell>*Main> texprint . inf . cnf . prenex . skolemize 
−  . pushNotInwards . elimImp $ foodFact</ 
+  . pushNotInwards . elimImp $ foodFact</haskell> 
<math>\begin{array}{ll} 
<math>\begin{array}{ll} 

& ([Person(x1) \wedge Food(x2)] \Rightarrow [Food(Skol2(x1)) \vee Person(Skol6(x2))]) \\ 
& ([Person(x1) \wedge Food(x2)] \Rightarrow [Food(Skol2(x1)) \vee Person(Skol6(x2))]) \\ 

Line 588:  Line 587:  
Our running example has already been pushed all the way through, so now we can relax and enjoy writing normalize. 
Our running example has already been pushed all the way through, so now we can relax and enjoy writing normalize. 

−  < 
+  <haskell class="haskell">normalize :: Formula Input > INF 
normalize = 
normalize = 

−  inf . cnf . prenex . skolemize . pushNotInwards . elimImp</ 
+  inf . cnf . prenex . skolemize . pushNotInwards . elimImp</haskell> 
== Remarks == 
== Remarks == 

−  Freely manipulating coproducts is a great way to make extensible data types as well as to express the structure of your data and computation. Though there is some syntactic overhead, it quickly becomes routine and readable. There does appear to be additional opportunity for scrapping boilerplate code. Ideally, we could elminate both the cases for uninteresting constructors and all the 
+  Freely manipulating coproducts is a great way to make extensible data types as well as to express the structure of your data and computation. Though there is some syntactic overhead, it quickly becomes routine and readable. There does appear to be additional opportunity for scrapping boilerplate code. Ideally, we could elminate both the cases for uninteresting constructors and all the �glue� instances for the coproduct of two functors. Perhaps given more firstclass manipulation of type classes and instances {typeclasses} we could express that a coproduct has only one reasonable implementation for ''any'' type class that is an implemention of a functor algebra, and never write an algebra instance for (:+:) again. 
Finally, Data Types à la Carte is not the only way to implement coproducts. In Objective Caml, polymorphic variants {ocamlvariants} serve a similar purpose, allowing free recombination of variant tags. The HList library {hlist} also provides an encoding of polymorphic variants in Haskell. 
Finally, Data Types à la Carte is not the only way to implement coproducts. In Objective Caml, polymorphic variants {ocamlvariants} serve a similar purpose, allowing free recombination of variant tags. The HList library {hlist} also provides an encoding of polymorphic variants in Haskell. 

Line 603:  Line 602:  
{Kenn} 
{Kenn} 

−  == Appendix 
+  == Appendix � Printing == 
−  We need to lift all the document operators into the freshness monad. I wrote all this starting with a pretty printer, so I just reuse the combinators and spit out TeX (which 
+  We need to lift all the document operators into the freshness monad. I wrote all this starting with a pretty printer, so I just reuse the combinators and spit out TeX (which doesn�t need to actually be pretty in source form). 
−  < 
+  <haskell class="haskell">sepBy str = hsep . punctuate (text str) 
−  ( 
+  (<++>) = liftM2 (<+>) 
−  ( 
+  (<>) = liftM2 (<>) 
textM = return . text 
textM = return . text 

parensM = liftM parens 
parensM = liftM parens 

−  class Functor f = 
+  class Functor f => TeXAlg f where 
−  texAlg :: f (Supply Doc)  
+  texAlg :: f (Supply Doc) > Supply Doc 
instance TeXAlg Atom where 
instance TeXAlg Atom where 

Line 620:  Line 619:  
instance TeXAlg NAtom where 
instance TeXAlg NAtom where 

−  texAlg (NAtom p ts) = textM 
+  texAlg (NAtom p ts) = textM "\\neg" <++> (return . tex $ Const p ts) 
instance TeXAlg TT where 
instance TeXAlg TT where 

−  texAlg _ = textM 
+  texAlg _ = textM "TT" 
instance TeXAlg FF where 
instance TeXAlg FF where 

−  texAlg _ = textM 
+  texAlg _ = textM "FF" 
instance TeXAlg Not where 
instance TeXAlg Not where 

−  texAlg (Not a) = textM 
+  texAlg (Not a) = textM "\\neg" <> parensM a 
instance TeXAlg Or where 
instance TeXAlg Or where 

texAlg (Or a b) = parensM a 
texAlg (Or a b) = parensM a 

−  +  <++> textM "\\vee" 

−  +  <++> parensM b 

instance TeXAlg And where 
instance TeXAlg And where 

texAlg (And a b) = parensM a 
texAlg (And a b) = parensM a 

−  +  <++> textM "\\wedge" 

−  +  <++> parensM b 

instance TeXAlg Impl where 
instance TeXAlg Impl where 

texAlg (Impl a b) = parensM a 
texAlg (Impl a b) = parensM a 

−  +  <++> textM "\\Rightarrow" 

−  +  <++> parensM b 

instance TeXAlg Forall where 
instance TeXAlg Forall where 

−  texAlg (Forall t) = do uniq 
+  texAlg (Forall t) = do uniq < fresh 
−  let name = 
+  let name = "x_{" ++ show uniq ++ "}" 
−  textM 
+  textM "\\forall" 
−  +  <++> textM name 

−  +  <> textM ".\\," 

−  +  <++> parensM (t (Var name)) 

instance TeXAlg Exists where 
instance TeXAlg Exists where 

−  texAlg (Exists t) = do uniq 
+  texAlg (Exists t) = do uniq < fresh 
−  let name = 
+  let name = "c_{" ++ show uniq ++ "}" 
−  textM 
+  textM "\\exists" 
−  +  <++> textM name 

−  +  <> textM ".\\," 

−  +  <++> parensM (t (Var name))</haskell> 

−  < 
+  <haskell class="haskell">instance (TeXAlg f, TeXAlg g) => TeXAlg (f :+: g) where 
texAlg (Inl x) = texAlg x 
texAlg (Inl x) = texAlg x 

texAlg (Inr x) = texAlg x 
texAlg (Inr x) = texAlg x 

class TeX a where 
class TeX a where 

−  tex :: a  
+  tex :: a > Doc 
−  instance TeXAlg f = 
+  instance TeXAlg f => TeX (Formula f) where 
tex formula = evalState 
tex formula = evalState 

(foldFormula texAlg formula) 
(foldFormula texAlg formula) 

initialUniqueSupply 
initialUniqueSupply 

−  instance (Functor f, TeXAlg f) = 
+  instance (Functor f, TeXAlg f) => TeX (f ()) where 
tex x = evalState 
tex x = evalState 

−  (texAlg . fmap (const (textM 
+  (texAlg . fmap (const (textM "")) $ x) 
initialUniqueSupply 
initialUniqueSupply 

instance TeX CNF where 
instance TeX CNF where 

−  tex formula = sepBy 
+  tex formula = sepBy "\\wedge" 
−  $ fmap (parens . sepBy 
+  $ fmap (parens . sepBy "\\vee" . fmap tex) formula 
instance TeX IClause where 
instance TeX IClause where 

−  tex (IClause p q) = (brackets $ sepBy 
+  tex (IClause p q) = (brackets $ sepBy "\\wedge" $ fmap tex $ p) 
−  +  <+> text "\\Rightarrow" 

−  +  <+> (brackets $ sepBy "\\vee" $ fmap tex $ q) 

instance TeX INF where 
instance TeX INF where 

−  tex formula = sepBy 
+  tex formula = sepBy "\\wedge" $ fmap (parens . tex) $ formula 
Line 695:  Line 694:  
tex (Var x) = text x 
tex (Var x) = text x 

tex (Const c []) = text c 
tex (Const c []) = text c 

−  tex (Const c args) = text c 
+  tex (Const c args) = text c <> parens (sepBy "," (fmap tex args)) 
−  texprint :: TeX a = 
+  texprint :: TeX a => a > IO () 
−  texprint = putStrLn . render . tex</ 
+  texprint = putStrLn . render . tex</haskell> 
Latest revision as of 22:09, 20 September 2008
Classical firstorder logic has the pleasant property that a formula can be reduced to an elegant implicative normal form through a series of syntactic simplifications. Using these transformations as a vehicle, this article demonstrates how to use Haskell�s type system, specifically a variation on Swierstra�s �Data Types à la Carte� method, to enforce the structural correctness property that these constructors are, in fact, eliminated by each stage of the transformation.
{# LANGUAGE RankNTypes,TypeOperators,PatternSignatures #} {# LANGUAGE UndecidableInstances,IncoherentInstances #} {# LANGUAGE MultiParamTypeClasses,TypeSynonymInstances #} {# LANGUAGE FlexibleContexts,FlexibleInstances #} import Text.PrettyPrint.HughesPJ import Control.Monad.State import Prelude hiding (or,and,not){LANGUAGE pragma and module imports}
Contents
 1 FirstOrder Logic
 2 Stage 1 � Eliminate Implications
 3 Stage 2 � Move Negation Inwards
 4 Stage 3 � Standardize variable names
 5 Stage 4 � Skolemization
 6 Stage 5 � Prenex Normal Form
 7 Stage 6 � Conjunctive Normal Form
 8 Stage 7 � Implicative Normal Form
 9 Voilà
 10 Remarks
 11 About the Author
 12 Appendix � Printing
FirstOrder Logic
Consider the optimistic statement �Every person has a heart.� If we were asked to write this formally in a logic or philosophy class, we might write the following formula of classical firstorder logic:
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \forall p.\, Person(p) \Rightarrow \exists h.\, Heart(h) \wedge Has(p,h) \\
If asked to write the same property for testing by QuickCheck {quickcheck}, we might instead produce this code:
heartFact :: Person > Bool
heartFact p = has p (heart p)
where heart :: Person > Heart
...
These look rather different. Ignoring how some of the predicates moved into our types, there are two other transformations involved. First, the universally quantified has been made a parameter, essentially making it a free variable of the formula. Second, the existentially quantified has been replaced by a function heart that, for any person, returns their heart. How did we know to encode things this way? We have performed firstorder quantifier elimination in our heads!
This idea has an elegant instantiation for classical firstorder logic which (along with some other simple transformations) yields a useful normal form for any formula. This article is a tour of the normalization process, implemented in Haskell, using a number of Haskell programming tricks. We will begin with just a couple of formal definitions, but quickly move on to �all code, all the time.�
First, we need the primitive set of terms , which are either variables or function symbols applied to a list of terms (constants are functions of zero arguments).
Next, we add atomic predicates over terms, and the logical constructions to combine atomic predicates. Since we are talking about classical logic, many constructs have duals, so they are presented sidebyside.
We will successively convert a closed (no free variables) firstorder logic formula into a series of equivalent formulae, eliminating many of the above constructs. Eventually the result will be in implicative normal form, in which the placement of all the logical connectives is strictly dictated, such that it does not even require a recursive specification. Specifically, an implicative normal form is the conjunction of a set of implications, each of which has a conjunction of terms on the left and a disjunction of terms on the right:
The normal form may be very large compared to the input formula, but it is convenient for many purposes, such as using Prolog�s resolution procedure or an SMT (Satisfiability Modulo Theories) solver. The following process for normalizing a formula is described by Russell and Norvig {Russell2003} in seven steps:
 Eliminate implications.
 Move negations inwards.
 Standardize variable names.
 Eliminate existential quantification, reaching Skolem normal form.
 Eliminate universal quantification, reaching prenex formal form.
 Distribute boolean connectives, reaching conjunctive normal form.
 Gather negated atoms, reaching implicative normal form.
Keeping in mind the pattern of systematically simplifying the syntax of a formula, let us consider some Haskell data structures for representing firstorder logic.
A Natural Encoding
Experienced Haskellers may translate the above definitions into the following Haskell data types immediately upon reading them:
data Term = Const String [Term]
 Var String
We will reuse the constructor names from FOL later, though, so this is not part of the code for the demonstration.
data FOL = Impl FOL FOL
 Atom String [Term]  Not FOL
 TT  FF
 Or FOL FOL  And FOL FOL
 Exists String FOL  Forall String FOL
To make things more interesting, let us work with the formula representing the statement �If there is a person that eats every food, then there is no food that noone eats.�
foodFact =
(Impl
(Exists "p"
(And (Atom "Person" [Var "p"])
(Forall "f"
(Impl (Atom "Food" [Var "f"])
(Atom "Eats" [Var "p", Var "f"])))))
(Not (Exists "f"
(And (Atom "Food" [Var "f"])
(Not (Exists "p"
(And (Atom "Person" [Var "p"])
(Atom "Eats" [Var "f"]))))))))
HigherOrder Abstract Syntax
While the above encoding is natural to write down, it has drawbacks for actual work. The first thing to notice is that we are using the String type to represent variables, and may have to carefully manage scoping. But what do variables range over? Terms. And Haskell already has variables that range over the data type Term, so we can reuse Haskell�s implementation; this technique is known as higherorder abstract syntax (HOAS).
data FOL = Impl FOL FOL
 Atom String [Term]  Not FOL
 TT  FF
 Or FOL FOL  And FOL FOL
 Exists (Term > FOL)  Forall (Term > FOL)
In a HOAS encoding, the binder of the object language (the quantifiers of firstorder logic) are implemented using the binders of the metalanguage (Haskell). For example, where in the previous encoding we would represent as Exists "x" (Const "P" [Var "x"]) we now represent it with Exists ( > (Const "P" [x])). And our example becomes:
foodFact =
(Impl
(Exists $ \p >
(And (Atom "Person" [p])
(Forall $ \f >
(Impl (Atom "Food" [f])
(Atom "Eats" [p, f])))))
(Not (Exists $ \f >
(And (Atom "Food" [f])
(Not (Exists $ \p >
(And (Atom "Person" [p])
(Atom "Eats" [f]))))))))
Since the variables p and f have taken the place of the String variable names, Haskell�s binding structure now ensures that we cannot construct a firstorder logic formula with unbound variables, unless we use the Var constructor, which is still present because we will need it later. Another important benefit is that the type now expresses that the variables range over the Term data type, while before it was up to us to properly interpret the String variable names.
Modify the code of this article so that the Var constructor is not introduced until it is required in stage 5.
Data Types à la Carte
But even using this improved encoding, all our transformations will be of type FOL > FOL. Because this type does not express the structure of the computation very precisely, we must rely on human inspection to ensure that each stage is written correctly. More importantly, we are not making manifest the requirement of certain stages that the prior stages� transformations have been performed. For example, our elimination of universal quantification is only a correct transformation when existentials have already been eliminated. A good goal for refining our type structure is to describe our data with types that reflect which connectives may be present.
Swierstra proposes a technique {dtalc} whereby a variant data type is built up by mixing and matching constructors of different functors using their coproduct (:+:), which is the �smallest� functor containing both of its arguments.
data (f :+: g) a = Inl (f a)  Inr (g a)
infixr 6 :+:
instance (Functor f, Functor g) => Functor (f :+: g) where
fmap f (Inl x) = Inl (fmap f x)
fmap f (Inr x) = Inr (fmap f x)
The :+: constructor is like Either but it operates on functors. This difference is crucial � if f and g represent two constructors that we wish to combine into a larger recursive data type, then the type parameter a represents the type of their subformulae.
To work conveniently with coproducts, we define a type class :<: that implements subtyping by explicitly providing an injection from one of the constructors to the larger coproduct data type. There are some technical aspects to making sure current Haskell implementations can figure out the needed instances of :<:, but in this example we need only Swierstra�s original subsumption instances, found in Figure fig:Subsumption. For your own use of the technique, discussion on Phil Wadler�s blog {wadlerdtalc} and the HaskellCafe mailing list {haskellcafedtalc} may be helpful.
class (Functor sub, Functor sup) => sub :<: sup where inj :: sub a > sup a instance Functor f => (:<:) f f where inj = id instance (Functor f, Functor g) => (:<:) f (f :+: g) where inj = Inl instance (Functor f, Functor g, Functor h, (f :<: g)) => (:<:) f (h :+: g) where inj = Inr . inj{Subsumption instances} (fig:Subsumption)
If the above seems a bit abstract or confusing, it will become very clear when we put it into practice. Let us immediately do so by encoding the constructors of firstorder logic in this modular fashion.
data TT a = TT
data FF a = FF
data Atom a = Atom String [Term]
data Not a = Not a
data Or a = Or a a
data And a = And a a
data Impl a = Impl a a
data Exists a = Exists (Term > a)
data Forall a = Forall (Term > a)
Each constructor is parameterized by a type a of subformulae; TT, FF, and Atom do not have any subformulae so they ignore their parameter. Logical operations such as And have two subformulae. Correspondingly, the And constructor takes two arguments of type a.
The compound functor Input is now the specification of which constructors may appear in a firstorder logic formula.
type Input = TT :+: FF :+: Atom
:+: Not :+: Or :+: And :+: Impl
:+: Exists :+: Forall
The final step is to �tie the knot� with the following Formula data type, which generates a recursive formula over whatever constructors are present in its functor argument f.
data Formula f = In { out :: f (Formula f) }
If you have not seen this trick before, that definition may be hard to read and understand. Consider the types of In and out.
In :: f (Formula f) > Formula f
out :: Formula f > f (Formula f)
Observe that . This pair of inverses allows us to �roll� and �unroll� one layer of a formula in order to operate on the outermost constructor. Haskell does this same thing when you patternmatch against �normal� recursive data types. Like Haskell, we want to hide this rolling and unrolling. To hide the rolling, we define some helper constructors, found in Figure fig:FOLboilerplate, that inject a constructor into an arbitrary supertype, and then apply In to yield a Formula.
To hide the unrolling, we use the fact that a fixpoint of a functor comes with a fold operation, or catamorphism, which we will use to traverse a formula�s syntax. The function foldFormula takes as a parameter an algebra of the functor f. Intuitively, algebra tells us how to fold �one layer� of a formula, assuming all subformulae have already been processed. The fixpoint then provides the recursive structure of the computation once and for all.
foldFormula :: Functor f => (f a > a) > Formula f > a
foldFormula algebra = algebra . fmap (foldFormula algebra) . out
We are already reaping some of the benefit of our �à la carte� technique: The boilerplate Functor instances in Figure fig:FOLboilerplate are not much larger than the code of foldFormula would have been, and they are defined modularly! Unlike a monolithic foldFormula implementation, this one function will work no matter which constructors are present. If the definition of foldFormula is unfamiliar, it is worth imagining a Formula f flowing through the three stages: First, out unrolls the formula one layer, then fmap recursively folds over all the subformulae. Finally, the results of the recursion are combined by algebra.
instance Functor TT where fmap _ _ = TT instance Functor FF where fmap _ _ = FF instance Functor Atom where fmap _ (Atom p ts) = Atom p ts instance Functor Not where fmap f (Not phi) = Not (f phi) instance Functor Or where fmap f (Or phi1 phi2) = Or (f phi1) (f phi2) instance Functor And where fmap f (And phi1 phi2) = And (f phi1) (f phi2) instance Functor Impl where fmap f (Impl phi1 phi2) = Impl (f phi1) (f phi2) instance Functor Forall where fmap f (Forall phi) = Forall (f . phi) instance Functor Exists where fmap f (Exists phi) = Exists (f . phi) inject :: (g :<: f) => g (Formula f) > Formula f inject = In . inj tt :: (TT :<: f) => Formula f tt = inject TT ff :: (FF :<: f) => Formula f ff = inject FF atom :: (Atom :<: f) => String > [Term] > Formula f atom p ts = inject (Atom p ts) not :: (Not :<: f) => Formula f > Formula f not = inject . Not or :: (Or :<: f) => Formula f > Formula f > Formula f or phi1 phi2 = inject (Or phi1 phi2) and :: (And :<: f) => Formula f > Formula f > Formula f and phi1 phi2 = inject (And phi1 phi2) impl :: (Impl :<: f) => Formula f > Formula f > Formula f impl phi1 phi2 = inject (Impl phi1 phi2) forall :: (Forall :<: f) => (Term > Formula f) > Formula f forall = inject . Forall exists :: (Exists :<: f) => (Term > Formula f) > Formula f exists = inject . Exists{Boilerplate for FirstOrder Logic Constructors} (fig:FOLboilerplate)
Here is what our running example looks like with this encoding:
foodFact :: Formula Input
foodFact = (exists $ \p > atom "Person" [p]
`and` (forall $ \f > atom "Food" [f]
`impl` atom "Eats" [p,f]))
`impl`
(not (exists $ \f > atom "Food" [f]
`and` (not (exists $ \p > atom "Person" [p]
`and` atom "Eats" [p,f]))))
A TeX prettyprinter is included as an appendix to this article. To make things readable, though, I�ll doctor its output into a nice table, and remove extraneous parentheses. But I won�t rewrite the variable names, since variables and binding are a key aspect of managing formulae. By convention, the printer uses for existentially quantified variables and for universally quantified variables.
*Main> texprint foodFact
Stage 1 � Eliminate Implications
The first transformation is an easy one, in which we �desugar� into . The highlevel overview is given by the type and body of elimImp.
type Stage1 = TT :+: FF :+: Atom :+: Not :+: Or :+: And :+: Exists :+: Forall
elimImp :: Formula Input > Formula Stage1
elimImp = foldFormula elimImpAlg
We take a formula containing all the constructors of firstorder logic, and return a formula built without use of Impl. The way that elimImp does this is by folding the algebras elimImpAlg for each constructor over the recursive structure of a formula.
The function elimImpAlg we provide by making each constructor an instance of the ElimImp type class. This class specifies for a given constructor how to eliminate implications � for most constructors this is just the identity function, though we must rebuild an identical term to alter its type. Perhaps there is a way to use generic programming to eliminate the uninteresting cases.
class Functor f => ElimImp f where
elimImpAlg :: f (Formula Stage1) > Formula Stage1
instance ElimImp Impl where elimImpAlg (Impl phi1 phi2) = (not phi1) `or` phi2
instance ElimImp TT where elimImpAlg TT = tt
instance ElimImp FF where elimImpAlg FF = ff
instance ElimImp Atom where elimImpAlg (Atom p ts) = atom p ts
instance ElimImp Not where elimImpAlg (Not phi) = not phi
instance ElimImp Or where elimImpAlg (Or phi1 phi2) = phi1 `or` phi2
instance ElimImp And where elimImpAlg (And phi1 phi2) = phi1 `and` phi2
instance ElimImp Exists where elimImpAlg (Exists phi) = exists phi
instance ElimImp Forall where elimImpAlg (Forall phi) = forall phi
We extend ElimImp in the natural way over coproducts, so that whenever all our constructors are members of the type class, then their coproduct is as well.
instance (ElimImp f, ElimImp g) => ElimImp (f :+: g) where
elimImpAlg (Inr phi) = elimImpAlg phi
elimImpAlg (Inl phi) = elimImpAlg phi
Our running example is now
*Main> texprint . elimImp $ foodFact
Design a solution where only the Impl case of elimImpAlg needs to be written.
Stage 2 � Move Negation Inwards
Now that implications are gone, we are left with entirely symmetrical constructions, and can always push negations in or out using duality: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \neg(\neg \phi) \Leftrightarrow \phi \\ \neg(\phi_1 \wedge \phi_2) \Leftrightarrow \neg\phi_1 \vee \neg\phi_2 \\ \neg(\phi_1 \vee \phi_2) \Leftrightarrow \neg\phi_1 \wedge \neg\phi_2 \\ \neg(\exists x.\, \phi) \Leftrightarrow \forall x.\, \neg\phi \\ \neg(\forall x.\, \phi) \Leftrightarrow \exists x.\, \neg\phi
Our eventual goal is to move negation all the way inward so it is only applied to atomic predicates. To express this structure in our types, we define a new constructor for negated atomic predicates as well as the type for the output of Stage 2:
data NAtom a = NAtom String [Term]
instance Functor NAtom where fmap f (NAtom p ts) = NAtom p ts
natom :: (NAtom :<: f) => String > [Term] > Formula f
natom p ts = inject (NAtom p ts)
type Stage2 = TT :+: FF :+: Atom
:+: NAtom
:+: Or :+: And
:+: Exists :+: Forall
One could imagine implementing duality with a multiparameter type class that records exactly the dual of each constructor, as
class (Functor f, Functor g) => Dual f g where
dual :: f a > g a
Unfortunately, this leads to a situation where our subtyping must use the commutativity of coproducts, which it is incapable of doing as written. For this article, let us just define an algebra to dualize a whole formula at a time.
dualize :: Formula Stage2 > Formula Stage2
dualize = foldFormula dualAlg
class Functor f => Dualize f where
dualAlg :: f (Formula Stage2) > Formula Stage2
instance Dualize TT where dualAlg TT = ff
instance Dualize FF where dualAlg FF = tt
instance Dualize Atom where dualAlg (Atom p ts) = natom p ts
instance Dualize NAtom where dualAlg (NAtom p ts) = atom p ts
instance Dualize Or where dualAlg (Or phi1 phi2) = phi1 `and` phi2
instance Dualize And where dualAlg (And phi1 phi2) = phi1 `or` phi2
instance Dualize Exists where dualAlg (Exists phi) = forall phi
instance Dualize Forall where dualAlg (Forall phi) = exists phi
instance (Dualize f, Dualize g) => Dualize (f :+: g) where
dualAlg (Inl phi) = dualAlg phi
dualAlg (Inr phi) = dualAlg phi
Now perhaps the pattern of these transformations is becoming clear. It is remarkably painless, involving just a little type class syntax as overhead, to write these functor algebras. The definition of pushNotInwards is another straightforward fold, with a helper type class PushNot. I�ve separated the instance for Not since it is the only one that does anything.
class Functor f => PushNot f where
pushNotAlg :: f (Formula Stage2) > Formula Stage2
instance PushNot Not where pushNotAlg (Not phi) = dualize phi
instance PushNot TT where pushNotAlg TT = tt
instance PushNot FF where pushNotAlg FF = ff
instance PushNot Atom where pushNotAlg (Atom p ts) = atom p ts
instance PushNot Or where pushNotAlg (Or phi1 phi2) = phi1 `or` phi2
instance PushNot And where pushNotAlg (And phi1 phi2) = phi1 `and` phi2
instance PushNot Exists where pushNotAlg (Exists phi) = exists phi
instance PushNot Forall where pushNotAlg (Forall phi) = forall phi
instance (PushNot f, PushNot g) => PushNot (f :+: g) where
pushNotAlg (Inr phi) = pushNotAlg phi
pushNotAlg (Inl phi) = pushNotAlg phi
All we have to do now is define a fold that calls pushNotAlg.
pushNotInwards :: Formula Stage1 > Formula Stage2
pushNotInwards = foldFormula pushNotAlg
Our running example now becomes:
*Main> texprint . pushNotInwards . elimImp $ foodFact
Instead of the NAtom constructor, define the composition of two functors f `O` g and rewrite Stage2 = TT :+: FF :+: Atom :+: (Not `O` Atom) :+: Or :+: And :+: Exists :+: Forall
Encode a form of subtyping that can reason using commutativity of coproducts, and rewrite the Dualize algebra using a twoparameter Dual type class as described above.
Stage 3 � Standardize variable names
To �standardize� variable names means to choose nonconflicting names for all the variables in a formula. Since we are using higherorder abstract syntax, Haskell is handling name conflicts for now. We can immediately jump to stage 4!
Stage 4 � Skolemization
It is interesting to arrive at the definition of Skolemization via the CurryHoward correspondence. You may be familiar with the idea that terms of type a > b are proofs of the proposition � implies �, assuming a and b are interpreted as propositions as well. This rests on a notion that a proof of a > b must be some process that can take a proof of a and generate a proof of b, a very computational notion. Rephrasing the above, a function of type a > b is a guarantee that for all elements of type a, there exists a corresponding element of type b. So a function type expresses an alternation of a universal quantifier with an existential. We will use this to replace all the existential quantifiers with freshlygenerated functions. We can of course, pass a unit type to a function, or a tuple of many arguments, to have as many universal quantifiers as we like.
Suppose we have , then in general there may be many choices for , given a particular and . By the axiom of choice, we can create a function that associates each pair with a corresponding arbitrarily, and then rewrite the above formula as . Technically, this formula is only equisatisfiable, but by convention I�m assuming constants to be existentially quantified.
So we need to traverse the syntax tree gathering free variables and replacing existentially quantified variables with functions of a fresh name. Since we are eliminating a binding construct, we now need to reason about fresh unique names.
Today�s formulas are small, so let us use a naïve and wasteful splittable unique identifier supply. Our supply is an infinite binary tree, where moving left prepends a 0 to the bit representation of the current counter, while moving right prepends a 1. Hence, the left and right subtrees are both infinite, nonoverlapping supplies of identifiers. The code for our unique identifier supplies is in Figure fig:unq.
Launchbury and PeytonJones {launchbury95state} have discussed how to use the ST monad to implement a much more sophisticated and spaceefficient version of the same idea.
type Unique = Int data UniqueSupply = UniqueSupply Unique UniqueSupply UniqueSupply initialUniqueSupply :: UniqueSupply initialUniqueSupply = genSupply 1 where genSupply n = UniqueSupply n (genSupply (2*n)) (genSupply (2*n+1)) splitUniqueSupply :: UniqueSupply > (UniqueSupply, UniqueSupply) splitUniqueSupply (UniqueSupply _ l r) = (l,r) getUnique :: UniqueSupply > (Unique, UniqueSupply) getUnique (UniqueSupply n l r) = (n,l) type Supply a = State UniqueSupply a fresh :: Supply Int fresh = do supply < get let (uniq,rest) = getUnique supply put rest return uniq freshes :: Supply UniqueSupply freshes = do supply < get let (l,r) = splitUniqueSupply supply put r return l{Unique supplies} (fig:unq)
The helper algebra for Skolemization is more complex than before because a Formula Stage2 is not directly transformed into Formula Stage4 but into a function from its free variables to a new formula. On top of that, the final computation takes place in the Supply monad because of the need to generate fresh names for Skolem functions. Correspondingly, we choose the return type of the algebra to be [Term] > Supply (Formula Stage4). Thankfully, many instances are just boilerplate.
type Stage4 = TT :+: FF :+: Atom :+: NAtom :+: Or :+: And :+: Forall
class Functor f => Skolem f where
skolemAlg :: f ([Term] > Supply (Formula Stage4))
> [Term] > Supply (Formula Stage4)
instance Skolem TT where
skolemAlg TT xs = return tt
instance Skolem FF where
skolemAlg FF xs = return ff
instance Skolem Atom where
skolemAlg (Atom p ts) xs = return (atom p ts)
instance Skolem NAtom where
skolemAlg (NAtom p ts) xs = return (natom p ts)
instance Skolem Or where
skolemAlg (Or phi1 phi2) xs = liftM2 or (phi1 xs) (phi2 xs)
instance Skolem And where
skolemAlg (And phi1 phi2) xs = liftM2 and (phi1 xs) (phi2 xs)
instance (Skolem f, Skolem g) => Skolem (f :+: g) where
skolemAlg (Inr phi) = skolemAlg phi
skolemAlg (Inl phi) = skolemAlg phi
In the case for a universal quantifier , any existentials contained within are parameterized by the variable , so we add to the list of free variables and Skolemize the body . Implementing this in Haskell, the algebra instance must be a function from Forall (Term > [Term] > Supply (Formula Stage4)) to [Term] > Supply (Forall (Term > Formula Stage4)), which involves some juggling of the unique supply.
instance Skolem Forall where
skolemAlg (Forall phi) xs =
do supply < freshes
return (forall $ \x > evalState (phi x (x:xs)) supply)
From the recursive result phi, we need to construct a new body for the forall constructor that has a pure body: It must not run in the Supply monad. Yet the body must contain only names that do not conflict with those used in the rest of this fold. This is why we need a moderately complex UniqueSupply data structure: To break off a disjointyetinfinite supply for use by evalState in the body of a forall, restoring purity to the body by running the Supply computation to completion.
Finally, the key instance for existentials is actually quite simple � just generate a fresh name and apply the Skolem function to all the arguments xs. The application phi (Const name xs) is how we express replacement of the existentially bound term with Const name xs with higherorder abstract syntax.
instance Skolem Exists where
skolemAlg (Exists phi) xs =
do uniq < fresh
phi (Const ("Skol" ++ show uniq) xs) xs
After folding the Skolemization algebra over a formula, Since we are assuming the formula is closed, we apply the result of folding skolemAlg to the empty list of free variables. Then the resulting Supply (Formula Stage4) computation is run to completion starting with the initialUniqueSupply.
skolemize :: Formula Stage2 > Formula Stage4
skolemize formula = evalState (foldResult []) initialUniqueSupply
where foldResult :: [Term] > Supply (Formula Stage4)
foldResult = foldFormula skolemAlg formula
And the output is starting to get interesting:
*Main> texprint . skolemize . pushNotInwards . elimImp $ foodFact
In the first line, maps a person to a food they don�t eat. In the second line, maps a food to a person who doesn�t eat it.
In the definition of skolemAlg, we use liftM2 to thread the Supply monad through the boring cases, but the (>) [Term] monad is managed manually. Augment the (>) [Term] monad to handle the Forall and Exists cases, and then combine this monad with Supply using either StateT or the monad coproduct {monadcoproduct}.
Stage 5 � Prenex Normal Form
Now that all the existentials have been eliminated, we can also eliminate the universally quantified variables. A formula is in prenex normal form when all the quantifiers have been pushed to the outside of other connectives. We have already removed existential quantifiers, so we are dealing only with universal quantifiers. As long as variable names don�t conflict, we can freely push them as far out as we like and commute all binding sites. By convention, free variables are universally quantifed, so a formula is valid if and only if the body of its prenex form is valid. Though this may sound technical, all we have to do to eliminate universal quantification is choose fresh names for all the variables and forget about their binding sites.
type Stage5 = TT :+: FF :+: Atom :+: NAtom :+: Or :+: And
prenex :: Formula Stage4 > Formula Stage5
prenex formula = evalState (foldFormula prenexAlg formula)
initialUniqueSupply
class Functor f => Prenex f where
prenexAlg :: f (Supply (Formula Stage5)) > Supply (Formula Stage5)
instance Prenex Forall where
prenexAlg (Forall phi) = do uniq < fresh
phi (Var ("x" ++ show uniq))
instance Prenex TT where
prenexAlg TT = return tt
instance Prenex FF where
prenexAlg FF = return ff
instance Prenex Atom where
prenexAlg (Atom p ts) = return (atom p ts)
instance Prenex NAtom where
prenexAlg (NAtom p ts) = return (natom p ts)
instance Prenex Or where
prenexAlg (Or phi1 phi2) = liftM2 or phi1 phi2
instance Prenex And where
prenexAlg (And phi1 phi2) = liftM2 and phi1 phi2
instance (Prenex f, Prenex g) => Prenex (f :+: g) where
prenexAlg (Inl phi) = prenexAlg phi
prenexAlg (Inr phi) = prenexAlg phi
Since prenex is just forgetting the binders, our example is mostly unchanged.
*Main> texprint . prenex . skolemize . pushNotInwards
. elimImp $ foodFact
Stage 6 � Conjunctive Normal Form
Now all we have left is possiblynegated atomic predicates connected by and . This secondtolast stage distributes these over each other to reach a canonical form with all the conjunctions at the outer layer, and all the disjunctions in the inner layer.
At this point, we no longer have a recursive type for formulas, so there�s not too much point to reusing the old constructors.
type Literal = (Atom :+: NAtom) ()
type Clause = [Literal]  implicit disjunction
type CNF = [Clause]  implicit conjunction
(\/) :: Clause > Clause > Clause
(\/) = (++)
(/\) :: CNF > CNF > CNF
(/\) = (++)
cnf :: Formula Stage5 > CNF
cnf = foldFormula cnfAlg
class Functor f => ToCNF f where
cnfAlg :: f CNF > CNF
instance ToCNF TT where
cnfAlg TT = []
instance ToCNF FF where
cnfAlg FF = [[]]
instance ToCNF Atom where
cnfAlg (Atom p ts) = [[inj (Atom p ts)]]
instance ToCNF NAtom where
cnfAlg (NAtom p ts) = [[inj (NAtom p ts)]]
instance ToCNF And where
cnfAlg (And phi1 phi2) = phi1 /\ phi2
instance ToCNF Or where
cnfAlg (Or phi1 phi2) = [ a \/ b  a < phi1, b < phi2 ]
instance (ToCNF f, ToCNF g) => ToCNF (f :+: g) where
cnfAlg (Inl phi) = cnfAlg phi
cnfAlg (Inr phi) = cnfAlg phi
And we can now watch our formula get really large and ugly, as our running example illustrates:
*Main> texprint . cnf . prenex . skolemize
. pushNotInwards . elimImp $ foodFact
Stage 7 � Implicative Normal Form
There is one more step we can take to remove all those aethetically displeasing negations in the CNF result above, reaching the particularly elegant implicative normal form. We just gather all negated literals and push them to left of an implicit implication arrow, i.e. utilize this equivalence:
data IClause = IClause  implicit implication
[Atom ()]  implicit conjunction
[Atom ()]  implicit disjunction
type INF = [IClause]  implicit conjuction
inf :: CNF > INF
inf formula = map toImpl formula
where toImpl disj = IClause [ Atom p ts  Inr (NAtom p ts) < disj ]
[ t  Inl t@(Atom _ _ ) < disj ]
This form is especially useful for a resolution procedure because one always resolves a term on the left of an IClause with a term on the right.
*Main> texprint . inf . cnf . prenex . skolemize
. pushNotInwards . elimImp $ foodFact
Voilà
Our running example has already been pushed all the way through, so now we can relax and enjoy writing normalize.
normalize :: Formula Input > INF
normalize =
inf . cnf . prenex . skolemize . pushNotInwards . elimImp
Remarks
Freely manipulating coproducts is a great way to make extensible data types as well as to express the structure of your data and computation. Though there is some syntactic overhead, it quickly becomes routine and readable. There does appear to be additional opportunity for scrapping boilerplate code. Ideally, we could elminate both the cases for uninteresting constructors and all the �glue� instances for the coproduct of two functors. Perhaps given more firstclass manipulation of type classes and instances {typeclasses} we could express that a coproduct has only one reasonable implementation for any type class that is an implemention of a functor algebra, and never write an algebra instance for (:+:) again.
Finally, Data Types à la Carte is not the only way to implement coproducts. In Objective Caml, polymorphic variants {ocamlvariants} serve a similar purpose, allowing free recombination of variant tags. The HList library {hlist} also provides an encoding of polymorphic variants in Haskell.
About the Author
Kenneth Knowles is a graduate student at the University of California, Santa Cruz, studying type systems, concurrency, and parallel programming. He maintains a blog of mathematical musings in Haskell at http://kennknowles.com/blog
{Kenn}
Appendix � Printing
We need to lift all the document operators into the freshness monad. I wrote all this starting with a pretty printer, so I just reuse the combinators and spit out TeX (which doesn�t need to actually be pretty in source form).
sepBy str = hsep . punctuate (text str)
(<++>) = liftM2 (<+>)
(<>) = liftM2 (<>)
textM = return . text
parensM = liftM parens
class Functor f => TeXAlg f where
texAlg :: f (Supply Doc) > Supply Doc
instance TeXAlg Atom where
texAlg (Atom p ts) = return . tex $ Const p ts
instance TeXAlg NAtom where
texAlg (NAtom p ts) = textM "\\neg" <++> (return . tex $ Const p ts)
instance TeXAlg TT where
texAlg _ = textM "TT"
instance TeXAlg FF where
texAlg _ = textM "FF"
instance TeXAlg Not where
texAlg (Not a) = textM "\\neg" <> parensM a
instance TeXAlg Or where
texAlg (Or a b) = parensM a
<++> textM "\\vee"
<++> parensM b
instance TeXAlg And where
texAlg (And a b) = parensM a
<++> textM "\\wedge"
<++> parensM b
instance TeXAlg Impl where
texAlg (Impl a b) = parensM a
<++> textM "\\Rightarrow"
<++> parensM b
instance TeXAlg Forall where
texAlg (Forall t) = do uniq < fresh
let name = "x_{" ++ show uniq ++ "}"
textM "\\forall"
<++> textM name
<> textM ".\\,"
<++> parensM (t (Var name))
instance TeXAlg Exists where
texAlg (Exists t) = do uniq < fresh
let name = "c_{" ++ show uniq ++ "}"
textM "\\exists"
<++> textM name
<> textM ".\\,"
<++> parensM (t (Var name))
instance (TeXAlg f, TeXAlg g) => TeXAlg (f :+: g) where
texAlg (Inl x) = texAlg x
texAlg (Inr x) = texAlg x
class TeX a where
tex :: a > Doc
instance TeXAlg f => TeX (Formula f) where
tex formula = evalState
(foldFormula texAlg formula)
initialUniqueSupply
instance (Functor f, TeXAlg f) => TeX (f ()) where
tex x = evalState
(texAlg . fmap (const (textM "")) $ x)
initialUniqueSupply
instance TeX CNF where
tex formula = sepBy "\\wedge"
$ fmap (parens . sepBy "\\vee" . fmap tex) formula
instance TeX IClause where
tex (IClause p q) = (brackets $ sepBy "\\wedge" $ fmap tex $ p)
<+> text "\\Rightarrow"
<+> (brackets $ sepBy "\\vee" $ fmap tex $ q)
instance TeX INF where
tex formula = sepBy "\\wedge" $ fmap (parens . tex) $ formula
instance TeX Term where
tex (Var x) = text x
tex (Const c []) = text c
tex (Const c args) = text c <> parens (sepBy "," (fmap tex args))
texprint :: TeX a => a > IO ()
texprint = putStrLn . render . tex