177 lines
9.6 KiB
Haskell
177 lines
9.6 KiB
Haskell
module EvaluationSpec (spec) where
|
|
|
|
import Data.Set (Set, empty, fromList)
|
|
import Evaluation (beta, bind, freeIn, freeVars, freshName, subst)
|
|
import Parser (Expr (Abstraction, Application, Variable), parse)
|
|
import Test.Hspec
|
|
|
|
spec :: Spec
|
|
spec = do
|
|
describe "freeIn" $ do
|
|
it "variable matches name" $
|
|
"x" `freeIn` Variable "x" `shouldBe` True
|
|
it "variable does not match name" $
|
|
"y" `freeIn` Variable "x" `shouldBe` False
|
|
it "free in abstraction body with different binder" $
|
|
"x" `freeIn` Abstraction "y" (Variable "x") `shouldBe` True
|
|
it "not free when abstraction binder shadows it" $
|
|
"x" `freeIn` Abstraction "x" (Variable "x") `shouldBe` False
|
|
it "outer binder shields entire subtree" $
|
|
"x" `freeIn` Abstraction "x" (Abstraction "y" (Variable "x")) `shouldBe` False
|
|
it "free through nested abstractions when no binder shadows it" $
|
|
"x" `freeIn` Abstraction "y" (Abstraction "z" (Variable "x")) `shouldBe` True
|
|
it "free in left side of application" $
|
|
"x" `freeIn` Application (Variable "x") (Variable "y") `shouldBe` True
|
|
it "free in right side of application" $
|
|
"x" `freeIn` Application (Variable "y") (Variable "x") `shouldBe` True
|
|
it "not free in either side of application" $
|
|
"x" `freeIn` Application (Variable "y") (Variable "z") `shouldBe` False
|
|
it "free on right side even when bound on left side" $
|
|
-- left: λx. x (x is bound); right: x (x is free)
|
|
"x" `freeIn` Application (Abstraction "x" (Variable "x")) (Variable "x") `shouldBe` True
|
|
|
|
describe "freeVars" $ do
|
|
it "single variable" $
|
|
freeVars (Variable "x") `shouldBe` fromList ["x"]
|
|
it "abstraction binds its variable" $
|
|
freeVars (Abstraction "x" (Variable "x")) `shouldBe` empty
|
|
it "abstraction with free variable in body" $
|
|
freeVars (Abstraction "x" (Variable "y")) `shouldBe` fromList ["y"]
|
|
it "application collects from both sides" $
|
|
freeVars (Application (Variable "x") (Variable "y")) `shouldBe` fromList ["x", "y"]
|
|
it "application deduplicates the same variable on both sides" $
|
|
freeVars (Application (Variable "x") (Variable "x")) `shouldBe` fromList ["x"]
|
|
it "abstraction removes its binder from body's free vars" $
|
|
freeVars (Abstraction "x" (Application (Variable "x") (Variable "y"))) `shouldBe` fromList ["y"]
|
|
it "only the unbound occurrence is free across an application" $
|
|
-- left: λx. x (x bound); right: x (x free)
|
|
freeVars (Application (Abstraction "x" (Variable "x")) (Variable "x")) `shouldBe` fromList ["x"]
|
|
|
|
describe "freshName" $ do
|
|
it "returns name unchanged when not in the set" $
|
|
freshName "x" empty `shouldBe` "x"
|
|
it "returns name unchanged when other names are in the set" $
|
|
freshName "x" (fromList ["y", "z"]) `shouldBe` "x"
|
|
it "appends one apostrophe when name is taken" $
|
|
freshName "x" (fromList ["x"]) `shouldBe` "x'"
|
|
it "appends two apostrophes when name and name' are both taken" $
|
|
freshName "x" (fromList ["x", "x'"]) `shouldBe` "x''"
|
|
it "appends three apostrophes when name, name', and name'' are all taken" $
|
|
freshName "x" (fromList ["x", "x'", "x''"]) `shouldBe` "x'''"
|
|
|
|
describe "subst" $ do
|
|
it "cannot substitute mismatched variables" $
|
|
subst (Variable "x") "y" (Variable "z") `shouldBe` Variable "x"
|
|
it "can substitute matched variables" $
|
|
subst (Variable "x") "x" (Variable "z") `shouldBe` Variable "z"
|
|
it "can substitute nested variables" $
|
|
subst absWithZ "z" absI `shouldBe` absWithZMaker absI
|
|
-- substitution in Application nodes
|
|
it "substitutes in both sides of an application" $
|
|
subst (Application (Variable "x") (Variable "x")) "x" (Variable "z")
|
|
`shouldBe` Application (Variable "z") (Variable "z")
|
|
it "substitutes when content is an abstraction" $
|
|
subst (Variable "x") "x" absI `shouldBe` absI
|
|
it "substitutes complex content into both sides of application" $
|
|
subst (Application (Variable "x") (Variable "x")) "x" absI
|
|
`shouldBe` Application absI absI
|
|
-- bound variable must shadow the substitution
|
|
it "does not substitute a variable that is bound in an abstraction (same binder name)" $
|
|
-- subst (λx. x) "x" z should be λx. x (x is bound, not free)
|
|
subst (Abstraction "x" (Variable "x")) "x" (Variable "z")
|
|
`shouldBe` Abstraction "x" (Variable "x")
|
|
it "does not substitute free variable in body when binder shadows it" $
|
|
-- subst (λx. x y) "x" z should be λx. x y (the x in the body is bound)
|
|
subst (Abstraction "x" (Application (Variable "x") (Variable "y"))) "x" (Variable "z")
|
|
`shouldBe` Abstraction "x" (Application (Variable "x") (Variable "y"))
|
|
-- variable capture: replacement's free variable must not be captured by a binder
|
|
it "does not capture free variables in the replacement (variable capture)" $
|
|
-- subst (λz. x) "x" z must rename the binder: result is λz'. z (or similar)
|
|
-- The replacement 'z' is free, but the abstraction binds 'z' — naive subst captures it.
|
|
-- Expected: the inner z stays free (some fresh binder wraps it, not 'z').
|
|
-- We test this by asserting the result is NOT λz. z (which would be a capture).
|
|
subst (Abstraction "z" (Variable "x")) "x" (Variable "z")
|
|
`shouldNotBe` Abstraction "z" (Variable "z")
|
|
describe "bind" $ do
|
|
it "bind on non-abstraction returns Nothing" $
|
|
bind (Variable "x") (Variable "y") `shouldBe` Nothing
|
|
it "bind identity lambda with argument returns the argument" $
|
|
bind absI (Variable "y") `shouldBe` Just (Variable "y")
|
|
it "bind constant lambda returns the constant" $
|
|
-- (λx. z) applied to y → z
|
|
bind (Abstraction "x" (Variable "z")) (Variable "y") `shouldBe` Just (Variable "z")
|
|
it "bind replaces all free occurrences of the bound variable" $
|
|
-- (λx. x x) applied to y → y y
|
|
bind selfApp (Variable "y") `shouldBe` Just (Application (Variable "y") (Variable "y"))
|
|
it "bind with abstraction argument substitutes correctly" $
|
|
-- (λx. x) applied to (λi. i) → (λi. i)
|
|
bind absI absI `shouldBe` Just absI
|
|
describe "beta reduction" $ do
|
|
-- Normal forms (no redex)
|
|
it "returns Nothing for a bare variable" $
|
|
beta (Variable "x") `shouldBe` Nothing
|
|
it "returns Nothing for a bare abstraction" $
|
|
beta absI `shouldBe` Nothing
|
|
it "returns Nothing when application has no lambda on left" $
|
|
beta (Application (Variable "x") (Variable "y")) `shouldBe` Nothing
|
|
it "returns Nothing with beta normal expression of great complexity" $
|
|
case parse "λa. λb. λc. λd. a (b (c d)) (c (a d b)) (b (a (c d) b) (c d))" of
|
|
Right exp -> beta exp `shouldBe` Nothing
|
|
Left err -> expectationFailure err
|
|
|
|
-- Basic reductions
|
|
it "reduces identity: (λi. i) y → y" $
|
|
beta (Application absI (Variable "y"))
|
|
`shouldBe` Just (Variable "y")
|
|
it "reduces constant function: (λx. z) y → z" $
|
|
beta (Application (Abstraction "x" (Variable "z")) (Variable "y"))
|
|
`shouldBe` Just (Variable "z")
|
|
|
|
-- Self-application
|
|
it "(λx. x x)(λi. i) → (λi. i)(λi. i)" $
|
|
beta (Application selfApp absI)
|
|
`shouldBe` Just (Application absI absI)
|
|
it "(λi. i)(λi. i) → (λi. i)" $
|
|
beta (Application absI absI)
|
|
`shouldBe` Just absI
|
|
|
|
-- Redex not at top level
|
|
it "reduces redex in function position: ((λi. i) y) z → y z" $
|
|
beta (Application (Application absI (Variable "y")) (Variable "z"))
|
|
`shouldBe` Just (Application (Variable "y") (Variable "z"))
|
|
it "reduces redex inside lambda body: λz. (λi. i) y → λz. y" $
|
|
beta (Abstraction "z" (Application absI (Variable "y")))
|
|
`shouldBe` Just (Abstraction "z" (Variable "y"))
|
|
|
|
-- Redex in argument position (function side is already normal)
|
|
it "reduces redex in argument position when function is a variable: x ((λi. i) y) → x y" $
|
|
beta (Application (Variable "x") (Application absI (Variable "y")))
|
|
`shouldBe` Just (Application (Variable "x") (Variable "y"))
|
|
it "reduces redex in argument position when function is a normal abstraction: (λa. a) ((λi. i) y) reduces function first" $
|
|
-- leftmost-outermost: the outer application is the redex, fires first
|
|
beta (Application absI (Application absI (Variable "y")))
|
|
`shouldBe` Just (Application absI (Variable "y"))
|
|
|
|
-- Omega combinator (divergent): (λx. x x)(λx. x x) → (λx. x x)(λx. x x)
|
|
it "omega combinator steps back to itself" $
|
|
beta (Application omega omega) `shouldBe` Just (Application omega omega)
|
|
|
|
-- Multi-step: applying beta repeatedly converges
|
|
it "iterating beta on (λi. i)((λi. i) y) reaches y in 2 steps" $
|
|
-- step 1: (λi. i)((λi. i) y) → (λi. i) y [outer redex fires first]
|
|
-- step 2: (λi. i) y → y
|
|
let step1 = beta (Application absI (Application absI (Variable "y")))
|
|
step2 = step1 >>= beta
|
|
in step2 `shouldBe` Just (Variable "y")
|
|
|
|
-- Substitution must not be performed under a shadowing binder
|
|
it "beta does not substitute into a shadowed variable: (λx. λx. x) y → λx. x" $
|
|
-- The inner λx re-binds x, so the outer x should not be substituted into it
|
|
beta (Application (Abstraction "x" (Abstraction "x" (Variable "x"))) (Variable "y"))
|
|
`shouldBe` Just (Abstraction "x" (Variable "x"))
|
|
where
|
|
absWithZMaker z = Abstraction "x" (Application z (Variable "x"))
|
|
absWithZ = absWithZMaker (Variable "z")
|
|
absI = Abstraction "i" (Variable "i")
|
|
selfApp = Abstraction "x" (Application (Variable "x") (Variable "x"))
|
|
omega = Abstraction "x" (Application (Variable "x") (Variable "x"))
|