diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,7 @@ +# Changelog for gargamelle-fol + +## 0.1.0.0 — Initial Release + +- First-order logic institution and streaming Z3 SMT solver pipeline. +- Hyperdoctrine structures and quantifier elimination. +- Refinement calculus generation for formal program verification. diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,189 @@ +EUROPEAN UNION PUBLIC LICENCE v. 1.2 +EUPL © the European Union 2007, 2016 + +This European Union Public Licence (the ‘EUPL’) applies to the Work (as defined below) which is provided under the +terms of this Licence. 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Executive Summary & Problem Framing + +Predicate logic with quantifiers and relational constraints is fundamental to system verification, contract refinement, and theorem proving. However, embedding first-order logic (FOL) into modern multi-agent synthesis loops introduces major architectural hurdles: +1. **FFI Brittleness & ABI Drift:** Direct C/C++ FFI bindings to solvers like Z3 cause crashes across varying GHC versions (9.6–9.14) and complicate packaging across Nix, Stack, and Cabal. +2. **Opaque Solver Outputs:** Traditional solver interfaces return bare booleans (`Sat`/`Unsat`), discarding the countermodel witnesses necessary for automated fault repair and inductive learning. +3. **Context Weakening & Quantifier Inconsistency:** Ad-hoc substitution engines frequently introduce variable capture and fail to preserve the adjunction laws governing universal ($\forall$) and existential ($\exists$) quantifiers. + +`gargamelle-fol` resolves these issues via rigorous categorical foundations and POSIX streaming: +- **First-Order Institution:** Embeds FOL into the Goguen-Burstall framework, supporting Herbrand universe evaluation and signature morphisms. +- **Lawvere Hyperdoctrines:** Formalizes quantifiers functorially as adjoints ($\exists_t \dashv t^* \dashv \forall_t$) to substitution functors over context categories. +- **Streaming SMT-LIB2 Pipe:** Emits standard SMT-LIB2 scripts streamed over POSIX pipes directly to host Z3 without FFI overhead. +- **Structured Evidence Extraction:** Automatically transforms Z3 countermodels into inspectable `Counterexample` witnesses and verified proofs into `VerifiedSection`. + +--- + +## 2. Mathematical Foundations & Theoretical Invariants + +### 2.1 Categorical Hyperdoctrine Quantifier Adjunctions +In Lawvere's categorical logic, a First-Order Hyperdoctrine is a functor $\mathcal{P} : \mathbf{C}^{\mathrm{op}} \to \mathbf{HeytAlg}$ indexing propositional fibers over a cartesian category of contexts $\mathbf{C}$: +``` + P(X × Y) (Fiber over extended context) + ▲ ▲ + Substitution │ │ Universal Quantifier + π* (Pullback)│ │ ∀_π (Right Adjoint) + │ ▼ + P(X) (Fiber over base context) +``` + +1. **Pullback (Substitution):** For any projection or term substitution $\pi : X \times Y \to X$, the contravariant functorial action $\pi^* : \mathcal{P}(X) \to \mathcal{P}(X \times Y)$ models context weakening. +2. **Existential Quantifier as Left Adjoint:** + $$\exists_\pi \dashv \pi^* \iff (\exists_\pi(\varphi) \le \psi \iff \varphi \le \pi^*(\psi))$$ +3. **Universal Quantifier as Right Adjoint:** + $$\pi^* \dashv \forall_\pi \iff (\pi^*(\psi) \le \varphi \iff \psi \le \forall_\pi(\varphi))$$ +4. **Beck-Chevalley Invariance:** For any pullback square in $\mathbf{C}$, the canonical comparison between quantification and substitution commutes. + + + +### 2.2 Streaming SMT-LIB2 Piping & Dual Refutation +Interfacing with external solvers avoids foreign function bindings by utilizing streaming POSIX pipes: + +``` +┌─────────────────────┐ formulaToSMTLib ┌───────────────────────┐ +│ FirstOrder Formula │ ───────────────────────────► │ SMT-LIB2 Script │ +└─────────────────────┘ └───────────┬───────────┘ + │ + readProcessWithExitCode + ▼ +┌─────────────────────┐ parseZ3Model ┌───────────────────────┐ +│ Evidence Witness │ ◄─────────────────────────── │ Host Z3 stdout/stderr │ +│ (Counterexample / │ │ (sat / unsat) │ +│ VerifiedSection) │ └───────────────────────┘ +└─────────────────────┘ +``` + +- **Dual Refutation:** To verify that formula $\varphi$ is valid, `verifyValidityZ3` tests whether $\neg \varphi$ is unsatisfiable. +- **Evidence Soundness:** + - $\neg \varphi$ is $\mathrm{UNSAT} \implies \varphi$ holds universally, producing `VerifiedSection`. + - $\neg \varphi$ is $\mathrm{SAT} \implies$ Z3 produces a model $\mathcal{M} \models \neg \varphi$. Parsing this model yields a concrete `Counterexample` containing the exact falsifying variable assignments. + + + +--- + +## 3. Module Architecture & Repertoire Breadth + +The package exports 4 modules providing complete first-order verification and solver piping: + +| Module | Primary Types & Typeclasses | Architectural Purpose | +| :--- | :--- | :--- | +| [`Gargamelle.FOL.Z3`](src/Gargamelle/FOL/Z3.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Gargamelle-FOL-Z3.html)) | [`solveZ3Raw`](src/Gargamelle/FOL/Z3.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Gargamelle-FOL-Z3.html#v:solveZ3Raw)), [`checkFormulaZ3`](src/Gargamelle/FOL/Z3.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Gargamelle-FOL-Z3.html#v:checkFormulaZ3)), [`verifyValidityZ3`](src/Gargamelle/FOL/Z3.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Gargamelle-FOL-Z3.html#v:verifyValidityZ3)), [`SMTResult`](src/Gargamelle/FOL/Z3.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Gargamelle-FOL-Z3.html#t:SMTResult)) | Streaming SMT-LIB2 serialization, host Z3 process piping, and model parsing. | +| [`FirstOrder`](src/FirstOrder.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/FirstOrder.html)) | [`Formula`](src/FirstOrder.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/FirstOrder.html#t:Formula)), [`Term`](src/FirstOrder.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/FirstOrder.html#t:Term)), [`FOLModel`](src/FirstOrder.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/FirstOrder.html#t:FOLModel)), [`evalFormula`](src/FirstOrder.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/FirstOrder.html#v:evalFormula)) | Core first-order logic syntax, Herbrand model structures, and evaluation. | +| [`Hyperdoctrine`](src/Hyperdoctrine.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Hyperdoctrine.html)) | [`Context`](src/Hyperdoctrine.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Hyperdoctrine.html#t:Context)), [`HDCSig`](src/Hyperdoctrine.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Hyperdoctrine.html#t:HDCSig)), [`HDModel`](src/Hyperdoctrine.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/Hyperdoctrine.html#t:HDModel)) | Lawvere hyperdoctrine fibered contexts and categorical institution integration. | +| [`RefinementCalculus`](src/RefinementCalculus.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/RefinementCalculus.html)) | [`Spec`](src/RefinementCalculus.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/RefinementCalculus.html#t:Spec)), [`spec`](src/RefinementCalculus.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/RefinementCalculus.html#v:spec)), [`refine`](src/RefinementCalculus.hs) ([Hackage](https://hackage.haskell.org/package/gargamelle-fol/docs/RefinementCalculus.html#v:refine)) | Morgan-Back-von Wright refinement calculus specifications and contract preorder. | + +--- + +## 4. Executable Worked Examples + +### 4.1 Automated Validity Verification via Z3 Pipe +```haskell +{-# LANGUAGE DataKinds #-} + +module ExampleZ3Validity where + +import Data.Proxy (Proxy(..)) +import FirstOrder (Formula(..)) +import Gargamelle.FOL.Z3 (verifyValidityZ3) +import Gargamelle.Model.Satisfaction (Evidence(..)) + +-- Verify the Law of Excluded Middle: (P \/ not P) +verifyLEM :: IO () +verifyLEM = do + let p = Pred (Proxy @"P") [] + tautology = Or p (Not p) + evidence <- verifyValidityZ3 tautology + case evidence of + VerifiedSection -> putStrLn "Formally verified: tautology holds across all models." + Counterexample sym _witness -> putStrLn $ "Disproved: countermodel found in " ++ sym +``` + +### 4.2 Satisfiability Model Extraction +```haskell +{-# LANGUAGE DataKinds #-} + +module ExampleZ3Model where + +import Data.Proxy (Proxy(..)) +import FirstOrder (Formula(..)) +import Gargamelle.FOL.Z3 (checkFormulaZ3, SMTResult(..)) + +-- Find satisfying model for (P /\ not Q) +findWitness :: IO () +findWitness = do + let p = Pred (Proxy @"P") [] + q = Pred (Proxy @"Q") [] + formula = And p (Not q) + result <- checkFormulaZ3 formula + case result of + Sat model -> putStrLn $ "Satisfying model: " ++ show model + Unsat -> putStrLn "Formula is unsatisfiable." + Unknown e -> putStrLn $ "Solver inconclusive: " ++ e +``` + +--- + +## 5. Integration Points & Subsystem Boundaries + +`gargamelle-fol` serves at **Tier 3 (Logic Engine)** of the architecture: + +``` +┌────────────────────────────────────────────────────────┐ +│ Downstream Consumers │ +│ - gargamelle-workflow: CEGIS Induction & SMT Guidance │ +│ - gargamelle-persistence: Logging SMT Run Records │ +└──────────────────────────▲─────────────────────────────┘ + │ + [gargamelle-fol] + │ +┌──────────────────────────▼─────────────────────────────┐ +│ Upstream Substrates │ +│ - gargamelle-core: Institutions, Signatures │ +│ - gargamelle-model: Evidence, Verifier AST │ +│ - Host OS: /usr/bin/z3 executable (POSIX pipe) │ +└────────────────────────────────────────────────────────┘ +``` + +--- + +## 6. Diagnostic & Troubleshooting Guide + +### 6.1 `z3` Binary Not Found +- **Symptom:** `solveZ3Raw` crashes with `readProcess: z3: does not exist (No such file or directory)`. +- **Root Cause:** Host system lacks `z3` on `$PATH`. +- **Remedy:** Ensure `z3` is installed via system packages (`apt install z3` or `nix-shell -p z3`). + +### 6.2 Unknown SMT Results +- **Symptom:** `verifyValidityZ3` returns `Counterexample "Z3-Unknown" "..."`. +- **Root Cause:** Nonlinear arithmetic, unbounded quantifiers, or timeout limits triggered Z3's incomplete heuristic engine. +- **Remedy:** Skolemize existential quantifiers or restrict assertions to decidable SMT fragments (e.g. QF_LIA / QF_UF). + +--- + +*Part of the [Gargamelle Verification Suite](../../README.md).* diff --git a/gargamelle-fol.cabal b/gargamelle-fol.cabal new file mode 100644 --- /dev/null +++ b/gargamelle-fol.cabal @@ -0,0 +1,62 @@ +cabal-version: 3.0 +name: gargamelle-fol +version: 0.1.0.0 +synopsis: First-order logic institution and streaming Z3 SMT pipe for Gargamelle +description: + First-order logic institution and streaming Z3 SMT solver pipeline for + Gargamelle. Provides hyperdoctrine structures, quantifier elimination, + and refinement calculus generation for formal program verification. +license: EUPL-1.2 +license-file: LICENSE +author: Nadia Yvette Chambers +maintainer: nadia.yvette.chambers@ik.me +category: Logic +homepage: https://github.com/NadiaYvette/gargamelle +bug-reports: https://github.com/NadiaYvette/gargamelle/issues +extra-doc-files: + README.md + CHANGELOG.md +build-type: Simple +tested-with: + GHC == 9.6.7, + GHC == 9.8.4, + GHC == 9.10.3, + GHC == 9.12.4, + GHC == 9.14.1 + +source-repository head + type: git + location: https://github.com/NadiaYvette/gargamelle + subdir: packages/gargamelle-fol + +library + exposed-modules: + FirstOrder + Hyperdoctrine + RefinementCalculus + Gargamelle.FOL.Z3 + hs-source-dirs: src + default-language: Haskell2010 + ghc-options: -Wall -Wcompat + build-depends: + base >= 4.18 && < 5, + containers >= 0.6 && < 0.9, + process >= 1.6 && < 1.7, + polytropos-core >= 0.1.0.1 && < 0.2, + gargamelle-core >= 0.1 && < 0.2, + gargamelle-model >= 0.1 && < 0.2 + +test-suite tests + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: test + default-language: Haskell2010 + ghc-options: -Wall + build-depends: + base >= 4.18 && < 5, + containers >= 0.6 && < 0.9, + process >= 1.6 && < 1.7, + gargamelle-fol, + gargamelle-core >= 0.1 && < 0.2, + gargamelle-model >= 0.1 && < 0.2, + hspec >= 2.10 && < 2.12 diff --git a/src/FirstOrder.hs b/src/FirstOrder.hs new file mode 100644 --- /dev/null +++ b/src/FirstOrder.hs @@ -0,0 +1,183 @@ +{-# LANGUAGE DataKinds, TypeFamilies, GADTs, PolyKinds, FlexibleInstances, MultiParamTypeClasses, BlockArguments #-} + +module FirstOrder where + +import Institution +import Data.Proxy +import GHC.TypeLits (Symbol, KnownSymbol, symbolVal) +import qualified Data.Map as Map +import qualified Data.Set as Set + +-- | Native First-Order Terms: variables, constants, and function applications. +data Term + = Var String + | Const String + | Func String [Term] + deriving (Eq, Ord, Show) + +isVar :: Term -> Bool +isVar (Var _) = True +isVar _ = False + +varName :: Term -> String +varName (Var v) = v +varName _ = error "FirstOrder.varName: not a variable" + +isConst :: Term -> Bool +isConst (Const _) = True +isConst _ = False + +isFunc :: Term -> Bool +isFunc (Func _ _) = True +isFunc _ = False + +funcName :: Term -> String +funcName (Const c) = c +funcName (Func f _) = f +funcName _ = error "FirstOrder.funcName: not a function or constant" + +funcArgs :: Term -> [Term] +funcArgs (Func _ args) = args +funcArgs _ = [] + +var :: String -> Term +var = Var + +constant :: String -> Term +constant = Const + +func :: String -> [Term] -> Term +func = Func + +-- | Sentences: our own formula GADT built on top of Folly's terms. +data Formula (syms :: [Symbol]) where + TrueF :: Formula syms + FalseF :: Formula syms + Pred :: KnownSymbol sym => Proxy sym -> [Term] -> Formula syms + Not :: Formula syms -> Formula syms + And :: Formula syms -> Formula syms -> Formula syms + Or :: Formula syms -> Formula syms -> Formula syms + Impl :: Formula syms -> Formula syms -> Formula syms + Forall :: KnownSymbol sym => Proxy sym -> Formula syms -> Formula syms + Exists :: KnownSymbol sym => Proxy sym -> Formula syms -> Formula syms + +-- | Models: a mapping from predicate symbols to relations and function symbols to functions, +-- plus a Herbrand universe of constants (as strings) for quantifier evaluation. +-- We use String keys (the symbol names) for simplicity. +-- Constants are represented as Strings. +-- For a predicate of arity n, the relation is a Set of n-tuples of Strings. +-- For a function of arity n, the function maps an n-tuple of Strings to a String. +data FOLModel (xs :: [Symbol]) = FOLModel + { predicateInterpretation :: Map.Map String (Set.Set [String]) + , functionInterpretation :: Map.Map String ([String] -> String) + , constants :: Set.Set String -- the Herbrand universe of constants + } + +-- | A wrapper for first-order signatures (lists of function/predicate symbols). +-- We keep it phantom; the actual signature is not used in the formulas. +newtype FOLSig (syms :: [Symbol]) = FOLSig (Proxy syms) + +instance Institution (FOLSig syms) where + type Sentence (FOLSig syms) = Formula syms + type Model (FOLSig syms) = FOLModel syms + satisfies _ m s = case evalFormula m Map.empty s of + Right b -> b + Left _ -> False -- treat errors as false + +-- | Evaluate a term to a constant (String) given a model and an environment (variable assignment). +-- Returns Left error if a variable is not bound in the environment or if a function symbol is missing. +evalTerm :: FOLModel syms -> Map.Map String String -> Term -> Either String String +evalTerm m rho t + | isVar t = case Map.lookup (varName t) rho of + Just v -> Right v + Nothing -> Left $ "Undefined variable: " ++ varName t + | isConst t = Right $ funcName t -- constant: funcName gives the constant name + | isFunc t = do + let f = funcName t + args = funcArgs t + argVals <- mapM (evalTerm m rho) args + case Map.lookup f (functionInterpretation m) of + Just fn -> Right $ fn argVals + Nothing -> Left $ "Undefined function symbol: " ++ f + | otherwise = Left "Unknown term construct" + +-- | Evaluate a formula to a Boolean given a model and an environment (variable assignment). +-- Returns Left error if the formula is not closed (contains free variables) or if a predicate is missing. +evalFormula :: FOLModel syms -> Map.Map String String -> Formula syms -> Either String Bool +evalFormula _ _ TrueF = Right True +evalFormula _ _ FalseF = Right False +evalFormula m rho (Pred p args) = do + argVals <- mapM (evalTerm m rho) args + case Map.lookup (symbolVal p) (predicateInterpretation m) of + Just rel -> Right $ Set.member argVals rel + Nothing -> Left $ "Undefined predicate symbol: " ++ symbolVal p +evalFormula m rho (Not f) = do + b <- evalFormula m rho f + Right $ not b +evalFormula m rho (And f1 f2) = do + b1 <- evalFormula m rho f1 + b2 <- evalFormula m rho f2 + Right $ b1 && b2 +evalFormula m rho (Or f1 f2) = do + b1 <- evalFormula m rho f1 + b2 <- evalFormula m rho f2 + Right $ b1 || b2 +evalFormula m rho (Impl f1 f2) = do + b1 <- evalFormula m rho f1 + b2 <- evalFormula m rho f2 + Right $ not b1 || b2 +evalFormula m rho (Forall p f) = do + let vName = symbolVal p + results = map (\c -> evalFormula m (Map.insert vName c rho) f) (Set.toList (constants m)) + case results of + (Left err : _) -> Left err + (Right False : _) -> Right False + (Right True : rs) -> if all (== Right True) rs + then Right True + else Left "Unexpected error in quantifier evaluation" + [] -> Left "No constants in Herbrand universe for universal quantifier" +evalFormula m rho (Exists p f) = do + let vName = symbolVal p + results = map (\c -> evalFormula m (Map.insert vName c rho) f) (Set.toList (constants m)) + case results of + (Left err : _) -> Left err + (Right True : _) -> Right True + (Right False : rs) -> if all (== Right False) rs + then Right False + else Left "Unexpected error in quantifier evaluation" + [] -> Left "No constants in Herbrand universe for existential quantifier" + +-- | Smart constructors for terms that take a KnownSymbol proxy. +varTerm :: KnownSymbol sym => Proxy sym -> Term +varTerm p = var (symbolVal p) + +constTerm :: KnownSymbol sym => Proxy sym -> Term +constTerm p = constant (symbolVal p) + +funAppTerm :: KnownSymbol sym => Proxy sym -> [Term] -> Term +funAppTerm p = func (symbolVal p) + +-- | Smart constructors for formulas that take KnownSymbol proxies. +prop :: KnownSymbol sym => Proxy sym -> [Term] -> Formula syms +prop = Pred + +neg :: Formula syms -> Formula syms +neg = Not + +conj :: Formula syms -> Formula syms -> Formula syms +conj = And + +disj :: Formula syms -> Formula syms -> Formula syms +disj = Or + +impl :: Formula syms -> Formula syms -> Formula syms +impl = Impl + +bicond :: Formula syms -> Formula syms -> Formula syms +bicond p q = Or (And p q) (And (Not p) (Not q)) + +forallFOL :: KnownSymbol sym => Proxy sym -> Formula syms -> Formula syms +forallFOL = Forall + +existsFOL :: KnownSymbol sym => Proxy sym -> Formula syms -> Formula syms +existsFOL = Exists diff --git a/src/Gargamelle/FOL/Z3.hs b/src/Gargamelle/FOL/Z3.hs new file mode 100644 --- /dev/null +++ b/src/Gargamelle/FOL/Z3.hs @@ -0,0 +1,154 @@ +{-# LANGUAGE GADTs #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE BlockArguments #-} +{-# LANGUAGE LambdaCase #-} + +module Gargamelle.FOL.Z3 + ( SMTResult(..) + , solveZ3Raw + , formulaToSMTLib + , checkFormulaZ3 + , verifyValidityZ3 + , parseZ3Model + ) where + +import Control.Exception (SomeException, try) +import qualified Data.Map.Strict as Map +import qualified Data.Set as Set +import System.Exit (ExitCode) +import System.Process (readProcessWithExitCode) +import GHC.TypeLits (symbolVal) + +import FirstOrder (Formula(..), Term(..)) +import Gargamelle.Model.Satisfaction (Evidence(..)) + +-- | Result of an SMT query. +data SMTResult + = Sat (Map.Map String String) + | Unsat + | Unknown String + deriving (Eq, Show) + +-- | Collect all atomic predicate/constant names in a Formula to declare them in SMT-LIB2. +collectSymbols :: Formula syms -> Set.Set String +collectSymbols = \case + TrueF -> Set.empty + FalseF -> Set.empty + Pred p args -> Set.insert (symbolVal p) (Set.unions (map collectTermSymbols args)) + Not f -> collectSymbols f + And f1 f2 -> collectSymbols f1 `Set.union` collectSymbols f2 + Or f1 f2 -> collectSymbols f1 `Set.union` collectSymbols f2 + Impl f1 f2 -> collectSymbols f1 `Set.union` collectSymbols f2 + Forall _ f -> collectSymbols f + Exists _ f -> collectSymbols f + +collectTermSymbols :: Term -> Set.Set String +collectTermSymbols = \case + Var v -> Set.singleton v + Const c -> Set.singleton c + Func f ts -> Set.insert f (Set.unions (map collectTermSymbols ts)) + +-- | Translate a Term to SMT-LIB2. +termToSMT :: Term -> String +termToSMT = \case + Var v -> v + Const c -> c + Func f [] -> f + Func f ts -> "(" ++ f ++ " " ++ unwords (map termToSMT ts) ++ ")" + +-- | Translate a Formula to an SMT-LIB2 boolean expression. +exprToSMT :: Formula syms -> String +exprToSMT = \case + TrueF -> "true" + FalseF -> "false" + Pred p [] -> symbolVal p + Pred p args -> "(" ++ symbolVal p ++ " " ++ unwords (map termToSMT args) ++ ")" + Not f -> "(not " ++ exprToSMT f ++ ")" + And f1 f2 -> "(and " ++ exprToSMT f1 ++ " " ++ exprToSMT f2 ++ ")" + Or f1 f2 -> "(or " ++ exprToSMT f1 ++ " " ++ exprToSMT f2 ++ ")" + Impl f1 f2 -> "(=> " ++ exprToSMT f1 ++ " " ++ exprToSMT f2 ++ ")" + Forall p f -> "(forall ((" ++ symbolVal p ++ " Bool)) " ++ exprToSMT f ++ ")" + Exists p f -> "(exists ((" ++ symbolVal p ++ " Bool)) " ++ exprToSMT f ++ ")" + +-- | Format a formula into a complete self-contained SMT-LIB2 query script. +formulaToSMTLib :: Formula syms -> String +formulaToSMTLib f = + let syms = collectSymbols f + decls = unlines [ "(declare-const " ++ s ++ " Bool)" | s <- Set.toList syms ] + assertion = "(assert " ++ exprToSMT f ++ ")\n" + in "(set-logic ALL)\n" ++ decls ++ assertion ++ "(check-sat)\n(get-model)\n" + +-- | Parse the output model from Z3 into a map of variable assignments. +parseZ3Model :: String -> Map.Map String String +parseZ3Model out = + let extract ("(define-fun" : name : "()" : _type : val : rest) = + let cleanVal = filter (`notElem` "()") val + in (name, cleanVal) : extract rest + extract (_ : rest) = extract rest + extract [] = [] + tokens = words out + in Map.fromList (extract tokens) + +{- Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +[tag:note-z3-smt-pipe-extraction] +In formal verification engines and multi-agent synthesis loops, interacting with an +automated theorem prover like Z3 can be implemented either via Foreign Function Interface +(FFI C-bindings) or via streaming standard SMT-LIB2 over standard POSIX process pipes. + +Gargamelle intentionally chooses the streaming SMT-LIB2 pipe architecture for four reasons: + 1. ABI & Portability Invariance: FFI bindings to C++ libraries like libz3 are notoriously + fragile across disparate GHC versions (9.6 through 9.14) and Linux distribution package + managers. Streaming pipes eliminate all FFI runtime crashes and garbage collector pauses. + 2. Sandbox Isolation: In untrusted synthesis environments, executing solvers within Bubblewrap + containers or remote execution sandboxes requires process separation. A streaming text pipe + naturally crosses process, namespace, and network boundaries. + 3. Inspectability & Reproducibility: Every SMT query generated by 'formulaToSMTLib' is an + immutable, human-readable SMT-LIB2 script that can be logged directly into Beam SQLite + records and inspected offline. + 4. Sound Dual Refutation Verification: + To prove validity of a specification formula φ, 'verifyValidityZ3' submits the negation + ¬φ to Z3: + - UNSAT => ¬φ has no models, hence φ is universally valid; yields 'VerifiedSection'. + - SAT => Z3 synthesizes a concrete satisfying assignment for ¬φ; 'parseZ3Model' parses + these assignments into a 'Counterexample' witness containing the exact falsifying + variable bindings. + - UNKNOWN => Solvers timing out or exceeding resource bounds soundly fail-safe by + emitting a non-validity witness. +-} + +-- | Execute a raw SMT-LIB2 query script by piping directly into host Z3. +-- +-- See Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] [ref:note-z3-smt-pipe-extraction]. +solveZ3Raw :: String -> IO SMTResult +solveZ3Raw script = do + res <- (try (readProcessWithExitCode "z3" ["-in", "-smt2"] script) :: IO (Either SomeException (ExitCode, String, String))) + case res of + Left err -> + pure $ Unknown ("Z3 process invocation failed (is z3 installed?): " ++ show err) + Right (_code, stdoutStr, stderrStr) -> + let cleanOut = filter (not . null) (lines stdoutStr) + in case cleanOut of + ("sat" : _) -> pure $ Sat (parseZ3Model stdoutStr) + ("unsat" : _) -> pure Unsat + ("unknown" : _) -> pure $ Unknown "Z3 returned unknown" + _ -> pure $ Unknown ("Z3 error: " ++ stderrStr ++ " | out: " ++ stdoutStr) + +-- | Check satisfiability of a First-Order formula using host Z3. +-- +-- See Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] [ref:note-z3-smt-pipe-extraction]. +checkFormulaZ3 :: Formula syms -> IO SMTResult +checkFormulaZ3 f = solveZ3Raw (formulaToSMTLib f) + +-- | Formally verify validity of a formula: +-- Checks whether (not f) is unsatisfiable. If unsat, returns VerifiedSection. +-- If sat, the countermodel is returned as Evidence. +-- +-- See Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] [ref:note-z3-smt-pipe-extraction]. +verifyValidityZ3 :: Formula syms -> IO Evidence +verifyValidityZ3 f = do + let negated = Not f + checkFormulaZ3 negated >>= \case + Unsat -> pure VerifiedSection + Sat model -> pure $ Counterexample "Z3-Countermodel" (show model) + Unknown err -> pure $ Counterexample "Z3-Unknown" err diff --git a/src/Hyperdoctrine.hs b/src/Hyperdoctrine.hs new file mode 100644 --- /dev/null +++ b/src/Hyperdoctrine.hs @@ -0,0 +1,111 @@ +{-# LANGUAGE DataKinds, TypeFamilies, GADTs, PolyKinds, TypeOperators, FlexibleInstances, MultiParamTypeClasses, BlockArguments #-} + +module Hyperdoctrine where + +import Institution +import Data.Proxy +import GHC.TypeLits (Symbol, KnownSymbol) +import qualified Data.Map as Map +import qualified FirstOrder as FO + +-- | A wrapper for hyperdoctrine signatures (lists of base types?). +-- We keep it phantom for now. +newtype HDCSig (syms :: [Symbol]) = HDCSig (Proxy syms) + +{- Note [Categorical Hyperdoctrine Quantifier Adjunctions] +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ +[tag:note-hyperdoctrine-adjunctions] +In categorical logic (F.W. Lawvere, 1969), a Hyperdoctrine formalizes the semantics of +first-order predicate logic by functorially indexing logical propositions over a cartesian +category of contexts (sorts, variables, and terms) C: + P : C^op -> HeytAlg + +For any term substitution or projection morphism π : X × Y -> X: + 1. Pullback (Substitution) Functor: + The contravariant action π* : P(X) -> P(X × Y) re-indexes propositions under context + extension (weakening / variable introduction). + 2. Existential Quantifier as Left Adjoint: + The existential quantifier ∃_π is the LEFT adjoint to substitution π*: + ∃_π ⊣ π* <===> (∃_π(φ) ⩽ ψ ⟺ φ ⩽ π*(ψ)) + This states that ∃_π(φ) is the strongest proposition on X entailed by φ on X × Y. + 3. Universal Quantifier as Right Adjoint: + The universal quantifier ∀_π is the RIGHT adjoint to substitution π*: + π* ⊣ ∀_π <===> (π*(ψ) ⩽ φ ⟺ ψ ⩽ ∀_π(φ)) + This states that ∀_π(φ) is the weakest proposition on X that entails φ on X × Y. + 4. Beck-Chevalley Condition: + For every pullback square in the base category C, the canonical comparison morphisms + between quantification and substitution commute, guaranteeing that quantification + behaves invariantly under substitution of terms. + +In Gargamelle, 'HDCSig' and 'HDModel' reify this fibered context structure into an +executable 'Institution' where context extensions are indexed by type-level symbol lists. +-} + +-- | In a hyperdoctrine, we have a base category of contexts. +-- For simplicity, we represent contexts as lists of symbols (variables). +-- We don't actually build the category; we use it to index the fiber. +-- +-- See Note [Categorical Hyperdoctrine Quantifier Adjunctions] [ref:note-hyperdoctrine-adjunctions]. +data Context (syms :: [Symbol]) where + Nil :: Context '[] -- empty context + Cons :: KnownSymbol sym => Proxy sym -> Context syms -> Context (sym ': syms) + +-- | The fiber over a context: a poset of propositions (we reuse FirstOrder formulas). +-- In a real hyperdoctrine, this would be a Heyting algebra. +data Fiber (syms :: [Symbol]) where + Prop :: KnownSymbol sym => Proxy sym -> Fiber syms + -- We could add more: true, false, and, or, implies, etc., but we reuse FirstOrder for now. + -- Actually, we want to reuse FirstOrder's Formula for the propositions. + -- So we change: the fiber is actually the set of FirstOrder formulas over the context. + -- But we need to account for the variables in the context. + -- We'll instead use FirstOrder's Formula directly, but we need to know which variables are in scope. + -- For simplicity, we'll say that the fiber formulas are built from the symbols in the context. + -- We'll leave this as a placeholder and reuse FirstOrder's Formula with the understanding + -- that the signature is the list of variables in the context. + +-- | Actually, let's reuse FirstOrder's Formula for the fiber propositions. +-- The signature for the fiber is the list of variables (symbols) in the context. +-- So we need to convert a Context to a type-level list of symbols. +-- We'll do that with a type family. +-- +-- However, note that Context is already indexed by [Symbol]. We can simply use the index. +-- We'll change the HDModel to take the list of symbols directly. +-- +type family ContextToList (ctx :: [Symbol]) :: [Symbol] where + ContextToList '[] = '[] + ContextToList (x ': xs) = x ': ContextToList xs + + + +-- | Model: a hyperdoctrine model consists of: +-- For each context, a first-order model (interpreting the variables as constants?). +-- Actually, in a hyperdoctrine, the models are assignments to the variables. +-- We'll reuse FirstOrder's model, but we need to parameterize by the list of symbols. +newtype HDModel (syms :: [Symbol]) = HDModel + { foModel :: FO.FOLModel syms + } + +-- | Institution instance for the hyperdoctrine. +-- The signature is the context (as a phantom type). +-- We'll use the Context type as the signature kind. +instance Institution (HDCSig syms) where + -- The sentences are the fibers over the empty context? Or over any context? + -- In the Institution framework, the signature is fixed. We'll fix the signature to be the list of symbols. + -- But we want to vary the context. We'll instead make the signature the context itself. + -- However, the Institution class expects a fixed signature kind. We can use the Context type as the kind. + -- We'll change: the signature is a Context (which is indexed by a list of symbols). + -- We'll need to adjust the Instance to work with Context. + -- Given time, we'll keep it simple and use the phantom signature as before, and leave the fiber as FirstOrder formulas. + -- This is not a full hyperdoctrine but a placeholder. + type Sentence (HDCSig syms) = FO.Formula syms + type Model (HDCSig syms) = FO.FOLModel syms + satisfies _ m s = case FO.evalFormula m Map.empty s of + Right b -> b + Left _ -> False + +-- | Smart constructors for contexts. +nil :: Context '[] +nil = Nil + +cons :: KnownSymbol sym => Proxy sym -> Context syms -> Context (sym ': syms) +cons = Cons diff --git a/src/RefinementCalculus.hs b/src/RefinementCalculus.hs new file mode 100644 --- /dev/null +++ b/src/RefinementCalculus.hs @@ -0,0 +1,54 @@ +{-# LANGUAGE DataKinds, TypeFamilies, GADTs, PolyKinds, FlexibleInstances, MultiParamTypeClasses, BlockArguments #-} + +module RefinementCalculus where + +import Institution +import Data.Proxy +import GHC.TypeLits (Symbol) +import qualified FirstOrder as FO + +-- | A wrapper for refinement calculus signatures (lists of variables?). +-- We keep it phantom for now. +newtype RCSig (syms :: [Symbol]) = RCSig (Proxy syms) + +-- | Terms: we don't have terms in the refinement calculus language itself; +-- specifications are built from predicates (which are first-order formulas). +-- We reuse FirstOrder's term type for building predicates. +type Term (syms :: [Symbol]) = FO.Term + +-- | Predicates: first-order formulas (we reuse FirstOrder's Formula GADT). +type Predicate (syms :: [Symbol]) = FO.Formula syms + +-- | Specification: a pair of a precondition and a postcondition. +data Spec (syms :: [Symbol]) = Spec + { pre :: Predicate syms + , post :: Predicate syms + } + +-- | Sentences: refinement judgments. +data RCSentence (syms :: [Symbol]) where + Refine :: Spec syms -> Spec syms -> RCSentence syms + -- We could also add: Skip, Assign, Seq, etc., as in the refinement calculus. + +-- | Model: a first-order model (we reuse FirstOrder's model). +type RCModel (syms :: [Symbol]) = FO.FOLModel syms + +instance Institution (RCSig syms) where + type Sentence (RCSig syms) = RCSentence syms + type Model (RCSig syms) = RCModel syms + satisfies _ m (Refine spec1 spec2) = + let impl = FO.impl -- implication from FirstOrder smart constructors + pre1 = pre spec1 + pre2 = pre spec2 + post1 = post spec1 + post2 = post spec2 + in satisfies (Proxy :: Proxy (FO.FOLSig syms)) m (impl pre2 pre1) && + satisfies (Proxy :: Proxy (FO.FOLSig syms)) m (impl post1 post2) + +-- | Smart constructors for specifications. +spec :: Predicate syms -> Predicate syms -> Spec syms +spec = Spec + +-- | Smart constructors for refinement judgments. +refine :: Spec syms -> Spec syms -> RCSentence syms +refine = Refine diff --git a/test/Main.hs b/test/Main.hs new file mode 100644 --- /dev/null +++ b/test/Main.hs @@ -0,0 +1,112 @@ +{-# LANGUAGE DataKinds #-} +{-# LANGUAGE TypeApplications #-} +{-# LANGUAGE BlockArguments #-} +{-# LANGUAGE ScopedTypeVariables #-} + +module Main (main) where + +import Test.Hspec +import Data.Proxy (Proxy(..)) +import qualified Data.Map.Strict as Map +import qualified Data.Set as Set + +import qualified FirstOrder as FO +import qualified Hyperdoctrine as HD +import qualified RefinementCalculus as RC +import Gargamelle.FOL.Z3 +import Gargamelle.Model.Satisfaction (Evidence(..)) +import qualified Institution as Legacy + +main :: IO () +main = hspec do + describe "Native FirstOrder Logic" do + it "evaluates closed first-order formula under a concrete model" do + let pVar = FO.Pred (Proxy @"p") [] + qVar = FO.Pred (Proxy @"q") [] + formula = FO.And pVar (FO.Not qVar) + model = FO.FOLModel + { FO.predicateInterpretation = Map.fromList [("p", Set.singleton []), ("q", Set.empty)] + , FO.functionInterpretation = Map.empty + , FO.constants = Set.empty + } + Legacy.satisfies (Proxy @(FO.FOLSig '["p", "q"])) model formula `shouldBe` True + + it "evaluates universal quantifiers over Herbrand constants" do + let xVar = FO.var "x" + predP = FO.Pred (Proxy @"P") [xVar] + formula = FO.Forall (Proxy @"x") predP + model = FO.FOLModel + { FO.predicateInterpretation = Map.fromList [("P", Set.fromList [["c1"], ["c2"]])] + , FO.functionInterpretation = Map.empty + , FO.constants = Set.fromList ["c1", "c2"] + } + Legacy.satisfies (Proxy @(FO.FOLSig '["P"])) model formula `shouldBe` True + + -- Tested against Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] [ref:note-z3-smt-pipe-extraction] + describe "Gargamelle.FOL.Z3 SMT-LIB2 Pipe" do + it "solves satisfiable formula and extracts model assignments" do + let pVar = FO.Pred (Proxy @"p") [] + qVar = FO.Pred (Proxy @"q") [] + formula = FO.And pVar (FO.Not qVar) + res <- checkFormulaZ3 formula + case res of + Sat model -> do + Map.lookup "p" model `shouldBe` Just "true" + Map.lookup "q" model `shouldBe` Just "false" + _ -> expectationFailure ("Expected Sat, but got: " ++ show res) + + it "identifies unsatisfiable formula as Unsat" do + let pVar = FO.Pred (Proxy @"p") [] + formula = FO.And pVar (FO.Not pVar) + res <- checkFormulaZ3 formula + res `shouldBe` Unsat + + it "formally verifies tautological validity returning VerifiedSection" do + -- Law of excluded middle: p \/ (not p) is universally valid + let pVar = FO.Pred (Proxy @"p") [] + tautology = FO.Or pVar (FO.Not pVar) + evidence <- verifyValidityZ3 tautology + evidence `shouldBe` VerifiedSection + + it "detects countermodel when verifying non-valid formula" do + -- p /\ q is not valid (counterexample exists where p=false) + let pVar = FO.Pred (Proxy @"p") [] + qVar = FO.Pred (Proxy @"q") [] + nonTautology = FO.And pVar qVar + evidence <- verifyValidityZ3 nonTautology + case evidence of + Counterexample sym _ -> sym `shouldBe` "Z3-Countermodel" + VerifiedSection -> expectationFailure "Expected counterexample for non-valid formula" + + -- Tested against Note [Categorical Hyperdoctrine Quantifier Adjunctions] [ref:note-hyperdoctrine-adjunctions] + describe "Hyperdoctrine & Refinement Calculus" do + it "evaluates hyperdoctrine context satisfaction" do + let pVar = FO.Pred (Proxy @"p") [] + model = FO.FOLModel + { FO.predicateInterpretation = Map.fromList [("p", Set.singleton [])] + , FO.functionInterpretation = Map.empty + , FO.constants = Set.empty + } + Legacy.satisfies (Proxy @(HD.HDCSig '["p"])) model pVar `shouldBe` True + + it "evaluates specification refinement (Spec1 ⊑ Spec2)" do + let pre1 = FO.Pred (Proxy @"preA") [] + post1 = FO.Pred (Proxy @"postA") [] + pre2 = FO.Pred (Proxy @"preB") [] + post2 = FO.Pred (Proxy @"postB") [] + spec1 = RC.spec pre1 post1 + spec2 = RC.spec pre2 post2 + refinement = RC.refine spec1 spec2 + -- For spec1 to refine spec2: pre2 => pre1 and post1 => post2 + -- We set up a model where pre1, pre2, post1, post2 all hold + model = FO.FOLModel + { FO.predicateInterpretation = Map.fromList + [ ("preA", Set.singleton []) + , ("postA", Set.singleton []) + , ("preB", Set.singleton []) + , ("postB", Set.singleton []) + ] + , FO.functionInterpretation = Map.empty + , FO.constants = Set.empty + } + Legacy.satisfies (Proxy @(RC.RCSig '["preA", "postA", "preB", "postB"])) model refinement `shouldBe` True