gargamelle-fol (empty) → 0.1.0.0

raw patch · 9 files changed

+1043/−0 lines, 9 filesdep +basedep +containersdep +gargamelle-core

Dependencies added: base, containers, gargamelle-core, gargamelle-fol, gargamelle-model, hspec, polytropos-core, process

Files

• CHANGELOG.md +7/−0
• LICENSE +189/−0
• README.md +171/−0
• gargamelle-fol.cabal +62/−0
• src/FirstOrder.hs +183/−0
• src/Gargamelle/FOL/Z3.hs +154/−0
• src/Hyperdoctrine.hs +111/−0
• src/RefinementCalculus.hs +54/−0
• test/Main.hs +112/−0

@@ -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.

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@@ -0,0 +1,171 @@+# `gargamelle-fol` — First-Order Logic Institution, Hyperdoctrines & Streaming Z3 SMT-LIB2++[![Haskell](https://img.shields.io/badge/Language-Haskell2010-purple.svg)](https://haskell.org)+[![License: EUPL-1.2](https://img.shields.io/badge/License-EUPL--1.2-blue.svg)](https://spdx.org/licenses/EUPL-1.2.html)+[![Tier 3 Logic Engine](https://img.shields.io/badge/Tier-3%20Logic%20%26%20Institution%20Engine-teal.svg)](../../README.md#architectural-tier-map)++---++## 1. 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.++<!-- See Note [Categorical Hyperdoctrine Quantifier Adjunctions] [ref:note-hyperdoctrine-adjunctions] -->++### 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.++<!-- See Note [Streaming Z3 SMT-LIB2 Piping and Model Extraction] [ref:note-z3-smt-pipe-extraction] -->++---++## 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).*

@@ -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

@@ -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

@@ -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

@@ -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

@@ -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

@@ -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