unicode-fol-kit 0.4.0

Parser and toolkit for first-order logic formulas using Unicode operators

unicode-fol-kit

A Python toolkit for parsing and working with first-order logic (FOL) formulas written with Unicode operators. The single parser class MSFLParser supports four modes — classical FOL, many-sorted FOL (MSFOL), many-sorted fuzzy logic (MSFL), and single-sorted fuzzy logic (FL, Łukasiewicz) — selected by constructor flags.

Features

• Four parser modes — FOL, many-sorted FOL (MSFOL), many-sorted fuzzy/Łukasiewicz logic (MSFL), and single-sorted fuzzy/Łukasiewicz logic (FL), all from one class
• Unicode surface syntax — natural symbols (∀ ∃ ∧ ∨ ¬ → ↔ ⊕ ⊗ = ≠ ≤ ≥) with no ASCII fallbacks needed
• Sorted quantifiers and constants — ∀x:Human P(x), P(alice:Human) in MSFOL and MSFL modes
• Łukasiewicz operators — weak ∧ / ∨ (min/max), strong ⊗ / ⊕ (t-norm/t-conorm), and Łukasiewicz ¬ → ↔ in MSFL mode
• Full AST — all standard FOL constructs plus MSFL-specific nodes, all as Python dataclasses
• Reductions — to_msfol() lowers Łukasiewicz operators to classical nodes; to_fol() further eliminates sorts via relativisation
• Serialisation — convert formulas to/from JSON dictionaries; round-trip safe
• Tree view — render any formula as a readable ASCII tree
• Unicode round-trip — to_unicode_str() renders any node back to a parseable Unicode formula; re-parsing in the matching mode yields a structurally equal AST
• LaTeX export — to_latex() renders any node as LaTeX math-mode markup, with the same precedence-aware parenthesisation as the Unicode renderer
• Normal forms — to_nnf(), to_pnf(), to_cnf(), to_dnf() (equivalence-preserving), to_tseitin_cnf() (equisatisfiable, no blow-up), and skolemize() (satisfiability-preserving) for classical FOL
• Horn check — is_horn() reports whether a formula's clausal form consists of Horn clauses
• Traversal API — walk(), subformulas(), atoms(), variables(), count(), depth() on every node
• Graphviz export — to_dot() renders the AST as a Graphviz DOT digraph
• Z3 export — translate formulas to Z3 expressions for SMT solving
• Satisfiability / validity / models — is_satisfiable(), is_valid(), and get_model() (counterexample extraction) via Z3
• Prover9 export — translate formulas to Prover9 syntax for automated theorem proving
• TPTP export — translate formulas to TPTP syntax
• Equivalence checking — check if two formulas are logically equivalent via Z3
• Entailment checking — check if a conclusion follows from premises via Prover9
• Built-in resolution prover — prove() and is_valid_resolution() decide entailment/validity in-process (sound first-order resolution, no external solver needed)
• Lambda abstraction — λx. φ syntax in all four parser modes; parameters can be variables (λx.), named constants (λfoo.), or predicate symbols (λP.); body extends rightward through all connectives
• Higher-order predicate application — (func)(arg) explicit application; λP. P(x) writes the body naturally and is automatically scope-resolved to Application(LambdaVar("P"), Variable("x"))
• Lambda-calculus operations — free-variable computation, capture-avoiding substitution, beta-reduction (normal-order, step-limited), eta-reduction, combined beta-eta normalisation to fixpoint, and lexical scope resolution
• Lambda elimination — eliminate_lambdas() beta-eta-normalises and checks the result is lambda-free (ready for export / normal forms); reduce_trace() returns the step-by-step reduction sequence
• Tarskian semantics — define a Structure (a world with a domain of individuals and interpretations of constants, functions, predicates, and sorts) and compute a formula's truth value with satisfies() (FOL and MSFOL)
• Fuzzy evaluation — fuzzy_evaluate() computes the Łukasiewicz truth degree in [0, 1] of an FL/MSFL formula under a valuation (∀ = inf, ∃ = sup)
• Fuzzy satisfiability / validity — fuzzy_is_satisfiable(), fuzzy_is_valid(), fuzzy_get_model() decide Łukasiewicz formulas via Z3 real arithmetic
• Arithmetic-aware solving — is_satisfiable_arith(), is_valid_arith(), get_model_arith() interpret + - * / < > ≤ ≥ over Z3 reals/integers (unlike the default uninterpreted-sort to_z3)
• Unification — unify() computes the most general unifier of two terms/atoms (Robinson, with occurs-check); apply_subst() applies a substitution
• Command-line interface — python -m unicode_fol_kit "∀x P(x)" --to latex parses and renders in any export format

Installation

Via pip

pip install unicode-fol-kit

Via git clone

git clone https://github.com/fvossel/unicode-fol-kit.git
cd unicode-fol-kit
pip install .

Parser modes

MSFLParser is instantiated with two boolean flags:

MSFLParser(many_sorted=False, fuzzy=False)   # FOL   (default)
MSFLParser(many_sorted=True,  fuzzy=False)   # MSFOL
MSFLParser(many_sorted=True,  fuzzy=True)    # MSFL
MSFLParser(many_sorted=False, fuzzy=True)    # FL

many_sorted	fuzzy	Mode	Quantifiers	Constants	Connectives
False	False	FOL	unsorted ∀x	unsorted	classical ∧ ∨ ⊕ ¬ → ↔
True	False	MSFOL	sorted ∀x:Sort	sorted alice:Sort	classical ∧ ∨ ¬ → ↔ (no ⊕)
True	True	MSFL	sorted ∀x:Sort	sorted alice:Sort	weak ∧ ∨, strong ⊗ ⊕, Łuk ¬ → ↔
False	True	FL	unsorted ∀x	unsorted	weak ∧ ∨, strong ⊗ ⊕, Łuk ¬ → ↔

Usage

FOL mode (default)

from unicode_fol_kit import MSFLParser

parser = MSFLParser()
formula = parser.parse("∀x (Human(x) → Mortal(x))")

MSFOL mode

Quantifiers and ground constants must carry a sort annotation. The colon can be written with or without a space before it:

parser = MSFLParser(many_sorted=True)

# Sorted quantifier
q = parser.parse("∀x:Human (Mortal(x) ∧ ¬Immortal(x))")
# SortedQuantifier(type='∀', variable=Variable('x'), sort='Human', formula=…)

# Sorted constant — both spacing forms are accepted
parser.parse("P(alice:Human)")
parser.parse("P(alice :Human)")

MSFL mode

Uses Łukasiewicz logic: ∧/∨ become weak (min/max), ⊗/⊕ become strong (t-norm/t-conorm), and ¬/→/↔ map to their Łukasiewicz counterparts:

parser = MSFLParser(many_sorted=True, fuzzy=True)

parser.parse("P(x) ∧ Q(x)")    # WeakConjunction (min)
parser.parse("P(x) ⊗ Q(x)")    # StrongConjunction (t-norm: max{0, x+y−1})
parser.parse("P(x) ⊕ Q(x)")    # StrongDisjunction (t-conorm: min{1, x+y})
parser.parse("¬P(x)")           # LukNegation (1−x)
parser.parse("P(x) → Q(x)")    # LukImplication (min{1, 1−x+y})
parser.parse("∀x:Human P(x)")  # SortedQuantifier

FL mode

Łukasiewicz operators with unsorted quantifiers and plain constants — same connectives as MSFL, same quantifier/constant syntax as FOL:

parser = MSFLParser(many_sorted=False, fuzzy=True)

parser.parse("P(x) ∧ Q(x)")          # WeakConjunction (min)
parser.parse("P(x) ⊗ Q(x)")          # StrongConjunction (t-norm: max{0, x+y−1})
parser.parse("P(x) ⊕ Q(x)")          # StrongDisjunction (t-conorm: min{1, x+y})
parser.parse("¬P(x)")                 # LukNegation (1−x)
parser.parse("P(x) → Q(x)")          # LukImplication (min{1, 1−x+y})
parser.parse("∀x P(x)")              # unsorted Quantifier (no sort annotation)
parser.parse("P(alice)")             # plain Constant (no sort annotation)

# Lambda works in FL mode too
parser.parse("λx. P(x) ⊗ Q(x)")
# Lambda(LambdaVar("x"), StrongConjunction(Atom("P",[LambdaVar("x")]), Atom("Q",[LambdaVar("x")])))

ASCII tree view

formula = MSFLParser().parse("∀x (Human(x) → Mortal(x))")
print(formula.tree_str())
# ∀ x
# └── →
#     ├── Atom: Human
#     │   └── Variable: x
#     └── Atom: Mortal
#         └── Variable: x

Round-trip to Unicode

to_unicode_str() is the inverse of parsing: it renders any node back to a Unicode formula string. Re-parsing that string in the same mode reproduces a structurally equal AST. The renderer is precedence-aware and only inserts the parentheses the grammar requires — including the no-mixing rule for same-level connectives and the tight-binding rule for quantifiers.

parser = MSFLParser()

ast = parser.parse("∀x P(x) ∧ Q(x)")
ast.to_unicode_str()              # '∀x P(x) ∧ Q(x)'
parser.parse(ast.to_unicode_str()) == ast   # True

# Precedence-driven parentheses are reconstructed, not the original spelling:
parser.parse("((P(x) ∧ Q(x)))").to_unicode_str()   # 'P(x) ∧ Q(x)'
parser.parse("P(x) ∧ (Q(x) ∨ R(x))").to_unicode_str()  # 'P(x) ∧ (Q(x) ∨ R(x))'

Available on every node, so subformulas render too. The output targets parseable ASTs; alpha-renamed variables introduced by beta_reduce (e.g. x_0) are not valid surface tokens and will not round-trip.

Exporting to other formats

formula.to_prover9()   # '(all x (Human(x) -> Mortal(x)))'
formula.to_tptp()      # '(![X]: (human(X) => mortal(X)))'
formula.to_latex()     # '\\forall x\\, (Human(x) \\rightarrow Mortal(x))'
formula.to_dict()      # JSON-serialisable dict

to_latex() uses the same precedence-aware parenthesisation as to_unicode_str(). Sorts render as \forall x{:}\mathrm{Human}\,; strong Łukasiewicz operators as \otimes / \oplus. Symbol and predicate names are emitted verbatim (no \mathrm wrapping).

Traversal and inspection

Every node exposes a small traversal API:

f = MSFLParser().parse("∀x (Human(x) → Mortal(x))")

list(f.walk())        # pre-order: every node and descendant
f.subformulas()       # every sub-node that is a formula (terms excluded)
f.atoms()             # [Atom("Human", …), Atom("Mortal", …)]
f.variables()         # {Variable("x")}  — set of logical variables (free + bound)
f.count()             # total node count
f.count(Atom)         # nodes of a given type
f.depth()             # tree height (a leaf has depth 1)

Graphviz export

print(f.to_dot())
# digraph AST {
#   node [shape=box];
#   n0 [label="∀ x"];
#   n1 [label="→"];
#   ...
# }

to_dot() mirrors the tree_str() view (the quantifier's bound variable is folded into its node label). Pipe the output to dot -Tpng to render an image.

Serialisation

from unicode_fol_kit import Node

d = formula.to_dict()
formula2 = Node.from_dict(d)  # round-trip

Lambda-calculus

All four parser modes support lambda abstraction and application. parse() automatically applies scope resolution, so the returned AST is always fully resolved.

from unicode_fol_kit import (
    MSFLParser,
    LambdaVar, Lambda, Application,
    free_variables, substitute,
    beta_reduce, eta_reduce, beta_eta_normalize,
    resolve_lambda_scope,
    ReductionLimitError,
)

parser = MSFLParser()

# Parse — scope resolution is applied automatically
term = parser.parse("λP. λx. P(x)")
# Lambda(LambdaVar("P"), Lambda(LambdaVar("x"), Application(LambdaVar("P"), LambdaVar("x"))))

# Application
app = parser.parse("(λP. P(x))(Q)")
# Application(Lambda(LambdaVar("P"), Application(LambdaVar("P"), Variable("x"))),
#             Atom("Q", []))

Free variables

term = parser.parse("λP. P(x)")
free_variables(term)
# {Variable("x")}  — x is free; P is lambda-bound and does not appear

The result is a mixed set that may contain both Variable (logical) and LambdaVar (lambda-bound) objects.

Beta-reduction

beta_reduce reduces to beta-normal form using a normal-order (leftmost-outermost) strategy with full capture-avoiding substitution. It raises ReductionLimitError after 10 000 steps if the term does not normalise.

# (λP. λx. P(x))(Q) → λx. Application(Atom("Q",[]), LambdaVar("x"))
result = beta_reduce(parser.parse("(λP. λx. P(x))(Q)"))

# Full pipeline: parse → resolve → reduce
reduced = beta_reduce(parser.parse("(λP. P(x))(λy. Q(y))"))
# Atom("Q", [Variable("x")])

Eta-reduction

eta_reduce performs a single bottom-up pass contracting all eta-redexes: λp. f(p) → f when p is not free in f. Quantifiers are recursed into but never treated as eta-redexes.

from unicode_fol_kit import LambdaVar, Lambda, Application, Atom, Variable

f = Atom("P", [Variable("x")])                  # some formula
term = Lambda(LambdaVar("p"), Application(f, LambdaVar("p")))  # λp. f(p)
eta_reduce(term)   # → f  (the Atom node, not the Lambda)

Beta-eta normalisation

beta_eta_normalize alternates beta_reduce and eta_reduce to fixpoint (up to 100 rounds). The alternation loop is a genuine necessity: eta-reduction can expose fresh beta-redexes, requiring another beta pass.

normal = beta_eta_normalize(parser.parse("(λP. P(x))(Q)"))

ReductionLimitError is raised if the inner beta-reduction limit or the outer round limit is exceeded.

Scope resolution (standalone)

resolve_lambda_scope is also available as a standalone function for hand-built ASTs:

from unicode_fol_kit import resolve_lambda_scope, Lambda, LambdaVar, Atom, Variable

raw = Lambda(LambdaVar("x"), Atom("P", [Variable("x")]))
resolved = resolve_lambda_scope(raw)
# Lambda(LambdaVar("x"), Atom("P", [LambdaVar("x")]))

Reducing MSFL formulas to classical FOL

to_fol() performs a two-phase reduction: it first lowers Łukasiewicz operators to classical ones (to_msfol()), then eliminates sort annotations via relativisation (_relativize()):

from unicode_fol_kit import MSFLParser, to_fol

parser = MSFLParser(many_sorted=True, fuzzy=True)
formula = parser.parse("∀x:Human (P(x) ∧ ¬Q(x))")

classical = to_fol(formula)
# Quantifier(∀, x, Implies(Atom(Human, [x]), And(Atom(P,[x]), Not(Atom(Q,[x])))))

# Optionally, conjoin sort-membership facts for constants at the top level:
classical_with_facts = to_fol(formula, include_sort_facts=True)

Note — this is a classical (Boolean) projection, not a fuzzy-preserving translation. to_msfol() maps both the strong (⊗/⊕) and the weak (∧/∨) Łukasiewicz connectives to the same classical And/Or. On crisp truth values {0, 1} the strong and weak operators coincide, so the reduction is sound as the two-valued projection of the formula — but the genuinely many-valued content is discarded. To compute the real-valued Łukasiewicz degree instead, use fuzzy_evaluate() or the fuzzy Z3 solver (see below).

Normal forms

to_nnf(), to_pnf(), to_cnf(), and skolemize() operate on classical FOL. They accept FOL, MSFOL, MSFL, and FL inputs — sorts and Łukasiewicz operators are reduced via to_fol() first. (Lambda terms must be beta-reduced and lambda-eliminated beforehand.)

from unicode_fol_kit import MSFLParser, to_nnf, to_pnf, to_cnf, skolemize

parser = MSFLParser()

to_nnf(parser.parse("P → Q"))        # eliminates → ↔ ⊕, pushes ¬ down to atoms
to_pnf(parser.parse("∀x P(x) ∧ ∃y Q(y)"))   # quantifier prefix + quantifier-free matrix
to_cnf(parser.parse("P ∨ (Q ∧ R)"))  # matrix as a conjunction of clauses
skolemize(parser.parse("∀x ∃y Loves(x, y)"))
# ∀v0 Loves(v0, sk0(v0)) — the existential becomes a Skolem function of x

• to_nnf / to_pnf / to_cnf are equivalence-preserving: the result is logically equivalent to the (classical) input.
• skolemize is satisfiability-preserving (not equivalence-preserving): existentials are replaced by Skolem terms over the universals in scope, and the universal prefix is retained. Bound variables are standardised apart (renamed to fresh v0, v1, …); Skolem symbols are named sk0, sk1, ….

to_dnf() is the dual of to_cnf(): a prenex form whose matrix is a disjunction of conjunctive clauses, and it is likewise equivalence-preserving.

from unicode_fol_kit import MSFLParser, to_dnf, to_tseitin_cnf

parser = MSFLParser()
to_dnf(parser.parse("P ∧ (Q ∨ R)"))   # (P ∧ Q) ∨ (P ∧ R)

to_tseitin_cnf() produces an equisatisfiable CNF using the Tseitin/definitional encoding: it introduces fresh auxiliary atoms (ts0, ts1, …) for compound subformulas, so the result grows linearly instead of risking the exponential blow-up of the distributive to_cnf. It is not logically equivalent to the input (the auxiliaries are existentially fresh), but the input is satisfiable iff its Tseitin CNF is. It operates on quantifier-free (propositional / ground) formulas and raises ValueError on quantified input.

from unicode_fol_kit import MSFLParser, to_tseitin_cnf, is_satisfiable

parser = MSFLParser()
phi = parser.parse("(P ∨ Q) ∧ (¬P ∨ R)")
is_satisfiable(phi) == is_satisfiable(to_tseitin_cnf(phi))   # True (equisatisfiable)

Horn check

is_horn() reports whether a formula's clausal form consists of Horn clauses — each clause has at most one positive literal. The formula is skolemised, its universal prefix dropped, and the matrix put into CNF before the clauses are checked.

from unicode_fol_kit import MSFLParser, is_horn

parser = MSFLParser()
is_horn(parser.parse("∀x (Body(x) → Head(x))"))     # True  (definite clause)
is_horn(parser.parse("P → (Q ∧ R)"))                # True  (splits into two Horn clauses)
is_horn(parser.parse("P → (Q ∨ R)"))                # False (clause has two positive literals)

Equivalence checking (Z3)

from unicode_fol_kit import MSFLParser, formulas_are_equivalent

parser = MSFLParser()
f1 = parser.parse("¬(P(x) ∧ Q(x))")
f2 = parser.parse("¬P(x) ∨ ¬Q(x)")

formulas_are_equivalent(f1, f2)  # True

Satisfiability, validity, and counterexamples (Z3)

from unicode_fol_kit import MSFLParser, is_satisfiable, is_valid, get_model, Not

parser = MSFLParser()

is_satisfiable(parser.parse("P ∧ Q"))     # True
is_satisfiable(parser.parse("P ∧ ¬P"))    # False
is_valid(parser.parse("P ∨ ¬P"))          # True

get_model(parser.parse("P ∧ Q"))          # {'P': 'True', 'Q': 'True'}
get_model(parser.parse("P ∧ ¬P"))         # None  (unsatisfiable)

# Counterexample to a claimed equivalence: a model of its negation.
get_model(Not(parser.parse("P ↔ Q")))     # e.g. {'P': 'True', 'Q': 'False'}

get_model returns a dict mapping each Z3 declaration (constants, uninterpreted predicates/functions) to its interpretation, or None when the formula is unsatisfiable or Z3 returns unknown within the timeout.

Entailment checking (Prover9)

from unicode_fol_kit import MSFLParser, check_logical_entailment

parser = MSFLParser()
premises = [
    parser.parse("∀x (Human(x) → Mortal(x))"),
    parser.parse("Human(socrates)"),
]
conclusion = parser.parse("Mortal(socrates)")

check_logical_entailment(premises, conclusion, prover9_path="/usr/bin/prover9")  # True

Entailment and validity (built-in resolution prover)

For entailment and validity without an external prover, the package ships a self-contained first-order resolution prover. It clausifies the input (skolemise → drop ∀ prefix → CNF → clauses), then refutes premises ∧ ¬conclusion by binary resolution and factoring, deriving the empty clause iff the entailment holds.

from unicode_fol_kit import MSFLParser, prove, is_valid_resolution

parser = MSFLParser()

premises = [parser.parse("∀x (Human(x) → Mortal(x))"), parser.parse("Human(socrates)")]
prove(premises, parser.parse("Mortal(socrates)"))   # True

prove([parser.parse("Human(socrates)")], parser.parse("Mortal(socrates)"))  # False (no entailment)

is_valid_resolution(parser.parse("P ∨ ¬P"))          # True
is_valid_resolution(parser.parse("∃x ∀y L(x, y) → ∀y ∃x L(x, y)"))  # True

• Sound, deliberately incomplete. First-order resolution is only semi-decidable, so prove/is_valid_resolution take a max_steps bound (default 10 000). They return True only when the empty clause is actually derived, and False both when the clause set saturates (genuinely no entailment) and when the bound is reached ("not proved within the bound") — they never report a non-theorem as proved.
• Equality is uninterpreted. = is treated as an ordinary predicate (no built-in reflexivity/congruence). Entailments that rely on the theory of equality (e.g. a = b, P(a) ⊨ P(b)) need those axioms supplied as explicit premises, or use the Z3 backend instead.
• to_clauses(formula) exposes the clausal form, and refute(clauses) runs the saturation directly.

Tarskian semantics (FOL / MSFOL)

Instead of calling an external solver, you can build a concrete world (a first-order Structure) and compute the truth value of a formula directly, following Tarski's recursive definition of satisfaction. A structure is a non-empty domain of individuals together with interpretations of the constants, functions, and predicates (and, for MSFOL, the named sorts).

from unicode_fol_kit import MSFLParser, Structure, satisfies

parser = MSFLParser()

# A world with two individuals where everyone loves someone else.
world = Structure(
    domain={"alice", "bob"},
    predicates={("Loves", 2): {("alice", "bob"), ("bob", "alice")}},
)

satisfies(parser.parse("∀x ∃y Loves(x, y)"), world)              # True
satisfies(parser.parse("∃x Loves(x, x)"), world)                 # False
satisfies(parser.parse("∀x ∀y (Loves(x, y) → Loves(y, x))"), world)  # True

• predicates maps (name, arity) to the relation's extension (a set of argument tuples); a missing predicate is the empty relation (false). A nullary predicate maps (name, 0) to a bool.
• functions maps (name, arity) to a Python callable or a {arg_tuple: value} dict; constants maps a name to an individual.
• Equality = (and ≠) is built in as identity on the domain — you do not interpret it. The order comparisons < > ≤ ≥ are ordinary relations whose extension you supply.
• satisfies(formula, structure, assignment=None) takes an optional assignment of free variables; models(formula, structure) is the closed-formula convenience alias.

For MSFOL, declare each sort's universe and quantify over it:

msfol = MSFLParser(many_sorted=True)

zoo = Structure(
    domain={"alice", "rex"},
    sorts={"Human": {"alice"}, "Dog": {"rex"}},
    predicates={("Barks", 1): {("rex",)}},
)

satisfies(msfol.parse("∀x:Dog Barks(x)"), zoo)     # True
satisfies(msfol.parse("∀x:Human Barks(x)"), zoo)   # False

Tarski semantics is two-valued: Łukasiewicz operators and lambda nodes are rejected (use the fuzzy evaluator below, and eliminate_lambdas() respectively).

Fuzzy evaluation (Łukasiewicz)

fuzzy_evaluate() computes the truth degree in [0, 1] of an FL/MSFL formula under a valuation — a mapping from each ground atom (keyed by its Unicode rendering, e.g. "P(alice)") to a degree. The Łukasiewicz operator semantics are applied exactly (weak ∧/∨ = min/max, strong ⊗/⊕ = t-norm/t-conorm, ¬ = 1−x, → and ↔ as usual); quantifiers are infimum (∀) and supremum (∃) over the domain.

from unicode_fol_kit import MSFLParser, fuzzy_evaluate

fl = MSFLParser(fuzzy=True)

fuzzy_evaluate(fl.parse("P ⊗ ¬P"), {"P": 0.6})            # 0.0   (strong: max(0, 0.6+0.4−1))
fuzzy_evaluate(fl.parse("P ⊕ ¬P"), {"P": 0.6})            # 1.0   (strong: min(1, 0.6+0.4))
fuzzy_evaluate(fl.parse("P ↔ Q"), {"P": 0.6, "Q": 0.7})   # 0.9   (1 − |0.6 − 0.7|)

# Quantifiers need a domain of constant-name strings; atoms are grounded to P(a), P(b), …
fuzzy_evaluate(fl.parse("∀x P(x)"), {"P(a)": 0.3, "P(b)": 0.8}, domain={"a", "b"})  # 0.3 (min)
fuzzy_evaluate(fl.parse("∃x P(x)"), {"P(a)": 0.3, "P(b)": 0.8}, domain={"a", "b"})  # 0.8 (max)

Fuzzy satisfiability and validity (Z3)

For Łukasiewicz formulas you can also ask the solver, rather than fixing a valuation, whether some (or every) assignment reaches a given degree. The atoms become Z3 reals in [0, 1] and the operators their real-arithmetic definitions.

from unicode_fol_kit import MSFLParser, fuzzy_is_valid, fuzzy_is_satisfiable, fuzzy_get_model

fl = MSFLParser(fuzzy=True)

fuzzy_is_valid(fl.parse("P ⊕ ¬P"))                     # True  (degree is 1 for every P)
fuzzy_is_satisfiable(fl.parse("P ⊗ ¬P"), threshold=0.5)  # False (max degree is 0)
fuzzy_is_satisfiable(fl.parse("P ∧ ¬P"), threshold=0.5)  # True  (weak: max degree is 0.5)
fuzzy_get_model(fl.parse("P → Q"), threshold=1.0)        # {'P': ..., 'Q': ..., 'degree': ...}

These cover quantifier-free (propositional) Łukasiewicz formulas; quantified input raises NotImplementedError.

Arithmetic-aware solving (Z3)

The default is_satisfiable() / to_z3() treat everything as one uninterpreted sort, so arithmetic terms are opaque. The *_arith variants instead interpret + - * / and the comparisons over a numeric sort ("real" by default, or "int"), so the solver can actually reason about numbers.

from unicode_fol_kit import MSFLParser, is_satisfiable_arith, is_valid_arith, get_model_arith

parser = MSFLParser()

is_satisfiable_arith(parser.parse("x + 1 = 2 ∧ x > 0"))   # True   (x = 1)
is_satisfiable_arith(parser.parse("x > 0 ∧ x < 0"))       # False
is_valid_arith(parser.parse("∀x (x * 2 = x + x)"))        # True
get_model_arith(parser.parse("x + 1 = 2 ∧ x > 0"))        # {'x': '1'}
is_satisfiable_arith(parser.parse("x + x = 1"), sort="int")  # False (no integer solution)

Lambda elimination and reduction trace

eliminate_lambdas() beta-eta-normalises a term and verifies the result is lambda-free, so it can be fed to the exporters or normal-form functions (which otherwise reject lambda nodes). reduce_trace() returns the intermediate steps, and beta_reduce_step() performs a single leftmost-outermost step.

from unicode_fol_kit import MSFLParser, eliminate_lambdas, reduce_trace, has_lambdas

parser = MSFLParser()

term = parser.parse("(λP. P(x))(Q)")
has_lambdas(term)                 # True
reduced = eliminate_lambdas(term) # Atom("Q", [Variable("x")])
has_lambdas(reduced)              # False
reduced.to_tptp()                 # now exportable: 'q(X)'

steps = reduce_trace(parser.parse("(λP. λx. P(x))(Q)"))  # [original, …, normal form]

A term that is stuck or only partially applied (no further redex but lambdas remain) raises ValueError from eliminate_lambdas.

Unification

unify() computes the most general unifier (Robinson's algorithm with occurs-check) of two terms or two atoms, returned as a substitution dict mapping variable names to terms; apply_subst() applies it. Note that single lowercase letters parse as variables, so build genuine constants with Constant.

from unicode_fol_kit import unify, apply_subst, Variable, Constant, Function

t1 = Function("f", [Variable("x"), Constant("a")])
t2 = Function("f", [Constant("b"), Variable("y")])

mgu = unify(t1, t2)        # {'x': Constant('b'), 'y': Constant('a')}
apply_subst(t1, mgu) == apply_subst(t2, mgu)   # True

unify(Variable("x"), Function("f", [Variable("x")]))   # None  (occurs-check fails)

Command-line interface

The package is runnable as a module for quick parsing and conversion:

python -m unicode_fol_kit "∀x (P(x) → Q(x))" --to tptp
# (![X]: (p(X) => q(X)))

python -m unicode_fol_kit "P(x) ⊗ Q(x)" --mode msfl --to latex
# P(x) \otimes Q(x)

--mode is one of fol (default), msfol, msfl, fl; --to is one of tree (default), unicode, latex, tptp, prover9, json, dot. Parse errors print a SYNTAX_ERROR message to stderr and exit with status 1.

Syntax reference

This section describes the full surface syntax accepted by the parser. Because the four modes share the same term and atom layer, most of the syntax is identical across modes; differences are called out explicitly.

Tokens

The lexer distinguishes the following token kinds. Because the patterns are mutually exclusive, a given identifier is unambiguously a variable, a constant, a function/predicate name, a number, or a sort annotation.

Token	Pattern	Examples	Meaning
Variable	one lowercase letter, optional trailing digits	x, y, x1, z42	a (possibly bound) logical variable
Name	lowercase, at least two letters, may contain digits and uppercase after the first letter	socrates, distance, centerOf, foo1	a bare constant or a function symbol
Constant (c_)	c_ followed by letters/digits	c_a, c_zero, c_42	an explicitly marked constant
Predicate	one uppercase letter, then letters/digits	P, Human, OnSurfaceOf	a predicate symbol
Number	digits, optional decimal part	0, 42, 3.14	a numeric literal
Sort annotation	: followed by an uppercase letter and letters/digits	:Human, :Sort1	a sort tag (MSFOL and MSFL modes only)

The c_ form exists so that single-letter constants can be written without colliding with variables. A bare a is always a variable; if you need the constant a, write c_a.

A function or predicate is recognised by being immediately followed by a parenthesised argument list, e.g. distance(x, y) or Human(socrates). The same token class (Name) serves both as a bare constant and, when applied, as a function symbol.

The sort annotation token always begins with :, which makes it lexically disjoint from all other tokens. Whitespace before the colon is optional: ∀x:Human P(x) and ∀x :Human P(x) are both valid and produce identical parse trees.

Terms

A term is one of:

• a variable (x, x1)
• a constant (socrates, c_a) or number (42, 3.14)
• in MSFOL / MSFL modes: a sort-annotated constant (alice:Human, c_a:Sort1)
• a function application (f(t1, …, tn), e.g. centerOf(x))
• an arithmetic combination of terms using +, -, *, /
• a parenthesised term ((t))

Arithmetic follows the usual precedence: * and / bind tighter than + and -, and both groups are left-associative. For example x + y * z parses as x + (y * z).

Sort rules in MSFOL / MSFL modes: variables are sorted implicitly by the quantifier that binds them; ground constants must carry an explicit sort annotation. An unsorted constant (e.g. bare alice) is a syntax error in sorted modes.

Atomic formulas

An atomic formula is either:

• a predicate applied to terms: P, Human(socrates), OnSurfaceOf(y, x) (a predicate may be nullary, i.e. used without arguments)
• an infix comparison between two terms: =, ≠, <, >, ≤, ≥, e.g. x1 + 1 = y1 or distance(y, c) > distance(z, c)

Compound formulas

Atomic formulas are combined with connectives and quantifiers. The available connectives and their interpretations depend on the mode:

FOL mode

Syntax	Operator	Interpretation
¬φ	negation	classical
φ ∧ ψ	conjunction	classical
φ ∨ ψ	disjunction	classical
φ ⊕ ψ	exclusive or	classical
φ → ψ	implication	classical
φ ↔ ψ	biconditional	classical
∀x φ	universal	unsorted
∃x φ	existential	unsorted

MSFOL mode

Same connectives as FOL except ⊕ (exclusive or) is not available. Quantifiers require a sort annotation:

Syntax	Operator
¬φ, φ ∧ ψ, φ ∨ ψ, φ → ψ, φ ↔ ψ	classical (as FOL)
∀x:Sort φ, ∃x:Sort φ	sorted quantifiers

MSFL mode

Connectives are reinterpreted as Łukasiewicz operators:

Syntax	Operator	Semantics
¬φ	Łuk. negation	1 − φ
φ ∧ ψ	weak conjunction	min(φ, ψ)
φ ∨ ψ	weak disjunction	max(φ, ψ)
φ ⊗ ψ	strong conjunction	max(0, φ + ψ − 1)
φ ⊕ ψ	strong disjunction	min(1, φ + ψ)
φ → ψ	Łuk. implication	min(1, 1 − φ + ψ)
φ ↔ ψ	Łuk. equivalence	1 − |φ − ψ|
∀x:Sort φ, ∃x:Sort φ	sorted quantifiers

FL mode

Same Łukasiewicz connectives as MSFL, but with unsorted quantifiers and plain constants (no :Sort required):

Syntax	Operator	Semantics
¬φ	Łuk. negation	1 − φ
φ ∧ ψ	weak conjunction	min(φ, ψ)
φ ∨ ψ	weak disjunction	max(φ, ψ)
φ ⊗ ψ	strong conjunction	max(0, φ + ψ − 1)
φ ⊕ ψ	strong disjunction	min(1, φ + ψ)
φ → ψ	Łuk. implication	min(1, 1 − φ + ψ)
φ ↔ ψ	Łuk. equivalence	1 − |φ − ψ|
∀x φ, ∃x φ	unsorted quantifiers

A formula may be wrapped in parentheses ( … ) or square brackets [ … ]; the two are interchangeable for grouping.

Operator precedence

The precedence levels are the same across all four modes (MSFL/FL use the same syntactic structure with Łukasiewicz semantics):

Precedence	Operators	Associativity
1 (highest)	¬, quantifiers ∀ / ∃	prefix
2	∧ ∨ ⊕ (FOL) / ∧ ∨ (MSFOL) / ∧ ∨ ⊗ ⊕ (MSFL / FL)	left
3	→	right
4 (lowest)	↔	right

Worked examples (parenthesised to show how the parser groups them):

• ¬P(x) ∧ Q(x) → (¬P(x)) ∧ Q(x) — negation binds tighter than conjunction
• P(x) ∧ Q(x) → R(x) → (P(x) ∧ Q(x)) → R(x) — conjunction binds tighter than implication
• P(x) → Q(x) ↔ R(x) → (P(x) → Q(x)) ↔ R(x) — implication binds tighter than biconditional
• P(x) → Q(x) → R(x) → P(x) → (Q(x) → R(x)) — implication is right-associative
• P(x) ∧ Q(x) ∧ R(x) → (P(x) ∧ Q(x)) ∧ R(x) — conjunction is left-associative

Mixing same-level operators

The same-level connectives (level 2 above) cannot be mixed without explicit parentheses. This is deliberate: it avoids the silent, easy-to-misread grouping that a default precedence would impose.

FOL mode — ∧, ∨, ⊕ cannot be mixed:

P(x) ∧ Q(x) ∨ R(x)      # rejected
(P(x) ∧ Q(x)) ∨ R(x)    # accepted

MSFOL mode — ∧ and ∨ cannot be mixed:

P(x) ∧ Q(x) ∨ R(x)      # rejected
(P(x) ∧ Q(x)) ∨ R(x)    # accepted

MSFL / FL mode — ∧, ∨, ⊗, ⊕ cannot be mixed:

P(x) ∧ Q(x) ⊗ R(x)        # rejected
(P(x) ∧ Q(x)) ⊗ R(x)      # accepted

A chain of the same operator is always fine: P ∧ Q ∧ R, P ⊗ Q ⊗ R, etc.

Quantifier scope

A quantifier binds only the immediately following (tightly bound) formula, not the rest of the line:

∀x P(x) ∧ Q(x)      # parses as (∀x P(x)) ∧ Q(x)
∀x P(x) → Q(x)      # parses as (∀x P(x)) → Q(x)

If you intend the quantifier to range over the whole formula — which is usually what is meant — add parentheses:

∀x (P(x) → Q(x))    # quantifier ranges over the implication
∀x (P(x) ∧ Q(x))    # quantifier ranges over the conjunction

Quantifiers can be stacked directly: ∀x:H ∀y:H ∃z:A φ.

Supported symbols

Category	FOL	MSFOL	MSFL	FL
Quantifiers	∀ ∃ (unsorted)	∀ ∃ (sorted :Sort)	∀ ∃ (sorted :Sort)	∀ ∃ (unsorted)
Connectives	∧ ∨ ⊕ ¬ → ↔	∧ ∨ ¬ → ↔	∧ ∨ ⊗ ⊕ ¬ → ↔	∧ ∨ ⊗ ⊕ ¬ → ↔
Lambda	λ	λ	λ	λ
Sort annotations	—	:Sort	:Sort	—
Equality / comparison	= ≠ < > ≤ ≥	same	same	same
Arithmetic	+ - * /	same	same	same
Grouping	( ) [ ]	same	same	same
Argument separator	,	same	same	same

Whitespace is insignificant and may be used freely between tokens — including before sort annotation colons.

Lambda abstraction and application (all modes)

A lambda abstraction is written λ followed by a parameter name, a literal ., and a body formula. All four parser modes support identical lambda surface notation.

Parameter types

Parameter form	Example	Typical use
Single lowercase letter	λx. P(x)	value variable
Multi-letter lowercase name	λfoo. P(foo(x))	named-constant parameter
Uppercase predicate symbol	λP. P(x)	predicate / higher-order parameter

All three token classes become a LambdaVar in the AST. Scope resolution (applied automatically by parse()) then rewrites body occurrences of the lambda-bound name:

• Variable occurrence — λx. P(x): the x in P(x) becomes LambdaVar("x").
• Predicate-application occurrence — λP. P(x): the P(x) in the body becomes Application(LambdaVar("P"), Variable("x")). Multi-argument atoms curry left: P(x, y) → Application(Application(LambdaVar("P"), x), y).
• Named-function occurrence — λfoo. P(foo(x)): the foo(x) in P's argument list (a term-level function call) becomes Application(LambdaVar("foo"), Variable("x")).

The scope obeys the innermost-binder rule: a quantifier removes the quantified name from the lambda-bound set. Inside λx. ∀x P(x), the x in P(x) is logical (stays Variable).

Body scope

The body extends rightward through all connectives — lambda has lower precedence than every binary operator:

λx. P(x) ∧ Q(x)      # body is the And node P(x) ∧ Q(x)
λx. P(x) → Q(x)      # body is the Implies node P(x) → Q(x)

Multi-parameter lambdas are written by nesting: λP. λx. P(x).

Application syntax

A lambda application requires both sides to be parenthesised: (func)(arg).

(λx. P(x))(a)         # arg is variable a
(λx. P(x))(alice)     # arg is constant alice
(λP. P(x))(Q)         # arg is the zero-arity atom Q
(λP. P(x))(Q(y))      # arg is the atom Q(y)

Higher-order application inside the body — a predicate parameter applied to arguments — is written in the natural P(x) notation, not as (P)(x). Scope resolution handles the rewrite automatically.

Parse examples

parser = MSFLParser()

parser.parse("λx. P(x)")
# Lambda(LambdaVar("x"), Atom("P", [LambdaVar("x")]))

parser.parse("λP. P(x)")
# Lambda(LambdaVar("P"), Application(LambdaVar("P"), Variable("x")))

parser.parse("λP. λx. P(x)")
# Lambda(LambdaVar("P"), Lambda(LambdaVar("x"), Application(LambdaVar("P"), LambdaVar("x"))))

parser.parse("λx. ∀x P(x)")
# Lambda(LambdaVar("x"), Quantifier("∀", Variable("x"), Atom("P", [Variable("x")])))
# x inside ∀ is quantifier-bound — NOT rewritten to LambdaVar

parser.parse("(λP. P(x))(Q)")
# Application(Lambda(LambdaVar("P"), Application(LambdaVar("P"), Variable("x"))), Atom("Q", []))

A complete FOL example

∀x ((Object(x) ∧ HasThreeDimensionalShape(x) ∧
     ∀y ∀z ((Point(y) ∧ OnSurfaceOf(y, x) ∧ Point(z) ∧ OnSurfaceOf(z, x))
            → distance(y, centerOf(x)) = distance(z, centerOf(x))))
    → Sphere(x))

A complete MSFOL example

∀x:Person ∀y:Person (Knows(x, y) ∧ Trusted(y)) → Shares(x, y)

A complete MSFL example

∀x:Patient ∀y:Treatment
    (Effective(y) ⊗ Tolerable(x, y))
    → Recommended(x, y)

A complete FL example

∀x ∀y
    (Effective(y) ⊗ Tolerable(x, y))
    → Recommended(x, y)

AST nodes

All nodes are frozen Python dataclasses and can be imported from unicode_fol_kit. Being frozen, every node is immutable and hashable, so nodes can be put in sets, used as dict keys, and deduplicated. Function and Atom store their args as a tuple (a list passed to the constructor is accepted and coerced), which is what makes them hashable.

Shared term and atom nodes (all modes)

Class	Fields	Notes
Variable	name: str	bound or free variable
Constant	name: str	bare constant or c_-prefixed
Number	value: int | float	numeric literal
Function	name: str, args: tuple	function application and arithmetic ops
Atom	predicate: str, args: tuple	predicate or infix comparison

Classical formula nodes (FOL / MSFOL)

Class	Fields
Not	formula
And	left, right
Or	left, right
Xor	left, right (FOL only)
Implies	left, right
Iff	left, right
Quantifier	type: str, variable, formula (FOL / FL — the unsorted modes)

MSFOL / MSFL nodes

Class	Fields	Notes
SortedQuantifier	type: str, variable, sort: str, formula	sort annotation without leading :
SortedConstant	name: str, sort: str	sort annotation without leading :

MSFL Łukasiewicz nodes

Class	Fields	Semantics
LukNegation	formula	1 − φ
WeakConjunction	left, right	min(φ, ψ)
WeakDisjunction	left, right	max(φ, ψ)
StrongConjunction	left, right	max(0, φ + ψ − 1)
StrongDisjunction	left, right	min(1, φ + ψ)
LukImplication	left, right	min(1, 1 − φ + ψ)
LukEquivalence	left, right	1 − |φ − ψ|

Lambda-calculus nodes (all modes)

Class	Fields	Notes
LambdaVar	name: str	lambda-bound variable; frozen and hashable — distinct from Variable
Lambda	param: LambdaVar, body: Node	lambda abstraction λparam. body
Application	func: Node, arg: Node	lambda application func(arg)

LambdaVar is kept separate from Variable so that logical binding (by quantifiers) and lambda binding never get confused. free_variables() returns a mixed set that may contain both.

Reductions

Every MSFL node implements two reduction steps:

• to_msfol() — lowers Łukasiewicz connectives to classical nodes while preserving sort annotations (SortedQuantifier and SortedConstant survive unchanged).
• _relativize(facts) — eliminates sort annotations by replacing ∀x:S φ with ∀x (S(x) → φ) and ∃x:S φ with ∃x (S(x) ∧ φ), and replacing SortedConstant(name, sort) with a plain Constant(name).

The top-level helper to_fol(node, include_sort_facts=False) chains both steps and optionally conjoins sort-membership atoms for all ground constants at the top level.

Error handling

Parse errors are reported with human-readable messages rather than raw parser internals. Lexer-level problems (an invalid character, a malformed name or number, or an attempt to mix same-level connectives without parentheses) raise NamingError; structural problems (an incomplete formula or a misplaced operator) raise ParsingError. Both report the offending position and, where useful, a hint. The hint text is mode-aware:

from unicode_fol_kit import MSFLParser

# FOL mode — hint names ∧, ∨, and ⊕
MSFLParser().parse("P(x) ∧ Q(x) ∨ R(x)")
# SYNTAX_ERROR: … Hint: Cannot mix conjunction (∧), disjunction (∨), and exclusive or (⊕) without parentheses

# MSFOL mode — hint names only ∧ and ∨
MSFLParser(many_sorted=True).parse("P(x) ∧ Q(x) ∨ R(x)")
# SYNTAX_ERROR: … Hint: Cannot mix conjunction (∧) and disjunction (∨) without parentheses

# MSFL mode — hint names all four Łukasiewicz connectives
MSFLParser(many_sorted=True, fuzzy=True).parse("P(x) ∧ Q(x) ⊗ R(x)")
# SYNTAX_ERROR: … Hint: Cannot mix weak conjunction (∧), weak disjunction (∨),
#                        strong conjunction (⊗), and strong disjunction (⊕) without parentheses

# FL mode — same hint as MSFL (Łukasiewicz connectives, unsorted)
MSFLParser(many_sorted=False, fuzzy=True).parse("P(x) ∧ Q(x) ⊗ R(x)")
# SYNTAX_ERROR: … Hint: Cannot mix weak conjunction (∧), weak disjunction (∨),
#                        strong conjunction (⊗), and strong disjunction (⊕) without parentheses

Citation

If you use this toolkit in academic work, please cite the accompanying preprint:

@misc{vossel2025advancingnaturallanguageformalization,
      title={Advancing Natural Language Formalization to First Order Logic with Fine-tuned LLMs},
      author={Felix Vossel and Till Mossakowski and Björn Gehrke},
      year={2025},
      eprint={2509.22338},
      archivePrefix={arXiv},
      primaryClass={cs.CL},
      url={https://arxiv.org/abs/2509.22338},
}

Vossel, F., Mossakowski, T., & Gehrke, B. (2025). Advancing Natural Language Formalization to First Order Logic with Fine-tuned LLMs. arXiv preprint arXiv:2509.22338.

License

MIT