pyNMMS 0.1.3

Non-Monotonic Multi-Succedent sequent calculus — propositional NMMS from Hlobil & Brandom 2025

pyNMMS

An automated reasoner for the Non-Monotonic Multi-Succedent (NMMS) propositional sequent calculus from Hlobil & Brandom 2025, Ch. 3.

Documentation | PyPI | GitHub

Installation

pip install pyNMMS

For development:

git clone https://github.com/bradleypallen/nmms-reasoner.git
cd nmms-reasoner
pip install -e ".[dev]"

Quick Start

from pynmms import MaterialBase, NMMSReasoner

# Create a material base with defeasible inferences
base = MaterialBase(
    language={"A", "B", "C"},
    consequences={
        (frozenset({"A"}), frozenset({"B"})),  # A |~ B
        (frozenset({"B"}), frozenset({"C"})),  # B |~ C
    },
)

reasoner = NMMSReasoner(base)

# A derives B (base consequence)
result = reasoner.derives(frozenset({"A"}), frozenset({"B"}))
assert result.derivable  # True

# A does NOT derive C (nontransitivity — no [Mixed-Cut])
result = reasoner.derives(frozenset({"A"}), frozenset({"C"}))
assert not result.derivable  # False

# A, C does NOT derive B (nonmonotonicity — no [Weakening])
result = reasoner.derives(frozenset({"A", "C"}), frozenset({"B"}))
assert not result.derivable  # False

# Classical tautologies still hold (supraclassicality)
result = reasoner.derives(frozenset(), frozenset({"A | ~A"}))
assert result.derivable  # True

CLI

# Create a base and add consequences
pynmms tell -b base.json --create "A |~ B"
pynmms tell -b base.json "B |~ C"

# Query derivability
pynmms ask -b base.json "A => B"        # DERIVABLE
pynmms ask -b base.json "A => C"        # NOT DERIVABLE
pynmms ask -b base.json "A, C => B"     # NOT DERIVABLE

# Interactive REPL
pynmms repl -b base.json

Key Properties

• Nonmonotonicity: Adding premises can defeat inferences (no Weakening)
• Nontransitivity: Chaining good inferences can yield bad ones (no Mixed-Cut)
• Supraclassicality: All classically valid sequents are derivable
• Conservative Extension: Logical vocabulary doesn't change base-level relations
• Explicitation Conditions: DD, II, AA, SS biconditionals hold

Implementation

Proof search strategy

The reasoner uses root-first backward proof search with memoization and backtracking. This is related to but distinct from the deterministic proof-search procedure in Definition 20 of the Ch. 3 appendix. Definition 20 specifies a deterministic decomposition: find the first complex sentence (alphabetically, left side first), apply the corresponding rule, repeat until all leaves are atomic, then check axioms. Our implementation instead tries each complex sentence in sorted order with backtracking — if decomposing one sentence fails to produce a proof, it backtracks and tries the next. Both approaches are correct because all NMMS rules are invertible (Proposition 27): if a sequent is derivable, any order of rule application will find the proof. Our approach adds memoization and depth-limiting as practical safeguards.

• 8 Ketonen-style propositional rules with third top sequent (compensates for working with sets rather than multisets, per Proposition 21)
• Memoization keyed on (frozenset, frozenset) pairs; cycle detection via pre-marking entries as False before recursion
• Depth-limited (default 25) to guarantee termination
• Deterministic rule application order (sorted iteration) for reproducible results

Design decisions

• Propositional fragment only; ALC restricted quantifiers deferred
• Sets (frozensets), not multisets — Contraction is built in (per Proposition 21)
• Sentences represented as strings, parsed on demand by a recursive descent parser producing frozen Sentence dataclass AST nodes
• Base consequences use exact syntactic match — no subset/superset matching, which is what enforces the no-Weakening property
• Containment (Γ ∩ Δ ≠ ∅) checked automatically as an axiom schema
• No runtime dependencies beyond the Python standard library

Known limitations

• Depth limit can cause false negatives for deeply nested valid sequents
• No incremental/persistent cache between queries
• Multi-premise rules ([L→], [L∨], [R∧]) each generate 3 subgoals, giving worst-case exponential branching
• Flat proof trace only — no structured proof tree or proof certificates
• Formula strings re-parsed at each proof step (no pre-compilation)
• Does not implement NMMS\ctr (contraction-free variant, Section 3.2.3), Monotonicity Box (□, Section 3.3.1), or classicality operator (⌈cl⌉, Section 3.3.2)

Test suite

214 tests across 11 test files:

• Syntax parsing (29 tests), MaterialBase construction/serialization (21), individual rule correctness (17), axiom derivability (6), structural properties with deterministic bases (18: nonmonotonicity, nontransitivity, supraclassicality, DD/II/AA/SS), soundness audit checking each rule for containment-leak false positives (17), CLI integration (17), logging/tracing (9)
• 63 tests from every concrete worked example in Ch. 3 (Toy Base T, monotonicity/transitivity failures, explicitating theorems, distribution failure, meta-modus-ponens failure, Mingle-Mix failure, conservative extension)
• 17 Hypothesis property-based tests against randomly generated material bases verifying: Containment, supraclassicality schemas, DDT biconditional, no-Weakening, no-Cut, conservativity, base consequence derivability, idempotency, and serialization roundtrip

Theoretical Background

This implements the NMMS sequent calculus from:

• Hlobil, U. & Brandom, R. B. (2025). Reasons for Logic, Logic for Reasons. Ch. 3: "Introducing Logical Vocabulary."

NMMS codifies open reason relations — consequence relations where Monotonicity and Transitivity can fail. The material base encodes defeasible material inferences among atomic sentences, and the Ketonen-style logical rules extend this to compound sentences while preserving nonmonotonicity.

License

MIT