LogicMark

A procedurally generated benchmark for evaluating symbolic logic in language models. Each problem presents a set of variable equality/inequality premises and asks the model to identify which conclusion necessarily follows.

Unlike knowledge-based benchmarks, LogicMark contains no facts a model could have memorised from pretraining. Every problem is generated fresh from abstract variable names (a, b, c, ...), so a model cannot pattern-match to training data - it must actually reason. This makes LogicMark a direct probe of intrinsic reasoning capability: the logical structure that has been built into the model's weights through training, independent of world knowledge or surface-level heuristics.

Evaluation is log-likelihood multiple-choice — no chain-of-thought, no prompting tricks. Models are scored purely on how well they assign probability to the correct completion.

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Benchmark Results (10000 examples)

Evaluated using average log-likelihood over ending tokens, normalised by length. Random chance = 25%.

Company	Model	Params	1-hop	2-hop	3-hop	4-hop	5-hop	Avg
Alibaba	Qwen2.5-Math-1.5B	1.5B	51.10%	56.47%	50.40%	44.60%	47.40%	51.73%
Axiomic Labs	GPT-X2-125M	125M	69.00%	48.98%	48.76%	41.07%	42.50%	49.09%
HuggingFace	SmolLM2-135M	135M	78.70%	50.72%	43.16%	38.47%	39.10%	48.6%
EleutherAI	pythia-2.8b	2.8B	79.90%	51.73%	40.56%	37.93%	37.50%	48.26%
Alibaba	Qwen2.5-3B	3.1B	47.10%	53.33%	47.68%	41.13%	40.80%	48.21%
Alibaba	Qwen2.5-1.5B	1.5B	48.30%	50.42%	46.36%	42.00%	43.00%	47.19%
HuggingFace	SmolLM2-1.7B	1.7B	70.50%	51.08%	37.64%	35.13%	37.10%	45.87%
Alibaba	Qwen2.5-Coder-1.5B	1.5B	46.70%	49.93%	43.48%	38.73%	42.60%	45.58%
EleutherAI	GPT-neo-125m	125M	72.10%	50.58%	36.96%	34.47%	36.00%	45.45%
Facebook	MobileLLM-R1-140M-base	140M	68.00%	48.98%	38.96%	34.00%	38.10%	45.04%
EleutherAI	pythia-31m	30M	63.90%	49.60%	37.84%	35.40%	37.40%	44.74%
OpenAI	gpt2	124M	56.60%	51.42%	36.88%	35.47%	37.50%	44.52%
OpenAI	gpt2-medium	355M	57.50%	51.42%	36.92%	35.47%	36.10%	44.48%
OpenAI	gpt2-xl	1.6B	58.60%	49.55%	37.76%	36.00%	38.00%	44.32%
HuggingFace	SmolLM-135M	135M	58.80%	50.28%	36.36%	34.93%	35.90%	43.91%
Alibaba	Qwen2.5-0.5B	494M	48.60%	49.10%	39.60%	37.53%	38.40%	43.87%
Facebook	OPT-125M	125M	59.40%	49.53%	36.68%	35.40%	36.20%	43.85%
Axiomic Labs	GPT-X-125M	125M	57.30%	49.45%	37.16%	35.33%	37.10%	43.81%
EleutherAI	pythia-160m	162M	56.10%	41.33%	32.68%	30.93%	34.20%	38.37%
EleutherAI	pythia-70m	70M	44.00%	38.70%	30.48%	28.67%	29.90%	34.79%

Task Format

If:
a = g
e != f
b != g
d = h
c = g
a != b
d = e
a = e

Then
A. c = h
B. c != h
C. c != e
D. b = h

The model must select the option that is logically entailed by the premises. Distractors include the flipped version of the correct answer and false statements drawn from the same variable set.

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Hop Depth

Each problem is tagged with a hop depth — the minimum number of inference steps required to derive the correct answer from the premises:

• 1-hop: answer is directly stated as a premise
• 2-hop: requires one transitive step (e.g. a=b, b=c → a=c)
• 3-hop: requires two transitive steps
• 4-hop+: longer inference chains

The current dataset (1000 problems) targets the following distribution:

Hop	Target
1	10%
2	40%
3	25%
4	15%
5	10%

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Graph Styles

Premises are generated using six equality graph topologies, each producing different reasoning patterns:

Style	Description
chain	Variables linked in a linear sequence
star	One central variable connected to many others
clusters	Groups of variables internally linked
tree	Binary tree topology — maximises chain depth per variable
bipartite	Edges only cross between two halves — same-side equalities are structurally impossible
mixed	Random combination of the above

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Generator

baseloggen_v2.py — full generator with hop depth control and equality type balancing.

from baseloggen_v2 import Config, generate_dataset, to_benchmark_format

cfg = Config(
    limit=1000,
    min_vars=4,
    max_vars=8,
    min_eq=2,
    max_eq=5,
    min_neq=1,
    max_neq=3,
    num_choices=4,
    min_answer_hop=2,
    target_hop_dist={1: 0.10, 2: 0.40, 3: 0.25, 4: 0.15, 5: 0.10},
    styles=("chain", "star", "clusters", "tree", "bipartite", "mixed"),
    seed=42,
)

dataset = generate_dataset(cfg)
formatted = [to_benchmark_format(p, i) for i, p in enumerate(dataset)]

Key Config Parameters

Parameter	Description
min_answer_hop	Minimum hop depth for the correct answer (suppresses trivial problems)
target_hop_dist	Dict mapping hop depth → fraction. Generator samples each bucket exactly.
styles	Tuple of graph topologies to sample from

Design Decisions

Equality type balance — correct answers are 50/50 sampled from = and != statements to prevent models from exploiting the observation that correct answers tend to be equality statements.

Exact hop targeting — when target_hop_dist is set, each bucket is generated with rejection sampling targeting exactly that hop depth, rather than relying on natural distribution (which heavily favours 2-hop).

Flipped distractor — the negation of the correct answer (e.g. a = b → a != b) is always included as a distractor to ensure the model can't win by ignoring inequality structure.

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Dataset Format

{
  "id": "symbolic_00042",
  "domain": "Symbolic",
  "context": "If:\na = b\nb != c\na = d\n\nThen",
  "options": ["a != c", "d = c", "b = d", "a = c"],
  "answer_index": 0,
  "answer": "a != c",
  "hop_depth": 2
}

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