Retrieval Axioms¶

Constraints on document rankings for a query

Retrieval axioms compare two documents \(d_1\), \(d_2\) for a query \(q\) and implement the interface Axiom[Query, Document]. The original formulations of most axioms are restricted to artificial settings, e.g., single-term queries or documents of exactly the same length. Following Hagen et al. [Hagen et al., 2016] and Völske et al. [Völske et al., 2021], the implementations in ir_axioms incorporate an axiom's formalized idea while also making it applicable in practical settings [Bondarenko et al., 2022]: they are reformulated to work for arbitrary queries, to express pairwise preferences, and they include parameters to relax or strengthen pre- and filter conditions (e.g., a relative margin_fraction within which values are considered equal).

The API reference documents all classes and parameters; the integrations guide shows how to apply the axioms to a retrieval index.

Below, each axiom is listed with its original intuition and formulation, followed by the changes made in ir_axioms. In the formulations, \(\mathrm{tf}(t, d)\) denotes the frequency of term \(t\) in document \(d\), \(|d|\) the document length, \(\mathrm{idf}(t)\) the inverse document frequency of \(t\), and \(\mathrm{sim}(t, t')\) a term similarity.

Axioms requiring a retrieval index

Axioms using collection statistics (TFC3, M-TDC, LEN-M-TDC, ASPECT-REG) require an index, provided by the PyTerrier or Pyserini integration. LB1 requires retrieval scores, and ORIG and ORACLE require ranks/scores and relevance labels, respectively (see utility axioms).

Term Frequency¶

TFC1¶

TFC1 Tfc1Axiom — "Prefer documents with more query term occurrences." [Fang et al., 2004]

Original formulation: Given \(q = \{t\}\) and \(d_1, d_2\) with \(|d_1| = |d_2|\): if \(\mathrm{tf}(t, d_1) > \mathrm{tf}(t, d_2)\), then \(d_1 >_\text{TFC1} d_2\).
Changes in ir_axioms: Term frequencies are summed over all query terms, to allow queries with more than one term. Term frequency sums within a relative margin are considered equal, i.e., the inequality requires at least a 10 % difference [Völske et al., 2021].
Parameters: margin_fraction=0.1 (term frequency sum margin), precondition=LEN() (document length precondition with 10 % margin).

TFC2¶

TFC2 — "Additional query term occurrences yield smaller retrieval score improvements." [Fang et al., 2011]

Not implemented, because the axiom compares score differences of three documents and does not induce a pairwise preference.

TFC3¶

TFC3 Tfc3Axiom — "Prefer documents with occurrences of more query terms." [Fang et al., 2011]

Original formulation: Given \(q = \{t_1, t_2\}\) and \(d_1, d_2\) with \(|d_1| = |d_2|\) and \(\mathrm{idf}(t_1) = \mathrm{idf}(t_2)\): if \(\mathrm{tf}(t_1, d_2) = \mathrm{tf}(t_1, d_1) + \mathrm{tf}(t_2, d_1)\) and \(\mathrm{tf}(t_2, d_2) = 0\) and \(\mathrm{tf}(t_1, d_1) \neq 0\) and \(\mathrm{tf}(t_2, d_1) \neq 0\), then \(d_1 >_\text{TFC3} d_2\).
Changes in ir_axioms: Term discrimination uses the inverse document frequency with a relative margin. Preferences are counted over all pairs of query terms, to allow queries with more than two terms [Völske et al., 2021].
Parameters: margin_fraction=0.1 (IDF margin), precondition=LEN().

TDC and M-TDC¶

M_TDC ModifiedTdcAxiom — "Prefer documents with more discriminative query terms."

Original formulation (TDC): Given \(q = \{t_1, t_2\}\) and \(d_1, d_2\) with \(|d_1| = |d_2|\), let \(d\) be some document. If \(\mathrm{idf}(t_1) > \mathrm{idf}(t_2)\) and \(t_1 \in d_1\), \(t_2 \notin d_1\), \(t_1 \notin d_2\), \(t_2 \in d_2\), then \(d \circ d_1 >_\text{TDC} d \circ d_2\) [Fang et al., 2004; Fang et al., 2011].
Changes in ir_axioms: The original TDC is not implemented. Instead, ir_axioms implements the modified TDC by Shi et al. [Shi et al., 2005], which compares documents with the same total query term frequency (but differently distributed over the query terms) and, for each pair of query terms with different IDF, gives a point to the document with more occurrences of the more discriminative term. The document with more points is preferred.
Variants: LEN_M_TDC LenModifiedTdcAxiom: M-TDC with an additional document length precondition (precondition=LEN()).

Document Length¶

LNC1¶

LNC1 Lnc1Axiom — "Penalize longer documents for non-relevant terms." [Fang et al., 2004]

Original formulation: Given \(q\) and \(d_1, d_2\), let \(t \notin q\) and \(t'\) denote some arbitrary terms. If \(\mathrm{tf}(t, d_1) + 1 = \mathrm{tf}(t, d_2)\) and \(\mathrm{tf}(t', d_1) = \mathrm{tf}(t', d_2)\) for all \(t' \neq t\), then \(d_1 \geq_\text{LNC1} d_2\).
Changes in ir_axioms: If all query terms occur equally often in both documents (within a relative margin), prefer the shorter document. The margin allows for relative term frequency implementations.
Parameters: margin_fraction=0.1.

LNC2¶

LNC2 — "Do not prefer shorter documents when the matched query term ratio is the same." [Fang et al., 2004]

Not implemented, because the precondition (\(d_1\) being a \(c\)-fold concatenation of \(d_2\)) practically never holds for real documents.

TF-LNC¶

TF_LNC TfLncAxiom — "Reward additional query terms more than document length is penalized." [Fang et al., 2004]

Original formulation: Given \(q = \{t\}\) and \(d_1, d_2\): if \(\mathrm{tf}(t, d_1) > \mathrm{tf}(t, d_2)\) and \(|d_1| = |d_2| + \mathrm{tf}(t, d_1) - \mathrm{tf}(t, d_2)\), then \(d_1 >_\text{TF-LNC} d_2\).
Changes in ir_axioms: For each query term, if both documents have the same length when ignoring that term, the document with more occurrences of the term gets a point. Points are counted over all query terms, to allow queries with more terms.

Lower-Bounding Term Frequency¶

LB1¶

LB1 Lb1Axiom — "Do not override the term presence–absence gap with length normalization." [Lv and Zhai, 2011]

Original formulation: Given \(t \in q\) and \(d_1, d_2\) with \(\mathrm{score}(q \setminus \{t\}, d_1) = \mathrm{score}(q \setminus \{t\}, d_2)\): if \(\mathrm{tf}(t, d_1) > 0\) and \(\mathrm{tf}(t, d_2) = 0\), then \(d_1 >_\text{LB1} d_2\).
Changes in ir_axioms: Following Hagen et al. [Hagen et al., 2016], uses the documents' retrieval scores (with 10 % softening) instead of the scores without term \(t\). Prefers the document that contains a query term that the other document does not contain.
Parameters: margin_fraction=0.1 (score margin). Requires documents with a score.

LB2¶

LB2 — "Repeated query term occurrence is less important than the first occurrence." [Lv and Zhai, 2011]

Not implemented, because the axiom requires modifying documents (adding/removing terms) and does not induce a practical pairwise preference.

Query Aspects¶

REG¶

REG RegAxiom — "Prefer documents covering more different query aspects." [Zheng and Fang, 2010]

Original formulation: Given \(q = \{t_1, t_2, t_3\}\) and \(d_1, d_2\) with \(|d_1| = |d_2|\), \(\mathrm{idf}(t_2) = \mathrm{idf}(t_3)\), \(\mathrm{sim}(t_1, t_2) > \mathrm{sim}(t_1, t_3)\), and \(\mathrm{sim}(t_1, t_2) > \mathrm{sim}(t_2, t_3)\): if \(\mathrm{tf}(t_1, d_1) = \mathrm{tf}(t_1, d_2) > 0\), \(\mathrm{tf}(t_3, d_1) = \mathrm{tf}(t_2, d_2) > 0\), and \(\mathrm{tf}(t_2, d_1) = \mathrm{tf}(t_3, d_2) = 0\), then \(d_1 >_\text{REG} d_2\).
Changes in ir_axioms: Determines the query term(s) with the minimum average similarity to the other query terms (i.e., the most "different" aspect) and prefers the document with more occurrences of these terms, as described by Völske et al. [Völske et al., 2021]. The term similarity is configurable (WordNet synonym sets by default, fastText if installed).
Parameters: precondition=LEN().
Variants: ANTI_REG AntiRegAxiom: uses the query term(s) with the maximum average similarity instead, as in the modification of REG by Hagen et al. [Hagen et al., 2016].; ASPECT_REG AspectRegAxiom: follows the query aspect clustering of Zheng and Fang [Zheng and Fang, 2010] (requires equally discriminative query terms) and prefers documents covering at least one of the query aspects.

AND¶

AND AndAxiom — "Prefer documents containing all query terms." [Wu and Fang, 2012]

Original formulation: Given \(q = \{t_1, t_2\}\) with \(\mathrm{idf}(t_1) \geq \mathrm{idf}(t_2)\): if \(\mathrm{tf}(t_1, d_1) = 1\), \(\mathrm{tf}(t_2, d_1) = 1\), \(\mathrm{tf}(t_1, d_2) > 1\), and \(\mathrm{tf}(t_2, d_2) = 0\), then \(d_1 >_\text{AND} d_2\).
Changes in ir_axioms: Prefers the document that contains all query terms if the other document does not, for queries of arbitrary length [Hagen et al., 2016].
Variants: LEN_AND LenAndAxiom: AND with a document length precondition (precondition=LEN()).; M_AND ModifiedAndAxiom: prefers the document that contains a larger subset of the query terms.; LEN_M_AND LenModifiedAndAxiom: M-AND with a document length precondition.

DIV¶

DIV DivAxiom — "Prefer documents that are less redundant with the query." [Gollapudi and Sharma, 2009]

Changes in ir_axioms: The diversity axioms of Gollapudi and Sharma characterize result-set diversification functions and do not express pairwise document preferences. Following Hagen et al. [Hagen et al., 2016], ir_axioms reformulates the idea as a pairwise preference: prefer the document with the lower vocabulary overlap (Jaccard coefficient) with the query.
Variants: LEN_DIV LenDivAxiom: DIV with a document length precondition (precondition=LEN()).

Semantic Similarity¶

STMC1¶

STMC1 Stmc1Axiom — "Prefer documents with terms more similar to query terms." [Fang and Zhai, 2006]

Original formulation: Given \(q = \{t\}\), \(d_1 = \{t_1\}\), and \(d_2 = \{t_2\}\) with \(t \neq t_1\), \(t \neq t_2\): if \(\mathrm{sim}(t, t_1) > \mathrm{sim}(t, t_2)\), then \(d_1 >_\text{STMC1} d_2\).
Changes in ir_axioms: Following Hagen et al. [Hagen et al., 2016], compares the average similarity of all document terms to all query terms, to allow arbitrary queries and documents. The term similarity is configurable (WordNet synonym sets by default, fastText if installed).

STMC2¶

STMC2 Stmc2Axiom — "Do not reward similar terms more than exact matches." [Fang and Zhai, 2006]

Original formulation: Given \(q = \{t\}\) and \(d_1, d_2\) with \(|d_1| = 1\), \(|d_2| = c > 0\), and a term \(t'\) with \(\mathrm{sim}(t, t') > 0\): if \(\mathrm{tf}(t, d_1) = 1\) and \(\mathrm{tf}(t', d_2) = c\), then \(d_1 \geq_\text{STMC2} d_2\).
Changes in ir_axioms: Following Hagen et al. [Hagen et al., 2016], given the most similar pair of a query term \(t\) and a non-query term \(t'\), prefer the document matching \(t\) exactly if the other document's frequency of \(t'\) relative to its length is similar (within a margin) to the first document's frequency of \(t\) relative to its length.
Parameters: margin_fraction=0.2.

Term Proximity¶

The proximity axioms were proposed by Hagen et al. [Hagen et al., 2016], inspired by the proximity measures studied by Tao and Zhai [Tao and Zhai, 2007]. Tao and Zhai's own constraints describe properties of distance measures and do not induce ranking preferences, so Hagen et al. derived the following pairwise axioms from the measures instead. All of them require that both documents contain the same subset of query terms (PROX1–PROX3) or all query terms (PROX4–PROX5).

PROX1¶

PROX1 Prox1Axiom — "Prefer documents with shorter distance between query term pairs."

Given \(q\) and \(d_1, d_2\) with \(d_1 \cap q = d_2 \cap q\), let \(P = \{(t, t') : t, t' \in q, t \neq t'\}\) and \(\delta(d, t, t')\) the average number of words between \(t\) and \(t'\) in \(d\). If \(\sum_{(t, t') \in P} \delta(d_1, t, t') < \sum_{(t, t') \in P} \delta(d_2, t, t')\), then \(d_1 >_\text{PROX1} d_2\).

PROX2¶

PROX2 Prox2Axiom — "Prefer documents with earlier query term occurrences."

Given \(q\) and \(d_1, d_2\) with \(d_1 \cap q = d_2 \cap q\), let \(\mathrm{first}(t, d)\) denote the position of the first occurrence of \(t\) in \(d\). If \(\sum_{t \in q} \mathrm{first}(t, d_1) < \sum_{t \in q} \mathrm{first}(t, d_2)\), then \(d_1 >_\text{PROX2} d_2\).

PROX3¶

PROX3 Prox3Axiom — "Prefer documents where the query occurs earlier as a phrase."

Given \(q\) and \(d_1, d_2\) with \(d_1 \cap q = d_2 \cap q\), let \(\tau(d, q)\) denote the first position of the whole query \(q\) as one phrase in \(d\) (\(\infty\) if \(q\) does not occur as a phrase). If \(\tau(d_1, q) < \tau(d_2, q)\), then \(d_1 >_\text{PROX3} d_2\).

PROX4¶

PROX4 Prox4Axiom — "Prefer documents that contain all query terms in a shorter substring."

Given \(q\) and \(d_1, d_2\) with \(d_1 \cap q = d_2 \cap q = q\), let \(\omega(d, q)\) denote the length of the shortest substring of \(d\) that contains all query terms. If \(\omega(d_1, q) < \omega(d_2, q)\), then \(d_1 >_\text{PROX4} d_2\). Following Hagen et al., \(\omega(d, q)\) is the pair of the number of non-query words within the closest grouping of all query terms and the frequency of this gap, so ties are broken by the number of such shortest substrings.

PROX5¶

PROX5 Prox5Axiom — "Prefer documents where the query terms are closer together on average."

Given \(q\) and \(d_1, d_2\) with \(d_1 \cap q = d_2 \cap q = q\), let \(\bar{s}(d, q)\) denote the average length of the shortest substrings of \(d\) that contain all query terms (averaged over all occurrences of query terms in \(d\)). If \(\bar{s}(d_1, q) < \bar{s}(d_2, q)\), then \(d_1 >_\text{PROX5} d_2\).

Argumentativeness¶

The argumentativeness axioms aim at queries that probably require argumentative results (e.g., "mercy killing") [Bondarenko et al., 2018; Bondarenko et al., 2019]. They identify argumentative units (premises and claims) with the TARGER argument mining API and apply a document length precondition by default (precondition=LEN()). Later work extended them for comparative questions [Reimer et al., 2022] and argument retrieval with embeddings [Heinrich et al., 2025] (see related axioms).

ArgUC¶

ArgUC ArgumentativeUnitsCountAxiom — "Prefer more argumentative units." [Bondarenko et al., 2022; Reimer et al., 2022]

Intuition: Favors documents with more argument units, i.e., premises and claims [Bondarenko et al., 2022].

QTArg¶

QTArg QueryTermOccurrenceInArgumentativeUnitsAxiom — "Prefer more query terms in argumentative units." [Bondarenko et al., 2022; Reimer et al., 2022]

Intuition: Favors documents where the query terms appear in (i.e., close to) the argument units [Bondarenko et al., 2022].

QTPArg¶

QTPArg QueryTermPositionInArgumentativeUnitsAxiom — "Prefer earlier query terms in argumentative units." [Bondarenko et al., 2022; Reimer et al., 2022]

Intuition: Favors documents where the first occurrence of a query term in an argumentative unit is closer to the beginning of the document, based on the general observation that query terms occur "earlier" in relevant documents [Troy and Zhang, 2007; Mitra et al., 2017].
Parameters: penalty=100000: penalty for the average query term position if a query term is not found in any argumentative unit (None to use the maximum length of the two documents).

aSL¶

aSL AverageSentenceLengthAxiom (alias: aSLDoc) — "Prefer passages with 12–20 words per sentence." [Bondarenko et al., 2022; Reimer et al., 2022]

Intuition: Favors more "readable" documents [Bondarenko et al., 2022], based on the general observation for text readability and good writing style [Markel, 2010; Newell, 2014].
Parameters: min_sentence_length=12, max_sentence_length=20.

References¶

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}

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@misc{newell2014editing,
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  year         = {2014},
  howpublished = {Online},
  note         = {Not indexed in DBLP; reference as given in the ir{\_}axioms source code}
}

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  author       = {Jan Heinrich Reimer and
                  Johannes Huck and
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  title        = {Grimjack at Touch{\'{e}} 2022: Axiomatic Re-ranking and Query Reformulation},
  booktitle    = {Proceedings of the Working Notes of CLEF 2022 - Conference and Labs
                  of the Evaluation Forum, Bologna, Italy, September 5th - to - 8th, 2022},
  series       = {CEUR Workshop Proceedings},
  volume       = {3180},
  publisher    = {CEUR-WS.org},
  year         = {2022},
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  author       = {Shuming Shi and
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  booktitle    = {{SIGIR} 2005: Proceedings of the 28th Annual International {ACM} {SIGIR}
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  publisher    = {{ACM}},
  year         = {2005},
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                  Conference on Research and Development in Information Retrieval,
                  Amsterdam, The Netherlands, July 23-27, 2007},
  pages        = {295--302},
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  year         = {2007},
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  author       = {Adam D. Troy and
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  title        = {Enhancing relevance scoring with chronological term rank},
  booktitle    = {{SIGIR} 2007: Proceedings of the 30th Annual International {ACM} {SIGIR}
                  Conference on Research and Development in Information Retrieval,
                  Amsterdam, The Netherlands, July 23-27, 2007},
  pages        = {599--606},
  publisher    = {{ACM}},
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                  Avishek Anand},
  title        = {Towards Axiomatic Explanations for Neural Ranking Models},
  booktitle    = {{ICTIR} '21: The 2021 {ACM} {SIGIR} International Conference on the
                  Theory of Information Retrieval, Virtual Event, Canada, July 11, 2021},
  pages        = {13--22},
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  author       = {Hao Wu and
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  title        = {Relation Based Term Weighting Regularization},
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                  Research, {ECIR} 2012, Barcelona, Spain, April 1-5, 2012. Proceedings},
  series       = {Lecture Notes in Computer Science},
  volume       = {7224},
  pages        = {109--120},
  publisher    = {Springer},
  year         = {2012},
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@inproceedings{zheng2010query,
  author       = {Wei Zheng and
                  Hui Fang},
  title        = {Query Aspect Based Term Weighting Regularization in Information Retrieval},
  booktitle    = {Advances in Information Retrieval, 32nd European Conference on {IR}
                  Research, {ECIR} 2010, Milton Keynes, UK, March 28-31, 2010. Proceedings},
  series       = {Lecture Notes in Computer Science},
  volume       = {5993},
  pages        = {344--356},
  publisher    = {Springer},
  year         = {2010},
  url          = {https://doi.org/10.1007/978-3-642-12275-0\_31},
  doi          = {10.1007/978-3-642-12275-0\_31}
}