Semiring Evaluation
====================

ProvSQL represents query provenance as a circuit of gates. A *semiring evaluation* :cite:`DBLP:conf/pods/GreenKT07` maps this circuit
to a value in a chosen semiring by:

1. Assigning a *leaf value* to each input gate via a provenance mapping.
2. Propagating these values through ``plus`` and ``times`` gates (and
   ``monus`` gates for m-semirings :cite:`DBLP:journals/japll/GeertsP10`)
   according to the semiring operations.

The built-in evaluation functions all follow the same calling convention:

.. code-block:: postgresql

    sr_<name>(provenance(), 'mapping_name')

where ``provenance()`` returns the token for the current output row and
``mapping_name`` is the name of a provenance mapping table.
Semirings are also extended with a *monus* operation allowing to
represent the output of non-monotone queries.

ProvSQL Studio's :ref:`evaluation strip <studio-circuit-eval-strip>`
drives every compiled and custom semiring interactively: pick a
semiring and a provenance mapping in the dropdown, click :guilabel:`Run`,
and the result lands inline. The rest of this chapter is the SQL
reference for the same operations.

Boolean Semiring
-----------------

:sqlfunc:`sr_boolean` evaluates the provenance in the Boolean semiring
``({true, false}, ∨, ∧, false, true)``, answering the question "was this
result derivable at all, depending on the Boolean mapping of input
provenance tokens?":

.. code-block:: postgresql

    SELECT name, sr_boolean(provenance(), 'my_mapping')
    FROM mytable;

Boolean-Expression Semiring
-----------------------------

:sqlfunc:`sr_boolexpr` evaluates the provenance in the *Boolean-expression*
semiring, returning a human-readable propositional formula. The mapping
argument is optional: with one, leaves are labelled by the mapping's
``value`` column; without one, leaves are rendered as bare ``x<id>``
placeholders.

.. code-block:: postgresql

    SELECT name, sr_boolexpr(provenance(), 'my_mapping')
    FROM mytable;

    SELECT name, sr_boolexpr(provenance())
    FROM mytable;

This is used as the basis for probability computation.


Symbolic Representation (as a Formula)
----------------------------------------

:sqlfunc:`sr_formula` returns a symbolic representation of the provenance
as a human-readable formula using symbols ``⊕`` and ``⊗``:

.. code-block:: postgresql

    SELECT name, sr_formula(provenance(), 'witness_mapping')
    FROM suspects;

For example, a result supported by tuples labelled ``a`` and ``b`` through
a join would show as ``a⊗b``; one that could come from either ``a`` or
``b`` would show as ``a⊕b``.

Counting Semiring (m-semiring)
-------------------------------

:sqlfunc:`sr_counting` evaluates the provenance in the *counting* m-semiring,
counting the number of derivations of the result (its multiplicity
under bag semantics):

.. code-block:: postgresql

    SELECT name, sr_counting(provenance(), 'count_mapping')
    FROM suspects;

The mapping should assign integer values (typically ``1``) to leaf tokens.

Why-Provenance
---------------

:sqlfunc:`sr_why` returns the *why-provenance* of a result – the set of
witnesses (sets of input tuples) that support the result:

.. code-block:: postgresql

    SELECT name, sr_why(provenance(), 'my_mapping')
    FROM mytable;

Leaf values may be bare labels (e.g. ``Alice``, treated as the singleton
witness ``{{Alice}}``) or already-structured why-provenance values
(``{}`` for zero, ``{{}}`` for one, ``{{a},{b,c}}`` for a multi-witness
set), which lets the output of one ``sr_why`` query be reused as input
to another.

How-Provenance
---------------

:sqlfunc:`sr_how` returns the *how-provenance* of a result – the
canonical polynomial in :math:`\mathbb{N}[X]` over the input-tuple
labels :cite:`DBLP:conf/pods/GreenKT07`.  Each derivation contributes
a monomial; coefficients count
distinct derivations of the same monomial:

.. code-block:: postgresql

    SELECT name, sr_how(provenance(), 'my_mapping')
    FROM mytable;

The result is rendered in canonical sum-of-products form, e.g.
``2⋅Alice⋅Bob + Alice^2 + Bob^2``.  Multiplication is the dot
``⋅``; exponents use ``^k``; ``0`` and ``1`` denote the additive and
multiplicative identities.  Because the form is canonical, two
semantically-equivalent provenance circuits collapse to identical
strings, making :sqlfunc:`sr_how` suitable for provenance-aware query
equivalence (e.g. checking that two ETL pipelines produce the same
provenance, not just the same tuples).  The how-semiring is
:math:`\mathbb{N}[X]`, the universal commutative semiring for
provenance.

Leaf values may be bare labels (e.g. ``Alice``, treated as the
monomial ``Alice``), the literal ``0``, or already-structured
polynomials following the same canonical syntax as the output (e.g.
``2⋅Alice⋅Bob^2 + 3⋅Charlie``), so the output of one ``sr_how`` query
can be reused as input to another.

Which-Provenance (Lineage)
---------------------------

:sqlfunc:`sr_which` returns the *which-provenance* (also known as
*lineage*) of a result – a single set of input labels that contributed
to it, namely the union of all witnesses:

.. code-block:: postgresql

    SELECT name, sr_which(provenance(), 'my_mapping')
    FROM mytable;

The result is rendered as ``{a,b,c}`` for a non-empty derivation, or
``⊥`` if no derivation exists.  Compared to :sqlfunc:`sr_why`,
which-provenance is more compact (a flat set rather than a set of
sets) but loses the breakdown into individual derivations.

Leaf values may be bare labels (e.g. ``Alice``, treated as the
singleton ``{Alice}``), the literal ``⊥`` (the additive zero), or
already-structured sets (``{}`` for the multiplicative identity,
``{a,b,c}`` for a non-empty set), so the output of one ``sr_which``
query can be reused as input to another.

Tropical Semiring (m-semiring)
-------------------------------

:sqlfunc:`sr_tropical` evaluates the provenance in the *tropical* (min-plus)
m-semiring ``(ℝ ∪ {+∞}, min, +, +∞, 0)``, returning the cost of the
cheapest derivation:

.. code-block:: postgresql

    SELECT name, sr_tropical(provenance(), 'cost_mapping')
    FROM mytable;

The mapping should assign ``float8`` cost values to leaf tokens; use
``'Infinity'::float8`` to encode the additive identity. This is useful
for shortest-path or least-cost provenance, where ``plus`` selects the
cheaper alternative and ``times`` accumulates cost along a derivation.

With the optional third argument ``nonnegative => true``, input costs
are checked nonnegative and the semiring becomes *absorptive*
(:math:`\min(0, a) = 0` for :math:`a \ge 0`): evaluation then also
accepts circuits carrying the ``'absorptive'`` assumption marker --
notably recursive queries evaluated under
``provsql.provenance = 'absorptive'`` (see
:ref:`provsql-provenance-class`), giving exact min-cost reachability
on cyclic graphs:

.. code-block:: postgresql

    SET provsql.provenance = 'absorptive';
    WITH RECURSIVE reach(node) AS (
        SELECT 1
      UNION
        SELECT e.dst FROM edges e JOIN reach r ON e.src = r.node
    )
    SELECT node, sr_tropical(provenance(), 'cost_mapping',
                             nonnegative => true) AS min_cost
    FROM reach;

On bounded-treewidth data this is also *fast*: the recursive
reachability shapes compile along a tree decomposition of the data
graph into certified circuits of linear total size (see
:doc:`probabilities`), which min-plus evaluation -- like any
absorptive-semiring evaluation -- reads off exactly, in time linear
in the circuit.  Single-source shortest distances over thousands of
probabilistic edges, hop-bounded variants (the min cost of reaching a
vertex within :math:`k` hops), per-region minima (through a
``GROUP BY`` over a joined member relation) and **exact directed
Steiner costs** -- the cheapest subgraph covering :math:`k` terminals
at once, through a self-join of the reachability CTE -- all stay on
this route.
The same holds for every absorptive semiring on this page:
:sqlfunc:`sr_viterbi` gives the most reliable path,
:sqlfunc:`sr_maxmin` the widest path, :sqlfunc:`sr_lukasiewicz` the
best fuzzy path, and :sqlfunc:`sr_temporal` the exact set of instants
at which a vertex is reachable from edges carrying validity
multiranges (see :doc:`temporal`).

Viterbi Semiring (m-semiring)
------------------------------

:sqlfunc:`sr_viterbi` evaluates the provenance in the *Viterbi* (max-times)
m-semiring ``([0,1], max, ×, 0, 1)``, returning the probability of the
most likely derivation:

.. code-block:: postgresql

    SELECT name, sr_viterbi(provenance(), 'prob_mapping')
    FROM mytable;

The mapping should assign ``float8`` probability values in :math:`[0,1]`
to leaf tokens. Unlike :sqlfunc:`probability_evaluate`, which marginalises
over derivations, the Viterbi semiring keeps only the single most likely
derivation, making it suitable for *most-probable-explanation* style
queries.

Łukasiewicz Fuzzy Semiring (m-semiring)
----------------------------------------

:sqlfunc:`sr_lukasiewicz` evaluates the provenance in the *Łukasiewicz*
fuzzy m-semiring ``([0,1], max, ⊗_Ł, 0, 1)``, where the
multiplicative operation is the Łukasiewicz t-norm
:math:`a \otimes_{\text{Ł}} b = \max(a + b - 1, 0)`:

.. code-block:: postgresql

    SELECT name, sr_lukasiewicz(provenance(), 'evidence_mapping')
    FROM mytable;

The mapping should assign ``float8`` graded-truth values in
:math:`[0,1]` to leaf tokens. Compared to :sqlfunc:`sr_viterbi` (which
multiplies probabilities), the Łukasiewicz t-norm preserves crisp
truth (:math:`0.7 \otimes_{\text{Ł}} 1 = 0.7`) and avoids the
near-zero collapse of long product chains. This makes it the standard
choice for fuzzy graded conjunctions where inputs are degrees of
evidence rather than independent probabilities.

Temporal (Interval-Union) Semiring (m-semiring)
------------------------------------------------

:sqlfunc:`sr_temporal` evaluates the provenance in the *temporal*
(interval-union) m-semiring over PostgreSQL ``tstzmultirange`` values.
Addition is multirange union, multiplication is intersection, monus is
set difference; the additive identity is ``'{}'`` and the multiplicative
identity is ``'{(,)}'`` (the universal range):

.. code-block:: postgresql

    SELECT entity_id, sr_temporal(provenance(), 'validity_mapping')
    FROM mytable;

:sqlfunc:`union_tstzintervals` is a thin SQL alias for
:sqlfunc:`sr_temporal`, retained for backward compatibility; both
compute the same union of validity intervals.

Requires PostgreSQL ≥ 14 (for ``tstzmultirange``).

Numeric and Integer Interval-Union Semirings (m-semirings)
-----------------------------------------------------------

:sqlfunc:`sr_interval_num` and :sqlfunc:`sr_interval_int` are the
numeric and integer counterparts of :sqlfunc:`sr_temporal`: the same
interval-union algebra (union for ⊕, intersection for ⊗, set
difference for monus) over ``nummultirange`` and ``int4multirange``
respectively.

.. code-block:: postgresql

    SELECT name, sr_interval_num(provenance(), 'sensor_validity_mapping')
    FROM mytable;

    SELECT cite, sr_interval_int(provenance(), 'page_range_mapping')
    FROM mytable;

Typical use cases:

- ``sr_interval_num``: measurement-validity provenance in scientific
  data integration (a sensor reading or model coefficient is valid
  only over a parameter range; joins compute jointly-valid ranges).
- ``sr_interval_int``: page-range or line-range provenance in
  scholarly / source-code corpora ("supported by pages [12,18] of
  doc A and pages [3,5] ∪ [40,42] of doc B").

Both require PostgreSQL ≥ 14.

Min-Max and Max-Min Semirings (m-semirings)
--------------------------------------------

:sqlfunc:`sr_minmax` and :sqlfunc:`sr_maxmin` evaluate the provenance
in the min-max and max-min m-semirings over an arbitrary user-defined
PostgreSQL ``ENUM`` type. The carrier order comes from
``pg_enum.enumsortorder``; bottom and top are derived automatically.

- :sqlfunc:`sr_minmax`: ``⊕ = min``, ``⊗ = max``, zero is the top of
  the enum, one is the bottom. The *security* shape: alternative
  derivations combine to the least sensitive label, joins combine to
  the most sensitive label.
- :sqlfunc:`sr_maxmin`: dual ``⊕ = max``, ``⊗ = min``. The *fuzzy* /
  availability / trust shape: alternatives combine to the most
  permissive label, joins combine to the strictest label.

The third argument to both functions is a sample value of the carrier
enum, used only for type inference; its value is ignored. Given a
``classification_level`` enum ordered from ``unclassified`` to
``not_available``, where ``not_available`` is the top of the enum and
plays the role of the semiring 𝟘 (no derivation possible):

.. code-block:: postgresql

    SELECT create_provenance_mapping('personnel_level', 'personnel', 'classification');

    SELECT city, sr_minmax(provenance(), 'personnel_level',
                           'unclassified'::classification_level) AS clearance
    FROM (SELECT DISTINCT city FROM personnel) t;

This is the compiled replacement for the hand-rolled access-control
semiring previously documented as the *security semiring*: a single
implementation covers any user enum (security lattices, fuzzy-discrete
trust levels, three-valued logic, project-specific orderings).

.. _custom-semirings:

Custom Semirings with :sqlfunc:`provenance_evaluate`
------------------------------------------------------

Advanced users can define custom semirings in SQL and evaluate them
using :sqlfunc:`provenance_evaluate`.  The function takes a provenance token,
a mapping table, a one element, and the names of aggregate functions
implementing the semiring operations:

.. code-block:: postgresql

    provenance_evaluate(
        token UUID,           -- provenance token to evaluate
        token2value regclass, -- mapping table (token → value)
        element_one,          -- identity element (type determines the semiring value type)
        plus_function,        -- name of the ⊕ aggregate
        times_function,       -- name of the ⊗ aggregate
        monus_function,       -- name of the ⊖ function (optional, for m-semirings)
        delta_function        -- name of the δ function (optional, for δ-semirings)
    )

The plus and times operations must be defined as PostgreSQL aggregate
functions with a two-argument state transition function. The monus and
delta operations are plain functions.

As a worked example, consider the *capability* semiring over the
``bit(2)`` carrier interpreted as the diamond lattice
:math:`\{00, 01, 10, 11\}` (e.g., ``(can_read, can_write)``).
``⊕ = |`` (bitwise OR) combines alternative derivations permissively;
``⊗ = &`` (bitwise AND) combines joins restrictively;
``⊖ = a & ~b`` (bitwise AND-NOT) is the Boolean difference monus;
``δ`` is the identity, so an aggregated group carries the OR of the
capabilities of its supporting rows (e.g., a Paris group built from a
read-only and a write-only row is annotated ``B'11'``, while a group
built from two read-only rows stays ``B'01'``). Zero is ``B'00'``,
one is ``B'11'``:

.. code-block:: postgresql

    CREATE FUNCTION cap_or  (state bit(2), v bit(2)) RETURNS bit(2) IMMUTABLE
      LANGUAGE SQL AS $$ SELECT state | v $$;
    CREATE FUNCTION cap_and (state bit(2), v bit(2)) RETURNS bit(2) IMMUTABLE
      LANGUAGE SQL AS $$ SELECT state & v $$;
    CREATE FUNCTION cap_minus(a bit(2), b bit(2))    RETURNS bit(2) IMMUTABLE
      LANGUAGE SQL AS $$ SELECT a & ~b $$;
    CREATE FUNCTION cap_delta(v bit(2)) RETURNS bit(2) IMMUTABLE
      LANGUAGE SQL AS $$ SELECT v $$;

    CREATE AGGREGATE cap_plus (bit(2)) (
      sfunc=cap_or,  stype=bit(2), initcond='00');
    CREATE AGGREGATE cap_times(bit(2)) (
      sfunc=cap_and, stype=bit(2), initcond='11');

    SELECT name,
           provenance_evaluate(provenance(), 'capability_mapping',
                               B'11'::bit(2),
                               'cap_plus', 'cap_times', 'cap_minus', 'cap_delta')
    FROM mytable;


This is a commutative m-semiring on the four-element Boolean
lattice :math:`B^2`; the lattice is partial (the two middle elements
are incomparable), so it is not subsumed by :sqlfunc:`sr_minmax` or
:sqlfunc:`sr_maxmin`. Additional examples can be found in the test
suite: :download:`test/sql/capability.sql <../../../test/sql/capability.sql>`
(the capability semiring above) and
:download:`test/sql/formula.sql <../../../test/sql/formula.sql>`
(symbolic representation as a formula).

.. note::

    :sqlfunc:`provenance_evaluate` is a PL/pgSQL function that traverses
    the provenance circuit recursively.  It does not support ``cmp`` gates
    introduced by ``HAVING`` clauses; queries with ``HAVING`` will produce
    an error.
    The built-in compiled semirings (:sqlfunc:`sr_formula`,
    :sqlfunc:`sr_counting`, etc.) are implemented in C, support the
    aggregation and ``HAVING`` gate types (``agg``, ``semimod``,
    ``cmp``, ``value``), and are significantly faster.
    Prefer compiled semirings when available; use
    :sqlfunc:`provenance_evaluate` for semirings not covered by the
    built-in set.

Provenance Mappings
--------------------

All semiring functions take a mapping name as their second argument.
A mapping is created with :sqlfunc:`create_provenance_mapping`:

.. code-block:: postgresql

    SELECT create_provenance_mapping('mapping_name', 'table_name', 'column_name');

The mapping table has columns ``value`` and ``provenance`` (uuid). You
can also populate it manually for custom scenarios:

.. code-block:: sql

    INSERT INTO mapping_name(provenance, value)
    SELECT provenance(), my_label FROM mytable;

.. _semiring-boolean-compat:

Compatibility with Boolean-Provenance Rewriting
------------------------------------------------

When the provenance class is ``'boolean'`` (``provsql.provenance``) (see
:doc:`probabilities`), the planner rewrites hierarchical CQs to a
read-once form whose probability is computable in linear time, and
tags the root of the rewritten circuit so that semirings whose
algebra is not Boolean-faithful refuse to evaluate it.  The refusal
raises an explicit error rather than silently producing a wrong
value:

.. code-block:: text

    ERROR:  ProvSQL: provenance_evaluate_compiled: The requested
            semiring does not admit a homomorphism from Boolean
            functions; the wrapped sub-circuit was computed under a
            Boolean-provenance assumption (typically by the safe-query
            rewrite, provenance class 'boolean') and the evaluation is
            unsound under this semiring.  Re-run the query under a more
            general provenance class, or pick a Boolean-compatible
            semiring (boolean, boolexpr, formula, ...).

The following compiled semirings are Boolean-faithful and run on
any circuit, including rewritten ones: :sqlfunc:`sr_boolean`,
:sqlfunc:`sr_boolexpr`, :sqlfunc:`sr_formula`,
:sqlfunc:`sr_temporal`, :sqlfunc:`sr_interval_num`,
:sqlfunc:`sr_interval_int`.  The following are not, and refuse on
rewritten circuits: :sqlfunc:`sr_counting`, :sqlfunc:`sr_how`,
:sqlfunc:`sr_why`, :sqlfunc:`sr_which`, :sqlfunc:`sr_tropical`,
:sqlfunc:`sr_viterbi`, :sqlfunc:`sr_lukasiewicz`,
:sqlfunc:`sr_minmax`, :sqlfunc:`sr_maxmin`.

To run a refused semiring, switch
the provenance class below ``'boolean'`` (in the current session, or for
the role / database) and re-evaluate; the next provenance circuit
built for the query is unrewritten and accepts any semiring.

ProvSQL Studio's evaluation strip filters the semiring dropdown to
hide the refusing semirings whenever the root carries the rewrite
tag.
