Ehrenfeucht-Fraïssé Games in Semiring Semantics

• Ehrenfeucht-Fraïssé games in semiring semantics are an adaptation of classical model comparison games to evaluate first-order logic over commutative semirings with rich annotations.
• The framework assesses logical equivalence by leveraging algebraic properties like full and n-idempotence, influencing soundness and completeness of game strategies.
• Variants such as bijection, counting, and homomorphism games address limitations of the standard approach, offering nuanced ways to capture equivalences in diverse semiring settings.

Ehrenfeucht-Fraïssé games provide a central technique in finite model theory for characterizing elementary equivalence and quantifier-rank equivalence between relational structures. In semiring semantics, these model comparison methods are generalized to first-order logics that evaluate formulae over commutative semirings rather than the classical Boolean domain. This extension supports provenance analysis and rich value assignments, capturing information such as sets of contributing facts, evaluation costs, confidence scores, access levels, or the enumeration of successful strategies. The applicability of Ehrenfeucht-Fraïssé methods and their model comparison game variants depends crucially on the algebraic properties of the chosen semiring. Soundness and completeness results from the classical Boolean case do not always extend; instead, new game-theoretic notions and algebraic conditions must be considered to capture equivalences in semiring-valued logics (Brinke et al., 2023).

1. Semiring-Valued First-Order Logic: Background and Definitions

A commutative semiring is a tuple $K = (S, +, \cdot, 0, 1)$, where both $(S, +, 0)$ and $(S, \cdot, 1)$ are commutative monoids, with distributivity and $0$ as a zero element: $0\cdot s = s\cdot 0 = 0$ for all $s \in S$. A natural order $\leq$ is induced by $s \leq t$ if there exists $r$ with $s + r = t$. Rings (with additive inverses) are not considered in this framework. Semirings are further classified by algebraic properties:

• Idempotent: $(S, +, 0)$0 for all $(S, +, 0)$1.
• Multiplicatively idempotent: $(S, +, 0)$2 for all $(S, +, 0)$3.
• Fully idempotent: both additive and multiplicative idempotence hold.
• Absorptive: $(S, +, 0)$4 for all $(S, +, 0)$5.

A $(S, +, 0)$6-interpretation $(S, +, 0)$7 (with $(S, +, 0)$8 a finite relational signature) extends from atomic literals to all formulae in negation normal form by applying $(S, +, 0)$9 for $(S, \cdot, 1)$0, $(S, \cdot, 1)$1 for $(S, \cdot, 1)$2, $(S, \cdot, 1)$3 for $(S, \cdot, 1)$4, and $(S, \cdot, 1)$5 for $(S, \cdot, 1)$6. The fundamental property ensures that any homomorphism $(S, \cdot, 1)$7 commutes with semantic interpretation: $(S, \cdot, 1)$8.

Key types of equivalence:

• Isomorphism ($(S, \cdot, 1)$9): bijection $0$0 preserves all literal values.
• Elementary equivalence ($0$1): for all sentences $0$2, $0$3.
• $0$4-equivalence ($0$5): equality for all sentences up to quantifier rank $0$6.

2. Classical Ehrenfeucht-Fraïssé Games and Their Semiring Extensions

The $0$7-round Ehrenfeucht-Fraïssé (EF) game $0$8 classically determines ($0$9-)elementary equivalence in Boolean semantics. In the semiring-annotated setting, $0\cdot s = s\cdot 0 = 0$0 proceeds with the same move structure, but the Duplicator must ensure that, for all literals of arity at most $0\cdot s = s\cdot 0 = 0$1, the values accord: $0\cdot s = s\cdot 0 = 0$2.

Soundness holds if a Duplicator winning strategy implies $0\cdot s = s\cdot 0 = 0$3-equivalence of interpretations. In semiring semantics, full idempotence of $0\cdot s = s\cdot 0 = 0$4 characterizes soundness for all $0\cdot s = s\cdot 0 = 0$5. Completeness stipulates that $0\cdot s = s\cdot 0 = 0$6-equivalence ensures a winning strategy; completeness is achieved only in the Boolean semiring case. Failure examples are given for a wide class of semirings, including min–max, Viterbi, tropical, Łukasiewicz, and polynomial semirings (Brinke et al., 2023).

3. Analysis for Specific Semirings

The interaction between EF-style games and semiring structure is dictated by three principal criteria:

• $0\cdot s = s\cdot 0 = 0$7 is sound for $0\cdot s = s\cdot 0 = 0$8 iff $0\cdot s = s\cdot 0 = 0$9 is fully idempotent.
• $s \in S$0 is complete for $s \in S$1 only if $s \in S$2 (the classical Boolean case).
• Counting and bijection game variants expand the set of semirings for which soundness or completeness can be achieved:
  • Natural numbers $s \in S$3: $s \in S$4 fails soundness; bijection games (see below) are sound and complete.
  • Polynomial semiring $s \in S$5: similar outcome.
  • Viterbi semiring $s \in S$6 and tropical semiring $s \in S$7: neither game is sound nor complete; counterexamples arise even in one move.
• For semirings relevant to access-level analysis (min–max structure), the EF game may show the inexpressibility of certain properties (e.g., FO formulas cannot compute minimal access levels along a path if Duplicator can always win) (Brinke et al., 2023).

4. Variants of Model Comparison Games

To address the limitations of $s \in S$8 in general semiring settings, several game variants are defined:

• Bijection Games ($s \in S$9): In each round, Duplicator provides a bijection between universes. BG$\leq$0 is sound for all $\leq$1 and complete for $\leq$2. The completeness proof leverages a "high-power separation" lemma: appropriate characteristic sentences force multisets to be permutation-equal.
• Counting Games ($\leq$3): Spoiler picks subsets, Duplicator matches size, and responses must preserve literal values. $\leq$4 is sound for $\leq$5-idempotent $\leq$6 (where sums of $\leq$7 equal elements collapse).
• Homomorphism Games (see below) (Brinke et al., 2023).

Game Type	General Soundness Condition	General Completeness Condition
EF ($\leq$8)	$\leq$9 fully idempotent	$s \leq t$0 (Boolean)
Bijection (BG$s \leq t$1)	Always	On $s \leq t$2 where finite elementary equivalence implies isomorphism (e.g., $s \leq t$3)
Counting ($s \leq t$4)	$s \leq t$5-idempotence	Not generally
Homomorphism ($s \leq t$6)	Separating set exists for $s \leq t$7	On all lattice semirings

5. Homomorphism Games and Lattice Semirings

Homomorphism games ($s \leq t$8) leverage semiring homomorphisms into the Boolean semiring $s \leq t$9 to "forget" parts of the annotation and distinguish elements by sets of homomorphisms. Spoiler selects a homomorphism $r$0 from a separating set $r$1 and the play proceeds on Boolean-valued interpretations $r$2. The Duplicator must maintain a one-sided inequality $r$3 for all literals $r$4.

Soundness and completeness of $r$5 for $r$6-equivalence hold for any semiring $r$7 with an appropriate separating set $r$8 of homomorphisms. For finite or infinite lattice semirings (fully idempotent and absorptive), prime ideal constructions or nonzero $r$9-indecomposables provide such separating sets [$s + r = t$0].

• Finite lattice: for $s + r = t$1, homomorphisms $s + r = t$2 iff $s + r = t$3.
• Infinite lattice: for $s + r = t$4 a prime ideal, $s + r = t$5 iff $s + r = t$6.

HG$s + r = t$7 thus characterizes $s + r = t$8-equivalence on all lattice semirings, including cases where other game variants fail. The Birkhoff–Stone representation ensures separating homomorphisms in infinite cases.

6. Key Theorems, Inexpressibility, and Examples

Principal results include:

• Full-idempotence and Soundness: G$s + r = t$9 is sound for $(S, +, 0)$00-equivalence iff $(S, +, 0)$01 is fully idempotent.
• Completeness: Completeness of $(S, +, 0)$02 for all $(S, +, 0)$03 implies $(S, +, 0)$04.
• Bijection Games Soundness and Completeness: BG$(S, +, 0)$05 is always sound; complete for $(S, +, 0)$06.
• Counting Games: CG$(S, +, 0)$07 is sound iff $(S, +, 0)$08 is $(S, +, 0)$09-idempotent.
• Homomorphism Game Characterization: For any $(S, +, 0)$10 with a separating homomorphism set, $(S, +, 0)$11 fully characterizes $(S, +, 0)$12-equivalence.
• Access-Level Example: In min–max access-level semirings, $(S, +, 0)$13 demonstrates the inexpressibility of minimal access-level path computations by FO-formulae if Duplicator wins.

A counterexample with infinite star graphs and the tropical or Viterbi semiring demonstrates unsoundness of $(S, +, 0)$14 in a non-idempotent case, even for one-move games. Thus, the algebraic tabulation of semiring properties and matching game variants is crucial for correctly characterizing logical equivalence and expressiveness in semiring-valued logic.

Full proofs, additional examples (including connections to provenance semirings), k-pebble games, and logics with fixed points are detailed in the technical report (Brinke et al., 2023).