Relational KAT+Antidomain Overview

• The paper introduces Relational KAT+Antidomain, extending KAT with antidomain to express no-successor properties and modal logic in a relational setting.
• It formalizes binary relations using composition, Kleene star, and Boolean tests, applying ultrafilter constructions for finite and infinite model representations.
• The equational theory is characterized by ExpTime and coNP complexity bounds, with decision procedures leveraging loop-automata and 2-way alternating automata.

Relational Kleene Algebra with Tests and Antidomain (Relational KAT+Antidomain) is an algebraic framework for reasoning about binary relations, equipped with the operations of Kleene algebra, a Boolean test subalgebra, and the antidomain operator. This formalism extends Kleene algebra with tests (KAT) to include the antidomain, enabling expressive modeling of reachability, modal logic, and program semantics, particularly for “no successor” properties. Recent results characterize its algebraic properties, representation theory, and equational complexity, establishing tight results for both finite and infinite relational models (Nakamura, 28 Dec 2025, McLean, 2015).

1. Algebraic Structure, Signature, and Semantics

Relational KAT+Antidomain operates over signatures containing:

• Composition (·): Relational composition, where $(x, y) \in (R \cdot S)$ iff $\exists z \in X: (x,z) \in R \wedge (z,y) \in S$.
• Join (+): Set-union of binary relations.
• Kleene star (*): Reflexive-transitive closure $R^* = \bigcup_{n \geq 0} R^n$.
• Constants 0, 1: Empty relation and identity relation, respectively.
• Antidomain (A(·)) and Domain (D(·)): For $R \subseteq X \times X$, $D(R) = \{ (x,x) : \exists y \; (x,y) \in R \}$, $A(R) = \{ (x,x) : \neg \exists y \; (x,y) \in R \}$.
• Boolean Subalgebra of Tests: Distinguished elements closed under $\land, \lor, \neg$, interpreted as “modal” properties (as subsets of the domain).

Relational models interpret each algebraic element as a binary relation $R \subseteq X \times X$ on some set $X$. The antidomain $A(R)$ captures the diagonal pairs $(x,x)$ for which $x$ lacks an $R$-successor.

2. Equational and Quasiequational Axiomatization

The equational theory is given by layered axioms, refined from the multiplace-function treatment (McLean, 2015):

Fragment	Key Axioms	Notable Laws
Composition + Antidomain	Associativity, Left-zero, Twisted antidomain $x\cdot A(y) = A(xy)\cdot x$, $A(x)\cdot x = 0$	Domain complement $0 = A(x)\cdot x$
Boolean Tests	Idempotence, Distributivity, Separation	$$D(a)\cdot b = D(a)\cdot c \land A(a)\cdot b = A(a)\cdot c \Rightarrow b=c$$
Kleene Algebra Layer	$(A,+,\cdot,*,0,1)$ forms idempotent semiring, star axioms	$x^*$ is least solution to $z=1+xz$
Antidomain Interaction	$A(1)=0$, $A(0)=1$, $A(xy)=A(x)+xA(y)$, $A(x+y)=A(x)\cdot A(y)$	$A(x)\leq 1$, $A(x)\cdot x=0$

The antidomain axioms (AD1–AD5) are:

• (AD1) $a(x) \leq 1$
• (AD2) $a(x) \cdot x = 0$
• (AD3) $a(x\cdot y) = a(x) + x \cdot a(y)$
• (AD4) $a(x + y) = a(x)\cdot a(y)$
• (AD5) $a(1) = 0$, $a(0) = 1$

These force $A(R)$ as the complement (in the diagonal $\{(x,x): x\in X\}$) of $D(R)$ in the relational setting (Nakamura, 28 Dec 2025, McLean, 2015).

3. Relational Representation and Finite Models

A complete representation theorem states that every KAT+antidomain algebra satisfying the above axioms is isomorphic to a subalgebra of binary relations with the corresponding operations. The construction uses ultrafilters over the Boolean algebra of tests:

• For each ultrafilter $U$, define a right congruence $\equiv_U$ via existence of a test $t \in U$ with $t\cdot a = t\cdot b$.
• The base set is the set of nonzero congruence classes $[a]$; action is $[a] \mapsto [a\cdot b]$.
• Taking the product over all ultrafilters yields an embedding into a relational (possibly infinite) model.

Every finite KAT+antidomain algebra admits a representation on a finite base of size $O(|A|^3)$ (McLean, 2015).

4. Equational Theory: Complexity Results

The computational complexity of the equational theory for relational KAT+antidomain is tightly classified:

• ExpTime-completeness: For the full equational theory with antidomain, the word problem (deciding $s=t$ in all relational models) is ExpTime-complete (Nakamura, 28 Dec 2025). The lower bound uses reductions from ExpTime-hard problems such as PDL satisfiability or alternating Turing machine acceptance. The upper bound proceeds via reduction to inclusion for loop-automata (see below).
• coNP-completeness: In restricted signatures (e.g., composition and antidomain only), the equational theory is coNP-complete. One can nondeterministically guess a small countermodel (linear in the size of the equation) and verify inequation in polynomial time (McLean, 2015). Even in the absence of the star operation, coNP-completeness holds.
• PSPACE-completeness: When the Kleene star is present (but not antidomain), the star-equational theory for KAT is PSPACE-complete (Nakamura, 28 Dec 2025).

5. Automaton-Theoretic Approach and Loop-Automata

A central methodology in analyzing relational KAT+antidomain employs automaton-theoretic reductions:

• Loop-automata: These extend nondeterministic finite automata (NFA) over relational structures with special “loop-test” transitions, testing existence of loops at the current vertex.
• Language Inclusion: Validity of a KAT+antidomain equation is reduced, in polynomial time, to inclusion between languages recognized by corresponding loop-automata.
• 2-way Alternating Automata (2AFA): Given two KAT+antidomain terms $e, f$, normalization yields loop-automata $A_e, A_f$; construction of 2AFAs $M_e, M_f$ enables a decision procedure for $L(M_e) \subseteq L(M_f)$ in ExpTime.

The small-model property holds: any necessary countermodel can be found of size polynomial in the length of the equation (Nakamura, 28 Dec 2025).

6. Illustrative Laws, Examples, and Modal Intuition

Fundamental interaction laws for antidomain include:

• $a(x \cdot y) = a(x) + x\cdot a(y)$, expressing that “no $(x\cdot y)$-edge” entails “no $x$-edge” or “after an $x$-edge, no $y$-edge.”
• $a(x)\cdot x = 0$: after verifying the absence of an $x$-successor, $x$ cannot be traversed.

The domain operator $d(x)$ acts as a modal diamond (“there is at least one $x$-successor”), while antidomain $a(x)$ is its complement (“there is no $x$-successor”). Antidomain enables explicit reasoning about “no next move,” essential in expressing boundary conditions and reachability constraints in system and program models (Nakamura, 28 Dec 2025).

7. Representation Flexibility and Connections to Related Structures

The axiomatic foundation of relational KAT+antidomain generalizes classic relation algebras, algebras of partial functions, and modal logics. Finite and infinite representation flexibility (via ultrafilter constructions) ensures that all finite algebras have a concrete relational model. Preferential union, intersection, and fixset operators admit finite or quasiequational axiomatizations when present in the signature (McLean, 2015).

This framework subsumes and extends work on relation algebras, modal algebras, and algebraic representations for program semantics. Its equational theory matches or refines known complexity bounds for related modal, dynamic, and program logics.