Generalized Intervals on Vectors

• Generalized intervals on vectors are algebraic structures that extend classical scalar intervals to vector spaces using both numeric and symbolic representations.
• They enable a variety of operations including set-theoretic, linear-algebraic, and analytic manipulations, supporting model checking and validated numerics.
• Applications span from efficient state-space reduction in Petri nets to robust interval analysis in optimization, with proven empirical advantages.

A generalized interval on vectors provides an algebraic or combinatorial structure extending the classical notion of scalar intervals to the context of vectors and functions on vector spaces, enabling rich operations, model semantics, and optimization for both symbolic and numerical applications. Unlike simply taking Cartesian products of scalar intervals (vectors of intervals), generalized intervals on vectors enjoy deeper closure properties under set-theoretic operations, admit representation of infinite or partially ordered sets, and support sophisticated linear-algebraic, analytic, or logical manipulations.

1. Algebraic and Geometric Foundations

Generalized intervals on vectors arise in two principal frameworks: algebraic models for vectors whose coordinates are intervals and symbolic representations for sets of vectors using interval-like constraints.

In the algebraic setting, given the set of real intervals $I = \{ [a, b] \subset \mathbb{R} : a \leq b \}$, one constructs a real vector space $(I, +, \cdot)$ by considering formal differences of intervals and quotienting by the relation $(X,Y) \sim (Z,T)$ if $X + T = Y + Z$. Scalar multiplication is defined by

• If $\alpha \geq 0$, $\alpha \cdot (K,0) = (\alpha K,0)$ and $\alpha \cdot (0, K) = (0, \alpha K)$;
• If $\alpha < 0$, $\alpha \cdot (K,0) = (0, -\alpha K)$ and $\alpha \cdot (0, K) = (-\alpha K, 0)$.

Every interval $X = [a, b]$ admits a unique decomposition as $X = c e_1 + r e_2$, where $c = \frac{b + a}{2}$ (center) and $r = \frac{b - a}{2}$ (radius), with basis vectors $e_1 = ([1,1], 0)$ and $e_2 = (0, [1,1])$. The vector space $I$ has dimension $2$ over $\mathbb{R}$ (Goze, 2010).

For $n$-vectors of intervals, $I^n = \{ V = (X_1, ..., X_n) : X_i \in I \}$ is a $2n$-dimensional real vector space with a natural basis extension.

In the combinatorial/symbolic setting, as in model checking, a generalized interval is a symbolic vector $(a, b)$, where $a, b \subset \mathbb{N}^n$, denoting all vectors $q \in \mathbb{N}^n$ such that $\forall q_a \in a,\, q_a \leq_e q$ and $\forall q_b \in b,\, q_b \not\leq_e q$. Singleton or empty $a$ and $b$ reduce to ordinary componentwise intervals, but the general form allows complex constraints on vector entries, such as arbitrary conjunctions of lower bounds and disjunctions of upper bound exclusions (Morard et al., 3 Feb 2026).

2. Algebraic Operations and Structural Properties

In the algebraic approach, matrices over intervals, $\mathrm{gl}(n, I) = I^{n \times n}$, inherit operations from the base vector space:

• Matrix addition: $(A + B)_{ij} = X_{ij} + Y_{ij}$,
• Matrix multiplication: $(AB)_{ij} = \sum_{k=1}^n (X_{ik} \cdot Y_{kj})$, using the distributive product from interval arithmetic.

The resulting algebra is associative over $\mathbb{R}$, and the determinant is defined analogously to the classical case but with interval multiplication: $$\det A = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) X_{1\sigma(1)} \cdots X_{n\sigma(n)}$$. Invertibility is governed by whether the determinant interval excludes $0$, with the adjugate/Cramer method yielding the inverse (Goze, 2010).

For generalized intervals in the symbolic model-checking context, the set $SV$ (symbolic vectors) is partially ordered by set-inclusion of their denoted subsets, and is closed under intersection, difference, and complementation. The Boolean algebra of sets of symbolic vectors, $SVS$, admits set-wise union, intersection, and complement, and all operations correspond homomorphically to their concrete set-theoretic analogs in $\mathbb{N}^n$ (Morard et al., 3 Feb 2026).

3. Applications: Model Checking, Interval Analysis, Polytopes

Model Checking

Generalized intervals on vectors provide a powerful symbolic representation for sets of markings in Petri nets, crucial for handling state space explosion. In this usage, they support:

• Direct representation of infinite or complexly constrained subsets of states.
• Homomorphic support for all set-theoretic operations and for the pre-image operator under Petri net transitions, lifting to full CTL (Computation Tree Logic) fixpoint evaluation.
• Canonical (unique) representations, minimizing redundancy and enabling effective comparison, inclusion, and merging (Morard et al., 3 Feb 2026).

Optimizations such as saturation—iterative local fixpoints over increasing capacity—significantly reduce intermediate symbolic structure size, and clustering—partitioning the vector space for transitions that touch only a few components—yield dramatic computational efficiency.

Numerical Interval Analysis

In validated numerics, generalized intervals are central to computing verified enclosures or inner approximations of the range of vector-valued functions. The classical outer approximation $[f]([x])$ over-approximates the range, while the goal of inner approximation is to find a sub-box $G \subset f([x])$. In the non-square case, identification of a full-rank submatrix of the interval Jacobian enables a reduced mean-value theorem argument to certify $G$ is indeed contained in the image, subject to non-singularity conditions and auxiliary heuristics for submatrix selection (Olivier et al., 2013).

Interval-Vector Polytopes

Combinatorial perspectives on interval vectors yield interval-vector polytopes, where an interval vector is any $(0,1)$-vector for which all $1$'s appear consecutively. The convex hull of such vectors (for various constraints on interval lengths or patterns) leads to polytopes with explicitly described vertices, face vectors, and volume. Some classes have volume equal to Catalan numbers or have face numbers given by the Pascal 3-triangle (Beck et al., 2012).

4. Canonical Forms and Computational Algorithms

A critical issue in symbolic representations is canonicalization: for the symbolic vector framework, every $(a,b)$ can be transformed uniquely into a canonical pair $(a_0, b_0)$ such that $a_0$ is a singleton, $b_0$ contains mutually incomparable vectors w.r.t. the partial order, $b_0$ strictly exceeds $a_0$ componentwise, and empty denotations reduce to a unique null form. For sets of symbolic vectors, canonicalization involves iterative merging of shareable elements using merge and diff primitives, with termination and uniqueness ensured by a strictly decreasing measure. While the number of operations is factorially bounded in the worst case, in practical CTL model checking the approach effectively controls intermediate complexity (Morard et al., 3 Feb 2026).

In numerical interval analysis, inner approximation algorithms rely on test operators like the reduced Krawczyk operator, and on tractable heuristics (linear programming for strict diagonal dominance, randomized search for Rohn's criterion) to extract high-rank regular submatrices from interval Jacobians (Olivier et al., 2013).

5. Diagonalization, Spectral Theory, and Exponentials

Interval-valued matrices exhibit spectral properties generalizing real linear algebra, but with novel features:

• The characteristic polynomial with interval coefficients $C_A(X) = \det(A - X I)$ may have more than $n$ roots since intervals lack the factoriality of real numbers.
• The concept of a central eigenvalue, defined via proximity of the center of an interval root to the eigenvalues of the center matrix $A_c$, is used to formulate diagonalization criteria.
• Diagonalization over intervals requires that the sum of eigenspaces (as defined in the interval vector space) spans the whole space, allowing for block diagonalization and enabling computation of the interval exponential map $\exp A = \sum_{k=0}^\infty A^k/k!$ (Goze, 2010).

6. Empirical Results and Practical Impact

Generalized intervals on vectors, as implemented in symbolic model checking tools (e.g., SVSKit), have demonstrated significant performance advantages over traditional vector-of-intervals methods such as Interval Decision Diagrams (IDDs). Specifically, in Petri net contexts where transitions impose constraints across multiple places, generalized intervals yield more compact and exact representations. Empirical results from the Model Checking Contest (MCC 2022) indicate:

• Drastic reductions in the number of symbolic vectors required under saturation (from 954 to 29 in representative examples).
• Consistent ability to exactly represent infinite sets where IDD methods over-approximate.
• Superior runtime and coverage on complex CTL queries—solving all 16 randomly generated fireability queries on a large Circadian-Clock model in under 30 minutes, compared to 9/16 by state-of-the-art tools in one hour (Morard et al., 3 Feb 2026).

7. Extensions, Open Problems, and Connections

The framework of generalized intervals on vectors connects to multiple domains:

• Functional analysis, via the Banach space structure on interval vectors, differential calculus, and associative interval algebras (0809.5173).
• Polyhedral combinatorics, where the convex hull of interval vectors yields polytopes with tight enumerative invariants, bijections to root polytopes, and conjectural relationships to higher Catalan and Pascal triangle sequences (Beck et al., 2012).
• Optimization and control, as inner approximations via generalized intervals provide robust certification in verified global optimization, reachability in dynamic systems, and safe abstract interpretation.

Open questions remain regarding the full combinatorial characterization of generalized-interval polytopes, efficient algorithms for canonicalization at large scale, sharper complexity bounds, and bijective or geometric proofs in polytope volume/catalan correspondences (Beck et al., 2012). A plausible implication is that further expansion of these structures will continue to impact symbolic verification, verified numerics, and higher-dimensional combinatorial geometry.