arXiv subjects

Noah Abou El Wafa

Publications and source records attributed to Noah Abou El Wafa.

5 recordsLinked to original sources

arXiv2026

Differential Equations as Fixpoints and Games

Games and fixpoints are unified by proving that first-order game logic GL and the first-order modal mu-calculus L_mu are proved to be equiexpressive and equivalent, thereby fully aligning their expressive and deductive power. That is, there is a semantics-preserving translation from GL to L_mu, and vice versa. And both translations are provability-preserving, while equivalence with there-and-back-again roundtrip translations are provable in both calculi. This is to be contrasted with the propositional case, where game logic is strictly less expressive than the modal mu-calculus (without adding sabotage games). The extensions with differential equations, differential game logic (dGL) and differential modal mu-calculus, are also proved equiexpressive and equivalent. Moreover, as the continuous dynamics are definable by fixpoints or via games, ODEs can be axiomatized completely and, as a consequence, infinitesimally robust properties of ODEs can be decided via proof search. Rational gameplay provably collapses the games into single-player games to yield a strong arithmetical completeness theorem for dGL with rational-time ODEs.

cs.LO↗

arXiv2026

Subreflexive Logic: Completeness without Identity

This paper shows that the substructural logic without the identity principle A->A (i.e., subreflexive logic) has principled sound and complete semantics and supports a variety of applications. This decidable generalization of propositional logic naturally interprets implication as robust consequence. Subreflexive logic is proved to admit syntactic cut elimination. Heyting and Boolean semialgebras are introduced as generalizations of Heyting and Boolean algebras and are shown to provide complete algebraic semantics without inadvertently reintroducing reflexivity. Semi-adjunctions on semi-categories and (identity-free) (co-)units are defined to give complete semi-categorical semantics. In the classical case, denotational set semantics that interpret implication as robust material implication are proved complete for subreflexive logic.

cs.LO↗

arXiv2026

Complete Robust Hybrid Systems Reachability

This paper introduces robust differential dynamic logic (a fragment of differential dynamic logic) to specify and reason about robust hybrid systems. Practically meaningful syntactic restrictions naturally ensure that definable properties are topologically open and thus by construction robust with respect to infinitesimal perturbations, without explicit quantitative margins of error in the syntax or in proofs. The main result is a proof of absolute completeness of robust differential dynamic logic for reachability properties of general hybrid systems. This is the first absolute completeness proof for hybrid systems with exact semantics. The proof is constructive, self-contained, and demonstrates how robustly correct hybrid systems reachability specifications can be automatically verified through proof.

cs.LO↗

arXiv2024

Complete Game Logic with Sabotage

Game Logic with sabotage ($\mathsf{GL_s}$) is introduced as a simple and natural extension of Parikh's game logic with a single additional primitive, which allows players to lay traps for the opponent. $\mathsf{GL_s}$ can be used to model infinite sabotage games, in which players can change the rules during game play. In contrast to game logic, which is strictly less expressive, $\mathsf{GL_s}$ is exactly as expressive as the modal $μ$-calculus. This reveals a close connection between the entangled nested recursion inherent in modal fixpoint logics and adversarial dynamic rule changes characteristic for sabotage games. A natural Hilbert-style proof calculus for $\mathsf{GL_s}$ is presented and proved complete using syntactic equiexpressiveness reductions. The completeness of a simple extension of Parikh's calculus for game logic follows.

cs.LO↗

arXiv2022

First-Order Game Logic and Modal Mu-Calculus

This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable assignments. Unlike in the propositional case, both logics are shown to have the same expressive power and their proof calculi to have the same deductive power. Both calculi are also mutually relatively complete. In the presence of differential equations, corollaries obtain usable and complete translations between differential game logic, a logic for the deductive verification of hybrid games, and the differential mu-calculus, the modal mu-calculus for hybrid systems. The differential mu-calculus is complete with respect to first-order fixpoint logic and differential game logic is complete with respect to its ODE-free fragment.

cs.LO↗