Relating Structure and Power: Comonadic Semantics for Computational Resources

Comonadic semantics for guarded fragments

“…In our setting, this implies that coalgebra maps α : A → GA are embeddings of σ-structures. In [1,2] it was shown that coalgebras for the Ehrenfeucht-Fraïssé, pebbling and modal comonads correspond to various forms of combinatorial decompositions of σ-structures. This led to coalgebraic characterizations of important combinatorial parameters such as tree-depth and tree-width.…”

“…In this paper, we continue the program initiated in [1], and further developed in [2], of applying methods from categorical semantics to finite model theory, combinatorics, and descriptive complexity. The aim, as articulated in [2], is to relate structure to power, i.e.…”

“…In this paper, we continue the program initiated in [1], and further developed in [2], of applying methods from categorical semantics to finite model theory, combinatorics, and descriptive complexity. The aim, as articulated in [2], is to relate structure to power, i.e. to apply the structural and compositional methods of semantics to questions of expressiveness, resource bounds, and algorithmic complexity.…”

“…The previous work in [1,2] showed how a range of model comparison games which play a central role in finite model theory, including Ehrenfeucht-Fraïssé, pebbling, and bisimulation games, can be captured in terms of resource-indexed comonads on the category of relational structures. Moreover, the coalgebras for these comonads capture important combinatorial parameters such as tree-width and tree-depth.…”

“…These guarded fragments have proved to be a fruitful generalization of the modal fragment, inheriting its good computational properties, such as decidability and the tree model property, while significantly enhancing its expressive power. The basic modal fragment was studied from the comonadic point of view in [2]. The extension to richer guarded fragments which we undertake here involves significant new ideas.…”