88 research outputs found
Limitations of Game Comonads via Homomorphism Indistinguishability
Abramsky, Dawar, and Wang (2017) introduced the pebbling comonad for
k-variable counting logic and thereby initiated a line of work that imports
category theoretic machinery to finite model theory. Such game comonads have
been developed for various logics, yielding characterisations of logical
equivalences in terms of isomorphisms in the associated co-Kleisli category. We
show a first limitation of this approach by studying linear-algebraic logic,
which is strictly more expressive than first-order counting logic and whose
k-variable logical equivalence relations are known as invertible-map
equivalences (IM). We show that there exists no finite-rank comonad on the
category of graphs whose co-Kleisli isomorphisms characterise IM-equivalence,
answering a question of \'O Conghaile and Dawar (CSL 2021). We obtain this
result by ruling out a characterisation of IM-equivalence in terms of
homomorphism indistinguishability and employing the Lov\'asz-type theorems for
game comonads established by Dawar, Jakl, and Reggio (2021). Two graphs are
homomorphism indistinguishable over a graph class if they admit the same number
of homomorphisms from every graph in the class. The IM-equivalences cannot be
characterised in this way, neither when counting homomorphisms in the natural
numbers, nor in any finite prime field.Comment: Minor corrections in Section
Craig Interpolation for Decidable First-Order Fragments
We show that the guarded-negation fragment (GNFO) is, in a precise sense, the
smallest extension of the guarded fragment (GFO) with Craig interpolation. In
contrast, we show that the smallest extension of the two-variable fragment
(FO2), and of the forward fragment (FF) with Craig interpolation, is full
first-order logic. Similarly, we also show that all extensions of FO2 and of
the fluted fragment (FL) with Craig interpolation are undecidable.Comment: Submitted for FoSSaCS 2024. arXiv admin note: substantial text
overlap with arXiv:2304.0808
Model theory of monadic predicate logic with the infinity quantifier
This paper establishes model-theoretic properties of ME∞, a variation of monadic first-order logic that features the generalised quantifier ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality (ME and M, respectively). For each logic L∈ { M, ME, ME∞} we will show the following. We provide syntactically defined fragments of L characterising four different semantic properties of L-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence φ to a sentence φp belonging to the corresponding syntactic fragment, with the property that φ is equivalent to φp precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for L-sentences
Chain logic and Shelah’s infinitary logic
For a cardinal of the form kappa = (sic)(kappa), Shelah's logic L-kappa(1) has a characterisation as the maximal logic above boolean OR(lambda We then show that the chain logic gives a partial solution to Problem 1.4 from Shelah's [28], which asked whether for kappa singular of countable cofinality there was a logic strictly between L kappa+,omega and L kappa(+),kappa(+) having interpolation. We show that modulo accepting as the upper bound a model class of L-kappa,L-kappa, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not kappa-compact, a question that we have asked on various occasions. We contribute to further development of chain logic by proving the Union Lemma and identifying the chainindependent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic L-kappa(1) in satisfying interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a completeness theorem.Peer reviewe
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