75 research outputs found
Relating Structure and Power: Comonadic Semantics for Computational Resources
Combinatorial games are widely used in finite model theory, constraint
satisfaction, modal logic and concurrency theory to characterize logical
equivalences between structures. In particular, Ehrenfeucht-Fraisse games,
pebble games, and bisimulation games play a central role. We show how each of
these types of games can be described in terms of an indexed family of comonads
on the category of relational structures and homomorphisms. The index k is a
resource parameter which bounds the degree of access to the underlying
structure. The coKleisli categories for these comonads can be used to give
syntax-free characterizations of a wide range of important logical
equivalences. Moreover, the coalgebras for these indexed comonads can be used
to characterize key combinatorial parameters: tree-depth for the
Ehrenfeucht-Fraisse comonad, tree-width for the pebbling comonad, and
synchronization-tree depth for the modal unfolding comonad. These results pave
the way for systematic connections between two major branches of the field of
logic in computer science which hitherto have been almost disjoint: categorical
semantics, and finite and algorithmic model theory.Comment: To appear in Proceedings of Computer Science Logic 201
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
Extension Preservation in the Finite and Prefix Classes of First Order Logic
It is well known that the classic ?o?-Tarski preservation theorem fails in the finite: there are first-order definable classes of finite structures closed under extensions which are not definable (in the finite) in the existential fragment of first-order logic. We strengthen this by constructing for every n, first-order definable classes of finite structures closed under extensions which are not definable with n quantifier alternations. The classes we construct are definable in the extension of Datalog with negation and indeed in the existential fragment of transitive-closure logic. This answers negatively an open question posed by Rosen and Weinstein
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
On a Product of Finite Monoids
In this paper, for each positive integer m, we associate with a finite monoid S0 and m finite commutative monoids S1,…, Sm, a product &#x25CAm(Sm,…, S1, S0). We give a representation of the free objects in the pseudovariety &#x25CAm(Wm,…, W1, W0) generated by these (m + 1)-ary products where Si &#x2208 Wi for all 0 &#x2264 i &#x2264 m. We then give, in particular, a criterion to determine when an identity holds in &#x25CAm(J1,…, J1, J1) with the help of a version of the Ehrenfeucht-Fraïssé game (J1 denotes the pseudovariety of all semilattice monoids). The union &#x222Am>0&#x25CAm (J1,…, J1, J1) turns out to be the second level of the Straubing’s dot-depth hierarchy of aperiodic monoids
Two first-order logics of permutations
We consider two orthogonal points of view on finite permutations, seen as
pairs of linear orders (corresponding to the usual one line representation of
permutations as words) or seen as bijections (corresponding to the algebraic
point of view). For each of them, we define a corresponding first-order logical
theory, that we call (Theory Of Two Orders) and
(Theory Of One Bijection) respectively. We consider various expressibility
questions in these theories.
Our main results go in three different direction. First, we prove that, for
all , the set of -stack sortable permutations in the sense of West
is expressible in , and that a logical sentence describing this
set can be obtained automatically. Previously, descriptions of this set were
only known for . Next, we characterize permutation classes inside
which it is possible to express in that some given points form
a cycle. Lastly, we show that sets of permutations that can be described both
in and are in some sense trivial. This gives a
mathematical evidence that permutations-as-bijections and permutations-as-words
are somewhat different objects.Comment: v2: minor changes, following a referee repor
On the Expressiveness of LARA: A Unified Language for Linear and Relational Algebra
We study the expressive power of the Lara language - a recently proposed unified model for expressing relational and linear algebra operations - both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. We start by showing Lara to be expressive complete with respect to first-order logic with aggregation. Since Lara is parameterized by a set of user-defined functions which allow to transform values in tables, the exact expressive power of the language depends on how these functions are defined. We distinguish two main cases depending on the level of genericity queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning operations. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain
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