829,648 research outputs found
Independence-friendly cylindric set algebras
Independence-friendly logic is a conservative extension of first-order logic
that has the same expressive power as existential second-order logic. In her
Ph.D. thesis, Dechesne introduces a variant of independence-friendly logic
called IFG logic. We attempt to algebraize IFG logic in the same way that
Boolean algebra is the algebra of propositional logic and cylindric algebra is
the algebra of first-order logic.
We define independence-friendly cylindric set algebras and prove two main
results. First, every independence-friendly cylindric set algebra over a
structure has an underlying Kleene algebra. Moreover, the class of such
underlying Kleene algebras generates the variety of all Kleene algebras. Hence
the equational theory of the class of Kleene algebras that underly an
independence-friendly cylindric set algebra is finitely axiomatizable. Second,
every one-dimensional independence-friendly cylindric set algebra over a
structure has an underlying monadic Kleene algebra. However, the class of such
underlying monadic Kleene algebras does not generate the variety of all monadic
Kleene algebras. Finally, we offer a conjecture about which subvariety of
monadic Kleene algebras the class of such monadic Kleene algebras does
generate.Comment: 42 pages. Submitted to the Logic Journal of the IGPL. See also
http://math.colgate.edu/~amann
On the Expressive Power of TeamLTL and First-Order Team Logic over Hyperproperties
In this article we study linear temporal logics with team semantics (TeamLTL) that are novel logics for defining hyperproperties. We define Kamp-type translations of these logics into fragments of first-order team logic and second-order logic. We also characterize the expressive power and the complexity of model-checking and satisfiability of team logic and second-order logic by relating them to second- and third-order arithmetic. Our results set in a larger context the recent results of Luck showing that the extension of TeamLTL by the Boolean negation is highly undecidable under the so-called synchronous semantics. We also study stutter-invariant fragments of extensions of TeamLTL.Peer reviewe
Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure
String languages recognizable in (deterministic) log-space are characterized
either by two-way (deterministic) multi-head automata, or following Immerman,
by first-order logic with (deterministic) transitive closure. Here we elaborate
this result, and match the number of heads to the arity of the transitive
closure. More precisely, first-order logic with k-ary deterministic transitive
closure has the same power as deterministic automata walking on their input
with k heads, additionally using a finite set of nested pebbles. This result is
valid for strings, ordered trees, and in general for families of graphs having
a fixed automaton that can be used to traverse the nodes of each of the graphs
in the family. Other examples of such families are grids, toruses, and
rectangular mazes. For nondeterministic automata, the logic is restricted to
positive occurrences of transitive closure.
The special case of k=1 for trees, shows that single-head deterministic
tree-walking automata with nested pebbles are characterized by first-order
logic with unary deterministic transitive closure. This refines our earlier
result that placed these automata between first-order and monadic second-order
logic on trees.Comment: Paper for Logical Methods in Computer Science, 27 pages, 1 figur
Undecidability of a weak version of MSO+U
We prove the undecidability of MSO on ω-words extended with the second-order predicate U1(X) which says that the distance between consecutive positions in a set X⊆N is unbounded. This is achieved by showing that adding U1 to MSO gives a logic with the same expressive power as MSO+U, a logic on ω-words with undecidable satisfiability. As a corollary, we prove that MSO on ω-words becomes undecidable if allowing to quantify over sets of positions that are ultimately periodic, i.e., sets X such that for some positive integer p, ultimately either both or none of positions x and x+p belong to X
Courcelle's Theorem - A Game-Theoretic Approach
Courcelle's Theorem states that every problem definable in Monadic
Second-Order logic can be solved in linear time on structures of bounded
treewidth, for example, by constructing a tree automaton that recognizes or
rejects a tree decomposition of the structure. Existing, optimized software
like the MONA tool can be used to build the corresponding tree automata, which
for bounded treewidth are of constant size. Unfortunately, the constants
involved can become extremely large - every quantifier alternation requires a
power set construction for the automaton. Here, the required space can become a
problem in practical applications.
In this paper, we present a novel, direct approach based on model checking
games, which avoids the expensive power set construction. Experiments with an
implementation are promising, and we can solve problems on graphs where the
automata-theoretic approach fails in practice.Comment: submitte
Expressive power and complexity of a logic with quantifiers that count proportions of sets
We present a second-order logic of proportional quantifiers, SOLP, which is essentially a first-order language
extended with quantifiers that act upon second-order variables of a given arity r and count the fraction of elements in
a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of
different problems of complexity up to NP-hard as, for example, the problem of deciding if at least a fraction 1/n of
the set of vertices of a graph form a clique; and fragments within our logic capture complexity classes as NL and P,
with auxiliary ordering relation. When restricted to monadic second-order variables, our logic of proportional
quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers (restricted to almost orders), for it does not have the bounded number
of degrees property. Moreover, we show that, in this almost-ordered setting, different fragments of this logic vary in their expressive power, and show the existence of an infinite hierarchy inside our monadic language.
We extend our inexpressibility result of almost-ordered structure to a fragment of SOLP, which in the presence of full order captures P. To obtain all our inexpressibility results, we developed combinatorial games appropriate
for these logics, whose application could go beyond the almost-ordered models and hence are interesting by themselves.Peer ReviewedPreprin
Neo-Logicism and Its Logic
The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism
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