210 research outputs found
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
Transitive Closure Logic and Multihead Automata with Nested Pebbles
Several extensions of first-order logic are studied in descriptive complexity theory. These extensions include transitive closure logic and deterministic transitive closure logic, which extend first-order logic with transitive closure operators. It is known that deterministic transitive closure logic captures the complexity class of the languages that are decidable by some deterministic Turing machine using a logarithmic amount of memory space. An analogous result holds for transitive closure logic and nondeterministic Turing machines.
This thesis concerns the k-ary fragments of these two logics. In each k-ary fragment, the arities of transitive closure operators appearing in formulas are restricted to a nonzero natural number k. The expressivity of these fragments can be studied in terms of multihead finite automata. The type of automaton that we consider in this thesis is a two-way multihead automaton with nested pebbles.
We look at the expressive power of multihead automata and the k-ary fragments of transitive closure logics in the class of finite structures called word models. We show that deterministic twoway k-head automata with nested pebbles have the same expressive power as first-order logic with k-ary deterministic transitive closure. For a corresponding result in the case of nondeterministic automata, we restrict to the positive fragment of k-ary transitive closure logic. The two theorems and their proofs are based on the article ’Automata with nested pebbles capture first-order logic with transitive closure’ by Joost Engelfriet and Hendrik Jan Hoogeboom. In the article, the results are proved in the case of trees. Since word models can be viewed as a special type of trees, the theorems considered in this thesis are a special case of a more general result
First-Order and Temporal Logics for Nested Words
Nested words are a structured model of execution paths in procedural
programs, reflecting their call and return nesting structure. Finite nested
words also capture the structure of parse trees and other tree-structured data,
such as XML. We provide new temporal logics for finite and infinite nested
words, which are natural extensions of LTL, and prove that these logics are
first-order expressively-complete. One of them is based on adding a "within"
modality, evaluating a formula on a subword, to a logic CaRet previously
studied in the context of verifying properties of recursive state machines
(RSMs). The other logic, NWTL, is based on the notion of a summary path that
uses both the linear and nesting structures. For NWTL we show that
satisfiability is EXPTIME-complete, and that model-checking can be done in time
polynomial in the size of the RSM model and exponential in the size of the NWTL
formula (and is also EXPTIME-complete). Finally, we prove that first-order
logic over nested words has the three-variable property, and we present a
temporal logic for nested words which is complete for the two-variable fragment
of first-order.Comment: revised and corrected version of Mar 03, 201
Temporal Logics on Words with Multiple Data Values
The paper proposes and studies temporal logics for attributed words, that is, data words with a (finite) set of (attribute,value)-pairs at each position. It considers a basic logic which is a semantical fragment of the logic of Demri and Lazic with operators for navigation into the future and the past. By reduction to the emptiness problem for data automata it is shown that this basic logic is decidable. Whereas the basic logic only allows navigation to positions where a fixed data value occurs, extensions are studied that also allow navigation to positions with different data values. Besides some undecidable results it is shown that the extension by a certain UNTIL-operator with an inequality target condition remains decidable
The Arity Hierarchy in the Polyadic -Calculus
The polyadic mu-calculus is a modal fixpoint logic whose formulas define
relations of nodes rather than just sets in labelled transition systems. It can
express exactly the polynomial-time computable and bisimulation-invariant
queries on finite graphs. In this paper we show a hierarchy result with respect
to expressive power inside the polyadic mu-calculus: for every level of
fixpoint alternation, greater arity of relations gives rise to higher
expressive power. The proof uses a diagonalisation argument.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Limit Your Consumption! Finding Bounds in Average-energy Games
Energy games are infinite two-player games played in weighted arenas with
quantitative objectives that restrict the consumption of a resource modeled by
the weights, e.g., a battery that is charged and drained. Typically, upper
and/or lower bounds on the battery capacity are part of the problem
description. Here, we consider the problem of determining upper bounds on the
average accumulated energy or on the capacity while satisfying a given lower
bound, i.e., we do not determine whether a given bound is sufficient to meet
the specification, but if there exists a sufficient bound to meet it.
In the classical setting with positive and negative weights, we show that the
problem of determining the existence of a sufficient bound on the long-run
average accumulated energy can be solved in doubly-exponential time. Then, we
consider recharge games: here, all weights are negative, but there are recharge
edges that recharge the energy to some fixed capacity. We show that bounding
the long-run average energy in such games is complete for exponential time.
Then, we consider the existential version of the problem, which turns out to be
solvable in polynomial time: here, we ask whether there is a recharge capacity
that allows the system player to win the game.
We conclude by studying tradeoffs between the memory needed to implement
strategies and the bounds they realize. We give an example showing that memory
can be traded for bounds and vice versa. Also, we show that increasing the
capacity allows to lower the average accumulated energy.Comment: In Proceedings QAPL'16, arXiv:1610.0769
Computer aided synthesis: a game theoretic approach
In this invited contribution, we propose a comprehensive introduction to game
theory applied in computer aided synthesis. In this context, we give some
classical results on two-player zero-sum games and then on multi-player non
zero-sum games. The simple case of one-player games is strongly related to
automata theory on infinite words. All along the article, we focus on general
approaches to solve the studied problems, and we provide several illustrative
examples as well as intuitions on the proofs.Comment: Invitation contribution for conference "Developments in Language
Theory" (DLT 2017
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