5,699 research outputs found
Third Order Matching is Decidable
The higher order matching problem is the problem of determining whether a
term is an instance of another in the simply typed -calculus, i.e. to
solve the equation a = b where a and b are simply typed -terms and b
is ground. The decidability of this problem is still open. We prove the
decidability of the particular case in which the variables occurring in the
problem are at most third order
The variable containment problem
The essentially free variables of a term in some -calculus, FV , form the set ( FV}. This set is significant once we consider equivalence classes of -terms rather than -terms themselves, as for instance in higher-order rewriting. An important problem for (generalised) higher-order rewrite systems is the variable containment problem: given two terms and , do we have for all substitutions and contexts [] that FV FV?
This property is important when we want to consider as a rewrite rule and keep -step rewriting decidable. Variable containment is in general not implied by FV FV. We give a decision procedure for the variable containment problem of the second-order fragment of . For full we show the equivalence of variable containment to an open problem in the theory of PCF; this equivalence also shows that the problem is decidable in the third-order case
Can a Turing Player identify itself?
We show that the problem of whether two Turing Machines are functionally equivalent is undecidable and explain why this is significant for the theory of repeated play and evolution
Verifying Recursive Active Documents with Positive Data Tree Rewriting
This paper proposes a data tree-rewriting framework for modeling evolving
documents. The framework is close to Guarded Active XML, a platform used for
handling XML repositories evolving through web services. We focus on automatic
verification of properties of evolving documents that can contain data from an
infinite domain. We establish the boundaries of decidability, and show that
verification of a {\em positive} fragment that can handle recursive service
calls is decidable. We also consider bounded model-checking in our data
tree-rewriting framework and show that it is \nexptime-complete
Reachability Analysis of Communicating Pushdown Systems
The reachability analysis of recursive programs that communicate
asynchronously over reliable FIFO channels calls for restrictions to ensure
decidability. Our first result characterizes communication topologies with a
decidable reachability problem restricted to eager runs (i.e., runs where
messages are either received immediately after being sent, or never received).
The problem is EXPTIME-complete in the decidable case. The second result is a
doubly exponential time algorithm for bounded context analysis in this setting,
together with a matching lower bound. Both results extend and improve previous
work from La Torre et al
The Algebraic Intersection Type Unification Problem
The algebraic intersection type unification problem is an important component
in proof search related to several natural decision problems in intersection
type systems. It is unknown and remains open whether the algebraic intersection
type unification problem is decidable. We give the first nontrivial lower bound
for the problem by showing (our main result) that it is exponential time hard.
Furthermore, we show that this holds even under rank 1 solutions (substitutions
whose codomains are restricted to contain rank 1 types). In addition, we
provide a fixed-parameter intractability result for intersection type matching
(one-sided unification), which is known to be NP-complete.
We place the algebraic intersection type unification problem in the context
of unification theory. The equational theory of intersection types can be
presented as an algebraic theory with an ACI (associative, commutative, and
idempotent) operator (intersection type) combined with distributivity
properties with respect to a second operator (function type). Although the
problem is algebraically natural and interesting, it appears to occupy a
hitherto unstudied place in the theory of unification, and our investigation of
the problem suggests that new methods are required to understand the problem.
Thus, for the lower bound proof, we were not able to reduce from known results
in ACI-unification theory and use game-theoretic methods for two-player tiling
games
An Introduction to Mechanized Reasoning
Mechanized reasoning uses computers to verify proofs and to help discover new
theorems. Computer scientists have applied mechanized reasoning to economic
problems but -- to date -- this work has not yet been properly presented in
economics journals. We introduce mechanized reasoning to economists in three
ways. First, we introduce mechanized reasoning in general, describing both the
techniques and their successful applications. Second, we explain how mechanized
reasoning has been applied to economic problems, concentrating on the two
domains that have attracted the most attention: social choice theory and
auction theory. Finally, we present a detailed example of mechanized reasoning
in practice by means of a proof of Vickrey's familiar theorem on second-price
auctions
Two-variable Logic with Counting and a Linear Order
We study the finite satisfiability problem for the two-variable fragment of
first-order logic extended with counting quantifiers (C2) and interpreted over
linearly ordered structures. We show that the problem is undecidable in the
case of two linear orders (in the presence of two other binary symbols). In the
case of one linear order it is NEXPTIME-complete, even in the presence of the
successor relation. Surprisingly, the complexity of the problem explodes when
we add one binary symbol more: C2 with one linear order and in the presence of
other binary predicate symbols is equivalent, under elementary reductions, to
the emptiness problem for multicounter automata
Can A Turing Player Identify Itself?
We show that the problem of whether two Turing Machines are functionally equivalent is undecidable and explain why this is significant for the theory of repeated play and evolution.Economic Theory
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