629 research outputs found
Undecidability of Multiplicative Subexponential Logic
Subexponential logic is a variant of linear logic with a family of
exponential connectives--called subexponentials--that are indexed and arranged
in a pre-order. Each subexponential has or lacks associated structural
properties of weakening and contraction. We show that classical propositional
multiplicative linear logic extended with one unrestricted and two incomparable
linear subexponentials can encode the halting problem for two register Minsky
machines, and is hence undecidable.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441
The Identity Correspondence Problem and its Applications
In this paper we study several closely related fundamental problems for words
and matrices. First, we introduce the Identity Correspondence Problem (ICP):
whether a finite set of pairs of words (over a group alphabet) can generate an
identity pair by a sequence of concatenations. We prove that ICP is undecidable
by a reduction of Post's Correspondence Problem via several new encoding
techniques.
In the second part of the paper we use ICP to answer a long standing open
problem concerning matrix semigroups: "Is it decidable for a finitely generated
semigroup S of square integral matrices whether or not the identity matrix
belongs to S?". We show that the problem is undecidable starting from dimension
four even when the number of matrices in the generator is 48. From this fact,
we can immediately derive that the fundamental problem of whether a finite set
of matrices generates a group is also undecidable. We also answer several
question for matrices over different number fields. Apart from the application
to matrix problems, we believe that the Identity Correspondence Problem will
also be useful in identifying new areas of undecidable problems in abstract
algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the
published journal version of this article, see footnote 3 on page 1
Non-associative, Non-commutative Multi-modal Linear Logic
Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ
Timed Comparisons of Semi-Markov Processes
Semi-Markov processes are Markovian processes in which the firing time of the
transitions is modelled by probabilistic distributions over positive reals
interpreted as the probability of firing a transition at a certain moment in
time. In this paper we consider the trace-based semantics of semi-Markov
processes, and investigate the question of how to compare two semi-Markov
processes with respect to their time-dependent behaviour. To this end, we
introduce the relation of being "faster than" between processes and study its
algorithmic complexity. Through a connection to probabilistic automata we
obtain hardness results showing in particular that this relation is
undecidable. However, we present an additive approximation algorithm for a
time-bounded variant of the faster-than problem over semi-Markov processes with
slow residence-time functions, and a coNP algorithm for the exact faster-than
problem over unambiguous semi-Markov processes
Combining decision procedures for the reals
We address the general problem of determining the validity of boolean
combinations of equalities and inequalities between real-valued expressions. In
particular, we consider methods of establishing such assertions using only
restricted forms of distributivity. At the same time, we explore ways in which
"local" decision or heuristic procedures for fragments of the theory of the
reals can be amalgamated into global ones. Let Tadd[Q] be the
first-order theory of the real numbers in the language of ordered groups, with
negation, a constant 1, and function symbols for multiplication by
rational constants. Let Tmult[Q] be the analogous theory for the
multiplicative structure, and let T[Q] be the union of the two. We
show that although T[Q] is undecidable, the universal fragment of
T[Q] is decidable. We also show that terms of T[Q]can
fruitfully be put in a normal form. We prove analogous results for theories in
which Q is replaced, more generally, by suitable subfields F
of the reals. Finally, we consider practical methods of establishing
quantifier-free validities that approximate our (impractical) decidability
results.Comment: Will appear in Logical Methods in Computer Scienc
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