1,603 research outputs found
Integrating a Global Induction Mechanism into a Sequent Calculus
Most interesting proofs in mathematics contain an inductive argument which
requires an extension of the LK-calculus to formalize. The most commonly used
calculi for induction contain a separate rule or axiom which reduces the valid
proof theoretic properties of the calculus. To the best of our knowledge, there
are no such calculi which allow cut-elimination to a normal form with the
subformula property, i.e. every formula occurring in the proof is a subformula
of the end sequent. Proof schemata are a variant of LK-proofs able to simulate
induction by linking proofs together. There exists a schematic normal form
which has comparable proof theoretic behaviour to normal forms with the
subformula property. However, a calculus for the construction of proof schemata
does not exist. In this paper, we introduce a calculus for proof schemata and
prove soundness and completeness with respect to a fragment of the inductive
arguments formalizable in Peano arithmetic.Comment: 16 page
Deciding regular grammar logics with converse through first-order logic
We provide a simple translation of the satisfiability problem for regular
grammar logics with converse into GF2, which is the intersection of the guarded
fragment and the 2-variable fragment of first-order logic. This translation is
theoretically interesting because it translates modal logics with certain frame
conditions into first-order logic, without explicitly expressing the frame
conditions.
A consequence of the translation is that the general satisfiability problem
for regular grammar logics with converse is in EXPTIME. This extends a previous
result of the first author for grammar logics without converse. Using the same
method, we show how some other modal logics can be naturally translated into
GF2, including nominal tense logics and intuitionistic logic.
In our view, the results in this paper show that the natural first-order
fragment corresponding to regular grammar logics is simply GF2 without extra
machinery such as fixed point-operators.Comment: 34 page
Strong Turing Degrees for Additive BSS RAM's
For the additive real BSS machines using only constants 0 and 1 and order
tests we consider the corresponding Turing reducibility and characterize some
semi-decidable decision problems over the reals. In order to refine,
step-by-step, a linear hierarchy of Turing degrees with respect to this model,
we define several halting problems for classes of additive machines with
different abilities and construct further suitable decision problems. In the
construction we use methods of the classical recursion theory as well as
techniques for proving bounds resulting from algebraic properties. In this way
we extend a known hierarchy of problems below the halting problem for the
additive machines using only equality tests and we present a further
subhierarchy of semi-decidable problems between the halting problems for the
additive machines using only equality tests and using order tests,
respectively
Forward Analysis and Model Checking for Trace Bounded WSTS
We investigate a subclass of well-structured transition systems (WSTS), the
bounded---in the sense of Ginsburg and Spanier (Trans. AMS 1964)---complete
deterministic ones, which we claim provide an adequate basis for the study of
forward analyses as developed by Finkel and Goubault-Larrecq (Logic. Meth.
Comput. Sci. 2012). Indeed, we prove that, unlike other conditions considered
previously for the termination of forward analysis, boundedness is decidable.
Boundedness turns out to be a valuable restriction for WSTS verification, as we
show that it further allows to decide all -regular properties on the
set of infinite traces of the system
Real-time and Probabilistic Temporal Logics: An Overview
Over the last two decades, there has been an extensive study on logical
formalisms for specifying and verifying real-time systems. Temporal logics have
been an important research subject within this direction. Although numerous
logics have been introduced for the formal specification of real-time and
complex systems, an up to date comprehensive analysis of these logics does not
exist in the literature. In this paper we analyse real-time and probabilistic
temporal logics which have been widely used in this field. We extrapolate the
notions of decidability, axiomatizability, expressiveness, model checking, etc.
for each logic analysed. We also provide a comparison of features of the
temporal logics discussed
Computational Processes and Incompleteness
We introduce a formal definition of Wolfram's notion of computational process
based on cellular automata, a physics-like model of computation. There is a
natural classification of these processes into decidable, intermediate and
complete. It is shown that in the context of standard finite injury priority
arguments one cannot establish the existence of an intermediate computational
process
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