2,938 research outputs found
Completeness in Equational Hybrid Propositional Type Theory
Equational Hybrid Propositional Type Theory (EHPTT) is a combination of
propositional type theory, equational logic and hybrid modal logic. The structures used to
interpret the language contain a hierarchy of propositional types, an algebra (a nonempty
set with functions) and a Kripke frame.
The main result in this paper is the proof of completeness of a calculus specifically
defined for this logic. The completeness proof is based on the three proofs Henkin published
last century: (i) Completeness in type theory (ii) The completeness of the first-order
functional calculus and (iii) Completeness in propositional type theory. More precisely,
from (i) and (ii) we take the idea of building the model described by the maximal consistent
set; in our case the maximal consistent set has to be named, ♦- saturated and extensionally
algebraic-saturated due to the hybrid and equational nature of EHPTT. From (iii), we use
the result that any element in the hierarchy has a name. The challenge was to deal with
all the heterogeneous components in an integrated system.publishe
Propositional logic with short-circuit evaluation: a non-commutative and a commutative variant
Short-circuit evaluation denotes the semantics of propositional connectives
in which the second argument is evaluated only if the first argument does not
suffice to determine the value of the expression. Short-circuit evaluation is
widely used in programming, with sequential conjunction and disjunction as
primitive connectives.
We study the question which logical laws axiomatize short-circuit evaluation
under the following assumptions: compound statements are evaluated from left to
right, each atom (propositional variable) evaluates to either true or false,
and atomic evaluations can cause a side effect. The answer to this question
depends on the kind of atomic side effects that can occur and leads to
different "short-circuit logics". The basic case is FSCL (free short-circuit
logic), which characterizes the setting in which each atomic evaluation can
cause a side effect. We recall some main results and then relate FSCL to MSCL
(memorizing short-circuit logic), where in the evaluation of a compound
statement, the first evaluation result of each atom is memorized. MSCL can be
seen as a sequential variant of propositional logic: atomic evaluations cannot
cause a side effect and the sequential connectives are not commutative. Then we
relate MSCL to SSCL (static short-circuit logic), the variant of propositional
logic that prescribes short-circuit evaluation with commutative sequential
connectives.
We present evaluation trees as an intuitive semantics for short-circuit
evaluation, and simple equational axiomatizations for the short-circuit logics
mentioned that use negation and the sequential connectives only.Comment: 34 pages, 6 tables. Considerable parts of the text below stem from
arXiv:1206.1936, arXiv:1010.3674, and arXiv:1707.05718. Together with
arXiv:1707.05718, this paper subsumes most of arXiv:1010.367
The Structure of Differential Invariants and Differential Cut Elimination
The biggest challenge in hybrid systems verification is the handling of
differential equations. Because computable closed-form solutions only exist for
very simple differential equations, proof certificates have been proposed for
more scalable verification. Search procedures for these proof certificates are
still rather ad-hoc, though, because the problem structure is only understood
poorly. We investigate differential invariants, which define an induction
principle for differential equations and which can be checked for invariance
along a differential equation just by using their differential structure,
without having to solve them. We study the structural properties of
differential invariants. To analyze trade-offs for proof search complexity, we
identify more than a dozen relations between several classes of differential
invariants and compare their deductive power. As our main results, we analyze
the deductive power of differential cuts and the deductive power of
differential invariants with auxiliary differential variables. We refute the
differential cut elimination hypothesis and show that, unlike standard cuts,
differential cuts are fundamental proof principles that strictly increase the
deductive power. We also prove that the deductive power increases further when
adding auxiliary differential variables to the dynamics
Boole's Method I. A Modern Version
A rigorous, modern version of Boole's algebra of logic is presented, based
partly on the 1890s treatment of Ernst Schroder
Probabilistic Argumentation. An Equational Approach
There is a generic way to add any new feature to a system. It involves 1)
identifying the basic units which build up the system and 2) introducing the
new feature to each of these basic units.
In the case where the system is argumentation and the feature is
probabilistic we have the following. The basic units are: a. the nature of the
arguments involved; b. the membership relation in the set S of arguments; c.
the attack relation; and d. the choice of extensions.
Generically to add a new aspect (probabilistic, or fuzzy, or temporal, etc)
to an argumentation network can be done by adding this feature to each
component a-d. This is a brute-force method and may yield a non-intuitive or
meaningful result.
A better way is to meaningfully translate the object system into another
target system which does have the aspect required and then let the target
system endow the aspect on the initial system. In our case we translate
argumentation into classical propositional logic and get probabilistic
argumentation from the translation.
Of course what we get depends on how we translate.
In fact, in this paper we introduce probabilistic semantics to abstract
argumentation theory based on the equational approach to argumentation
networks. We then compare our semantics with existing proposals in the
literature including the approaches by M. Thimm and by A. Hunter. Our
methodology in general is discussed in the conclusion
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