We extend the meet-implication fragment of propositional intuitionistic logic
with a meet-preserving modality. We give semantics based on semilattices and a
duality result with a suitable notion of descriptive frame. As a consequence we
obtain completeness and identify a common (modal) fragment of a large class of
modal intuitionistic logics. We recognise this logic as a dialgebraic logic,
and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic
framework, we then investigate the extension of the meet-implication fragment
of propositional intuitionistic logic with a monotone modality and prove
completeness and expressivity-somewhere-else for it.
In probabilistic coherence spaces, a denotational model of probabilistic
functional languages, morphisms are analytic and therefore smooth. We explore
two related applications of the corresponding derivatives. First we show how
derivatives allow to compute the expectation of execution time in the weak head
reduction of probabilistic PCF (pPCF). Next we apply a general notion of
"local" differential of morphisms to the proof of a Lipschitz property of these
morphisms allowing in turn to relate the observational distance on pPCF terms
to a distance the model is naturally equipped with. This suggests that
extending probabilistic programming languages with derivatives, in the spirit
of the differential lambda-calculus, could be quite meaningful.
This paper is a contribution to the search for efficient and high-level
mathematical tools to specify and reason about (abstract) programming languages
or calculi. Generalising the reduction monads of Ahrens et al., we introduce
transition monads, thus covering new applications such as
lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential
lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus.
Moreover, we design a suitable notion of signature for transition monads.
We develop a uniform coalgebraic approach to Jónsson-Tarski and Thomason
type dualities for various classes of neighborhood frames and neighborhood
algebras. In the first part of the paper we construct an endofunctor on the
category of complete and atomic Boolean algebras that is dual to the double
powerset functor on $\mathsf{Set}$. This allows us to show that Thomason
duality for neighborhood frames can be viewed as an algebra-coalgebra duality.
We generalize this approach to any class of algebras for an endofunctor
presented by one-step axioms in the language of infinitary modal logic. As a
consequence, we obtain a uniform approach to dualities for various classes of
neighborhood frames, including monotone neighborhood frames, pretopological
spaces, and topological spaces.
In the second part of the paper we develop a coalgebraic approach to
Jónsson-Tarski duality for neighborhood algebras and descriptive
neighborhood frames. We introduce an analogue of the Vietoris endofunctor on
the category of Stone spaces and show that descriptive neighborhood frames are
isomorphic to coalgebras for this endofunctor. This allows us to obtain a
coalgebraic proof of the duality between descriptive neighborhood frames and
neighborhood algebras. Using one-step axioms in the language of finitary modal
logic, we restrict this duality to other classes of neighborhood algebras
studied in the literature, including monotone modal algebras and contingency
algebras.
We conclude the paper by […]
Adding propositional quantification to the modal logics K, T or S4 is known
to lead to undecidability but CTL with propositional quantification under the
tree semantics (tQCTL) admits a non-elementary Tower-complete satisfiability
problem. We investigate the complexity of strict fragments of tQCTL as well as
of the modal logic K with propositional quantification under the tree
semantics. More specifically, we show that tQCTL restricted to the temporal
operator EX is already Tower-hard, which is unexpected as EX can only enforce
local properties. When tQCTL restricted to EX is interpreted on N-bounded trees
for some N >= 2, we prove that the satisfiability problem is AExpPol-complete;
AExpPol-hardness is established by reduction from a recently introduced tiling
problem, instrumental for studying the model-checking problem for interval
temporal logics. As consequences of our proof method, we prove Tower-hardness
of tQCTL restricted to EF or to EXEF and of the well-known modal logics such as
K, KD, GL, K4 and S4 with propositional quantification under a semantics based
on classes of trees.
Inspired by a mathematical riddle involving fuses, we define the "fusible
numbers" as follows: $0$ is fusible, and whenever $x,y$ are fusible with
$|y-x|<1$, the number $(x+y+1)/2$ is also fusible. We prove that the set of
fusible numbers, ordered by the usual order on $\mathbb R$, is well-ordered,
with order type $\varepsilon_0$. Furthermore, we prove that the density of the
fusible numbers along the real line grows at an incredibly fast rate: Letting
$g(n)$ be the largest gap between consecutive fusible numbers in the interval
$[n,\infty)$, we have $g(n)^{-1} \ge F_{\varepsilon_0}(n-c)$ for some constant
$c$, where $F_\alpha$ denotes the fast-growing hierarchy. Finally, we derive
some true statements that can be formulated but not proven in Peano Arithmetic,
of a different flavor than previously known such statements: PA cannot prove
the true statement "For every natural number $n$ there exists a smallest
fusible number larger than $n$." Also, consider the algorithm "$M(x)$: if $x<0$
return $-x$, else return $M(x-M(x-1))/2$." Then $M$ terminates on real inputs,
although PA cannot prove the statement "$M$ terminates on all natural inputs."
Non-volatile memory (NVM), also known as persistent memory, is an emerging
paradigm for memory that preserves its contents even after power loss. NVM is
widely expected to become ubiquitous, and hardware architectures are already
providing support for NVM programming. This has stimulated interest in the
design of novel concepts ensuring correctness of concurrent programming
abstractions in the face of persistency and in the development of associated
verification approaches.
Software transactional memory (STM) is a key programming abstraction that
supports concurrent access to shared state. In a fashion similar to
linearizability as the correctness condition for concurrent data structures,
there is an established notion of correctness for STMs known as opacity. We
have recently proposed durable opacity as the natural extension of opacity to a
setting with non-volatile memory. Together with this novel correctness
condition, we designed a verification technique based on refinement. In this
paper, we extend this work in two directions. First, we develop a durably
opaque version of NOrec (no ownership records), an existing STM algorithm
proven to be opaque. Second, we modularise our existing verification approach
by separating the proof of durability of memory accesses from the proof of
opacity. For NOrec, this allows us to re-use an existing opacity proof and
complement it with a proof of the durability of accesses to shared state.
In the author's PhD thesis (2019) universal envelopes were introduced as a
tool for studying the continuously obtainable information on discontinuous
functions.
To any function $f \colon X \to Y$ between $\operatorname{qcb}_0$-spaces one
can assign a so-called universal envelope which, in a well-defined sense,
encodes all continuously obtainable information on the function. A universal
envelope consists of two continuous functions $F \colon X \to L$ and $\xi_L
\colon Y \to L$ with values in a $\Sigma$-split injective space $L$. Any
continuous function with values in an injective space whose composition with
the original function is again continuous factors through the universal
envelope. However, it is not possible in general to uniformly compute this
factorisation.
In this paper we propose the notion of uniform envelopes. A uniform envelope
is additionally endowed with a map $u_L \colon L \to \mathcal{O}^2(Y)$ that is
compatible with the multiplication of the double powerspace monad
$\mathcal{O}^2$ in a certain sense. This yields for every continuous map with
values in an injective space a choice of uniformly computable extension. Under
a suitable condition which we call uniform universality, this extension yields
a uniformly computable solution for the above factorisation problem.
Uniform envelopes can be endowed with a composition operation. We establish
criteria that ensure that the composition of two uniformly universal envelopes
is again uniformly universal. […]
In this paper, we investigate the problem of synthesizing computable
functions of infinite words over an infinite alphabet (data $\omega$-words).
The notion of computability is defined through Turing machines with infinite
inputs which can produce the corresponding infinite outputs in the limit. We
use non-deterministic transducers equipped with registers, an extension of
register automata with outputs, to describe specifications. Being
non-deterministic, such transducers may not define functions but more generally
relations of data $\omega$-words. In order to increase the expressive power of
these machines, we even allow guessing of arbitrary data values when updating
their registers.
For functions over data $\omega$-words, we identify a sufficient condition
(the possibility of determining the next letter to be outputted, which we call
next letter problem) under which computability (resp. uniform computability)
and continuity (resp. uniform continuity) coincide.
We focus on two kinds of data domains: first, the general setting of
oligomorphic data, which encompasses any data domain with equality, as well as
the setting of rational numbers with linear order; and second, the set of
natural numbers equipped with linear order. For both settings, we prove that
functionality, i.e. determining whether the relation recognized by the
transducer is actually a function, is decidable. We also show that the
so-called next letter problem is decidable, yielding equivalence […]
Turing machines and register machines have been used for decades in
theoretical computer science as abstract models of computation. Also the
$\lambda$-calculus has played a central role in this domain as it allows to
focus on the notion of functional computation, based on the substitution
mechanism, while abstracting away from implementation details. The present
article starts from the observation that the equivalence between these
formalisms is based on the Church-Turing Thesis rather than an actual encoding
of $\lambda$-terms into Turing (or register) machines. The reason is that these
machines are not well-suited for modelling $\lambda$-calculus programs.
We study a class of abstract machines that we call "addressing machine" since
they are only able to manipulate memory addresses of other machines. The
operations performed by these machines are very elementary: load an address in
a register, apply a machine to another one via their addresses, and call the
address of another machine. We endow addressing machines with an operational
semantics based on leftmost reduction and study their behaviour. The set of
addresses of these machines can be easily turned into a combinatory algebra. In
order to obtain a model of the full untyped $\lambda$-calculus, we need to
introduce a rule that bares similarities with the $\omega$-rule and the rule
$\zeta_\beta$ from combinatory logic.
The problem of model checking procedural programs has fostered much research
towards the definition of temporal logics for reasoning on context-free
structures. The most notable of such results are temporal logics on Nested
Words, such as CaRet and NWTL. Recently, the logic OPTL was introduced, based
on the class of Operator Precedence Languages (OPLs), more powerful than Nested
Words. We define the new OPL-based logic POTL and prove its FO-completeness.
POTL improves on NWTL by enabling the formulation of requirements involving
pre/post-conditions, stack inspection, and others in the presence of
exception-like constructs. It improves on OPTL too, which instead we show not
to be FO-complete; it also allows to express more easily stack inspection and
function-local properties. In a companion paper we report a model checking
procedure for POTL and experimental results based on a prototype tool developed
therefor. For completeness a short summary of this complementary result is
provided in this paper too.
Timed automata (TA) have been widely adopted as a suitable formalism to model
time-critical systems. Furthermore, contemporary model-checking tools allow the
designer to check whether a TA complies with a system specification. However,
the exact timing constants are often uncertain during the design phase.
Consequently, the designer is often able to build a TA with a correct
structure, however, the timing constants need to be tuned to satisfy the
specification. Moreover, even if the TA initially satisfies the specification,
it can be the case that just a slight perturbation during the implementation
causes a violation of the specification. Unfortunately, model-checking tools
are usually not able to provide any reasonable guidance on how to fix the model
in such situations. In this paper, we propose several concepts and techniques
to cope with the above mentioned design phase issues when dealing with
reachability and safety specifications.
The notion of comparison between system runs is fundamental in formal
verification. This concept is implicitly present in the verification of
qualitative systems, and is more pronounced in the verification of quantitative
systems. In this work, we identify a novel mode of comparison in quantitative
systems: the online comparison of the aggregate values of two sequences of
quantitative weights. This notion is embodied by comparator automata
(comparators, in short), a new class of automata that read two infinite
sequences of weights synchronously and relate their aggregate values.
We show that aggregate functions that can be represented with Büchi
automaton result in comparators that are finite-state and accept by the Büchi
condition as well. Such $\omega$-regular comparators further lead to generic
algorithms for a number of well-studied problems, including the quantitative
inclusion and winning strategies in quantitative graph games with incomplete
information, as well as related non-decision problems, such as obtaining a
finite representation of all counterexamples in the quantitative inclusion
problem.
We study comparators for two aggregate functions: discounted-sum and
limit-average. We prove that the discounted-sum comparator is $\omega$-regular
iff the discount-factor is an integer. Not every aggregate function, however,
has an $\omega$-regular comparator. Specifically, we show that the language of
sequence-pairs for which limit-average aggregates exist is […]
The program-over-monoid model of computation originates with Barrington's
proof that the model captures the complexity class $\mathsf{NC^1}$. Here we
make progress in understanding the subtleties of the model. First, we identify
a new tameness condition on a class of monoids that entails a natural
characterization of the regular languages recognizable by programs over monoids
from the class. Second, we prove that the class known as $\mathbf{DA}$
satisfies tameness and hence that the regular languages recognized by programs
over monoids in $\mathbf{DA}$ are precisely those recognizable in the classical
sense by morphisms from $\mathbf{QDA}$. Third, we show by contrast that the
well studied class of monoids called $\mathbf{J}$ is not tame. Finally, we
exhibit a program-length-based hierarchy within the class of languages
recognized by programs over monoids from $\mathbf{DA}$.
We define a point-free construction of real exponentiation and logarithms,
i.e.\ we construct the maps $\exp\colon (0, \infty)\times \mathbb{R}
\rightarrow \!(0,\infty),\, (x, \zeta) \mapsto x^\zeta$ and $\log\colon
(1,\infty)\times (0, \infty) \rightarrow\mathbb{R},\, (b, y) \mapsto
\log_b(y)$, and we develop familiar algebraic rules for them. The point-free
approach is constructive, and defines the points of a space as models of a
geometric theory, rather than as elements of a set - in particular, this allows
geometric constructions to be applied to points living in toposes other than
Set. Our geometric development includes new lifting and gluing techniques in
point-free topology, which highlight how properties of $\mathbb{Q}$ determine
properties of real exponentiation.
This work is motivated by our broader research programme of developing a
version of adelic geometry via topos theory. In particular, we wish to
construct the classifying topos of places of $\mathbb{Q}$, which will provide a
geometric perspective into the subtle relationship between $\mathbb{R}$ and
$\mathbb{Q}_p$, a question of longstanding number-theoretic interest.
While many applications of automata in formal methods can use
nondeterministic automata, some applications, most notably synthesis, need
deterministic or good-for-games (GFG) automata. The latter are nondeterministic
automata that can resolve their nondeterministic choices in a way that only
depends on the past. The minimization problem for deterministic Büchi and
co-Büchi word automata is NP-complete. In particular, no canonical minimal
deterministic automaton exists, and a language may have different minimal
deterministic automata. We describe a polynomial minimization algorithm for GFG
co-Büchi word automata with transition-based acceptance. Thus, a run is
accepting if it traverses a set $\alpha$ of designated transitions only
finitely often. Our algorithm is based on a sequence of transformations we
apply to the automaton, on top of which a minimal quotient automaton is
defined. We use our minimization algorithm to show canonicity for
transition-based GFG co-Büchi word automata: all minimal automata have
isomorphic safe components (namely components obtained by restricting the
transitions to these not in $\alpha$) and once we saturate the automata with
$\alpha$-transitions, we get full isomorphism.
Priced timed games are two-player zero-sum games played on priced timed
automata (whose locations and transitions are labeled by weights modelling the
cost of spending time in a state and executing an action, respectively). The
goals of the players are to minimise and maximise the cost to reach a target
location, respectively. We consider priced timed games with one clock and
arbitrary integer weights and show that, for an important subclass of them (the
so-called simple priced timed games), one can compute, in pseudo-polynomial
time, the optimal values that the players can achieve, with their associated
optimal strategies. As side results, we also show that one-clock priced timed
games are determined and that we can use our result on simple priced timed
games to solve the more general class of so-called negative-reset-acyclic
priced timed games (with arbitrary integer weights and one clock). The
decidability status of the full class of priced timed games with one-clock and
arbitrary integer weights still remains open.
We present a finitary version of Moss' coalgebraic logic for $T$-coalgebras,
where $T$ is a locally monotone endofunctor of the category of posets and
monotone maps. The logic uses a single cover modality whose arity is given by
the least finitary subfunctor of the dual of the coalgebra functor
$T_\omega^\partial$, and the semantics of the modality is given by relation
lifting. For the semantics to work, $T$ is required to preserve exact squares.
For the finitary setting to work, $T_\omega^\partial$ is required to preserve
finite intersections. We develop a notion of a base for subobjects of $T_\omega
X$. This in particular allows us to talk about the finite poset of subformulas
for a given formula. The notion of a base is introduced generally for a
category equipped with a suitable factorisation system.
We prove that the resulting logic has the Hennessy-Milner property for the
notion of similarity based on the notion of relation lifting. We define a
sequent proof system for the logic, and prove its completeness.
We introduce a generalization of the bisimulation game that finds
distinguishing Hennessy-Milner logic formulas from every finitary,
subformula-closed language in van Glabbeek's linear-time--branching-time
spectrum between two finite-state processes. We identify the relevant
dimensions that measure expressive power to yield formulas belonging to the
coarsest distinguishing behavioral preorders and equivalences; the compared
processes are equivalent in each coarser behavioral equivalence from the
spectrum. We prove that the induced algorithm can determine the best fit of
(in)equivalences for a pair of processes.
We identify a notion of reducibility between predicates, called instance
reducibility, which commonly appears in reverse constructive mathematics. The
notion can be generally used to compare and classify various principles studied
in reverse constructive mathematics (formal Church's thesis, Brouwer's
Continuity principle and Fan theorem, Excluded middle, Limited principle,
Function choice, Markov's principle, etc.). We show that the instance degrees
form a frame, i.e., a complete lattice in which finite infima distribute over
set-indexed suprema. They turn out to be equivalent to the frame of upper sets
of truth values, ordered by the reverse Smyth partial order. We study the
overall structure of the lattice: the subobject classifier embeds into the
lattice in two different ways, one monotone and the other antimonotone, and the
$\lnot\lnot$-dense degrees coincide with those that are reducible to the degree
of Excluded middle.
We give an explicit formulation of instance degrees in a relative
realizability topos, and call these extended Weihrauch degrees, because in
Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees
correspond precisely to Weihrauch degrees. The extended degrees improve the
structure of Weihrauch degrees by equipping them with computable infima and
suprema, an implication, the ability to control access to parameters and
computation of results, and by generally widening the scope of Weihrauch
reducibility.
It is well-known that typability, type inhabitation and type inference are
undecidable in the Girard-Reynolds polymorphic system F. It has recently been
proven that type inhabitation remains undecidable even in the predicative
fragment of system F in which all universal instantiations have an atomic
witness (system Fat). In this paper we analyze typability and type inference in
Curry style variants of system Fat and show that typability is decidable and
that there is an algorithm for type inference which is capable of dealing with
non-redundancy constraints.