An exercise on the generation of many-valued dynamic logics

Madeira, Alexandre; Neves, Renato; Martins, Manuel A.

doi:10.1016/j.jlamp.2016.03.004

Cited by 22 publications

(48 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore we believe that our game logic automata have the potential to help us understand a wider class of automata for families of dynamic logics such as coalgebraic dynamic logics [14] or many-valued dynamic logics as described in [21] or for a combination of these frameworks.…”

Section: Resultsmentioning

confidence: 99%

Parity Games and Automata for Game Logic

Hansen

Kupke

Marti

et al. 2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. Parikh's game logic is a PDL-like fixpoint logic interpreted on monotone neighbourhood frames that represent the strategic power of players in determined two-player games. Game logic translates into a fragment of the monotone µ-calculus, which in turn is expressively equivalent to monotone modal automata. Parity games and automata are important tools for dealing with the combinatorial complexity of nested fixpoints in modal fixpoint logics, such as the modal µ-calculus. In this paper, we (1) discuss the semantics a of game logic over neighbourhood structures in terms of parity games, and (2) use these games to obtain an automata-theoretic characterisation of the fragment of the monotone µ-calculus that corresponds to game logic. Our proof makes extensive use of structures that we call syntax graphs that combine the ease-of-use of syntax trees of formulas with the flexibility and succinctness of automata. They are essentially a graph-based view of the alternating tree automata that were introduced by Wilke in the study of modal µ-calculus.

show abstract

Section: Resultsmentioning

confidence: 99%

Parity Games and Automata for Game Logic

Hansen

Kupke

Marti

et al. 2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The satisfaction of formula (w |= CGDL π ρ) is given by the weight of some fuzzy set φ which is related with state w by some fuzzy multirelation, and that of ρ for every state of the domain of φ. Moreover, the satisfaction for the box operator follows [14], where every execution of the program must lead to a set of states all of which satisfy ρ.…”

Section: Parametric Construction Of Multi-valued Concurrent Dynamic Lmentioning

confidence: 99%

On the Construction of Multi-valued Concurrent Dynamic Logics

Gomes

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Dynamic logic is a powerful framework for reasoning about imperative programs. An extension with a concurrent operator [18] was introduced to formalise programs running in parallel. In other direction, other authors proposed a systematic method for generating multi-valued propositional dynamic logics to reason about weighted programs [14]. This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations. In the developed framework, a weight is assigned to each branch of the parellel execution, resulting in a (possible) asymmetric parallelism, inherent to fuzzy programming paradigm [21,2]. By adopting such an approach, a family of logics is obtained, called multi-valued concurrent propositional dynamic logics (CGDL(A)), parametric on an action lattice A specifying a notion of "weight" assigned to program execution. Additionally, the validity of some axioms of CPDL is discussed in the new family of generated logics. IntroductionOver time, the different variations of dynamic logics developed went hand-inhand with the very notion of its object, the program. This resulted in a diverse myriad of dynamic logics tailored to specific programming paradigms. Examples include probabilistic [11], concurrent [18], quantum [1] and continuous [19] computations, and combinations thereof. An example of another non-trivial paradigm is the fuzzy one [21,2], where the execution of a program differs from both classical and probabilistic scenarios: a conditional statement may act as a concurrent execution with a weight associated to each branch. The formalisation of such behaviour encompasses two non-trivial computational settings: concurrency and uncertainty. An extensive research can be found in the literature on diverse formalisms to reason about programs running in parallel [9,10] and to deal with uncertainty [11,5,20,4]. However, even when these two components are combined into a single framework [16], the uncertainty models probabilistic nondeterminism. Thus we are still missing a proper semantics to describe the behaviour of the fuzzy paradigm.Recently, reference [14] initiated a research agenda on the systematic development of multi-valued propositional dynamic logics, parametric on an action ⋆

show abstract

“…The approach taken in this paper, adding a residual as a logical implication to capture a multi-valued setting, is based on previous work [23], where an action lattice is adopted as the basic algebraic structure to generate many-valued dynamic logics. Originally derived from action algebras [19], an action lattice entails both a generic space of computations, with choice, composition and iteration, and, supported by residuation, a proper truth space for a non bivalent interpretation of assertions (as a residuated lattice).…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Generalising KAT to Verify Weighted Computations

Gomes¹,

Madeira²,

Barbosa³

2019

SACS

Self Cite

View full text Add to dashboard Cite

Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean tests. This paper introduces two generalisations of this structure able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a variant where tests are also idempotent (I-GKAT). On this context, and in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT [22], we discuss the encoding of a graded PHL in I-GKAT and of its while-free fragment in GKAT. Moreover, to establish semantics for these structures four new algebras are defined: F SET (T ), F REL(K, T ) and F LAN G(K, T ) over complete residuated lattices K and T , and M (n, A) over a GKAT or I-GKAT A. As a final exercise, the paper discusses some program equivalence proofs in a graded context.

show abstract

An exercise on the generation of many-valued dynamic logics

Cited by 22 publications

References 33 publications

Parity Games and Automata for Game Logic

Parity Games and Automata for Game Logic

On the Construction of Multi-valued Concurrent Dynamic Logics

Generalising KAT to Verify Weighted Computations

Contact Info

Product

Resources

About