Platinum: Reusing Constraint Solutions in Bounded Analysis of Relational Logic

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

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Automating first-order relational logic

Proceedings of the 8th ACM SIGSOFT international symposium on Foundations of software engineering twenty-first century applications - SIGSOFT '00/FSE-8 ◽

10.1145/355045.355063 ◽

2000 ◽

Cited By ~ 68

Author(s):

Daniel Jackson

Keyword(s):

First Order ◽

Relational Logic

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A Relational Logic for Spatial Contact Based on Rough Set Approximation

Fundamenta Informaticae ◽

10.3233/fi-2016-1430 ◽

2016 ◽

Vol 148 (1-2) ◽

pp. 191-206 ◽

Cited By ~ 1

Author(s):

Ivo Düntsch ◽

Ewa Orłowska ◽

Hui Wang

Keyword(s):

Rough Set ◽

Set Approximation ◽

Relational Logic

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Applications and extensions of Alloy: past, present and future

Mathematical Structures in Computer Science ◽

10.1017/s0960129512000291 ◽

2013 ◽

Vol 23 (4) ◽

pp. 915-933 ◽

Cited By ~ 9

Author(s):

EMINA TORLAK ◽

MANA TAGHDIRI ◽

GREG DENNIS ◽

JOSEPH P. NEAR

Keyword(s):

Automated Analysis ◽

Test Case Generation ◽

Test Case ◽

Recent Application ◽

Code Verification ◽

Constraint Solver ◽

First Order ◽

The Core ◽

Automated Simulation ◽

Relational Logic

Alloy is a declarative language for lightweight modelling and analysis of software. The core of the language is based on first-order relational logic, which offers an attractive balance between analysability and expressiveness. The logic is expressive enough to capture the intricacies of real systems, but is also simple enough to support fully automated analysis with the Alloy Analyzer. The Analyzer is built on a SAT-based constraint solver and provides automated simulation, checking and debugging of Alloy specifications. Because of its automated analysis and expressive logic, Alloy has been applied in a wide variety of domains. These applications have motivated a number of extensions both to the Alloy language and to its SAT-based analysis. This paper provides an overview of Alloy in the context of its three largest application domains, lightweight modelling, bounded code verification and test-case generation, and three recent application-driven extensions, an imperative extension to the language, a compiler to executable code and a proof-capable analyser based on SMT.

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Higher-order probabilistic adversarial computations: categorical semantics and program logics

Proceedings of the ACM on Programming Languages ◽

10.1145/3473598 ◽

2021 ◽

Vol 5 (ICFP) ◽

pp. 1-30

Author(s):

Alejandro Aguirre ◽

Gilles Barthe ◽

Marco Gaboardi ◽

Deepak Garg ◽

Shin-ya Katsumata ◽

...

Keyword(s):

Higher Order ◽

General Notion ◽

Memory Footprint ◽

Resource Policy ◽

Expected Values ◽

Error Probabilities ◽

The Cost ◽

Relational Logic ◽

Categorical Semantics ◽

Develop Program

Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several domains, including security, privacy and machine learning. In this paper, we develop program logics for reasoning about adversarial computations in a higher-order setting. Our logics are built on top of a simply typed λ-calculus extended with a graded monad for probabilities and state. The grading is used to model and restrict the memory footprint and the cost (in terms of oracle calls) of computations. Under this view, an adversary is a higher-order expression that expects as arguments the code of its oracles. We develop unary program logics for reasoning about error probabilities and expected values, and a relational logic for reasoning about coupling-based properties. All logics feature rules for adversarial computations, and yield guarantees that are valid for all adversaries that satisfy a fixed resource policy. We prove the soundness of the logics in the category of quasi-Borel spaces, using a general notion of graded predicate liftings, and we use logical relations over graded predicate liftings to establish the soundness of proof rules for adversaries. We illustrate the working of our logics with simple but illustrative examples.

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Relational Communication Spaces: Infrastructures and Discursive Practices

Media and Communication ◽

10.17645/mac.v9i3.3988 ◽

2021 ◽

Vol 9 (3) ◽

pp. 28-39 ◽

Cited By ~ 1

Author(s):

Alexa Keinert ◽

Volkan Sayman ◽

Daniel Maier

Keyword(s):

Power Relations ◽

Communication Technologies ◽

Public Communication ◽

Relational Communication ◽

Discursive Practices ◽

Analytical Perspective ◽

Communication Space ◽

Relational Concept ◽

New Perspective ◽

Relational Logic

Digital communication technologies, social web platforms, and mobile communication have fundamentally altered the way we communicate publicly. They have also changed our perception of space, thus making a re-calibration of a spatial perspective on public communication necessary. We argue that such a new perspective must consider the relational logic of public communication, which stands in stark contrast to the plain territorial notion of space common in communication research. Conceptualising the spatiality of public communication, we draw on Löw’s (2016) sociology of space. Her relational concept of space encourages us to pay more attention to (a) the infrastructural basis of communication, (b) the operations of synthesising the relational communication space through discursive practices, and (c) power relations that determine the accessibility of public communication. Thus, focusing on infrastructures and discursive practices means highlighting crucial socio-material preconditions of public communication and considering the effects of the power relations which are inherent in their spatialisation upon the inclusivity of public communication<em>.</em> This new approach serves a dual purpose: Firstly, it works as an analytical perspective to systematically account for the spatiality of public communication. Secondly, the differentiation between infrastructural spaces and spaces of discursive practices adds explanatory value to the perspective of relational communication spaces.

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