Embedding First-order Logic in a Pure Type System with Parameters

2001 ◽  
Vol 11 (4) ◽  
pp. 545-557 ◽  
Author(s):  
T. Laan
Keyword(s):  
Type System ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Pure Type

scholarly journals Modular specification and verification of closures in Rust

2021 ◽  
Vol 5 (OOPSLA) ◽  
pp. 1-29
Author(s):  
Fabian Wolff ◽  
Aurel Bílý ◽  
Christoph Matheja ◽  
Peter Müller ◽  
Alexander J. Summers
Keyword(s):  
Formal Verification ◽  
Functional Properties ◽  
Type System ◽  
Order Logic ◽  
First Order Logic ◽  
Formal Reasoning ◽  
First Order ◽  
Novel Technique ◽  
Smt Solvers ◽  
Specification And Verification

Closures are a language feature supported by many mainstream languages, combining the ability to package up references to code blocks with the possibility of capturing state from the environment of the closure's declaration. Closures are powerful, but complicate understanding and formal reasoning, especially when closure invocations may mutate objects reachable from the captured state or from closure arguments. This paper presents a novel technique for the modular specification and verification of closure-manipulating code in Rust. Our technique combines Rust's type system guarantees and novel specification features to enable formal verification of rich functional properties. It encodes higher-order concerns into a first-order logic, which enables automation via SMT solvers. Our technique is implemented as an extension of the deductive verifier Prusti, with which we have successfully verified many common idioms of closure usage.

An object-oriented calculus with term constraints

2007 ◽  
Vol 17 (3) ◽  
pp. 353-386
Author(s):  
GÁBOR M. SURÁNYI
Keyword(s):  
Type System ◽  
Object Oriented ◽  
Order Logic ◽  
First Order Logic ◽  
Engineering Process ◽  
Type Safety ◽  
First Order ◽  
Multiple Dispatch ◽  
Fundamental Requirement ◽  
Distinguished Object

AbstractSafety has become a fundamental requirement in all aspects of computer systems. Object-oriented calculi, such as Castagna's λ&-calculus and its variants (Castagna, 1997) ensure type safety in environments based on the distinguished object-oriented paradigm. Although for safety reasons object invariance and operation specifications are getting widely employed in all stages of the engineering process, they are not supported by these calculi. In this paper, a new calculus is presented which supports term (value) constraints besides the key object-oriented mechanisms (class types, inheritance, overloading with multiple dispatch and late binding). We also show how a type with constraints may realise a role, another useful object-oriented modelling element. The soundness of the type system and the confluence of the notion of reduction of the calculus are considered. The contribution also discusses computability issues partially arising from the use of first-order logic to formalise the constraints.

scholarly journals Using Vampire in Soundness Proofs of Type Systems

10.29007/22x6 ◽  
2018 ◽  
Author(s):  
Sylvia Grewe ◽  
Sebastian Erdweg ◽  
Mira Mezini
Keyword(s):  
Programming Languages ◽  
Type System ◽  
Lambda Calculus ◽  
Type Systems ◽  
Current Approach ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Domain Specific ◽  
Automated Proof

Type systems for programming languages shall detect type errors in programs before runtime. To ensure that a type system meets this requirement, its soundness must be formally verified. We aim at automating soundness proofs of type systems to facilitate the development of sound type systems for domain-specific languages.Soundness proofs for type systems typically require induction. However, many of the proofs of individual induction cases only require first-order reasoning. For the development of our workbench Veritas, we build on this observation by combining automated first-order theorem provers such as Vampire with automated proof strategies specific to type systems. In this paper, we describe how we encode type soundness proofs in first-order logic using TPTP. We show how we use Vampire to prove the soundness of type systems for the simply-typed lambda calculus and for parts of a typed SQL. We report on which parts of the proofs are handled well by Vampire, and what parts work less well with our current approach.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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