523,352 research outputs found
On Equivalence and Canonical Forms in the LF Type Theory
Decidability of definitional equality and conversion of terms into canonical
form play a central role in the meta-theory of a type-theoretic logical
framework. Most studies of definitional equality are based on a confluent,
strongly-normalizing notion of reduction. Coquand has considered a different
approach, directly proving the correctness of a practical equivalance algorithm
based on the shape of terms. Neither approach appears to scale well to richer
languages with unit types or subtyping, and neither directly addresses the
problem of conversion to canonical.
In this paper we present a new, type-directed equivalence algorithm for the
LF type theory that overcomes the weaknesses of previous approaches. The
algorithm is practical, scales to richer languages, and yields a new notion of
canonical form sufficient for adequate encodings of logical systems. The
algorithm is proved complete by a Kripke-style logical relations argument
similar to that suggested by Coquand. Crucially, both the algorithm itself and
the logical relations rely only on the shapes of types, ignoring dependencies
on terms.Comment: 41 page
Logical Semantics and Commonsense Knowledge: Where Did we Go Wrong, and How to Go Forward, Again
We argue that logical semantics might have faltered due to its failure in distinguishing between two fundamentally very different types of concepts: ontological concepts, that should be types in a strongly-typed ontology, and logical concepts, that are predicates corresponding to properties of and relations between objects of various ontological types. We will then show that accounting for these differences
amounts to the integration of lexical and compositional semantics in one coherent framework, and to an embedding in our logical semantics of a strongly-typed ontology that reflects our commonsense view of the world and the way we talk about it in ordinary language. We will show that in such a framework a number of challenges in natural language semantics can be adequately and systematically treated
ChatRule: Mining Logical Rules with Large Language Models for Knowledge Graph Reasoning
Logical rules are essential for uncovering the logical connections between
relations, which could improve the reasoning performance and provide
interpretable results on knowledge graphs (KGs). Although there have been many
efforts to mine meaningful logical rules over KGs, existing methods suffer from
the computationally intensive searches over the rule space and a lack of
scalability for large-scale KGs. Besides, they often ignore the semantics of
relations which is crucial for uncovering logical connections. Recently, large
language models (LLMs) have shown impressive performance in the field of
natural language processing and various applications, owing to their emergent
ability and generalizability. In this paper, we propose a novel framework,
ChatRule, unleashing the power of large language models for mining logical
rules over knowledge graphs. Specifically, the framework is initiated with an
LLM-based rule generator, leveraging both the semantic and structural
information of KGs to prompt LLMs to generate logical rules. To refine the
generated rules, a rule ranking module estimates the rule quality by
incorporating facts from existing KGs. Last, a rule validator harnesses the
reasoning ability of LLMs to validate the logical correctness of ranked rules
through chain-of-thought reasoning. ChatRule is evaluated on four large-scale
KGs, w.r.t. different rule quality metrics and downstream tasks, showing the
effectiveness and scalability of our method.Comment: 11 pages, 4 figure
Enhancing Reuse of Constraint Solutions to Improve Symbolic Execution
Constraint solution reuse is an effective approach to save the time of
constraint solving in symbolic execution. Most of the existing reuse approaches
are based on syntactic or semantic equivalence of constraints; e.g. the Green
framework is able to reuse constraints which have different representations but
are semantically equivalent, through canonizing constraints into syntactically
equivalent normal forms. However, syntactic/semantic equivalence is not a
necessary condition for reuse--some constraints are not syntactically or
semantically equivalent, but their solutions still have potential for reuse.
Existing approaches are unable to recognize and reuse such constraints.
In this paper, we present GreenTrie, an extension to the Green framework,
which supports constraint reuse based on the logical implication relations
among constraints. GreenTrie provides a component, called L-Trie, which stores
constraints and solutions into tries, indexed by an implication partial order
graph of constraints. L-Trie is able to carry out logical reduction and logical
subset and superset querying for given constraints, to check for reuse of
previously solved constraints. We report the results of an experimental
assessment of GreenTrie against the original Green framework, which shows that
our extension achieves better reuse of constraint solving result and saves
significant symbolic execution time.Comment: this paper has been submitted to conference ISSTA 201
RulE: Neural-Symbolic Knowledge Graph Reasoning with Rule Embedding
Knowledge graph (KG) reasoning is an important problem for knowledge graphs.
It predicts missing links by reasoning on existing facts. Knowledge graph
embedding (KGE) is one of the most popular methods to address this problem. It
embeds entities and relations into low-dimensional vectors and uses the learned
entity/relation embeddings to predict missing facts. However, KGE only uses
zeroth-order (propositional) logic to encode existing triplets (e.g., ``Alice
is Bob's wife."); it is unable to leverage first-order (predicate) logic to
represent generally applicable logical \textbf{rules} (e.g., ``''). On the other hand, traditional rule-based KG reasoning methods
usually rely on hard logical rule inference, making it brittle and hardly
competitive with KGE. In this paper, we propose RulE, a novel and principled
framework to represent and model logical rules and triplets. RulE jointly
represents entities, relations and logical rules in a unified embedding space.
By learning an embedding for each logical rule, RulE can perform logical rule
inference in a soft way and give a confidence score to each grounded rule,
similar to how KGE gives each triplet a confidence score. Compared to KGE
alone, RulE allows injecting prior logical rule information into the embedding
space, which improves the generalization of knowledge graph embedding. Besides,
the learned confidence scores of rules improve the logical rule inference
process by softly controlling the contribution of each rule, which alleviates
the brittleness of logic. We evaluate our method with link prediction tasks.
Experimental results on multiple benchmark KGs demonstrate the effectiveness of
RulE
A logical foundation for session-based concurrent computation
Linear logic has long been heralded for its potential of providing a logical basis for concurrency.
While over the years many research attempts were made in this regard, a Curry-Howard correspondence between linear logic and concurrent computation was only found recently, bridging the proof theory of linear logic and session-typed process calculus. Building upon this work, we have
developed a theory of intuitionistic linear logic as a logical foundation for session-based concurrent computation, exploring several concurrency related phenomena such as value-dependent session
types and polymorphic sessions within our logical framework in an arguably clean and elegant way, establishing with relative ease strong typing guarantees due to the logical basis, which ensure the fundamental properties of type preservation and global progress, entailing the absence of deadlocks
in communication.
We develop a general purpose concurrent programming language based on the logical interpretation, combining functional programming with a concurrent, session-based process layer through the form of a contextual monad, preserving our strong typing guarantees of type preservation and
deadlock-freedom in the presence of general recursion and higher-order process communication.
We introduce a notion of linear logical relations for session typed concurrent processes, developing an arguably uniform technique for reasoning about sophisticated properties of session-based concurrent computation such as termination or equivalence based on our logical approach, further supporting our goal of establishing intuitionistic linear logic as a logical foundation for sessionbased concurrency
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