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We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its…
The combination of higher-order theories and fuzzy logic can be useful in decision-making tasks that involve reasoning across abstract functions and predicates, where exact matches are often rare or unnecessary. Developing efficient…
We present a new approach to automated reasoning about higher-order programs by endowing symbolic execution with a notion of higher-order, symbolic values. Our approach is sound and relatively complete with respect to a first-order solver…
We formulate a framework for describing behaviour of effectful higher-order recursive programs. Examples of effects are implemented using effect operations, and include: execution cost, nondeterminism, global store and interaction with a…
Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent computations where code usage plays a central role. The theory of program equivalence for modal and coeffectful languages, however, is…
Language models are increasingly being used in important decision pipelines, so ensuring the correctness of their outputs is crucial. Recent work has proposed evaluating the "factuality" of claims decomposed from a language model generation…
Motivated by applications in automated verification of higher-order functional programs, we develop a notion of constrained Horn clauses in higher-order logic and a decision problem concerning their satisfiability. We show that, although…
The bisimulation proof method can be enhanced by employing `bisimulations up-to' techniques. A comprehensive theory of such enhancements has been developed for first-order (i.e., CCS-like) labelled transition systems (LTSs) and…
Proving compositionality of behavioral equivalence on state-based systems with respect to algebraic operations is a classical and widely studied problem. We study a categorical formulation of this problem, where operations on state-based…
Reachability Logic is a formalism that can be used, among others, for expressing partial-correctness properties of transition systems. In this paper we present three proof systems for this formalism, all of which are sound and complete and…
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…
Quantitative logic reasons about the degree to which formulas are satisfied. This paper studies the fundamental reasoning principles of higher-order quantitative logic and their application to reasoning about probabilistic programs and…
Recursive definitions of predicates are usually interpreted either inductively or coinductively. Recently, a more powerful approach has been proposed, called flexible coinduction, to express a variety of intermediate interpretations,…
Both syntax-phonology and syntax-semantics interfaces in Higher Order Grammar (HOG) are expressed as axiomatic theories in higher-order logic (HOL), i.e. a language is defined entirely in terms of provability in the single logical system.…
This paper proposes a novel higher-order multi-scale (HOMS) computational method, which is highly targeted for efficient, high-accuracy and low-computational-cost simulation of hygro-thermo-mechanical (H-T-M) coupling problems in…
Deductive verification of hybrid systems (HSs) increasingly attracts more attention in recent years because of its power and scalability, where a powerful specification logic for HSs is the cornerstone. Often, HSs are naturally modelled by…
We present a systematic approach to logical predicates based on universal coalgebra and higher-order abstract GSOS, thus making a first step towards a unifying theory of logical relations. We first observe that logical predicates are…
We prove a general congruence result for bisimilarity in higher-order languages, which generalises previous work to languages specified by a labelled transition system in which programs may occur as labels, and which may rely on operations…
Recent studies have shown that novel collective behaviors emerge in complex systems due to higher-order interactions. However, the way in which the structural correlations of these interactions shape such behaviors remains a significant gap…
In search for a foundational framework for reasoning about observable behavior of programs that may not terminate, we have previously devised a trace-based big-step semantics for While. In this semantics, both traces and evaluation…