Related papers: Coinductive Uniform Proofs
A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations,…
Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and co-induction. These proof principles…
After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference…
Proofs in propositional logic are typically presented as trees of derived formulas or, alternatively, as directed acyclic graphs of derived formulas. This distinction between tree-like vs. dag-like structure is particularly relevant when…
The original idea of proof nets can be formulated by means of interaction nets syntax. Additional machinery as switching, jumps and graph connectivity is needed in order to ensure correspondence between a proof structure and a correct proof…
We investigate the scaling properties of implicit deductive reasoning over Horn clauses in depth-bounded Transformers. By systematically decorrelating provability from spurious features and enforcing algorithmic alignment, we find that in…
Coinduction is a widely used technique for establishing behavioural equivalence of programs in higher-order languages. In recent years, the rise of languages with quantitative (e.g.~probabilistic) features has led to extensions of…
This paper presents a simple framework for Horn clause abduction, with probabilities associated with hypotheses. It is shown how this representation can represent any probabilistic knowledge representable in a Bayesian belief network. The…
In this paper we develop cyclic proof systems for the problem of inclusion between the least sets of models of mutually recursive predicates, when the ground constraints in the inductive definitions belong to the quantifier-free fragments…
We explore an inquisitive modal logic designed to reason about neighborhood models. This logic is based on an inquisitive strict conditional operator, which quantifies over neighborhoods, and which can be applied to both statements and…
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…
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,…
We introduce a generalized notion of inference system to support more flexible interpretations of recursive definitions. Besides axioms and inference rules with the usual meaning, we allow also coaxioms, which are, intuitively, axioms which…
We propose a new type-theoretic approach to SLD-resolution and Horn-clause logic programming. It views Horn formulas as types, and derivations for a given query as a construction of the inhabitant (a proof-term) for the type given by the…
Using a call-by-value functional language as an example, this article illustrates the use of coinductive definitions and proofs in big-step operational semantics, enabling it to describe diverging evaluations in addition to terminating…
A central problem in proof-theory is that of finding criteria for identity of proofs, that is, for when two distinct formal derivations can be taken as denoting the same logical argument. In the literature one finds criteria which are…
We consider constrained Horn clause solving from the more general point of view of solving formula equations. Constrained Horn clauses correspond to the subclass of Horn formula equations. We state and prove a fixed-point theorem for Horn…
We introduce an operational rewriting-based semantics for strictly positive nested higher-order (co)inductive types. The semantics takes into account the "limits" of infinite reduction sequences. This may be seen as a refinement and…
We develop a second-order extension of intuitionistic modal logic, allowing quantification over propositions, both syntactically and semantically. A key feature of second-order logic is its capacity to define positive connectives from the…
Coherence theorems for covariant structures carried by a category have traditionally relied on the underlying term rewriting system of the structure being terminating and confluent. While this holds in a variety of cases, it is not a…