Related papers: Enumerating proofs of positive formulae
Suppose we have been sold on the idea that formalised proofs in an LCF system should resemble their written counterparts, and so consist of formulas that only provide signposts for a fully verified proof. To be practical, most of the fully…
In this note, we use Kunen's notion of a signing to establish two theorems about the well-founded semantics of logic programs, in the case where we are interested in only (say) the positive literals of a predicate $p$ that are consequences…
One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and produce outputs in different formats. The project ProofCert promotes the…
Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning…
Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This…
We consider the untyped lambda calculus with constructors and recursively defined constants. We construct a domain-theoretic model such that any term not denoting bottom is strongly normalising provided all its `stratified approximations'…
We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and…
The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension…
We systematically investigate the complexity of model checking the existential positive fragment of first-order logic. In particular, for a set of existential positive sentences, we consider model checking where the sentence is restricted…
Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently…
The safety of automated driving systems must be justified by convincing arguments and supported by compelling evidence to persuade certification agencies, regulatory entities, and the general public to allow the systems on public roads.…
Propositional logics in general, considered as a set of sentences, can be undecidable even if they have "nice" representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already…
We present a straightforward embedding of quantified multimodal logic in simple type theory and prove its soundness and completeness. Modal operators are replaced by quantification over a type of possible worlds. We present simple…
It has been argued that reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the…
Understanding the behavior of learned classifiers is an important task, and various black-box explanations, logical reasoning approaches, and model-specific methods have been proposed. In this paper, we introduce probabilistic sufficient…
We investigate the extent to which Linear Temporal Logic (LTL) formulas can be uniquely characterized by a finite set of labeled examples. We consider different types of examples, ranging from finite words to transfinite words, as well as…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
We present a logically principled foundation for systematizing, in a way that works with any computational effect and evaluation order, SMT constraint generation seen in refinement type systems for functional programming languages. By…
We describe a remarkable relation between the notion of valid formula of predicate logic and the specification of network protocols. We give several examples such as the acknowledgement of one packet or of a sequence of packets. We show how…
The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a…