Related papers: Induction and Co-induction in Sequent Calculus
Chain of Thought (CoT) prompting can encourage language models to engage in multi-step logical reasoning. The quality of the provided demonstrations significantly influences the success of downstream inference tasks. Current unsupervised…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
In this paper, an application of automated theorem proving techniques to computational semantics is considered. In order to compute the presuppositions of a natural language discourse, several inference tasks arise. Instead of treating…
Programs with control are usually modeled using lambda calculus extended with control operators. Instead of modifying lambda calculus, we consider a different model of computation. We introduce continuation calculus, or CC, a deterministic…
Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in…
Uniform proofs are sequent calculus proofs with the following characteristic: the last step in the derivation of a complex formula at any stage in the proof is always the introduction of the top-level logical symbol of that formula. We…
The automated proof search system and decidability for logic of correlated knowledge is presented in this paper. The core of the proof system is the sequent calculus with the properties of soundness, completeness, admissibility of cut and…
Ordered logics and type systems have been used in a variety of applications including computational linguistics, memory allocation, stream processing, logical frameworks, parametricity, and enforcing security protocols. In most…
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
In a previously published ENTCS paper (Santos et al. (2016)), we introduced a sequent calculus called $\mathbf{LMT^{\rightarrow}}$ for Minimal Implicational Propositional Logic ($\mathbf{LMT^{\rightarrow}}$). This calculus provides a proof…
In answer set programming, two groups of rules are considered strongly equivalent if they have the same meaning in any context. Strong equivalence of two programs can be sometimes established by deriving rules of each program from rules of…
Selective rationalization has become a common mechanism to ensure that predictive models reveal how they use any available features. The selection may be soft or hard, and identifies a subset of input features relevant for prediction. The…
Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
Existing refinement calculi provide frameworks for the stepwise development of imperative programs from specifications. This paper presents a refinement calculus for deriving logic programs. The calculus contains a wide-spectrum logic…
In the former article "Formal mathematical systems including a structural induction principle" we have presented a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the…
We introduce real induction, a proof technique analogous to mathematical induction but applicable to statements indexed by an interval on the real line. More generally we give an inductive principle applicable in any Dedekind complete…
We study subsystems of open induction which are strongly connected to methods of automated inductive theorem proving. Specifically, we consider systems obtained from restricting induction to atoms, literals, clauses, and dual clauses. We…
Warning: This paper contains a mistake, rendering the proof of the main theorem invalid. The logic of Bunched Implications (BI) combines both additive and multiplicative connectives, which include two primitive intuitionistic implications.…
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is…