A Curry–Howard View of Basic Justification Logic

Justification logics are epistemic logics that include explicit justifications for an agent’s knowledge. In the present paper, we introduce a justification logic \(\mathcal{JALC}\) over the description logic \(\mathcal{ALC}\). We provide a deductive system and a semantics for our logic and we establish soundness and completeness results. Moreover, we show that our logic satisfies the so-called internalization property stating that it internalizes its own notion of proof. We then sketch two applications of \(\mathcal{JALC}\): (i) the justification terms can be used to generate natural language explanations why an \(\mathcal{ALC}\) statement holds and (ii) the terms can be used to study data privacy issues for description logic knowledge bases.

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

An important issue when defining a rule-based agent programming language is the design of interpreters for these programming languages. Since these languages are all based on some notion of rule, an interpreter must provide some means of selection from a set of such rules. We provide a concrete and intuitive ordering on rules on which this selection can be based. This ordering is inspired by the common sense metaphor of intelligent agents in Artificial Intelligence. Furthermore, we provide a language with a formal semantics for programming agent interpreters. The main idea is not to integrate this language into the agent language itself, but to provide the constructs for building interpreters at another (meta) level of specification. The operational semantics is accordingly specified at the meta level, by means of a (meta) transition system. Using this language we make a comparison between several interpreters for agent languages in the literature.

Defining the semantics of programming languages by axioms and rules of inference yields a deduction system within which proofs may be given that programs satisfy specifications. The deduction system herein is shown to be consistent and also deduction complete with respect to Hoare's system. A subgoaler for the deduction system is described whose input is a significant subset of Pascal programs plus inductive assertions. The output is a set of verification conditions or lemmas to be proved. Several non-trivial arithmetic and sorting programs have been shown to satisfy specifications by using an interactive theorem prover to automatically generate proofs of the verification conditions. Additional components for a more powerful verification system are under construction.

Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, called generalized multiary λ-calculus , is a rich extension of the λ-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability to form lists of arguments) and generality (the ability to prescribe a kind of continuation). The system integrates in a modular way the multiary λ-calculus and an isomorphic copy of the λ-calculus with generalized application, Λ J (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules — either the μ-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary β-rule. This article establishes the metatheory of the system, with emphasis on the role of the μ-rule, and including a study of the interaction of reduction and permutative conversions.

Logics for ‘generally’ were introduced as extensions of First-Order Logic (FOL) for handling assertions with vague notions (e.g., ‘generally,’ ‘most,’ and ‘several’) expressed by formulas with generalized quantifiers. Deductive systems have been developed for such logics. Here, we characterize the structure of derivations in natural deduction style for filter logic. This characterization extends the familiar one for FOL.

In this thesis we give an algebraic characterization of the syntax and semantics of simply–typed languages. More precisely, we characterize simply–typed binding syntax equipped with reduction rules via a universal property, namely as the initial object of some category. We specify a language by a 2–signature (Σ, A), that is, a signature on two levels: the syntactic level Σ specifies the sorts and terms of the language, and associates a sort to each term. The semantic level A specifies, through inequations, reduction rules on the terms of the language. To any given 2–signature (Σ, A) we associate a category of “models” of (Σ, A). We prove that this category has an initial object, which integrates the terms freely generated by Σ and the reduction relation — on those terms — generated by A. We call this object the programming language generated by (Σ, A). Initiality provides an iteration principle which allows to specify translations on the syntax, possibly to a language over different sorts. Furthermore, translations specified via the iteration principle are by construction type–safe and faithful with respect to reduction. To illustrate our results, we consider two examples extensively: firstly, we specify a double negation translation from classical to intuitionistic propositional logic via the category–theoretic iteration principle. Secondly, we specify a translation from PCF to the untyped lambda calculus which is faithful with respect to reduction in the source and target languages. In a second part, we formalize some of our initiality theorems in the proof assistant Coq. The implementation yields a machinery which, when given a 2–signature, returns an implementation of its associated abstract syntax together with certified substitution operation, iteration operator and a reduction relation generated by the specified reduction rules.

A database interface based on Montague's approachto the interpretation of natural language

The main result of this paper is a normalizing system of natural deduction for the full language of intuitionistic linear logic. No explicit weakening or contraction rules for \(\)-formulas are needed. By the systematic use of general elimination rules a correspondence between normal derivations and cut-free derivations in sequent calculus is obtained. Normalization and the subformula property for normal derivations follow through translation to sequent calculus and cut-elimination.

Language analysis has always been a popular subject in computer science; henceforth a vast knowledge base for compiler construction is available nowadays and consequently one can devise several models that provide efficient methods to determine the acceptance of a programming language, either at a lexical, syntactic or semantic level. By evaluating and analyzing existing techniques, we have noticed that certain algorithm complexity or grammar adaptation was required to accomplish an efficient solution. Therefore here lies the motivation for the current research. We are going to present an implementation based on a lexical and syntactic analysis model seeking to fulfill the following requisites: simplicity and expressiveness, algorithm efficiency and reduced grammar adaptation. Lexical resolution was mainly supported by regular language acceptance, whereas the subsequent syntactic analysis was performed with a pushdown automaton allowing an efficient method to recognize a context free language, avoiding additional elements, such as action-goto tables, often used in bottom-up parsers. The proposed model highlights the underlying relationship between language interpretation and automata theory. Then you may encounter an interesting approach which will provide a way to go through the most basic concepts of formal language theory and how these are wielded for solving syntactic analysis problems.

Gentzen's systems of natural deduction and sequent calculus were byproducts in his program of proving the consistency of arithmetic and analysis. It is suggested that the central component in his results on logical calculi was the use of a tree form for derivations. It allows the composition of derivations and the permutation of the order of application of rules, with a full control over the structure of derivations as a result. Recently found documents shed new light on the discovery of these calculi. In particular, Gentzen set up five different forms of natural calculi and gave a detailed proof of normalization for intuitionistic natural deduction. An early handwritten manuscript of his thesis shows that a direct translation from natural deduction to the axiomatic logic of Hilbert and Ackermann was, in addition to the influence of Paul Hertz, the second component in the discovery of sequent calculus. A system intermediate between the sequent calculus LI and axiomatic logic, denoted LIG in unpublished sources, is implicit in Gentzen's published thesis of 1934–35. The calculus has half rules, half “groundsequents,” and does not allow full cut elimination. Nevertheless, a translation from LI to LIG in the published thesis gives a subformula property for a complete class of derivations in LIG. After the thesis, Gentzen continued to work on variants of sequent calculi for ten more years, in the hope to find a consistency proof for arithmetic within an intuitionistic calculus.

This is a companion to a paper by the authors entitled “Godel on deduction”, which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Godel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel’s Notre Dame course, and is followed by comments concerning Godel’s natural deduction system.

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

A decision procedure revisited: notes on direct logic, linear logic and its implementation

Direct consistency proof of Gentzen's system of natural deduction.