An internal language for autonomous categories

We present an internal language for symmetric monoidal closed (autonomous) categories analogous to the typed lambda calculus as an internal language for cartesian closed categories. The language we propose is the term assignment to the multiplicative fragment of Intuitionistic Linear Logic, which possesses exactly the right structure for an autonomous theory. We prove that this language is an internal language and show as an application the coherence theorem of Kelly and Mac Lane, which becomes straightforward to state and prove. Finally, we extend the language with the natural numbers and show that this corresponds to a weak Natural Numbers Object in an autonomous category.     

A parallel game semantics for Linear Logic

We describe the constructive content of proofs in a fragment of propositional Infinitary Linear Logic in terms of strategies for a suitable class of games. Such strategies interpret linear proofs as parallel algorithms as long as the asymmetry of the connectives ? and ! allows it. Received December 5, 1994     

THE JUDGEMENT CALCULUS FOR INTUITIONISTIC LINEAR LOGIC: PROOF THEORY AND SEMANTICS

In this paper we propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic; these rules ease the problem of defining a suitable mathematical semantics. A proof of the canonical form theorem for this new system is given: it assures, beside the consistency of the calculus, the termination of the evaluation process of every well-typed element. The definition of the mathematical semantics and a completeness theorem, that turns out to be a representation theorem, follow. This semantics is the basis to obtain a semantics for the evaluation process of every well-typed program. 1991 MSC: 03B20, 03B40.     

Wellfoundedness proofs by means of non-monotonic inductive definitions II: First order operators

In this paper, we give two proofs of the wellfoundedness of a recursive notation system for Π_N-reflecting ordinals. One is based on distinguished classes, and the other is based on -inductive definitions.     

A modal logic framework for reasoning about comparative distances and topology

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S4 and conditional logics. We provide finite (Hilbert-style) axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.     

The Logic of Uncertain Justifications

In Artemov?s Justification Logic, one can make statements interpreted as “t is evidence for the truth of formula F.” We propose a variant of this logic in which one can say “I have degree r of confidence that t is evidence for the truth of formula F.” After defining both an axiomatic approach and a semantics for this Logic of Uncertain Justifications, we will prove the usual soundness and completeness theorems.     

Monotone Proofs of the Pigeon Hole Principle

We study the complexity of proving the Pigeon Hole Principle (PHP)in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We prove a size‐depth trade‐off upper bound for monotone proofs of the standard encoding of the PHP as a monotone sequent. At one extreme of the trade‐off we get quasipolynomia ‐size monotone proofs, and at the other extreme we get subexponential‐size bounded‐depth monotone proofs. This result is a consequence of deriving the basic properties of certain monotone formulas computing the Boolean threshold functions. We also consider the monotone sequent expressing the Clique‐Coclique Principle (CLIQUE) defined by Bonet, Pitassi and Raz [9]. We show that monotone proofs for this sequent can be easily reduced to monotone proofs of the one‐to‐one and onto PHP, and so CLIQUE also has quasipolynomia ‐size monotone proofs. As a consequence of our results, Resolution, Cutting Planes with polynomially bounded coefficients, and Bounded‐Depth Frege are exponentially separated from the monotone Gentzen Calculus. Finally, a simple simulation argument implies that these results extend to the Intuitionistic Gentzen Calculus. Our results partially answer some questions left open by P. Pudlák.     

Possible world semantics for first-order logic of proofs

In the tech report Artemov and Yavorskaya (Sidon) (2011) [4] an elegant formulation of the first-order logic of proofs was given, FOLP. This logic plays a fundamental role in providing an arithmetic semantics for first-order intuitionistic logic, as was shown. In particular, the tech report proved an arithmetic completeness theorem, and a realization theorem for FOLP. In this paper we provide a possible-world semantics for FOLP, based on the propositional semantics of Fitting (2005) [5]. We also give an Mkrtychev semantics. Motivation and intuition for FOLP can be found in Artemov and Yavorskaya (Sidon) (2011) [4], and are not fully discussed here.     

Extended Curry‐Howard terms for second‐order logic

In order to allow the use of axioms in a second‐order system of extracting programs from proofs, we define constant terms, a form of Curry‐Howard terms, whose types are intended to correspond to those axioms. We also define new reduction rules for these new terms so that all consequences of the axioms can be represented. We finally show that the extended Curry‐Howard terms are strongly normalizable.     

Glivenko theorems revisited

Glivenko-type theorems for substructural logics (over FL) are comprehensively studied in the paper [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384]. Arguments used there are fully algebraic, and based on the fact that all substructural logics are algebraizable (see [N. Galatos, H. Ono, Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica 83 (2006) 279-308] and also [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated Lattices: An Algebraic Glimpse at Substructural Logics, in: Studies in Logic and the Foundations of Mathematics, vol. 151, Elsevier, 2007] for the details).As a complementary work to the algebraic approach developed in [N. Galatos, H. Ono, Glivenko theorems for substructural logics over FL, Journal of Symbolic Logic 71 (2006) 1353-1384], we present here a concise, proof-theoretic approach to Glivenko theorems for substructural logics. This will show different features of these two approaches.     

Modal characterisation theorems over special classes of frames

We investigate model theoretic characterisations of the expressive power of modal logics in terms of bisimulation invariance. The paradigmatic result of this kind is van Benthem’s theorem, which says that a first-order formula is invariant under bisimulation if, and only if, it is equivalent to a formula of basic modal logic. The present investigation primarily concerns ramifications for specific classes of structures. We study in particular model classes defined through conditions on the underlying frames, with a focus on frame classes that play a major role in modal correspondence theory and often correspond to typical application domains of modal logics. Classical model theoretic arguments do not apply to many of the most interesting classes-for instance, rooted frames, finite rooted frames, finite transitive frames, well-founded transitive frames, finite equivalence frames-as these are not elementary. Instead we develop and extend the game-based analysis (first-order Ehrenfeucht-Fraïssé versus bisimulation games) over such classes and provide bisimulation preserving model constructions within these classes. Over most of the classes considered, we obtain finite model theory analogues of the classically expected characterisations, with new proofs also for the classical setting. The class of transitive frames is a notable exception, with a marked difference between the classical and the finite model theory of bisimulation invariant first-order properties. Over the class of all finite transitive frames in particular, we find that monadic second-order logic is no more expressive than first-order as far as bisimulation invariant properties are concerned — though both are more expressive here than basic modal logic. We obtain ramifications of the de Jongh-Sambin theorem and a new and specific analogue of the Janin-Walukiewicz characterisation of bisimulation invariant monadic second-order for finite transitive frames.     

The axiomatic power of Kolmogorov complexity

The famous Gödel incompleteness theorem states that for every consistent, recursive, and sufficiently rich formal theory T there exist true statements that are unprovable in T. Such statements would be natural candidates for being added as axioms, but how can we obtain them? One classical (and well studied) approach is to add to some theory T an axiom that claims the consistency of T  . In this paper we discuss another approach motivated by Chaitin's version of Gödel's theorem where axioms claiming the randomness (or incompressibility) of some strings are probabilistically added, and show that it is not really useful, in the sense that this does not help us prove new interesting theorems. This result answers a question recently asked by Lipton. The situation changes if we take into account the size of the proofs: randomly chosen axioms may help making proofs much shorter (unless NP=PSPACE$\mathrm{NP}=\mathrm{PSPACE}$).     

Games with 1-backtracking

We associate with any game G another game, which is a variant of it, and which we call . Winning strategies for have a lower recursive degree than winning strategies for G: if a player has a winning strategy of recursive degree 1 over G, then it has a recursive winning strategy over , and vice versa. Through we can express in algorithmic form, as a recursive winning strategy, many (but not all) common proofs of non-constructive Mathematics, namely exactly the theorems of the sub-classical logic Limit Computable Mathematics (Hayashi (2006) [6], Hayashi and Nakata (2001) [7]).     

Reasoning about proof and knowledge

In previous work [15], we presented a hierarchy of classical modal systems, along with algebraic semantics, for the reasoning about intuitionistic truth, belief and knowledge. Deviating from Gödel's interpretation of IPC in S4, our modal systems contain IPC in the way established in [13]. The modal operator can be viewed as a predicate for intuitionistic truth, i.e. proof. Epistemic principles are partially adopted from Intuitionistic Epistemic Logic IEL [4]. In the present paper, we show that the S5-style systems of our hierarchy correspond to an extended Brouwer–Heyting–Kolmogorov interpretation and are complete w.r.t. a relational semantics based on intuitionistic general frames. In this sense, our S5-style logics are adequate and complete systems for the reasoning about proof combined with belief or knowledge. The proposed relational semantics is a uniform framework in which also IEL can be modeled. Verification-based intuitionistic knowledge formalized in IEL turns out to be a special case of the kind of knowledge described by our S5-style systems.     

Two applications of Boolean models

Semantical arguments, based on the completeness theorem for first-order logic, give elegant proofs of purely syntactical results. For instance, for proving a conservativity theorem between two theories, one shows instead that any model of one theory can be extended to a model of the other theory. This method of proof, because of its use of the completeness theorem, is a priori not valid constructively. We show here how to give similar arguments, valid constructively, by using Boolean models. These models are a slight variation of ordinary first-order models, where truth values are now regular ideals of a given Boolean algebra. Two examples are presented: a simple conservativity result and Herbrand's theorem. Received December 5, 1995     

Typed lambda-calculus in classical Zermelo-Frænkel set theory

system of simple types  , which uses the intuitionistic propositional calculus, with the only connective →. It is very important, because the well known Curry-Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property: every typed term is strongly normalizable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard [4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin [6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare [1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages. There are now many type systems which are based on classical logic; among the best known are the system LC of J.-Y. Girard [5] and the λμ-calculus of M. Parigot [11]. We shall use below a system closely related to the latter, called the λ _c -calculus [8, 9]. Both systems use classical second order logic and have the normalization property. In the sequel, we shall extend the λ _c -calculus to the Zermelo-Fr?nkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF _ɛ . Received: 6 September 1999 / Published online: 25 January 2001     

Unification in linear temporal logic LTL

We prove that a propositional Linear Temporal Logic with Until and Next (LTL) has unitary unification. Moreover, for every unifiable in LTL formula A there is a most general projective unifier, corresponding to some projective formula B, such that A is derivable from B in LTL. On the other hand, it can be shown that not every open and unifiable in LTL formula is projective. We also present an algorithm for constructing a most general unifier.     

Proof complexity of intuitionistic implicational formulas

We study implicational formulas in the context of proof complexity of intuitionistic propositional logic (IPC). On the one hand, we give an efficient transformation of tautologies to implicational tautologies that preserves the lengths of intuitionistic extended Frege (EF) or substitution Frege (SF) proofs up to a polynomial. On the other hand, EF proofs in the implicational fragment of IPC polynomially simulate full intuitionistic logic for implicational tautologies. The results also apply to other fragments of other superintuitionistic logics under certain conditions.In particular, the exponential lower bounds on the length of intuitionistic EF proofs by Hrube? (2007), generalized to exponential separation between EF and SF systems in superintuitionistic logics of unbounded branching by Je?ábek (2009), can be realized by implicational tautologies.