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 intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various ways can (...) be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)

This book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate (...) or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic. (shrink)

I discuss a collection of problems in relevance logic. The main problems discussed are: the decidability of the positive semilattice system, decidability of the fragments of R in a restricted number of variables, and the complexity of the decision problem for the implicational fragment of R. Some related problems are discussed along the way.

We show that provability in the implicational fragment of relevance logic is complete for doubly exponential time, using reductions to and from coverability in branching vector addition systems.

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models (...) extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

In this paper, we show by a proof-theoretical argument that in a logic without structural rules, that is in noncommutative linear logic with exponentials, every formula A for which exchange rules (and weakening and contraction as well) are admissible is provably equivalent to ?A. This property shows that the expressive power of "noncommutative exponentials" is much more important than that of "commutative exponentials".

Hereditary structural completeness is established for a range of substructural logics, mainly without the weakening rule, including fragments of various relevant or many-valued logics. Also, structural completeness is disproved for a range of systems, settling some previously open questions.

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives /,,\, together with a package of structural postulates characterizing the resource management properties of the connective.Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms (...) of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system.The simple systems earch have their virtues in linguistic analysis. But none of them in isolation provides a basis for a full theory of grammar. In the second part of the paper, we consider two types of mixed Lambek systems. (shrink)

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and (...) a multimodal version incorporating structural rules and unary modes. (shrink)

We present a sequent calculus for the modal logic S4, and building on some relevant features of this system we show how S4 can easily be translated into full propositional linear logic, extending the Grishin-Ono translation of classical logic into linear logic. The translation introduces linear modalities only in correspondence with S4 modalities. We discuss the complexity of the decision problem for several classes of linear formulas naturally arising from the proposed translations.

An embedding of the implicational propositional intuitionistic logic into the nonmodal fragment of intuitionistic linear logic is given. The embedding preserves cut-free proofs in a proof system that is a variant of IIL. The embedding is efficient and provides an alternative proof of the PSPACE-hardness of IMALL. It exploits several proof-theoretic properties of intuitionistic implication that analyze the use of resources in IIL proofs.

§ 1. Introduction. Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣x∣ for the length of a string x, a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally given by (...) a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P, where ∣y∣ ≤ p. Intuitively, we can think of a string y as a possible proof that x ϵ L, with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L, and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y, where y is presented in a certain way, flips at most O coins and reads at most a constant number of bits of x and y. Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L, there exists y with ∣y∣ ≤ q and with V accepting with probability 1 and, for x ∉ L, there is no y with ∣y∣ ≤ q and with V accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”. (shrink)

§ 1. Introduction. Perhaps the most surprising recent development in complexity theory is the discovery that the class NP can be characterized using a form of randomized proof checker that only examines a constant number of bits of the “proof” that a string is in a language [6, 5, 31, 3, 4]. More specifically, writing ∣x∣ for the length of a string x, a language L in the class NP of languages recognizable in Nondeterministic polynomial time is traditionally given by (...) a polynomial p and a polynomial-time predicate P such that a string x is in L iff there is some string y satisfying P, where ∣y∣ ≤ p. Intuitively, we can think of a string y as a possible proof that x ϵ L, with the predicate P some kind of proof checker that distinguishes good proofs from bad ones. A string x is therefore in a language L ϵ NP if there is a short proof that x ϵ L, and not in L otherwise. The surprising discovery is that the deterministic proof checker that reads the entire input and proof can be replaced by a probabilistic verifier that on input x and possible proof y, where y is presented in a certain way, flips at most O coins and reads at most a constant number of bits of x and y. Obviously, a probabilistic verifier that does not read the whole proof will sometimes make mistakes. However, the surprising and essentially non-intuitive mathematical fact is that for each language L in NP, it is possible to find a polynomial q and verifier V flipping a logarithmic number of coins and reading a constant number of bits such that, for any x ϵ L, there exists y with ∣y∣ ≤ q and with V accepting with probability 1 and, for x ∉ L, there is no y with ∣y∣ ≤ q and with V accepting with probability ≥ 1/4. This idea canalsobeextended to PSPACE [10, 9], using a game that alternates between two players instead of a proof presented by a “single player.”. (shrink)

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. This accounting is made possible by removing the ‘structural’ rules of contraction and weakening, adding a modal operator and adding finer versions of the propositional connectives. Linear logic has fundamental logical interest and applications to computer science, particularly to Petri nets, concurrency, storage allocation, garbage collection and the control structure of logic programs. In addition, there is a direct correspondence between polynomial-time computation (...) and proof normalization in a bounded form of linear logic. In this paper we show that unlike most other propositional logics, full propositional linear logic is undecidable. Further, we prove that without the modal storage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-complete. We also establish membership in NP for the multiplicative fragment, NP-completeness for the multiplicative fragment extended with unrestricted weakening, and undecidability for fragments of noncommutative propositional linear logic. (shrink)

Recently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is (...) proved by means of the phase semantics. (shrink)

Infinitary action logic can be naturally expanded by adding exponential and subexponential modalities from linear logic. In this article we shall develop infinitary action logic with a subexponential that allows multiplexing (instead of contraction). Both non-commutative and commutative versions of this logic will be considered, presented as infinitary sequent calculi. We shall prove cut admissibility for these calculi, and estimate the complexity of the corresponding derivability problems: in both cases it will turn out to be between complete first-order arithmetic and (...) the \(\omega ^\omega \) level of the hyperarithmetical hierarchy. Here the complexity upper bound is much lower than that for the system with a subexponential that allows contraction. The complexity lower bound in turn is much higher than that for infinitary action logic. (shrink)

In this paper the questions remaining open about NP-completeness of multiplicative and Horn fragments of the Linear Logic and the Linear Logic with the weakening rule are answered.

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative connectives, additive conjunction and disjunction, brackets and bracket modalities, and the! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut. We also prove algorithmic undecidability for both (...) calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages. (shrink)

The question at issue is to develop a computational interpretation of Girard's Linear Logic [Girard, 1987] and to obtain efficient decision algorithms for this logic, based on the bottom-up approach. It involves starting with the simplest natural fragment of linear logic and then expanding it step-by-step. We give a complete computational interpretation for the Horn fragment of Linear Logic and some natural generalizations of it enriched by the two additive connectives: and &. Within the framework of this interpretation, it becomes (...) possible to explicitly formalize and clarify the computational aspects of the fragments of Linear Logic in question and establish exactly the complexity level of these fragments. In particular, the simplest natural Horn fragment of Linear Logic is proved to be NP-complete. As a corollary, we obtain the affirmative solution for the problem ): whether the multiplicative fragment of linear logic is NP-complete. (shrink)

Linear Logic was introduced by Girard as a resource-sensitive refinement of classical logic. In this paper we establish strong connections between natural fragments of Linear Logic and a number of basic concepts related to different branches of Computer Science such as Concurrency Theory, Theory of Computations, Horn Programming and Game Theory. In particular, such complete correlations allow us to introduce several new semantics for Linear Logic and to clarify many results on the complexity of natural fragments of Linear Logic. As (...) a main corollary, any non-deterministic and concurrent computation is proved to be simulated directly within the framework of each of these systems. (shrink)

A Linear Logic automaton is a hybrid of a finite automaton and a non-deterministic Petri net. LL automata commands are represented by propositional Horn Linear Logic formulas. Computations performed by LL automata directly correspond to cut-free derivations in Linear Logic.A programming language of LL automata is developed in which typical sequential, non-deterministic and parallel programming constructs are expressed in the natural way.All non-deterministic computations, e.g. computations performed by programs built up of guarded commands in the Dijkstra's approach to non-deterministic programming, (...) are directly simulated within the framework of Linear Logic automata, and thereby within the Horn framework of Linear Logic. (shrink)

The question at issue is to develop a computational interpretation of Linear Logic [8] and to establish exactly its expressive power. We follow the bottom-up approach. This involves starting with the simplest of the systems we are interested in, and then expanding them step-by-step. We begin with the !-Horn fragment of Linear Logic, which uses only positive literals, the linear implication ⊸, the tensor product ⊗, and the modal storage operator !. We give a complete computational interpretation for the !-Horn (...) fragment of Linear Logic and for some natural generalizations of it formed by introducing additive connectives. Here we use the well-known ‘ or ’-like connective ⊕, and, for the sake of the computational duality, we introduce a new ‘ and ’-like connective @ For !-Horn sequents, we prove that their derivability problem is directly equivalent to the reachability problem for Petri nets, which is known to be decidable [19]. For the -Horn fragment of Linear Logic, which uses only positive literals, the linear implication ⊸, the tensor product ⊗, the modal storage operator!, and the additive ‘disjunction’ ⊕, we prove that standard Minsky machines [21] can be directly encoded in this -Horn fragment. Standard Minsky machines can be directly encoded in the corresponding ‘dual’ -Horn fragment of Linear Logic, as well. As a corollary, both these fragments of Linear Logic are proved to be undecidable. (shrink)

To express fine-grained resource-sensitive reasoning, a temporal soft linear logic (TSLL) is introduced as an extension of both Girard's (propositional classical) linear logic (CLL) and Lafont's (propositional classical) soft linear logic (SLL). It is known that the linear exponential operator in CLL can express a specific infinitely reusable resource, i.e. it is reusable not only for any number, but also many times. In contrast, the soft exponential operator in SLL, which is a weak version of the linear exponential operator, can (...) express a specific usable resource, i.e. it is usable in any number, but only once (i.e. it is consumed after use). In TSLL, the resource operators (i.e. linear and soft exponentials) and some temporal operators are combined based on the interpretation that “time” is regarded as a “resource”. The completeness theorem (with respect to phase semantics) for TSLL and the cut-elimination and decidability theorems for some subsystems of TSLL are proved as the main results of this paper. A decidable subsystem, called a bounded soft linear logic (BSLL), which has a restricted soft exponential operator, can represent a specific finitely usable resource. (shrink)

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.

All known structural extensions of the substructural logic $\textbf{FL}_{\textbf{e}}$, the Full Lambek calculus with exchange/commutativity (corresponding to subvarieties of commutative residuated lattices axiomatized by $\{\vee, \cdot, 1\}$ -equations), have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the (...) corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames. (shrink)

Completeness is shown for several versions of Girard's linear logic with respect to Petri nets as the class of models. One logic considered is the -free fragment of intuitionistic linear logic without the exponential !. For this fragment Petri nets form a sound and complete model. The strongest logic considered is intuitionistic linear logic, with ,&, and the exponential ! , and forms of quantification. This logic is shown sound and complete with respect to atomic nets , though only once (...) we add extra axioms specific to the Petri-net model. The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. Unfortunately, with respect to this logic, whether an assertion is true of a finite net becomes undecidable. (shrink)

We present a setting in which the search for a proof of B or a refutation of B can be carried out simultaneously: in contrast, the usual approach in automated deduction views proving B or proving ¬B as two, possibly unrelated, activities. Our approach to proof and refutation is described as a two-player game in which each player follows the same rules. A winning strategy translates to a proof of the formula and a counter-winning strategy translates to a refutation of (...) the formula. The game is described for multiplicative and additive linear logic . A game theoretic treatment of the multiplicative connectives is intricate and our approach to it involves two important ingredients. First, labeled graph structures are used to represent positions in a game and, second, the game playing must deal with the failure of a given player and with an appropriate resumption of play. This latter ingredient accounts for the fact that neither player might win. (shrink)

RASP is a recent extension to Answer Set Programming (ASP) that permits declarative specification and reasoning on the consumption and production of resources. ASP can be seen as a particular case of RASP. In this paper, we study the relationship between linear logic and RASP problem specification. We prove that RASP programs can be translated into (a fragment of) linear logic, and vice versa. In doing so, we introduce a linear logic representation of default negation as understood in ASP. We (...) are also able to establish a link between linear logic and here-and-there (HT) logic. (shrink)

The logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: γ, xn → y / γ, xm → y. It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite model property (...) with respect to its algebraic semantics and hence that the logic is decidable. (shrink)

We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cut-elimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of display-equivalence in our calculi in such a way that an exhaustive proof search need be only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We (...) also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist. (shrink)

In this paper we introduce a new type of nets with bounded types of distributed resources . Linear Logic to describe the behaviour of BR-nets is defined. It is based on Girard's Linear Logic but captures not only consumption of resources but their presence as well. Theorem of soundness and completeness of the proposed axiomatization is proved and the complexity of the provability problem is established for the general case and some particular ones.

In this paper we continue the study of Girard's Linear Logic and introduce a new Linear Logic with modalities. Our logic describes not only the consumption, but also the presence of resources. We introduce a new semantics and a new calculus for this logic. In contrast to the results of Lincoln [7] and Kanovich [4] about the NP-completeness of the problem of the construction of a proof for a given sequent in the multiplicative fragment of Girard's Linear Logic, we present (...) here a non-exponential algorithm to construct a proof for a given sequent and a given point of a given model in our Linear Logic. (shrink)

, from Stanford Encyclopaedia of Philosophy.

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order (...) logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the (...) traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)

The impossibility results in judgement aggregation show a clash between fair aggregation procedures and rational collective outcomes. In this paper, we are interested in analysing the notion of rational outcome by proposing a proof-theoretical understanding of collective rationality. In particular, we use the analysis of proofs and inferences provided by linear logic in order to define a fine-grained notion of group reasoning that allows for studying collective rationality with respect to a number of logics. We analyse the well-known paradoxes in (...) judgement aggregation and we pinpoint the reasoning steps that trigger the inconsistencies. Moreover, we extend the map of possibility and impossibility results in judgement aggregation by discussing the case of substructural logics. In particular, we show that there exist fragments of linear logic for which general possibility results can be obtained. (shrink)