A modal logic internalizing normal proofs

In the proof-theoretic study of logic, the notion of normal proof has been understood and investigated as a metalogical property. Usually we formulate a system of logic, identify a class of proofs as normal proofs, and show that every proof in the system reduces to a corresponding normal proof. This paper develops a system of modal logic that is capable of expressing the notion of normal proof within the system itself, thereby making normal proofs an inherent property of the logic. Using a modality △ to express the existence of a normal proof, the system provides a means for both recognizing and manipulating its own normal proofs. We develop the system as a sequent calculus with the implication connective ⊃ and the modality △, and prove the cut elimination theorem. From the sequent calculus, we derive two equivalent natural deduction systems.     

A Matrix Characterization for Multiplicative Exponential Linear Logic

We develop a matrix characterization of logical validity in MELL, the multiplicative fragment of propositional linear logic with exponentials and constants. To prove the correctness and completeness of our characterization, we use a purely proof-theoretical justification rather than semantical arguments. Our characterization is based on concepts similar to matrix characterizations proposed by Wallen for other nonclassical logics. It provides a foundation for developing proof search procedures for MELL by adopting techniques that are based on these concepts and also makes it possible to adopt algorithms that translate the machine-found proofs back into the usual sequent calculus for MELL.     

Contraction-Free Linear Depth Sequent Calculi for Intuitionistic Propositional Logic with the Subformula Property and Minimal Depth Counter-Models

In this paper we present LSJ, a contraction-free sequent calculus for Intuitionistic propositional logic whose proofs are linearly bounded in the length of the formula to be proved and satisfy the subformula property. We also introduce a sequent calculus RJ for intuitionistic unprovability with the same properties of LSJ. We show that from a refutation of RJ of a sequent σ we can extract a Kripke counter-model for σ. Finally, we provide a procedure that given a sequent σ returns either a proof of σ in LSJ or a refutation in RJ such that the extracted counter-model is of minimal depth.     

Eliminability of cut in hypersequent calculi for some modal logics of linear frames

Hypersequent calculi, introduced independently by Pottinger and Avron, provide a powerful generalization of ordinary sequent calculi. In the paper we present a proof of eliminability of cut in hypersequent calculi for three modal logics of linear frames: K4.3, KD4.3 and S4.3. Our cut-free calculus is based on Avron's HC formalization for Gödel–Dummett's logic. The presented proof of eliminability of cut is purely syntactical and based on Ciabattoni, Metcalfe, Montagna's proof of eliminability of cut for hypersequent calculi for some fuzzy logics with modalities.     

Cut-free sequent systems for temporal logic

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakening-free and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.     

A Framework for Proof Systems

Linear logic can be used as a meta-logic to specify a range of object-level proof systems. In particular, we show that by providing different polarizations within a focused proof system for linear logic, one can account for natural deduction (normal and non-normal), sequent proofs (with and without cut), and tableaux proofs. Armed with just a few, simple variations to the linear logic encodings, more proof systems can be accommodated, including proof system using generalized elimination and generalized introduction rules. In general, most of these proof systems are developed for both classical and intuitionistic logics. By using simple results about linear logic, we can also give simple and modular proofs of the soundness and relative completeness of all the proof systems we consider.     

Proof-Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification

We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid E-unification. We show that the problem of existence of a sequent proof with a given skeleton is polynomial-time equivalent to simultaneous rigid E-unifiability. This gives us a proof procedure for intuitionistic logic with equality modulo simultaneous rigid E-unification. We also show that simultaneous rigid E-unifiability is polynomial-time reducible to intuitionistic logic with equality. Thus, any proof procedure for intuitionistic logic with equality can be considered as a procedure for simultaneous rigid E-unifiability. In turn, any procedure for simultaneous rigid E-unifiability gives a procedure for establishing provability in intuitionistic logic with equality.     

Proof Search in Intuitionistic Logic with Equality, or Back to Simultaneous Rigid E-Unification

We characterize provability in intuitionistic logic with equality in terms of a constraint calculus. This characterization uncovers close connections between provability in intuitionistic logic with equality and solutions to simultaneous rigid E-unification. We show that the problem of existence of a sequent proof with a given skeleton is polynomial-time equivalent to simultaneous rigid E-unifiability. This gives us a proof procedure for intuitionistic logic with equality modulo simultaneous rigid E-unification. We also show that simultaneous rigid E-unifiability is polynomial-time reducible to intuitionistic logic with equality. Thus, any proof procedure for intuitionistic logic with equality can be considered as a procedure for simultaneous rigid E-unifiability. In turn, any procedure for simultaneous rigid E-unifiability gives a procedure for establishing provability in intuitionistic logic with equality.     

The <Emphasis Type="Italic">λ</Emphasis>-Calculus and the Unity of Structural Proof Theory

In the context of intuitionistic implicational logic, we achieve a perfect correspondence (technically an isomorphism) between sequent calculus and natural deduction, based on perfect correspondences between left-introduction and elimination, cut and substitution, and cut-elimination and normalisation. This requires an enlarged system of natural deduction that refines von Plato’s calculus. It is a calculus with modus ponens and primitive substitution; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. Both sequent calculus and natural deduction are presented as typing systems for appropriate extensions of the λ-calculus. The whole difference between the two calculi is reduced to the associativity of applicative terms (sequent calculus = right associative, natural deduction = left associative), and in fact the achieved isomorphism may be described as the mere inversion of that associativity. The novel natural deduction system is a “multiary” calculus, because “applicative terms” may exhibit a list of several arguments. But the combination of “multiarity” and left-associativity seems simply wrong, leading necessarily to non-local reduction rules (reason: normalisation, like cut-elimination, acts at the head of applicative terms, but natural deduction focuses at the tail of such terms). A solution is to extend natural deduction even further to a calculus that unifies sequent calculus and natural deduction, based on the unification of cut and substitution. In the unified calculus, a sequent term behaves like in the sequent calculus, whereas the reduction steps of a natural deduction term are interleaved with explicit steps for bringing heads to focus. A variant of the calculus has the symmetric role of improving sequent calculus in dealing with tail-active permutative conversions.     

Constructive linear-time temporal logic: Proof systems and Kripke semantics

In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry-Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination. Moreover, we also consider Kripke semantics and prove soundness and completeness. One distinguishing feature of this logic is that distributivity of the “next” operator over disjunction “?(A∨B)⊃?A∨?B” is rejected in view of a type-theoretic interpretation.     

Sequent calculi and decidability for intuitionistic hybrid logic

In this paper we study the proof theory of the first constructive version of hybrid logic called Intuitionistic Hybrid Logic (IHL) in order to prove its decidability. In this perspective we propose a sequent-style natural deduction system and then the first sequent calculus for this logic. We prove its main properties like soundness, completeness and also the cut-elimination property. Finally we provide, from our calculus, the first decision procedure for IHL and then prove its decidability.     

Theorem Proving Modulo

Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of general interest because it permits one to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof-theoretic account of the combination of computations and deductions. The congruence on propositions is handled through rewrite rules and equational axioms. Rewrite rules apply to terms but also directly to atomic propositions. The second contribution is to give a complete proof search method, called extended narrowing and resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory in deduction modulo. Applying the ENAR method to this presentation of higher-order logic subsumes full higher-order resolution. This revised version was published online in August 2006 with corrections to the Cover Date.     

A modular and parameterized presentation of pregroup calculus

The concept of pregroup was introduced by Lambek for natural language analysis, with a close link to non-commutative linear logic. We reformulate the pregroup calculus so as to extend it by composition with other logics and calculi. The cut elimination property and the decidability property of the sequent calculus proposed in the article are shown. Properties of composed calculi are also discussed.     

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

Craig interpolation has become a versatile tool in formal verification, used for instance to generate program assertions that serve as candidates for loop invariants. In this paper, we consider Craig interpolation for quantifier-free Presburger arithmetic (QFPA). Until recently, quantifier elimination was the only available interpolation method for this theory, which is, however, known to be potentially costly and inflexible. We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants. We prove our calculus to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks. The results document the robustness and efficiency of our interpolation procedure. Finally, we compare the procedure against alternative interpolation methods, both for QFPA and linear rational arithmetic.     

Theorem Prover for Intuitionistic Logic Based on the Inverse Method

The first-order intuitionistic logic is a formal theory from the family of constructive logics. In intuitionistic logic, it is possible to extract a particular example x = a and a proof of a formula P(a) from a proof of a formula ?xP(x). Owing to this feature, intuitionistic logic has many applications in mathematics and computer science. Many modern proof assistants include automated tactics for the first-order intuitionistic logic, which simplify the task of solving challenging problems, such as formal verification of software, hardware, and protocols. In this paper, a new theorem prover (called WhaleProver) for full first-order intuitionistic logic is presented. Testing on the ILTP benchmarking library has shown that WhaleProver performance is comparable with the state-of-the-art intuitionistic provers. Our prover has solved more than 800 problems from the ILTP version 1.1.2. Some of them are intractable for other provers. WhaleProver is based on the inverse method proposed by S.Yu. Maslov. We introduce an intuitionistic inverse method calculus which is, in turn, a special kind of sequent calculus. It is also described how to adopt for this calculus several existing proof search strategies proposed for different logical calculi by S.Yu. Maslov, V.P. Orevkov, A.A. Voronkov, and others. In addition, a new proof search strategy is proposed that allows one to avoid redundant inferences. The paper includes results of experiments with WhaleProver on the ILTP library. We believe that Whale- Prover can be used as a test bench for different inference procedures and strategies, as well as for educational purposes.     

Resolution with Order and Selection for Hybrid Logics

We investigate labeled resolution calculi for hybrid logics with inference rules restricted via selection functions and orders. We start by providing a sound and refutationally complete calculus for the hybrid logic H(@,ˉ,A)\mathcal{H}(@,{\downarrow},\mathsf{A}), even under restrictions by selection functions and orders. Then, by imposing further restrictions in the original calculus, we develop a sound, complete and terminating calculus for the H(@)\mathcal{H}(@) sublanguage. The proof scheme we use to show refutational completeness of these calculi is an adaptation of a standard completeness proof for saturation-based calculi for first-order logic that guarantees completeness even under redundancy elimination. In fact, one of the contributions of this article is to show that the general framework of saturation-based proving for first-order logic with equality can be naturally adapted to saturation-based calculi for other languages, in particular modal and hybrid logics.     

On BAN logic and hash functions or: how an unjustified inference rule causes problems

BAN logic, an epistemic logic for analyzing security protocols, contains an unjustifiable inference rule. The inference rule assumes that possession of H(X) (i.e., the cryptographic hash value of X) counts as a proof of possession of X, which is not the case. As a result, BAN logic exhibits a problematic property, which is similar to unsoundness, but not strictly equivalent to it. We will call this property ‘unsoundness’ (with quotes). The property is demonstrated using a specially crafted protocol, the two parrots protocol. The ‘unsoundness’ is proven using the partial semantics which is given for BAN logic. Because of the questionable character of the semantics of BAN logic, we also provide an alternative proof of ‘unsoundness’ which we consider more important.     

Investigations into a left-structural right-substructural sequent calculus

We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we call ipsilateral: applying such a rule does not make any (sub)formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions.     

On the calculus of positively constructed formulas for automated theorem proving

The paper deals with an expressive logic language LF and its calculus. Formulas of this language consist of some large-scale structural elements, such as type quantifiers. The language LF contains only two logic symbols—∀ and ∃, which form the set of logic connectives of the language. The logic calculus JF and complete strategy for automated proof search based on a single unary rule of inference are considered. This calculus has a number of other features which lead to the reduction of the combinatorial complexity of finding the deductions in comparison to the known systems for automated theorem proving as the Resolution method and Genzen calculus. Problems of effective implementation of JF as a program system for automated theorem proving are considered.     

Evolvable concurrent processes

This paper presents a formal model of concurrent processes whose functions can be modified from the environments during execution. In many applications, it takes too much cost to stop the whole of the concurrent system working in distributed environments to update its components. Features of dynamic evolution that make it possible to modify functions of processes without termination are important. In this paper, a formal model of evolvable concurrent processes is presented using logical formulas of linear logic. The operational semantics is defined using a formal proof in a fragment of the sequent calculus of linear logic. A method for replacing the continuations of working processes by applications of inference rules of linear logic is presented.