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.     

Formalizing process algebraic verifications in the calculus of constructions

This paper reports on the first steps towards the formal verification of correctness proofs of real-life protocols in process algebra. We show that such proofs can be verified, and partly constructed, by a general purpose proof checker. The process algebra we use isCRL, ACP augmented with data, which is expressive enough for the specification of real-life protocols. The proof checker we use is Coq, which is based on the Calculus of Constructions, an extension of simply typed lambda calculus. The focus is on the translation of the proof theory ofCRL andCRL-specifications to Coq. As a case study, we verified the Alternating Bit Protocol.     

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.     

Lower bounds for the polynomial calculus and the Gröbner basis algorithm

Razborov (1996) recently proved that polynomial calculus proofs of the pigeonhole principle must have degree at least ⌈n/2⌉ + 1 over any field. We present a simplified proof of the same result.?Furthermore, we show a matching upper bound on polynomial calculus proofs of the pigeonhole principle for any field of suficiently large characteristic, and an ⌈n/2⌉ + 1 lower bound for any subset sum problem over the field of reals.?We show that these degree lower bounds also translate into lower bounds on the number of monomials in any polynomial calculus proof, and hence on the running time of most implementations of the Gr?bner basis algorithm. Received: October 14, 1997.     

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.     

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.     

Coquand's calculus of constructions: A mathematical foundation for a proof development system

The calculus of constructions of Coquand, which is a version of higher order typed-calculus based on the dependent function type, is considered from the perspective of its use as the mathematical foundation for a proof development system. The paper considers formulations of the calculus, the underlying consistency of the formalism (i.e., the strong normalisation theorem), and the proof theory of adding assumptions for notions from logic and set theory. Proofs are not given, but references to them are.A preliminary version of this paper was presented at the Third Banff Higher Order Workshop, 23–28 September 1989.     

Uniqueness of Normal Proofs in Implicational Intuitionistic Logic

A minimal theorem in a logic L is an L-theorem which is not a non-trivial substitution instance of another L-theorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique -normal proof in NJ whenever A is provable without non-prime contraction. The non-prime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komori's problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique -normal proof in NJ.     

Unnatural Language Processing

Computer-based logic proofs are a form of unnatural language in which the process and structure of proof generation can be observed in considerable detail. We have been studying how students respond to multimodal logic teaching, and performance measures have already indicated that students' pre-existing cognitive styles have a significant impact on teaching outcome. Furthermore, a large corpus of proofs has been gathered via automatic logging of proof development. This paper applies a series of techniques, including corpus statistical methods, to the proof logs. The results indicate that students' cognitive styles influence the structure of their logical discourse, via their differing methods of handling abstract information in diagrams, and transferring information between modalities.     

Defaults as restrictions on classical Hilbert-style proofs

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hubert-style proofs of a given logicL. We propose to describe default logic as a logic where the juxtaposition of default proofs is subordinate to a restriction condition . Hence a default logic is a pair (L, ) where properties of the logic , like compactness, can be interpreted through the restriction condition . Different default systems are then given a common characterization through a specific condition on the logicL. We also prove cumulativity for any default logic (L, ) by slightly modifying the notion of default proof. We extend, in fact, the language ofL in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on and called linear logics, which lie between Reiter's and ukaszewicz' versions of default logic.Work carried out in the framework of the agreement between Italian PT Administration and FUBLaforia, Université Paris VI Pierre et Marie Curie, 4 Place Jussieu,Tour 46, 75252 Paris, France     

Formalization of the Resolution Calculus for First-Order Logic

I present a formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs. To prove the calculus sound, I use the substitution lemma, and to prove it complete, I use Herbrand interpretations and semantic trees. The correspondence between unsatisfiable sets of clauses and finite semantic trees is formalized in Herbrand’s theorem. I discuss the difficulties that I had formalizing proofs of the lifting lemma found in the literature, and I formalize a correct proof. The completeness proof is by induction on the size of a finite semantic tree. Throughout the paper I emphasize details that are often glossed over in paper proofs. I give a thorough overview of formalizations of first-order logic found in the literature. The formalization of resolution is part of the IsaFoL project, which is an effort to formalize logics in Isabelle/HOL.     

Conquering the Meredith Single Axiom

For more than three and one-half decades, beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems studied range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for two-valued sentential (or propositional) calculus and whose conclusion is the ukasiewicz three-axiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps contains a term of the form n(n(t)) for any term t. As evidence of the power of the new methodology, the article also discusses OTTER's success in obtaining the first known proof of a theorem concerning a single axiom of ukasiewicz.     

Short proofs for tricky formulas

Summary The object of this paper is to demonstrate how certain tricky mathematical arguments can be encoded as short formal proofs for the propositional tautologies representing the mathematical statements. Using resolution as a base proof system for the propositional calculus, we exhibit these short proofs under resolution augmented by one of two principles: the principle of extension, originally suggested by Tseitin, and the principle of symmetry, introduced in this paper. These short proofs illustrate the power of extension and symmetry in theorem proving.The principle of extension allows the introduction of auxiliary variables to represent intermediate formulas so that the length of a proof can be significantly reduced by manipulating these variables instead of the formulas that they stand for. Symmetry, on the other hand, allows one to recognize that a tautology remains invariant under certain permutations of variable names, and use that information to avoid repeated independent derivations of intermediate formulas that are merely permutational variants of one another.First we show that a number of inductive arguments can be encoded as short formal proofs using either extension or symmetry. We provide the details for the tautologies derived by encoding the statement, An acyclic digraph on n vertices must have a source. We then consider the familiar checkerboard puzzle which asserts that a checkerboard, two of whose diagonally opposite corner squares are removed, cannot be perfectly covered with dominoes. We demonstrate short proofs for the tautologies derived from the above assertion, using extension to mimic the tricky informal argument. Finally, we consider statements asserting the Ramsey property of numbers much larger than the critical Ramsey numbers. We show that the proof of Ramsey's theorem can be imitated using the principle of symmetry to yield short proofs for these tautologies.The main theme of the paper is that both extension and symmetry are very powerful augmentations to resolution. We leave open whether either extension or symmetry can polynomially simulate the other.This work was performed while the author was with the General Electric Research Center     

A Legacy Recalled and a Tradition Continued

Since the first presentation of classical sentential logic as an axiomatic system by Frege in 1879, the study of a variety of sentential calculi has flourished. One major area of investigation, initiated by ukasiewicz and his colleagues in the first half of the twentieth century and carried into the second half by Meredith, Thomas, Prior, et al., focuses on alternative axiom sets for such logics, and on formal proofs within them. This paper recalls a sampling of the results obtained heretofore, noting along the way how the papers in this special issue of the Journal of Automated Reasoning fit into that larger tradition of which they now form a part. It also suggests a number of further questions, open problems, and projects to which the methods developed in these papers seem ideally suited and might well be fruitfully applied.     

Practical Reflection for Sequent Logics

It is well-known that adding reflective reasoning can tremendously increase the power of a proof assistant. In order for this theoretical increase of power to become accessible to users in practice, the proof assistant needs to provide a great deal of infrastructure to support reflective reasoning. In this paper we explore the problem of creating a practical implementation of such a support layer.Our implementation takes a specification of a logical theory (which is identical to how it would be specified if we were simply going to reason within this logical theory, instead of reflecting it) and automatically generates the necessary definitions, lemmas, and proofs that are needed to enable the reflected meta-reasoning in the provided theory.One of the key features of our approach is that the structure of a logic is preserved when it is reflected. In particular, all variables, including meta-variables, are preserved in the reflected representation. This also allows the preservation of proof automation—there is a structure-preserving one-to-one map from proof steps in the original logic to proof step in the reflected logic.To enable reasoning about terms with sequent context variables, we develop a principle for context induction, called teleportation.This work is fully implemented in the MetaPRL theorem prover.     

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.     

Solution Validation and Extraction for QBF Preprocessing

In the context of reasoning on quantified Boolean formulas (QBFs), the extension of propositional logic with existential and universal quantifiers, it is beneficial to use preprocessing for solving QBF encodings of real-world problems. Preprocessing applies rewriting techniques that preserve (satisfiability) equivalence and that do not destroy the formula’s CNF structure. In many cases, preprocessing either solves a formula directly or modifies it such that information helpful to solvers becomes better visible and irrelevant information is hidden. The application of a preprocessor, however, prevented the extraction of proofs for the original formula in the past. Such proofs are required to independently validate the correctness of the preprocessor’s rewritings and the solver’s result as well as for the extraction of solutions in terms of Skolem functions. In particular for the important technique of universal expansion efficient proof checking and solution generation was not possible so far. This article presents a unified proof system with three simple rules based on quantified resolution asymmetric tautology (\(\mathsf {QRAT}\)). In combination with an extended version of universal reduction, we use this proof system to efficiently express current preprocessing techniques including universal expansion. Further, we develop an approach for extracting Skolem functions. We equip the preprocessor bloqqer with \(\mathsf {QRAT}\) proof logging and provide a proof checker for \(\mathsf {QRAT}\) proofs which is also able to extract Skolem functions.     

The foundation of a generic theorem prover

Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a meta-logic (or logical framework) in which the object-logics are formalized. Isabelle is now based on higher-order logic-a precise and well-understood foundation.Examples illustrate the use of this meta-logic to formalize logics and proofs. Axioms for first-order logic are shown to be sound and complete. Backwards proof is formalized by meta-reasoning about object-level entailment.Higher-order logic has several practical advantages over other meta-logics. Many proof techniques are known, such as Huet's higher-order unification procedure.     

Lifting lower bounds for tree-like proofs

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and “lift” it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I Δ₀(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy \({G_i^*}\) of quantified propositional proof systems does not collapse.     

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.