Proof Procedure Research Articles

Instantiation Schemes for Nested Theories

This article investigates under which conditions instantiation-based proof procedures can be combined in a nested way, in order to mechanically construct new instantiation procedures for richer theories. Interesting applications in the field of verification are emphasized, particularly for handling extensions of the theory of arrays.

That We See That Some Diagrammatic Proofs Are Perfectly Rigorous

Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of lack of rigor.

Scandinavian bilingual and L2 interaction: A view from afar

Who better to write a concluding commentary on a collection of Scandinavian studies than an Australian living in Japan? While I was honored to be asked, my immediate reaction was, What do I know about Scandinavia? My entire experience with the region to date consists of a week in Sweden and a 2-h stopover in Copenhagen. But the more I thought about it, the more I realized that an outsider could perhaps offer a broadening perspective on the question of how Scandinavian bilingual and second-language users interact; after all, sometimes, you have to stand back in order to get a better view. My own research is based on Japanese bilingual speakers and learners of English, but on reading the articles in this collection, I realized that there is a lot more in common with my situation than I first thought. While this is a collection of articles based in Finnish, Danish, Swedish, and Norwegian contexts, the microcontext of bilingual interaction has plenty of implications for other pairs and for second speakers all over the world.The authors make use of conversation analysis (CA)-an approach that began with the work of Sacks in the United States in the 1960s.1 Although the vast majority of CA research today has been carried out among L1 speakers of English, recent years have seen considerable advances in CA studies in non-English contexts, many of which have been made by Scandinavian researchers (e.g. Cromdal, 2003; Kangasharju, 1996; Kurhila, 2006; Lindstrom, 1994; Steensig, 2000). CA has its roots in sociology, so rather than focusing on per se, it is first and foremost interested in how people accomplish a variety of actions by and through talk. Some of the actions highlighted in the current collection, for example, include reformulations (Svennevig), receipts (Foegtmann), clarifications (Osvaldsson et al.), explanations (Slotte-Luttge et al.), and disagreements (Cekaite and Bjork-Willen).CA's radically emic approach enables analysts to observe talk from the perspective of speakers themselves. Central to the analysis is a holistic, bottom-up approach to looking at naturally occurring interaction, allowing the analyst to document a variety of practices that participants use to achieve intersubjectivity, or mutual understanding, including such features as the organization of sequence, turn-taking, repair, and preference.The CA approach has been used to examine bilingual interaction for over 25 years now, beginning with Peter Auer's groundbreaking work on code-switching (Auer, 1984). This generated a variety of CA investigations into other pairs, including those by Li Wei (1994), Alfonzetti (1998), Sebba and Wooffitt (1998), Bailey (2000, 2002), Torras (2005), Cashman (2005), Cromdal (2001, 2004), and Gafaranga (1999, 2000, 2001, 2005, 2007). Generally speaking, these authors have been interested in the way that alternation can be used to achieve various socio-interactional functions and the part it plays in the discursive display of identity. Justifying their findings through the next-turn proof procedure, CA studies of bilingual interaction have focused on real-time participant displays of understanding, which led Gafaranga and Torras (2002) to respecify code-switching as interactional otherness, suspending the notion of language until a point in the talk when the participants themselves orient to their languages.Applied linguists, on the other hand, have only relatively recently begun to recognize the potential of the CA approach for analyzing second-language talk. Two Denmark-based researchers, Firth and Wagner (1997), called for a reconceptualization of second-language learning-one that dedicates itself to the sociointeractional rather than the cognitive. This article was met with considerable debate, but it has also led to a renewed interest in the use of CA as a methodological approach for documenting second-language learner talk, including a variety of edited collections and special issues (Gardner & Wagner, 2004; Hua, Seedhouse, Wei, & Cook, 2007; Markee & Kasper, 2004; Richards & Seedhouse, 2005) and a recent special issue of The Modern Language Journal that looks at the effect of Firth and Wagners' call 10 years on (Wagner & Firth, 2007). …

Abstract argument games via modal logic

Inspired by some logical considerations, the paper proposes a novel perspective on the use of two-players zero-sum games in abstract argumentation. The paper first introduces a second-order modal logic, within which all main Dung-style semantics are shown to be formalizable, and then studies the model checking game of this logic. The model checking game is then used to provide a systematic game theoretic proof procedure to test membership with respect to all those semantics formalizable in the logic. The paper discusses this idea in detail and illustrates it by providing a game for the so-called skeptical preferred and skeptical semi-stable semantics.

The CHR-based Implementation of the SCIFF Abductive System

Abduction is a form of inference that supports hypothetical reasoning and has been applied to a number of domains, such as diagnosis, planning, protocol verification. Abductive Logic Programming ALP is the integration of abduction in logic programming. Usually, the operational semantics of an ALP language is defined as a proof procedure. The first implementations of ALP proof-procedures were based on the meta-interpretation technique, which is flexible but limits the use of the built-in predicates of logic programming systems. Another, more recent, approach exploits theoretical results on the similarity between abducibles and constraints. With this approach, which bears the advantage of an easy integration with built-in predicates and constraints, Constraint Handling Rules has been the language of choice for the implementation of abductive proof procedures. The first CHR-based implementation mapped integrity constraints directly to CHR rules, which is an efficient solution, but prevents defined predicates from being in the body of integrity constraints and does not allow a sound treatment of negation by default. In this paper, we describe the CHR-based implementation of the SCIFF abductive proof-procedure, which follows a different approach. The SCIFF implementation maps integrity constraints to CHR constraints, and the transitions of the proof-procedure to CHR rules, making it possible to treat default negation, while retaining the other advantages of CHR-based implementations of ALP proof-procedures.

Approximate Reasoning in Fuzzy Resolution

Resolution is an useful tool for mechanical theorem proving in modelling the refutation proof procedure, which is mostly used in constructing a “proof” of a “theorem”. An attempt is made to utilize approximate reasoning methodology in fuzzy resolution. Approximate reasoning is a methodology which can deduce a specific information from general knowledge and specific observation. It is dependent on the form of general knowledge and the corresponding deductive mechanism. In ordinary approximate reasoning, we derive from A→B and by some mechanism. In inverse approximate reasoning, we conclude from A→B and using an altogether different mechanism. An important observation is that similarity is inherent in fuzzy set theory. In approximate reasoning methodology-similarity relation is used in fuzzification while, similarity measure is used in fuzzy inference mechanism. This research proposes that similarity based approximate reasoning-modelling generalised modus ponens/generalised modus tollens—can be used to derive a resolution—like inference pattern in fuzzy logic. The proposal is well-illustrated with artificial examples.

A proof procedure for adaptive logics

In this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuNr is already presented in [7]. In this article a procedure for CLuNm is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes.

A proof-theoretic analysis of the classical propositional matrix method

The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press.

The axially symmetric flow-through problem for the Navier-Stokes equations in variables “vorticity-stream function”

The axially symmetric steady-state motion of a viscous incompressible fluid in a region of the type of a spherical layer is considered. The boundary problem for the Navier-Stokes equations is formulated in terms of the stream function and vorticity. The solvability of this problem was previously established provided that the fluid flow rate through each layer boundary is sufficiently small. The main result of this study is in proving the solvability of the axially symmetric flow-through problem without limitations on the magnitude of the flow rate. The proof is based on the a priori Dirichlet estimate of the flow velocity field. The asymptotic properties of the solution near the symmetry axis are established conditionally. The proof procedure admits the generalization to the case of axially symmetric problems in multiply-connected domains.

On the complexity of entailment in existential conjunctive first-order logic with atomic negation

We consider the entailment problem in the fragment of first-order logic (FOL) composed of existentially closed conjunctions of literals (without functions), denoted by FOL(∃,∧,¬a). This problem can be recast as several fundamental problems in artificial intelligence and databases, namely query containment for conjunctive queries with negation, clause entailment for clauses without functions and query answering with incomplete information for Boolean conjunctive queries with negation over a fact base. Entailment in FOL(∃,∧,¬a) is Π2P-complete, whereas it is only NP-complete when the formulas contain no negation. We investigate the role of specific literals in this complexity increase. These literals have the property of being “exchangeable”, with this notion taking the structure of the formulas into account. To focus on the structure of formulas, we shall see them as labeled graphs. Graph homomorphism, which provides a sound and complete proof procedure for positive formulas, is at the core of this study. Let Entailmentk be the following family of problems: given two formulas g and h in FOL(∃,∧,¬a), such that g has at most k pairs of exchangeable literals, is g entailed by h? The main results are that Entailmentk is NP-complete if k is less or equal to 1, and P‖NP-complete for any value of k greater or equal to 3. As a corollary of our proofs, we are able to classify exactly Entailmentk for any value of k≠2 when g is decomposable into a tree.

Natural Deduction System in Paraconsistent Setting: Proof Search for PCont

This paper continues a systematic approach to build natural deduction calculi and corresponding proof procedures for non-classical logics. Our attention is now paid to the framework of paraconsistent logics. These logics are used, in particular, for reasoning about systems where paradoxes do not lead to the `deductive explosion', i.e., where formulae of the type `A follows from false', for any A, are not valid. We formulate the natural deduction system for the logic PCont, explain its main concepts, define a proof searching technique and illustrate it by examples. The presentation is accompanied by demonstrating the correctness of these developments.

Compositional Shape Analysis by Means of Bi-Abduction

The accurate and efficient treatment of mutable data structures is one of the outstanding problem areas in automatic program verification and analysis. Shape analysis is a form of program analysis that attempts to infer descriptions of the data structures in a program, and to prove that these structures are not misused or corrupted. It is one of the more challenging and expensive forms of program analysis, due to the complexity of aliasing and the need to look arbitrarily deeply into the program heap. This article describes a method of boosting shape analyses by defining a compositional method, where each procedure is analyzed independently of its callers. The analysis algorithm uses a restricted fragment of separation logic, and assigns a collection of Hoare triples to each procedure; the triples provide an over-approximation of data structure usage. Our method brings the usual benefits of compositionality---increased potential to scale, ability to deal with incomplete programs, graceful way to deal with imprecision---to shape analysis, for the first time. The analysis rests on a generalized form of abduction (inference of explanatory hypotheses), which we call bi-abduction . Bi-abduction displays abduction as a kind of inverse to the frame problem: it jointly infers anti-frames (missing portions of state) and frames (portions of state not touched by an operation), and is the basis of a new analysis algorithm. We have implemented our analysis and we report case studies on smaller programs to evaluate the quality of discovered specifications, and larger code bases (e.g., sendmail, an imap server, a Linux distribution) to illustrate the level of automation and scalability that we obtain from our compositional method. This article makes number of specific technical contributions on proof procedures and analysis algorithms, but in a sense its more important contribution is holistic: the explanation and demonstration of how a massive increase in automation is possible using abductive inference.

An analytic tableau calculus for a temporalised belief logic

A tableau is a refutation-based decision procedure for a related logic, and is among the most popular proof procedures for modal logics. In this paper, we present a labelled tableau calculus for a temporalised belief logic called TML +, which is obtained by adding a linear-time temporal logic onto a belief logic by the temporalisation method of Finger and Gabbay. We first establish the soundness and the completeness of the labelled tableau calculus based on the soundness and completeness results of its constituent logics. We then sketch a resolution-type proof procedure that complements the tableau calculus and also propose a model checking algorithm for TML + based on the recent results for model checking procedures for temporalised logics. TML + is suitable for formalising trust and agent beliefs and reasoning about their evolution for agent-based systems. Based on the logic TML +, the proposed labelled tableau calculus could be used for analysis, design and verification of agent-based systems operating in dynamic environments.

Modular instantiation schemes

Instantiation schemes are proof procedures that test the satisfiability of clause sets by instantiating the variables they contain, and testing the satisfiability of the resulting ground set of clauses. Such schemes have been devised for several theories, including fragments of linear arithmetic or theories of data-structures. In this paper we investigate under what conditions instantiation schemes can be combined to solve satisfiability problems in unions of theories.

Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.

Between proof and truth

We consider two versions of truth as grounded in verification procedures: Dummett’s notion of proof as an effective way to establish the truth of a statement and Hintikka’s GTS notion of truth as given by the existence of a winning strategy for the game associated with a statement. Hintikka has argued that the two notions should be effective and that one should thus restrict one’s attention to recursive winning strategies. In the context of arithmetic, we show that the two notions do not coincide: on the one hand, proofs in PA do not yield recursive winning strategies for the associated game; on the other hand, there is no sound and effective proof procedure that captures recursive GTS truths. We then consider a generalized version of Game Theoretical Semantics by introducing games with backward moves. In this setting, a connection is made between proofs and recursive winning strategies. We then apply this distinction between two kinds of verificationist procedures to a recent debate about how we recognize the truth of Godelian sentences.

The Functional Interpretation of Direct Computations

The concept of direct computation used by Statman (1977) was instrumental in the development of a notion of normal form for proofs of equality. In order to find a functional (Curry–Howard style) interpretation of direct computations we take a closer look at proof procedures for first-order sentences with equality drawing the attention to the need for introducing (function) symbols for rewrites. This leads us to a proposal to the effect that the framework of labelled natural deduction gives the right tools to formulate a proof theory for the “logical connective” of propositional equality in the style of the so-called Curry–Howard interpretation. The basic idea is that when analysing an equality sentence into (i) proof conditions (introduction) and (ii) immediate consequences (elimination), it becomes clear that we need to bring in identifiers (i.e. function symbols) for sequences of rewrites, and this is what we claim is the missing entity in P. Martin-Löfʼs equality types (both intensional and extensional). For the proof system for equality we establish a normalisation procedure, proving that it is terminating and confluent.11Part of this material was presented in June 2006 at the Logical Methods in the Humanities Seminar, Stanford University, and the authors would like to thank Solomon Feferman and Grigori Mints for their comments and suggestions. The main results were produced while the first author was Edward Larocque Tinker Visiting Professor at the Department of Philosophy, Stanford Univ, and the second author was a Visiting Scholar at CSLI, Stanford.

Decidability and Undecidability Results for Propositional Schemata

We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions and iterated connectives ranging over intervals parameterized by arithmetic variables. The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called bound-linear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that bound-linear schemata represent a good compromise between expressivity and decidability.

Modelling defeasible reasoning by means of adaptive logic games

In this article, I present a dynamic logic game for defeasible reasoning. I argue that, as far as defeasible reasoning is concerned, one should distinguish between practical and ideal rationality. Starting from the adaptive logic framework, I formalize both rationality notions by means of logic games. The presented adaptive logic games are based on (i) standard logic games on the one hand and (ii) dynamic proof procedures for adaptive logic on the other hand. The games are similar to standard logic games, but have the extra property that some moves are revisable. This is handled by means of a main control game, which starts different standard logic games. I argue that the adaptive logic games form intuitive reasoning models for rationality in defeasible reasoning contexts. Moreover, I will also demonstrate that the games give a good insight in the computational complexity of defeasible reasoning forms.

The generation of abductive explanations from inconsistent theories

In this article I will show how the goal-directed proof procedure for the propositional fragment of the 'inconsistency-adaptive' logic CLuN(r)from Batens (2005, J. Appl. Logic, 3, 221-250) called pCLuN(r) allows for generating sensible abductive explanations from finite inconsistent theories. Before this is shown, I give a more general account on how goal-directed proof procedures contribute to the study of abduction as backward reasoning. Between these two parts, I describe the goal-directed proof procedure for the propositional fragment of the 'inconsistency-adaptive' logic CLuNr from Batens (2005, J. Appl. Logic, 3, 221-250).