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Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles that clarifies what it means for a formal logic to do plausible reasoning is presented. A new propositional logic, called Propositional Plausible Logic (PPL), is defined and applied to some important examples. PPL is the only non-numeric non-monotonic logic we know of that satisfies all the principles and correctly reasons with all the examples. Some important results about PPL are proved.

In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at http://www.cis.upenn.edu/~giorgi/cl.html and http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .

In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The present contribution gives a detailed exposition of a soundness and completeness proof for an axiomatization of one of the most basic fragments of computability logic. The logical vocabulary of this fragment contains operators for the so called parallel and choice operations, and its atoms represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper "Introduction to computability logic" [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, -- the context that determines the value of the present results. Online introduction to the subject is available at http://www.cis.upenn.edu/~giorgi/cl.html and http://www.csc.villanova.edu/~japaridz/CL/gsoll.html .

Bi-intuitionistic logic is the conservative extension of intuitionistic logic with a connective dual to implication. It is sometimes presented as a symmetric constructive subsystem of classical logic. In this paper, we compare three sequent calculi for bi-intuitionistic propositional logic: (1) a basic standard-style sequent calculus that restricts the premises of implication-right and exclusion-left inferences to be single-conclusion resp. single-assumption and is incomplete without the cut rule, (2) the calculus with nested sequents by Gore et al., where a complete class of cuts is encapsulated into special "unnest" rules and (3) a cut-free labelled sequent calculus derived from the Kripke semantics of the logic. We show that these calculi can be translated into each other and discuss the ineliminable cuts of the standard-style sequent calculus.

This paper explores the use of Propositional Dynamic Logic (PDL) as a suitable formal framework for describing Sign Language (SL), the language of deaf people, in the context of natural language processing. SLs are visual, complete, standalone languages which are just as expressive as oral languages. Signs in SL usually correspond to sequences of highly specific body postures interleaved with movements, which make reference to real world objects, characters or situations. Here we propose a formal representation of SL signs, that will help us with the analysis of automatically-collected hand tracking data from French Sign Language (FSL) video corpora. We further show how such a representation could help us with the design of computer aided SL verification tools, which in turn would bring us closer to the development of an automatic recognition system for these languages.

In this work we suggest the use of a set-theoretical interpretation of semantic tableaux for teaching propositional logic. If the student has previous notions of basic set theory, this approach to semantical tableaux can clarify her the way semantic trees operate, linking the syntactical and semantical sides of the process. Also, it may be useful for the introduction of more advanced topics in logic, like modal logic.

In this work we suggest the use of a set-theoretical interpretation of semantic tableaux for teaching propositional logic. If the student has previous notions of basic set theory, this approach to semantical tableaux can clarify her the way semantic trees operate, linking the syntactical and semantical sides of the process. Also, it may be useful for the introduction of more advanced topics in logic, like modal logic.

The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a canonical form. A computation becomes `stuck' when it reaches the point that it needs to evaluate a proof term that is an application of the univalence axiom. So we wish to find a way to compute with the univalence axiom. While this problem has been solved with the formulation of cubical type theory, where the computations are expressed using a nominal extension of lambda-calculus, it may be interesting to explore alternative solutions, which do not require such an extension. As a first step, we present here a system of propositional higher-order minimal logic (PHOML). There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over $\Omega$. There are proofs which inhabit propositions, which are the terms of type $\Omega$. The canonical propositions are those constructed from $\bot$ by implication $\supset$. Thirdly, there are paths which inhabit equations $M =_A N$, where $M$ and $N$ are terms of type $A$. There are two ways to prove an equality: reflexivity, and propositional extensionality - logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity. We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.

Modal logic is a paradigm for several useful and applicable formal systems in computer science. It generally retains the low complexity of classical propositional logic, but notable exceptions exist in the domains of description, temporal, and spatial logic, where the most expressive formalisms have a very high complexity or are even undecidable. In search of computationally well-behaved fragments, clausal forms and other sub-propositional restrictions of temporal and description logics have been recently studied. This renewed interest on sub-propositional logics, which mainly focus on the complexity of the various fragments, raise natural questions on their the relative expressive power, which we try to answer here for the basic multi-modal logic Kn. We consider the Horn and the Krom restrictions, as well as the combined restriction (known as the core fragment) of modal logic, and, orthogonally, the fragments that emerge by disallowing boxes or diamonds from positive literals. We study the problem in a very general setting, to ease transferring our results to other meaningful cases.

We find a translation with particularly nice properties from intuitionistic propositional logic in countably many variables to intuitionistic propositional logic in two variables. In addition, the existence of a possibly-not-as-nice translation from any countable logic into intuitionistic propositional logic in two variables is shown. The nonexistence of a translation from classical logic into intuitionistic propositional logic which preserves ``and'' and ``or'' but not necessarily ``true'' is proven. These results about translations follow from additional results about embeddings into free Heyting algebras.

Detection and elimination of redundant clauses from propositional formulas in Conjunctive Normal Form (CNF) is a fundamental problem with numerous application domains, including AI, and has been the subject of extensive research. Moreover, a number of recent applications motivated various extensions of this problem. For example, unsatisfiable formulas partitioned into disjoint subsets of clauses (so-called groups) often need to be simplified by removing redundant groups, or may contain redundant variables, rather than clauses. In this report we present a generalized theoretical framework of labelled CNF formulas that unifies various extensions of the redundancy detection and removal problem and allows to derive a number of results that subsume and extend previous work. The follow-up reports contain a number of additional theoretical results and algorithms for various computational problems in the context of the proposed framework.

We introduce BPDL, a combination of propositional dynamic logic PDL with the basic four-valued modal logic BK studied by Odintsov and Wansing (`Modal logics with Belnapian truth values', J. Appl. Non-Class. Log. 20, 279--301 (2010)). We modify the standard arguments based on canonical models and filtration to suit the four-valued context and prove weak completeness and decidability of BPDL.

We give an optimal (EXPTIME), sound and complete tableau-based algorithm for deciding satisfiability for propositional dynamic logic with converse (CPDL) which does not require the use of analytic cut. Our main contribution is a sound methodto combine our previous optimal method for tracking least fix-points in PDL with our previous optimal method for handling converse in the description logic ALCI. The extension is non-trivial as the two methods cannot be combined naively. We give sufficient details to enable an implementation by others. Our OCaml implementation seems to be the first theorem prover for CPDL.

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

In this paper, we present a propositional logic (called mixed logic) containing disjoint copies of minimal, intuitionistic and classical logics. We prove a completeness theorem for this logic with respect to a Kripke semantics. We establish some relations between mixed logic and minimal, intuitionistic and classical logics. We present at the end a sequent calculus version for this logic.

We show a reduction to propositional logic from a Boolean combination of inequalities of the form Vi is greater or equal Vj + C and Vi is less than Vj + C where C is a constant, and Vi, Vj are variables of type real or integer. Equalities and uninterpreted functions can be expressed in this logic as well. We discuss the advantages of using this reduction as compared to competing methods, and present experimental results that support our claims.

Large complexity classes, like the exponential time hierarchy, received little attention in terms of finding complete problems. In this work a generalization of propositional logic is investigated which fills this gap with the introduction of Boolean higher-order quantifiers or equivalently Boolean Skolem functions. This builds on the important results of Wrathall and Stockmeyer regarding complete problems, namely QBF and QBF-k, for the polynomial hierarchy. Furthermore it generalizes the Dependency QBF problem introduced by Peterson, Reif and Azhar which is complete for NEXP, the first level of the exponential hierarchy. Also it turns out that the hardness results do not collapse at the consideration of conjunctive and disjunctive normal forms, in contrast to plain QBF.

Recently, we had to realize that more and more game theoretical articles have been published in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that "if A then B" is valid, they suppose that the assumptions "A and not B" are valid to derive a contradiction in order to deduce "if A then B". Hence, they want to establish the equivalent proposition "A and not B implies A and not A" to conclude that "if A then B" is valid. In fact, they prove that a truth implies a falsehood, which is a wrong statement. As a consequence, "if A then B" is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles.

Recently, we had to realize that more and more game theoretical articles have been published in peer-reviewed journals with severe logical deficiencies. In particular, we observed that the indirect proof was not applied correctly. These authors confuse between statements of propositional logic. They apply an indirect proof while assuming a prerequisite in order to get a contradiction. For instance, to find out that "if A then B" is valid, they suppose that the assumptions "A and not B" are valid to derive a contradiction in order to deduce "if A then B". Hence, they want to establish the equivalent proposition "A and not B implies A and not A" to conclude that "if A then B" is valid. In fact, they prove that a truth implies a falsehood, which is a wrong statement. As a consequence, "if A then B" is invalid, disproving their own results. We present and discuss some selected cases from the literature with severe logical flaws, invalidating the articles.

Defeasible logic is a rule-based nonmonotonic logic, with both strict and defeasible rules, and a priority relation on rules. We show that inference in the propositional form of the logic can be performed in linear time. This contrasts markedly with most other propositional nonmonotonic logics, in which inference is intractable.

We study local-search satisfiability solvers for propositional logic extended with cardinality atoms, that is, expressions that provide explicit ways to model constraints on cardinalities of sets. Adding cardinality atoms to the language of propositional logic facilitates modeling search problems and often results in concise encodings. We propose two ``native'' local-search solvers for theories in the extended language. We also describe techniques to reduce the problem to standard propositional satisfiability and allow us to use off-the-shelf SAT solvers. We study these methods experimentally. Our general finding is that native solvers designed specifically for the extended language perform better than indirect methods relying on SAT solvers.

In this paper, we present the implementation of an automatic Sign Language (SL) sign annotation framework based on a formal logic, the Propositional Dynamic Logic (PDL). Our system relies heavily on the use of a specific variant of PDL, the Propositional Dynamic Logic for Sign Language (PDLSL), which lets us describe SL signs as formulae and corpora videos as labeled transition systems (LTSs). Here, we intend to show how a generic annotation system can be constructed upon these underlying theoretical principles, regardless of the tracking technologies available or the input format of corpora. With this in mind, we generated a development framework that adapts the system to specific use cases. Furthermore, we present some results obtained by our application when adapted to one distinct case, 2D corpora analysis with pre-processed tracking information. We also present some insights on how such a technology can be used to analyze 3D real-time data, captured with a depth device.

In this paper, we aim to initiate the study of enumeration complexity in the field of dependence logics. Consequently, as a first step, we investigate the problem of enumerating all satisfying teams of a given propositional dependence logic formula without the split junction operator. We distinguish between restricting the team size by arbitrary functions and the parametrised version where the parameter is the team size. We show that a polynomial delay can be reached for polynomials and otherwise, in the parametrised setting, we reach FPT delay. However, the constructed enumeration algorithm with polynomial delay requires exponential space. We show that an incremental polynomial delay algorithm exists which uses polynomial space only. Negatively, we show that for the general problem without restricting the team size, an enumeration algorithm running in polynomial space cannot exist.

The interpretation of propositional dynamic logic (PDL) through Kripke models requires the relations constituting the interpreting Kripke model to closely observe the syntax of the modal operators. This poses a significant challenge for an interpretation of PDL through stochastic Kripke models, because the programs' operations do not always have a natural counterpart in the set of stochastic relations. We use rewrite rules for building up an interpretation of PDL. It is shown that each program corresponds to an essentially unique irreducible tree, which in turn is assigned a predicate lifting, serving as the program's interpretation. The paper establishes and studies this interpretation. It discusses the expressivity of probabilistic models for PDL and relates properties like logical and behavioral equivalence or bisimilarity to the corresponding properties of a Kripke model for a closely related non-dynamic logic of the Hennessy-Milner type.

We show that the model checking problem for intuitionistic propositional logic with one variable is complete for logspace-uniform AC1. As basic tool we use the connection between intuitionistic logic and Heyting algebra, and investigate its complexity theoretical aspects. For superintuitionistic logics with one variable, we obtain NC1-completeness for the model checking problem.

We develop a timeout based extension of propositional linear temporal logic (which we call TLTL) to specify timing properties of timeout based models of real time systems. TLTL formulas explicitly refer to a running global clock together with static timing variables as well as a dynamic variable abstracting the timeout behavior. We extend LTL with the capability to express timeout constraints. From the expressiveness view point, TLTL is not comparable with important known clock based real-time logics including TPTL, XCTL, and MTL, i.e., TLTL can specify certain properties, which cannot be specified in these logics (also vice-versa). We define a corresponding timeout tableau for satisfiability checking of the TLTL formulas. Also a model checking algorithm over timeout Kripke structure is presented. Further we prove that the validity checking for such an extended logic remains PSPACE-complete even in the presence of timeout constraints and infinite state models. Under discrete time semantics, with bounded timeout increments, the model-checking problem that if a TLTL-formula holds in a timeout Kripke structure is also PSPACE complete. We further prove that when TLTL is interpreted over discrete time, it can be embedded in the monadic second order logic with time, and when TLTL is interpreted over dense time without the condition of non-zenoness, the resulting logic becomes $\Sigma_1^1$-complete.

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Game semantics aim at describing the interactive behaviour of proofs by interpreting formulas as games on which proofs induce strategies. In this article, we introduce a game semantics for a fragment of first order propositional logic. One of the main difficulties that has to be faced when constructing such semantics is to make them precise by characterizing definable strategies - that is strategies which actually behave like a proof. This characterization is usually done by restricting to the model to strategies satisfying subtle combinatory conditions such as innocence, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task which requires to combine tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of definable strategies by the means of generators and relations: those strategies can be generated from a finite set of ``atomic'' strategies and that the equality between strategies generated in such a way admits a finite axiomatization. These generators satisfy laws which are a variation of bialgebras laws, thus bridging algebra and denotational semantics in a clean and unexpected way.

We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the (n+1)-valued Kripke models and a corresponding language based on a modal extension of {\L}ukasiewicz many-valued logic. We illustrate the definitions by providing a framework for an analysis of the R\'enyi - Ulam searching game with errors. Our main result is the axiomatization of the theory of the (n+1)-valued Kripke models. This result is obtained through filtration of the canonical model of the smallest (n+1)-valued propositional dynamic logic.

We examine a bidirectional propositional dynamic logic (PDL) for finite and infinite message sequence charts (MSCs) extending LTL and TLC-. By this kind of multi-modal logic we can express properties both in the entire future and in the past of an event. Path expressions strengthen the classical until operator of temporal logic. For every formula defining an MSC language, we construct a communicating finite-state machine (CFM) accepting the same language. The CFM obtained has size exponential in the size of the formula. This synthesis problem is solved in full generality, i.e., also for MSCs with unbounded channels. The model checking problem for CFMs and HMSCs turns out to be in PSPACE for existentially bounded MSCs. Finally, we show that, for PDL with intersection, the semantics of a formula cannot be captured by a CFM anymore.

It is known that the set of tautologies of second order intuitionistic propositional logic, $\mathrm{IPC} 2$, is undecidable. Here, we prove that the sets of formulas of $\mathrm{IPC} 2$ which are true in the algebra of open subsets of reals or rationals are decidable.

In this paper we present a transformation of finite propositional default theories into so-called propositional argumentation systems. This transformation allows to characterize all notions of Reiter's default logic in the framework of argumentation systems. As a consequence, computing extensions, or determining wether a given formula belongs to one extension or all extensions can be answered without leaving the field of classical propositional logic. The transformation proposed is linear in the number of defaults.

Logic $L$ was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: $L$ is a conservative extension of CPC and it contains a copy of IPC via the embedding $\varphi\mapsto\square\varphi$. In this article, we consider $L3$, i.e. $L$ augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. The resulting logics combine classical knowledge and belief with intuitionistic truth. Some epistemic laws of Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [1] are reflected by classical modal principles. In particular, the implications "intuitionistic truth $\Rightarrow$ knowledge $\Rightarrow$ classical truth" are represented by the theorems $\square\varphi\rightarrow K\varphi$ and $K\varphi\rightarrow\varphi$ of our logic $EL3$, where we are dealing with classical instead of intuitionistic knowledge. Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1].

Interpolation is an important property of classical and many non classical logics that has been shown to have interesting applications in computer science and AI. Here we study the Interpolation Property for the propositional version of the non-monotonic system of equilibrium logic, establishing weaker or stronger forms of interpolation depending on the precise interpretation of the inference relation. These results also yield a form of interpolation for ground logic programs under the answer sets semantics. For disjunctive logic programs we also study the property of uniform interpolation that is closely related to the concept of variable forgetting.

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.

This paper presents a formal theory which describes propositional binary logic as a semantically closed formal language, and allows for syntactically and semantically well-formed formulae, formal proofs (demonstrability in Hilbertian acception), deduction (Gentzen's view of demonstrability), CNF-ization, and deconstruction to be expressed and tested in the same (computational) formal language, using the same data structure. It is also shown here that Cognitive Binary Logic is a self-described theory in which the Liar Paradox is deconstructed.

\emph{Canonical (logic) programs} (CP) refer to normal logic programs augmented with connective $not\ not$. In this paper we address the question of whether CP are \emph{succinctly incomparable} with \emph{propositional formulas} (PF). Our main result shows that the PARITY problem, which can be polynomially represented in PF but \emph{only} has exponential representations in CP. In other words, PARITY \emph{separates} PF from CP. Simply speaking, this means that exponential size blowup is generally inevitable when translating a set of formulas in PF into an equivalent program in CP (without introducing new variables). Furthermore, since it has been shown by Lifschitz and Razborov that there is also a problem that separates CP from PF (assuming $\mathsf{P}\nsubseteq \mathsf{NC^1/poly}$), it follows that CP and PF are indeed succinctly incomparable. From the view of the theory of computation, the above result may also be considered as the separation of two \emph{models of computation}, i.e., we identify a language in $\mathsf{NC^1/poly}$ which is not in the set of languages computable by polynomial size CP programs.

Performance data for 42 subjects pertaining to seven course objectives and ten propositional logic tasks were analyzed for hierarchical relationships using ordering theoretic technique. The resulting hierarchy indicated that simple implication and particular contraposition propositional logic tasks were prerequisite to some course objectives. This procedure was discussed as being useful in generating hierarchies in subject areas less amenable to this process and of value in the design and implementation of competency based teacher education and mastery learning systems. (Author)

The BHK interpretation interprets propositional statements as descriptions of the world of proofs; a world which is hierarchical in nature. It consists of different layers of the concept of proof; the proofs, the proofs about proofs and so on. To describe this hierarchical world, one approach is the Russellian approach in which we use a typed language to reflect this hierarchical nature in the syntax level. In this case, since the connective responsible for this hierarchical behavior is implication, we will use a typed language equipped with a hierarchy of implications, $\{\rightarrow_n\}_{n=0}^{\infty}$. In fact, using this typed propositional language, we will introduce the hierarchical counterparts of the logics $\mathbf{BPC}$, $\mathbf{EBPC}$, $\mathbf{IPC}$ and $\mathbf{FPL}$ and then by proving their corresponding soundness-completeness theorems with respect to their natural BHK interpretations, we will show how these different logics describe different worlds of proofs embodying different hierarchical behaviors.

Propositional Projection Temporal Logic (PPTL) is a useful formalism for reasoning about period of time in hardware and software systems and can handle both sequential and parallel compositions. In this paper, based on discrete time Markov chains, we investigate the probabilistic model checking approach for PPTL towards verifying arbitrary linear-time properties. We first define a normal form graph, denoted by NFG_inf, to capture the infinite paths of PPTL formulas. Then we present an algorithm to generate the NFG_inf. Since discrete-time Markov chains are the deterministic probabilistic models, we further give an algorithm to determinize and minimize the nondeterministic NFG_inf following the Safra's construction.

The logic of equality with uninterpreted functions (EUF) provides a means of abstracting the manipulation of data by a processor when verifying the correctness of its control logic. By reducing formulas in this logic to propositional formulas, we can apply Boolean methods such as Ordered Binary Decision Diagrams (BDDs) and Boolean satisfiability checkers to perform the verification. We can exploit characteristics of the formulas describing the verification conditions to greatly simplify the propositional formulas generated. We identify a class of terms we call "p-terms" for which equality comparisons can only be used in monotonically positive formulas. By applying suitable abstractions to the hardware model, we can express the functionality of data values and instruction addresses flowing through an instruction pipeline with p-terms. A decision procedure can exploit the restricted uses of p-terms by considering only "maximally diverse" interpretations of the associated function symbols, where every function application yields a different value except when constrained by functional consistency. We present two methods to translate formulas in EUF into propositional logic. The first interprets the formula over a domain of fixed-length bit vectors and uses vectors of propositional variables to encode domain variables. The second generates formulas encoding the conditions under which pairs of terms have equal valuations, introducing propositional variables to encode the equality relations between pairs of terms. Both of these approaches can exploit maximal diversity to greatly reduce the number of propositional variables that need to be introduced and to reduce the overall formula sizes. We present experimental results demonstrating the efficiency of this approach when verifying pipelined processors using the method proposed by Burch and Dill.

In [17], we introduced a modal logic, called $L$, which combines intuitionistic propositional logic $IPC$ and classical propositional logic $CPC$ and is complete w.r.t. an algebraic semantics. However, $L$ seems to be too weak for Kripke-style semantics. In this paper, we add positive and negative introspection and show that the resulting logic $L5$ has a Kripke semantics. For intermediate logics $I$, we consider the parametrized versions $L5(I)$ of $L5$ where $IPC$ is replaced by $I$. $L5(I)$ can be seen as a classical modal logic for the reasoning about truth in $I$. From our results, we derive a simple method for determining algebraic and Kripke semantics for some specific intermediate logics. We discuss some examples which are of interest for Computer Science, namely the Logic of Here-and-There, G\"odel-Dummett Logic and Jankov Logic. Our method provides new proofs of completeness theorems due to Hosoi, Dummett/Horn and Jankov, respectively.

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning.

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \forall p, \exists p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed. Kremer (1997) has shown that the quantified propositional intuitionistic logic H\pi+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable. It is shown that it is, in fact, decidable. The methods used can also be used to establish the decidability of modal S4 with propositional quantification on similar types of Kripke structures.

The logic of equality with uninterpreted functions (EUF) provides a means of abstracting the manipulation of data by a processor when verifying the correctness of its control logic. By reducing formulas in this logic to propositional formulas, we can apply Boolean methods such as Ordered Binary Decision Diagrams (BDDs) and Boolean satisfiability checkers to perform the verification. We can exploit characteristics of the formulas describing the verification conditions to greatly simplify the propositional formulas generated. In particular, we exploit the property that many equations appear only in positive form. We can therefore reduce the set of interpretations of the function symbols that must be considered to prove that a formula is universally valid to those that are ``maximally diverse.'' We present experimental results demonstrating the efficiency of this approach when verifying pipelined processors using the method proposed by Burch and Dill.

Isomorphism between formulae is defined with respect to categories formalizing equality of deductions in classical propositional logic and in the multiplicative fragment of classical linear propositional logic caught by proof nets. This equality is motivated by generality of deductions. Characterizations are given for pairs of isomorphic formulae, which lead to decision procedures for this isomorphism.

We present a hierarchical framework for analysing propositional linear-time temporal logic (PTL) to obtain standard results such as a small model property, decision procedures and axiomatic completeness. Both finite time and infinite time are considered and one consequent benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with both the operator Until and past time naturally reduces to that for PTL without either one. Our method utilises a low-level normal form for PTL called a "transition configuration". In addition, we employ reasoning about intervals of time. Besides being hierarchical and interval-based, the approach differs from other analyses of PTL typically based on sets of formulas and sequences of such sets. Instead we describe models using time intervals represented as finite and infinite sequences of states. The analysis relates larger intervals with smaller ones. Steps involved are expressed in Propositional Interval Temporal Logic (PITL) which is better suited than PTL for sequentially combining and decomposing formulas. Consequently, we can articulate issues in PTL model construction of equal relevance in more conventional analyses but normally only considered at the metalevel. We also describe a decision procedure based on Binary Decision Diagrams.

Both propositional dependence logic and inquisitive logic are expressively complete. As a consequence, every formula with intuitionistic disjunction or intuitionistic implication can be translated equivalently into a formula in the language of propositional dependence logic without these two connectives. We show that although such a (non-compositional) translation exists, neither intuitionistic disjunction nor intuitionistic implication is uniformly definable in propositional dependence logic.

We report complexity results about redundancy of formulae in 2CNF form. We first consider the problem of checking redundancy and show some algorithms that are slightly better than the trivial one. We then analyze problems related to finding irredundant equivalent subsets (I.E.S.) of a given set. The concept of cyclicity proved to be relevant to the complexity of these problems. Some results about Horn formulae are also shown.

This paper presents a soundness and completeness proof for propositional intuitionistic calculus with respect to the semantics of computability logic. The latter interprets formulas as interactive computational problems, formalized as games between a machine and its environment. Intuitionistic implication is understood as algorithmic reduction in the weakest possible -- and hence most natural -- sense, disjunction and conjunction as deterministic-choice combinations of problems (disjunction = machine's choice, conjunction = environment's choice), and "absurd" as a computational problem of universal strength. See http://www.cis.upenn.edu/~giorgi/cl.html for a comprehensive online source on computability logic.