# Computational Logic Seminar

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# Resource Sharing Linear Logic, I

Linear Logic, introduced by Girard ([Gir] 1987), is a kind of resource sensitive logic, since it lacks general forms of structural inferences. However, it restores controlled uses of structural inferences by its exponentials, which behaves like the S4 modality. In the usual resource semantics for Linear Logic, the formulas with exponentials are interpreted to mean the inexhaustibility of a resource. We study an extension of Linear Logic by relaxing the exponentials to the S5 type modality. Then, it is interpreted to mean the inexhaustibility and transferability of a resource. We show some basic properties of this logic: completeness, cut-eliminatbility, decidability. Also, we provide the realizability of our logic via Logic of proofs (aka, Justification Logic), introduced by Artemov [Art1, Art2].

This work is a continuation of our on-going research project with Professor Kurokawa on ’synthesizing substructural logics and logics of proofs’, including [KK], 2013.

References

[Art1] S. N. Artemov, Explicit provability and constructive semantics, Bulletin of Symbolic Logic 7(1), pp.1–36, 2001.

[Art2] S. N. Artemov, The Logic of Justification, Review of Symbolic Logic, 1(4), pp. 477-513, 2008.

[Gir] J. Girard, Linear Logic, Theoretical Computer Science 50, pp. 1–102, 1987.

[KK] H. Kurokawa and H. Kushida, Substructural Logic of Proofs, WoLLIC 2013: Logic, Language, Information, and Computation, LNCS 8071, pp.194-210, 2013.

# Sub-Kripkean epistemic models I

Kripke models provide a versatile semantical tool in modal logic. It appears that when Kripkean methods were adopted in formal epistemology “as is” a key hidden assumption has been overlooked: the very truth condition of a knowledge assertion presupposes that the model itself is common knowledge among the agents. Naturally, this covers only well-controlled epistemic scenarios and leaves off scope numerous situations with partial/asymmetric knowledge. In this talk, we

1) offer a theory of sub-Kripkean epistemic models for unimodal logics;

2) discuss research directions for multi-agent cases.

# Generic logical semantics of justifications III

Proofs and justification are gradually making their way from meta-logical notions into the formal logic itself and becoming mathematical logical objects. This makes the logic language (much) more precise and connects the logic apparatus to numerous new areas of interest.

Part III: Possible worlds with and without modalities. More epistemic examples. Open questions and research projects.

# Generic logical semantics of justifications II

Proofs and justification are gradually making their way from meta-logical notions into the formal logic itself and becoming mathematical logical objects. This makes the logic language (much) more precise and connects the logic apparatus to numerous new areas of interest.

Part I: we describe a generic logical semantics of justifications within the classical truth values logic framework: justifications here appear as sets of formulas with appropriate closure conditions.

Part II: introducing possible worlds.

# Generic logical semantics of justifications.

Proofs and justification are gradually making their way from meta-logical notions into the formal logic itself and becoming mathematical logical objects. This makes the logic language (much) more precise and connects the logic apparatus to numerous new areas of interest. In this talk, we describe a generic logical semantics of justifications within the classical truth values logic framework: justifications here appear as sets of formulas with appropriate closure conditions.

# Cut Elimination

Gentzen proved a famous cut elimination theorem using constructive methods. It is also possible to prove it non-constructively, with a conceptually simpler argument, but a constructive proof is important for a number of reasons. I will present the constructive argument for cut elimination, for classical propositional logic, formulated using tableaus. This is purely expository; there is nothing new. But it is a proof that every logician, or logician-to-be, should know, and I will cover the basics using a detailed set of slides which will be made available afterward, on my web site.

# Kripke completeness of strictly positive modal logics and related problems

We consider the fragment of propositional multi-modal logic that consists of implications whose premise and conclusion are strictly positive (SP) modal formulas constructed from propositional variables using conjunction and unary diamond operators. SP-logics can be interpreted over bounded meet-semilattices with normal monotone operators (SLOs, for short), and over first-order structures (aka Kripke frames). The former interpretation provides natural calculi for SP-logics, while reasoning over the latter can be supported by modern provers and description logic reasoners, and is often tractable. An SP-logic is called (Kripke) complete if it has the same consequences in each of these semantics. This talk reports on our first steps in developing completeness theory for SP-logics, which turns out to be quite different from classical completeness theory for modal logics. We also consider some related problems, in particular, axiomatisability and computational complexity.

# The unreasonable effectiveness of Nonstandard Analysis.

Nonstandard Analysis (NSA) was introduced around 1965 by Robinson as a formalization of the intuitive infinitesimal calculus which is in use to date in most of physics and historically in mathematics until the advent of Weierstrass’ epsilon-delta framework. Famous people like Connes and Bishop have derided NSA for its alleged utter lack of computational/effective/constructive content. In this talk I show that every theorem of ‘pure’ NSA can be (equivalently) converted to a theorem of computable mathematics. In many cases, the resulting theorem is even constructive in the sense of Bishop.

# Epistemic Updates on Algebras

The present talk reports on some recent developments in the mathematical theory of epistemic updates, a very fruitful line of research initiated in [4, 3] and further pursued in [2, 5, 1]. This line of research is motivated by the observations that ‘dynamic phenomena’ are independent of their underlying static logic being classical, and that this assumption is unrealistic in many important contexts; for instance, in all those contexts (such as scientific experiments, acquisition of legal evidence, verification of programs, etc.) where the notion of truth is procedural. Desirable and conceptually important as it is, the more general problem of identifying the right intuitionistic (or more generally nonclassical) counterparts of (modal-like) expansions of classical logic (such as modal logics themselves, or hybrid logics, etc.) has proven to be difficult, and for most of these logics, this question is still open. Indeed, different axiomatizations which in the presence of classical tautologies define the same logic become nonequivalent against a weaker propositional background. Hence, each classical axiomatization might have infinitely many nonequivalent nonclassical potential counterparts. We introduce a uniform methodology for defining the nonclassical counterparts of dynamic logics, which is grounded on semantics rather than on syntax. This methodology is based on the dual characterizations of the transformations of models which interpret the actions. This dual characterization makes it possible to define epistemic updates on algebras, thanks to which an algebraic semantics for (e.g.) the intuitionistic counterpart of Baltag-Moss-Solecki’s dynamic epistemic logic can be defined e.g. on Heyting algebras.

References

[1] Z. Bakhtiari, U. Rivieccio. Epistemic Updates on Bilattices. Proceedings of the Fifth International Workshop on Logic, Rationality and Interaction – LORI, LNCS 9394, pp. 426-8 (2015).

[2] W. Conradie, S. Frittella, A. Palmigiano, A. Tzimoulis, Probabilistic Epistemic Updates on Algebras, Proceedings of the Fifth International Workshop on Logic, Rationality and Interaction – LORI, LNCS 9394, pp. 64-76 (2015).

[3] Alexander Kurz and Alessandra Palmigiano. Epistemic updates on algebras. Logical Methods in Computer Science, 2013. arXiv:1307.0417. 1

[4] Minghui Ma, Alessandra Palmigiano, and Mehrnoosh Sadrzadeh. Algebraic semantics and model completeness for intuitionistic public announcement logic. Annals of Pure and Applied Logic, 165 (2014) 963-995.

[5] U. Rivieccio, Bilattice Public Announcement Logic, Proc. AiML 10 (2014).

# Some abstract versions of Goedel’s Second Incompleteness Theorem based on non-classical logics

We study abstract versions of Goedel’s second incompleteness theorem and formulate generalizations of Loeb’s derivability conditions that work for logics weaker than the classical one. We isolate the role of the contraction and weakening rules in Goedel’s theorem and give a (toy) example of a system based on modal logic without contraction invalidating Goedel’s argument. On the other hand, Goedel’s theorem is established for a version of Peano arithmetic without contraction. (Joint work with Daniyar Shamkanov)

# Dark Matter of Epistemology

We have already shown that only deductively complete epistemic scenarios admit traditional single-model characterizations. In this talk we show that the paradigmatic Muddy Children story is deductively complete and fairly represented by its standard Kripke model. However, the whole variety of its modifications with partial knowledge, asymmetric knowledge, etc., are not deductively complete and hence are invisible “Dark Matter” from the traditional single-model perspective. We show several examples of such “invisible” scenarios that admit natural syntactic analysis and resolution. Finally, we will discuss a version of Muddy Children, in which justifications become a key ingredient of the solution. These examples represent, in a nutshell, the corresponding classes of real world epistemic problems which lie off limits of the traditional single-model analysis but can be analyzed by a proper combination of syntactic and semantic methods.

# Epistemic Model Theory revisited II.

We provide a brief but rigorous review of the model theory for modal epistemic logic. In addition to the classical soundness and completeness results we will focus on features that are normally left hidden: necessitation in epistemic scenarios, internalization property of the model, knowledge of the model, etc. This should provide a solid base for further epistemic logic investigations.

In this talk, we take a fresh look at canonical models of the usual multi-modal systems, and discuss what they can and cannot do in the epistemic settings.

# Gödel/Artemov-Style Analyses for Variants of Intuitionistic Logic

Kurt Gödel’s provability interpretation of intuitionistic logic and Sergei Artemov’s Logic of Proofs (LP) have jointly shed a great deal of light on the inner machinery of intuitionistic logic. In this talk, I will give Gödel/Artemov-style analyses to several variants of intuitionistic logic: Cecylia Rauszer’s Heyting-Brouwer Logic (HB) and David Nelson’s Logics of Constructible Falsity (N3 and N4). Heinrich Wansing has sketched out Gödel-style provability interpretations of these systems by distinguishing between categories of justification, e.g., proof and refutation. I will show that identifying these categories with distinct agents in Tatyana Yavorskaya-Sidon’s two-agent Logic of Proofs LP^2 permits salient Gödel/Artemov-style translations of HB and N3 into extensions of LP^2. Finally, we will discuss the prospects for a similar analysis of N4, considering proposals of Melvin Fitting and Che-Ping Su and a conjecture concerning a translation of N4 into the modal logic S4.

# Towards Paraconsistent Game Theory

In this programmatic talk, I will talk about three major examples of paraconsistency and paradoxes in games. Paraconsistent logic is a family of non-classical logics in which contradictions do not entail everything. In this talk, first, I will discuss game theoretical semantics for paraconsistent logic and observe how semantic games change in different, especially in paraconsistent logics. Then, I will consider a self-referential paradox of epistemic game theory, called Brandenburger-Keisler Paradox, and present a model for it. Following, I will shift my attention non-self-referential paradoxes and suggest a non-self-referential paradox in games. These three major cases, I will argue, will be a call for the necessity of the use of non-classical logics in game theoretical reasoning.

# Epistemic Model Theory revisited.

We provide a brief but rigorous review of the model theory for modal epistemic logic. In addition to the classical soundness and completeness results we will focus on features that are normally left hidden: necessitation in epistemic scenarios, internalization property of the model, knowledge of the model, etc. This should provide a solid base for further epistemic logic investigations.

# Does knowledge of the model matter?

As we have already discussed, the traditional epistemic reading of Kripke models relies on the hidden assumption of the knowledge of the model by the agent. In the multi-agent setting, e.g. in Epistemic Game Theory, this becomes the assumption of the common knowledge of the model, which is not regarded realistic, and often is theoretically impossible. We sketch logician’s recommendations of how to handle epistemic scenarios without relying on common knowledge of the model.

# Justification Epistemic Models: technical details.

We will provide exact definitions and a rigorous formal treatment of justification epistemic models (JEM). The JEM for Russell’s prime minister example will be presented with all details.

# How (Not) To Aggregate Normative Reasons

Justification logic can be used to make sense of deontic modality, where justification terms are interpreted as normative reasons. Unlike proofs or epistemic evidence, which are (generally) factive, normative reasons must be ordered, since arguably one only ought to do just what one has most (more, stronger, etc) reason to do.

I make precise in what sense normative reasons are scalar, why an aggregation operation is needed, and introduce the most common types of scales. I then show that normative reasons cannot be (numerically) measured, and that the scale of normative reasons, if any, is therefore not ratio, interval, or ordinal (in a precise measurement-theoretic sense). I eventually discuss the consequences of these results for normative theorizing, and especially for normative particularism.

# Paraconsistent logic, evidence, and justification

In a forthcoming paper, Walter Carnielli and Abilio Rodriguez propose a Basic Logic of Evidence (BLE) whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodriguez understanding of evidence is informal. We provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which box-X can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit evidence interpretations in a formal sense.

# Evidence-based epistemic models

We will survey the standard modal logic approach to model the “knowledge vs justified true belief” scenarios, in particular its recent account in Williamson, Timothy “A note on Gettier cases in epistemic logic” Philosophical Studies 172.1 (2015): 129-140. We argue that the “old school” approach by Williamson to model justified belief as a modality without going deeper to the level of individual justifications is rather limited. Whereas it indeed suffices to satisfactory treat Gettier examples with a unique justification in the picture, it fails even on simplest examples with more than one justification, e.g., Russell’s Prime Minister Example. On the constructive side, we offer a principled way to building epistemic models entirely from systems of justifications; standard epistemic models are special cases of these evidence-based epistemic models. New models cover wide range of epistemic scenarios, including Russell’s example, in a natural way.

# Justification Logic and Aggregating Probabilistic Evidence

Evidence aggregation is a well-known problem which appears naturally in many areas. A classical approach to this problem is via Bayesian probabilistic evidence aggregation. Our approach is radically different. We consider the following situation: suppose a proposition X logically follows from a database — a set of propositions D which are supported by some known evidence, vector u of events in a probability space. We answer the question of what is the best aggregated evidence for X justified by the given data. We show that such aggregated evidence e(u) could be assembled algorithmically from the collection of all logical derivations of X from D. This approach can handle conflicting and inconsistent data and allows the gathering both positive and negative evidence for the same proposition. The problem is formalized in a version of justification logic and the conclusions are supported by corresponding completeness theorems.

# An Arithmetical Interpretation of Verification and Intuitionistic Knowledge

Intuitionistic epistemic logic, IEL, introduces to intuitionistic logic an epistemic operator which reflects the intended BHK semantics of intuitionism. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the Logic of Proofs, LP, and its arithmetical semantics. We show here that the Gödel embedding, realization, and an arithmetical interpretation can all be extended to S4 and LP extended with a verification modality, thereby providing intuitionistic epistemic logic with an arithmetical semantics too.

# Intuitionistic Analysis of Russell and Gettier Examples

Intuitionistic truth is proof-based, and intuitionistic knowledge is the result of verification. The corresponding logic IEL of intuitionistic knowledge (Artemov and Protopopescu, 2014) has offered a principled resolution of the Fitch-Church knowability paradox. In this talk we will analyze paradigmatic Russell and Gettier examples from intuitionistic epistemic positions, both informally, and formally within proper versions of IEL.

Time permitting, we will discuss the list of possible research projects.

# Epistemic Considerations on Extensive-Form Games

In this talk, we discuss several topics in extensive-form games. First, we consider perfect information games with belief revision with players who are tolerant of each other’s “hypothetical” errors. Second, we study imperfect information games with players who have no way of assigning probabilities to various possible outcomes, and we analyze how players might choose in such games. We then look into how a truthful manipulator whose motives are not known to players can create a certain knowledge situation about a game, and change the way the game will be played.

# Syntactic View of Strategic Games

We suggest an epistemic logic analysis of strategic games with ordinal payoffs, including Nash’s justification of the equilibrium solution concept: the solution is derived by the players from the game description. We show that most of the classes of games do not have Nash solutions, and for those that have, stronger solution concepts (e.g, domination) and/or refinements of Aumann rationality are needed.

# A general completeness theorem relating sequents and bivaluations

In this talk I will present a theorem justifying an intuitive semantical interpretation of rules of sequent proof systems. This theorem is based on the one hand on a generalization of Lindenbuam maximalization theorem and on the other hand on a general semantical theory of bivaluations, not necessarily truth-functional, originally developed by Newton da Costa for paraconsistent systems. I will show that this theorem gives a better understanding of sequent calculus and that it can fruitfully be applied to a wide class of logical systems providing instantaneous completeness results. This is a typical example of work in the line of the universal logic project, of which I will say a few words.

Reference: Logica Universalis, Special Double Issue – Scope of Logic Theorems. Volume 8, Issue 3-4, December 2014.

# Cut-Free Proofs for More and More Logics

Gentzen introduced cut-free proofs in 1934. In these, all formulas are subformulas of what is being proved (and with the same polarity). This is an important feature for many reasons, proof search among them. In 1957 Ohnishi and Matsumoto extended Gentzen’s ideas to several modal logics. Beth introduced tableaus in the 1950’s, with their modern version coming from Smullyan in the 1960’s. Roughly, tableaus are sequent calculi upside-down. I (probably) introduced tableau versions of Ohnishi and Matsumoto’s calculi in the 1970’s. All this treated a small number of modal logics.

Based on an idea of Fitch, I introduced prefixed tableaus in 1972. These add new machinery, but are able to handle many more modal logics. Today they are rather common in automated theorem proving.

In a recent (yet unpublished) paper I introduced set prefixed tableaus. We now have machinery that can handle an infinite family of modal logics, including all the most common ones. And they do this in a uniform way, separating logical rules from so-called structural rules.

The idea that tableaus and sequent calculi are dual to each other continues. Nested sequents have been much studied. In a paper from 2012 I showed that nested sequents and prefixed tableaus were dual, in this sense. And I have a new construct, indexed nested sequents, which is dual to set prefixed tableaus. The pattern continues.

I will cover as much of this as time permits. (My talk is intended to be a trial run for a talk to be given at UNILOG, In Istanbul, this summer.)

# NEXP-completeness and Universal Hardness Results for Justification Logic

We provide a lower complexity bound for the satisfiability problem of a multi-agent justification logic, establishing that there are certain NEXP-complete multi-agent justification logics with interactions. We then use a simple modification of the corresponding reduction to prove that satisfiability for all multi-agent justification logics in a general class we consider is $Sigma_2^p$-hard — given certain reasonable conditions. Our methods improve on these required conditions for the same lower bound for the single-agent justification logics, proven by Buss and Kuznets in 2009, thus answering one of their open questions.

Link to paper on arXiv:

http://arxiv.org/abs/1503.00362

# Bi-simulation of epistemic structures: a semantic proof of completeness for Muddy Children Puzzle.

In this talk we discuss truth preserving bi-simulations of multi-agent Kripke models. As an application, we present a semantic proof of the completeness of the syntactic formulation of Muddy Children Puzzle with respect to its standard model.

# Reasoning about reasoning about games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov’s knowledge based reasoning approach to game theory. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both syntactic proofs and semantical arguments. Formal proofs are used establish what is the case and what is needed to show this. Semantical machinery is used to establish what cannot be the case. All this makes a case that PDL + E can be a useful basis for the logical investigation of game theory.

# Syntactic Epistemic Logic and Games IV

In this talk we consider knowledge structures used to model epistemic components of extensive-form games. We show that the traditional semantic mode of specification by a single model cannot capture some natural epistemic notions, e.g., mutual knowledge of rationality, used in Epistemic Game Theory. In contrast, Syntactic Epistemic Logic handles this and many other conditions that lie outside the scope of the semantic tradition. We outline general principles of Syntactic Epistemic Logic in connection to Game Theory.

# Syntactic Epistemic Logic and Games III

We discuss the Muddy Children puzzle MC, identify and fix the gap in its solution: a completeness analysis is required if the problem is specified syntactically, but analyzed by reasoning on a specific model. We show that the syntactic description of MC is complete w.r.t. its model used in the standard solution. We will present two proofs of completeness: syntactic, by induction on a formulas, and semantic, which makes use of bounded morphisms, a special case of bisimulations. We then present a modification of MC, MClite, and show that it is not deductively complete, has no semantic characterization and cannot be handled by traditional semantic methods. We give a syntactic solution of MClite.

Time permitting, we will venture further into extensive games and their epistemic models

# Syntactic Epistemic Logic and Games II

The Syntactic Epistemic Logic, SEL, suggests making the syntactic formalization of an epistemic scenario its formal specification. Since SEL accommodates constructive model reasoning, we speak about extending the scope of formal epistemology. SEL offers a cure for two principal weaknesses of the traditional semantic approach.

1. The traditional semantic approach in epistemology covers only deductively complete scenarios in which all epistemic assertions of any depth are specified. However, intuitively, “most” of epistemic scenarios are not complete. SEL suggests a way to handle incomplete scenarios with reasonable syntactic descriptions.

2. In addition, SEL seals the conceptual and technical gap, identified by R. Aumann, between the syntactic character of game descriptions and the semantic method of analyzing games.

# Syntactic Epistemic Logic and Games I

We argue that the traditional way to specify an epistemic situation via a single model is unnecessarily restrictive: there are numerous scenarios which do not have conventional single-model characterizations: less-than-common knowledge (e.g., mutual knowledge), partial, asymmetric knowledge, etc. Syntactic Epistemic Logic, SEL, suggests a way out of this predicament by viewing an epistemic scenario as specified syntactically by a set of formulas which corresponds to the characterization via a class of models rather than a single model. In addition, in Game Theory, SEL helps to seal the conceptual and technical gap, identified by R.Aumann, between the syntactic character of game descriptions and the predominantly semantical way of analyzing games.

This will be the first in the series of talks on SEL. We will consider natural examples of epistemic or game theoretic scenarios which have manageable syntactic description but do not have acceptable single model characterizations.

# First-order justification logic JT45

Among the different propositional justification logics, JT45 is the justification counterpart of the modal logic S5. Although there is a growing literature about propositional justification logic, there are only a few papers about the first-order version of justification logic. And most of those papers focus on the first-order version of LP. The goal of this talk is to present a first approach to first-order JT45 (FOJT45). Using Fitting models we will present a possible worlds semantics for FOJT45 and with that establish a Completeness Theorem. We will also discuss some topics related to first-order modal logic such as the Barcan Formula and the failure of the Interpolation Theorem.

# What is Constructive Axiomatic Method?

The received notion of axiomatic theory, which stems from Hilbert, is that of set T of propositions (either contentful or non-interpreted aka propositional forms) with subset A of axioms provided with a notion of consequence, which generates T from A in the obvious way. I argue that this standard notion is too narrow for being an adequate theoretical model of many mathematical theories; the class of such counter-examples is apparently very large and it includes such different theories as the geometrical theory of Euclid’s Elements, Book 1, and the more recent Homotopy type theory. In order to fix this problem I introduce a more general notion of theory, which uses typing and a generalized notion of consequence applicable also to objects of other types than propositions. I call such a theory constructive axiomatic theory and show that this particular conception of being constructive indeed captures many important ideas concerning the mathematical constructivity found in the earlier literature from Hilbert to Kolmogorov to Martin-Lof. Finally I provide an epistemological argument intended to show that the notion of constructive axiomatic theory is more apt to be useful in natural sciences and other empirical contexts than the standard notion. Disclaimer: The notion of constructive axiomatic theory is not my invention. The idea and its technical implementation are found in Martin-Lof ‘s constructive type theory if not already in Euclid. My aim is to make this notion explicit and introduce it into the continuing discussions on axiomatic method and mathematical and logical constructivity.

# What ‘If’?

Solutions to the paradoxes of semantic self-reference which endorse the unrestricted T-schema (and cognate principles) normally assume that the conditional involved in the schema is a detachable one (i.e., one that satisfies modus ponens). In a paraconsistent setting, there is another possibility: to take it to be the material conditional. In this talk, I will discuss some of the ramifications of this possibility.

# Paraconsistent Justification Logic (II): Quasi-Realization Theorem

Paraconsistent justification logics are justification logic systems for inconsistent beliefs or commitments. In other words, these systems are intended to model the agent’s justification structure, when she has inconsistent beliefs or commitments.

Realization theorem is one important meta-theorem in the discourse of justification logic. Quasi-realization theorem can be taken to be a major step toward proving realization theorem. In the last semester, I introduced one paraconsistent justification logic, called PJ_b. The main focus of the talk on the coming Tuesday is quasi-realization. It will be proved that quasi-realization theorem – which holds for standard justification logic systems – also holds for PJ_b.

# Unified Logics

The presentation studies the propositional logical systems containing multiple-conclusion rules that admit the asserted and rejected propositions at the same time.

# Modeling Intuitionistic Epistemic Logic via Provability and Verification (joint work with S. Artemov)

In 1933, Goedel offered a provability semantics for intuitionsitic logic IPC by means of a syntactical embedding of IPC into the modal logic S4 which Goedel considered a calculus for classical provability. Goedel’s embedding later became a key ingredient of the BHK-style semantics of classical proofs for intuitionsitic logic via the Logic of Proofs. We will establish the embedding/interpretation of Intuitionistic Epistemic Logic IEL into a natural extension (which we suggest calling S4V) of Goedel’s provability logic S4 by a verification modality. Basically, we will explain IEL via classical provability and verification coded in S4V the way Goedel explained intuitionistic logic via classical provability by interpreting it in S4.

# On non-self-referential realizable fragments

Given the known fact that realizations of modal logics T, K4, S4, and intuitionistic propositional logic IPC (via S4) each requires justification formula that is self-referential, it is reasonable to consider the fragment of each that can be realized non-self-referentially, called the non-self-referential realizable (NSR) fragment. Based on prehistoric graphs of G3-style sequent proofs of modal theorems, we are able to define the loop-free-provable (LFP) fragment of each logic that turns out to be a decidable subset of the NSR fragment. In this talk, we will explore properties of LFP fragments, their applications in the study of NSR fragments, and other related issues.

# Separation Logic and Friends

Separation Logic is used as an assertion language for Hoare-style proof systems about programs with pointers, and there is an ongoing quest for understanding its complexity and expressive power. Alternatively, there are a lot of activities to develop verification methods with decision procedures for fragments of practical use.

Actually, there exist many versions or fragments of Separation Logic that can be viewed as fragments of Second-Order Logic, as well as variants of Modal or Temporal Logics in which models can be updated dynamically (as in Public Announcement Logic).

In this talk, after introducing first principles on Separation Logic, issues related to decidability, computational complexity and expressive power are discussed. We provide several tight relationships with Second-Order Logic, Interval Temporal Logic or Data Logics, depending on the variants of the logic and on the syntactic resources available.

Results presented during the talk span from standard ones about Separation Logic to more recent ones, some of them obtained in collaboration with Dr. Morgan Deters (NYU, CIMS).

# Height and Happiness

Non-rigid designators play a major role in modal logics, when applied to aspects of natural language. Not only constants can be non-rigid, but also function symbols, and this is sometimes essential. I will discuss some examples, formalize them, and provide a semantics. If time permits, I will also present a nested sequent proof system to go with all this.

# Parameters and the Complexity of Modal Satisfiability

The complexity of Modal Logic has been extensively studied. Ladner has established most of what are now considered classical results on the matter, determining that most of the usual modal logics are PSPACE-complete. Furthermore, as Chagrov and Rybakov showed, some of these logics remain PSPACE-complete even for their variable-free fragments. We present parameters of modal formulas whose effect on the complexity of modal logic has been studied – like modal depth – and we identify sources of complexity for modal satisfiability.

A big part of this talk is joint work with Michael Lampis and Valia Mitsou.

# From Intuitionistic Epistemic Logic to a constructive resolution of the Knowability Paradox (joint work with S. Artemov)

We outline an intuitionistic view of knowledge which maintains the Brouwer-Heyting-Kolmogorov semantics and is consistent with Williamson’s suggestion that intuitionistic knowledge be regarded as the result of verification. We argue that on this view A -> KA is valid and KA -> A is not. We show the latter is a distinctly classical principle, too strong as the intuitionistic truth condition for knowledge which can be more adequately expressed by other modal means, e.g. ~(KA & ~A) “false is not known.” We construct a system of intuitionistic epistemic logic, IEL, codifying this view of knowledge and prove the standard meta-theorems for it. From this it follows that previous outlines of intuitionistic knowledge, responding to the knowability paradox, are insufficiently intuitionistic; by endorsing KA -> A they implicitly adopt a classical view of knowledge, by rejecting A -> KA they reject the constructivity of truth. Within the framework of IEL, the knowability paradox is resolved in a constructive manner which, as we hope, reflects its intrinsic meaning.

# Models of Justification Logic III

In this lecture we will review topological models of modal and justification logics.

# On Models of Justification Logic II.

We will introduce a new, slightly more general class of models for Justification Logic and compare them with Mkrtychev, Fitting, and modular models. Completeness Theorem will be established, examples will be given and plausible areas of applications outlined.

# On models of Justification Logic

This will be a mostly survey talk in which we discuss a variety of models for Justification Logic: arithmetical models (Goedel, Artemov), Mkrtychev models, Fitting models, topological models (Artemov, Nogina), modular models (Artemov), Sedlar’s group models, and some others. We will continue discussing applications.

# Some new directions in Epistemic Logic.

We will review research avenues on which we will focus this semester.

1. Applications of Justification Logic: general theory of logical omniscience, new format of handling inconsistent information, argumentation theory, truth maintenance systems, tracking evidence and belief revision.

2. Syntactic Epistemic Logic. There is some well-known tension between a syntactic description of an epistemic scenario and its semantical analysis based on a specific model. Such analysis requires common knowledge of the model which does not usually follow from the assumptions. Syntactic Epistemic Logic offers an approach to epistemic situations which does not rely on the common knowledge of a model.

3. Generic Common Knowledge.The notion of common knowledge of F, as introduced by Lewis in 1969, refers to any situation in which each finite iteration of `agent 1 knows that agent 2 knows that … F’ holds. A similar approach has been offered by McCarthy and others. This situatuion is now called Generic Common Knowledge, GCK, of F. Now dominant notion of Common Knowledge CK was coined by Aumann in 1976 as the least of all GCK’s of F. Now there is a growing number of specific epistemic scenarios which require GCK rather than CK. We are developing the theory of GCK and applying GCK to CK in appropriate situations.

# Quantum Computing: 2nd trip report

# Large-data deduplication problem (a joint session with the Computer Science Colloquium).

Imagine that you have a long list of items, say a hundred thousands of items. For example, the items may be client addresses. Some of the addresses are essentially duplicates distinguished only by “St.” vs. “Street”, or “Bill” vs. “William”, or by little spelling errors, etc. You don’t want to miss any of your clients, and you don’t want to annoy them by sending them multiple copies of your communications. How do you clean up your item list? The problem is ubiquitous and hard. We analyze the problem and describe a fast probabilistic algorithm for it.

# Paraconsistent Justification Logic

In the literature of belief revision, there is one approach called belief base belief revision, where the belief set is not required to be closed under a consequence relation. According to Sven Ove Hansson, in belief base belief revision, there are two ways to define the revision operator:

revision = expansion + contraction

revision = contraction + expansion

Hansson has a result that these two ways of defining the revision operator do not collapse into the same operator. Hansson also thinks that in the first way, there is an intermediate inconsistent epistemic state that occurs after expansion. Paraconsistent justification logic is intended to model the agent’s justification structure, when the agent is in such an inconsistent epi-state. My hope is that this logic could help us better model belief revision.

In my talk, the motivation will be better clarified. And, a paraconsistent justification logic system will be introduced.

# Structures of Social Proof

“Social proof” means that single agents assume beliefs, norms or actions of other agents in an attempt to reflect the correct view, stance, behavior for a given situation. The structure and modularity of social proof is unravelled including formal characterizations of derived socio-informational phenomena like bystander-effects and cascades. Sometimes social proof may be responsible for information spinning out of control – in very unfortunate ways. Joint work with Rasmus K. Rendsvig.

# Realization Semantically

This talk continues my previous one from October 22. In that, I gave justification counterparts for S4.2 and for K4^3, possible world justification semantics, and a completeness proof. The completeness proof used a canonical model construction. Now I’l use that canonical model to prove, non-constructively, realization theorems for the two logics. The methodology is the same as in a paper of mine published in 2005, though the presentation has change somewhat. Primarily my motivation is to explore the range of modal logics having justification counterparts—to discover its extent and limits.

# Reflection Principles Involving Provability and Explicit Proofs

Reflection principles are classical objects in proof theory and the areas studying Gödel’s Incompleteness. Reflection principles based on provability predicates were introduced in the 1930s by Rosser and Turing, and later were explored by Feferman, Kreisel & Levi, Schmerl, Artemov, Beklemishev and others.

We study reflection principles of Peano Arithmetic involving both Proof and Provability predicates. We find a classification of these principles and establish their linear ordering with respect to their metamathematical strength.

# J-Calc: Justified Modal Types and Functional Programs

In this talk we give a computationally motivated presentation of J-Calc. J-Calc is an extension of simply typed lambda calculus with justifications and justified modal types.

Initiating from textbook examples of modular programs, we will present the modal type theory of J-Calc as a type system corresponding to programming with modular definitions. More specifically, we will show how justified modality offers a logical reading to separate compilation in modular functional programming.

# Justification Logic Semantics: A Little New, but Mostly Old

Possible world semantics was introduced for justification logic in 2005. Initially it was for the Logic of Proofs, but it quickly extended to “nearby” logics, and more slowly to a wider family. Eight years is a long time, and by now people may be generally familiar with the ideas without having gone through any of the details. I will try to remedy that.

# On the Complexity of Multi-agent Justification Logic Under Interacting Justifications II

We introduce a family of multi-agent justification logics with interactions between the agents’ justifications, by extending and generalizing the two-agent versions of LP introduced by Yavorskaya in 2008. Known concepts and tools from the single-agent justification setting are adjusted for this multiple agent case. We present tableau rules and some preliminary complexity results: in several important cases, the satisfiability problem for these logics remains in the second level of the polynomial hierarchy, while for others it is PSPACE or EXP-hard.

This week we will present the general tableau rules for the family. We look more closely at particular examples and complexity bounds and we identify common sources of jumps in their complexity.

# On the Complexity of Multi-agent Justification Logic Under Interacting Justifications

We introduce a family of multi-agent justification logics with interactions between the agents’ justifications, by extending and generalizing the two-agent versions of LP introduced by Yavorskaya in 2008. Known concepts and tools from the single-agent justification setting are adjusted for this multiple agent case. We present tableau rules and some preliminary complexity results: in several important cases, the satisfiability problem for these logics remains in the second level of the polynomial hierarchy, while for others it is PSPACE or EXP-hard.

# Schroedinger and Turing on the Logic of Life: from the “coding” to the “genesis” of forms.

Schroedinger’s and Turing’s analyses of life phenomena have a twofold aspects. They both follow, first, a “coding paradigm”, of embryogenesis or of human computations and deductions respectively, and then move towards a more “dynamicist” approach. Schroedinger, in the second part of his 1944 book, hints to biological organization as negentropy – a variant of Gibbs dynamical analysis of energy – that we revitalized as anti-entropy, see references. Turing, after stressing that “the nervous system is surely not a Discrete State machine” (1950), invents the mathematics for an action/reaction/diffusion process, a “continuous system” (1952), where chemical matter (an hardware with no software) organizes itself along morphogenesis.

We will hint to the paths for thought opened by Turing’s dynamics as continuous deformations at the core of Turing’s pioneering paper of 1952, where symmetry breakings are a key component of the bio-chemical processes.

References

Schrödinger, E. What Is Life?, Cambridge University Press, 1944.

Alan M. Turing, “On Computable Numbers with an Application to the Entscheidungsproblem”, Proc. London Math. Soc. 42, 230-265, 1936.

Alan M. Turing, “The Chemical Basis of Morphogenesis”, Philo. Trans. Royal Soc., B237, 37-72, 1952.

Francis Bailly, Giuseppe Longo. Mathematics and Natural Sciences : the Physical Singularity of Life, Imperial College Press, London, 2011.

Giuseppe Longo, Maël Montévil, Perspectives on Organisms: Biological Time, Symmetries and Singularities, Springer, 2013.

Papers in http://www.di.ens.fr/users/longo:

Giuseppe Longo, “From exact sciences to life phenomena: following Schrödinger and Turing on Programs, Life and Causality”. Information and Computation, 207, 5: 543-670, 2009.

Francis Bailly, Giuseppe Longo. Biological Organization and Anti-Entropy. In J. Biological Systems, Vol. 17, No. 1, pp. 63-96, 2009.

Giuseppe Longo. Incomputability in Physics and Biology. Invited Lecture, Proceedings of Computability in Europe, Azores, Pt, June 30 – July 4, LNCS 6158, Springer, 2010.

Longo G., P. A. Miquel, C. Sonnenschein, A. Soto. Is Information a proper observable for biological organization? Progress in Biophysics and Molecular Biology, Vol. 109, Issue 3, pp. 108-114, August 2012.

# The road from Leibniz to Turing: from syllogisms to computations. Tribute to Alan Turing.

A group of brilliant innovators spanning three centuries were concerned with the nature of human reason. This endeavor led to the all-purpose digital computer. Except for Turing, none of them had any idea that his work might have such a tremendous application to our modern world. Leibniz saw far, but not that far. Boole could hardly have imagined that his algebra of logic would be used to design complex electronic circuits. Frege would have been amazed to find equivalents of his logical rules incorporated into computer programs for carrying out deductions. Cantor certainly never anticipated the ramifications of his diagonal method. Hilbert’s program to secure the foundations of mathematics was pointed in a very different direction. And Goedel, living his life of the mind, hardly thought of application to mechanical devices (M. Davis). However, Turing’s great vision has now been realized.

# Towards Syntactic Epistemic Game Theory

The traditional semantic approach to consider a game as an Aumann structure, though flexible and convenient, is not foundationally satisfactory due to assumptions that a given Aumann structure adequately represents the game and that this structure itself is common knowledge for the players.

These assumptions leave a gap between the officially syntactic character of the game description that often admits multiple models and studying a game as a specific model that is somehow assumed to be commonly known. This gap has been largely ignored or covered up by using as examples simple epistemic scenarios with natural models that were tacitly used as definitions of the game instead of declared syntactic game descriptions. Among others, Aumman found this foundationally unsatisfactory and argued for using what he called ‘Syntactic epistemic logic’ for reasoning about games.

In this talk, we outline a systematic approach to epistemic game theory which we suggest calling ‘Syntactic Epistemic Game Theory’, SEGT, consistent with Aumann’s suggestions, that studies games as they are normally described, in their syntactic form. In SEGT, semantic methods should be properly justified from the original game description. As a case study, we offer a SEGT theory of definitive solutions of strategic games with ordinal payoffs.

# Coincidence of Two Solutions to Nash’s Bargaining Problem

John Nash in 1950 gave an elegant solution to the bargaining problem using his somewhat controversial IIA axiom. Ehud Kalai and Meir Smorodinsky in 1975 gave a different solution replacing the IIA condition by their own Monotonicity condition. While the two solutions do not coincide in general, they obviously do so when the problem is symmetric. Are there other cases where the two solutions coincide? Indeed there are and we give a complete characterization.

# Knowledge and the Passage of Time

This talk will discuss how knowledge, nested knowledge, and common knowledge are gained in the presence of clocks and time bound information. It will complement the previous talk, in providing the causal structure underlying knowledge gain, from which the causal structure underlying basic coordination follows.

The talk will be based on joint work with Ido Ben Zvi.

# Realization Implemented

Justification logics are connected to modal logics via realization theorems. These have both constructive and non-constructive proofs. Here we do two things. First we provide a new path to constructive realization proofs, going through an intermediate quasi-realization stage. Quasi-realizers are easier to produce than realizers, though like them they are constructed from cut-free proofs. Quasi-realizers in turn constructively convert to realizers, and this conversion is independent of the justification logic in question. The construction depends only on the structure of the formula involved.

The second thing we do is provide a Prolog implementation of quasi-realization, and quasi-realization to realization conversion, for the logic LP. Many other justification logics can obviously be treated similarly.

Our quasi-realization algorithm, and its implementation, assumes the underlying modal proof system S4 is based on tableaus. Since these may not be familiar to everybody, we provide a sketch of how tableaus work. Then we present our algorithms, our implementation, and a discussion of implementation behavior and design decisions.

We believe our algorithms are simple and straightforward. The original realization algorithm, for instance, needed the entire cut-free proof as input. Our quasi-realization algorithm works one formal proof step at a time. There is, in the literature, another realization construction that works one step at a time, but it requires extensive use of substitution while our quasi-realization algorithm does not. The conversion algorithm is, as noted above, independent of particular justification logics and so only needs to be understood once. It is only here that substitution is needed.

Both the program to compute realizations, and a tech report discussing the algorithm, necessary background, and the implementation, are available on the speaker’s web page:

http://comet.lehman.cuny.edu/fitting/

# On Time, Communication and Coordination

Recent work has shown that applications of the modal logic of knowledge allow a characterization of the interaction between communication and coordination in systems with clocks and timing information. This talk will survey some of these results and the underlying notions and techniques. Time permitting, some issues related to the interaction between time and causality will be discussed. The focus of the talk will be on the interface between logic (or semantic notions) and application, rather than on the properties of the logic itself.

# Justification Logic for Argumentation

Argumentation is the process of giving arguments, counter-arguments, counter-counter-arguments and so on. In artificial intelligence, since the late 1980s there has been research that tries to use formal methods to approach argumentation theory. One very young approach in formal argumentation theory is to study argumentation under the framework of modal logic. Two possible advantages of doing so are: (1) this is a new angle, from which to look at argumentation we might have new insights, and (2) techniques in modal logic might become available for studying argumentation. In this talk, I would like to provide a different way of connecting argumentation theory and modal logic. More specifically, I would like to try to locate formal argumentation theory within the framework of justification logic. The resulting logic is a justification logic for argumentation.

# Semantic Knowledge Base – system build in Association Oriented Database Model

Semantic Knowledge Base is a new approach towards building systems capable of storing and processing complex information also in the means of inference over it. The main assumption is to distinguish concepts (meanings) from terms (natural language identifiers). The information is held in two modules: ontological core and semantic network module. Ontological core is used for building a hypergraph, where each node is a concept and edges are links between them. Semantic networks are used to store more complex information (rules and facts) in a structure using predicate-object structures build over previously defined concepts. The system provides also support for inference in the means of modal and hybrid logic through custom build system of quantifiers and possibility of describing dimensions and spaces, where each and every part of information is valid or invalid (extended Kripke’s model of possible world theory).

The system has been designed in association oriented database model (AODB). In this model we assume two main categories: collections and associations. Collections are used to store information in objects holding the attributes values, while associations store information about n-ary relation between collections. Associations are consider to be containers for roles and each role is considered as the permission given to the collection to be part of an association. The model assumes that both, collections and association may build inheritance structures, where virtual, private and real inheritance of accordingly attributes and roles occur. Each of the structure elements of AODB has its own unique role therefore their semantic is strictly connected to the grammar of the model. Moreover such construction made it possible to unify conceptual and physical model of the database into one structure. AODB has its native object storage model. There have also been two languages designed and implemented – Association Modeling Language (AML) and Association Query Language (AQL). The latter is using graph templates as queries and returns list of graph.

# Computing Bounds from Arithmetical Proofs

We explore the role of the function a+2^x, and its generalizations to higher number classes, in analyzing and measuring the computational content of a broad spectrum of arithmetical theories. Newer formalizations of such theories, based on the well-known normal/safe variable separation of Bellantoni-Cook, enable uniform proof-theoretical treatments of poly-time arithmetics, through Peano arithmetic, and up to finitely iterated inductive definitions.

# Expanding the realm of the idea of da Costa

Non-classical logics that deny ex falso quod libet are said to be paraconsistent. Since the monumental work of Stanislaw Jaskowski in 1948, a large number of paraconsistent systems have been developed. Those include systems of da Costa, Nelson, Belnap-Dunn, Priest, Batens and Scotch-Jennings. In this talk, we focus on the consistency operator which is the characteristic connective of da Costa’s systems. We understand that the main idea of da Costa is to make explicit, within the system, the area in which you can infer classically. The aim of the talk is twofold. First, we review some of the results within the framework of Logics of Formal Inconsistency. Then, we introduce and present some results on normality operator which generalizes consistency operator so that one can deal not only with inconsistency, but also with incompleteness. Second, we show that the normality operator may be employed, in a sense to be specified, in developing other paraconsistent logics such as modal logics, Nelson’s systems and expansions of the four-valued logic of Belnap and Dunn.

# Lost in translation: a critical view of epistemic puzzles solutions.

There are two basic ways to specify an epistemic scenario:

1. Syntactic: a verbal description with some epistemic logic on the background and some additional formalizable assumptions; think of the Muddy Children puzzle.

2. Semantic: providing an epistemic model; think of Aumann structures – a typical way to define epistemic components of games.

Such classical examples as Muddy Children, Wise Men, Wise Girls, etc., are given by syntactic descriptions (type 1), each of which is “automatically” replaced by a simple finite model (type 2). Validity of such translations from (1) to (2) will be the subject of our study.

We argue that in reducing (1) to (2), it is imperative to check whether (1) is complete with respect to (2) without which solutions of puzzles by model reasoning in (2) are not complete, at best. We have already shown that such reductions can be justified in the Muddy Children puzzle MC due to its model categoricity: we have proved that MC has the unique “cube” model Q_n for each n. This fixes an obvious gap in the “textbook” solution of Muddy Children which did not provide a sufficient justification for using Q_n.

We also show that an adequate reduction of (1) to (2) is rather a lucky exception, which makes the requirement to check the completeness of (1) w.r.t. (2) necessary. To make this point, we provide a simplified version of Muddy Children (by dropping the assumption “no kid knows whether he is muddy”) which admits the usual deductive solution by reasoning from the syntactic description, but which cannot be reduced to any finite model.

# On model reasoning in epistemic scenarios

It was long noticed that the textbook “solution” of the Muddy Children puzzle MC via reasoning on the n-dimensional cube Q_n had a fundamental gap since it presupposed common knowledge of the Kripke model Q_n which was not assumed in the puzzle description. In the MC scenario, the verbal description and logical postulates are common knowledge, but this does not yield common knowledge of the model.

Of course, a rigorous solution of MC can be given by a formal deduction in an appropriate epistemic logic with updates after epistemic actions (public announcements). This way requires an advanced and generally accepted logical apparatus, certain deductive skills in modal logic, and, most importantly, steers away from the textbook “solution.”

We argue that the gap in the textbook “solution” of MC can be fixed. We establish that MC_n is complete with respect to Q_n and hence a reasoning on Q_n can be accepted as a substitute for the aforementioned deductive solution (given that kids are smart enough to establish that this completeness theorem is itself common knowledge). This yields the following clean solution of MC:

1. prove the completeness of MC_n w.r.t. Q_n;

2. argue that (1) is common knowledge;

3. use a properly worded version of the textbook “solution.”

This approach seems to work for some other well-known epistemic scenarios related to knowledge. However, it does not appear to be universal, e.g., it does not work in the case of beliefs. In general, we have to rely on the deductive solution which fits the syntactic description of the problem.

The completeness of MC_n w.r.t. Q_n was established by me some years ago, and shortly after my corresponding seminar presentation, Evan Goris offered an elegant and more general solution which will be presented at this talk.

# Geodesic Semantics for Belief Change

Any logical system that models human reasoning needs to address the possibility of error and the subsequent belief change once this error is recognized. The need to deal with error-prone reasoning has only been widely acknowledged in the last thirty years or so; witnesses the popularity of the AGM postulates for Belief Revision. Despite the variety of syntactical and semantical offerings, all seem to agree that we need to model a concept of minimal change by choosing the most similar epistemic state to the present one. The favorite choice mechanisms are preferential orderings and, their generalization, distance maps. Preferential orderings provide satisfactory representation results but fail to model iteration. Distance maps model iteration but fail to provide satisfactory completeness results. In this talk, I will introduce a third semantical approach using geodesic distance (length of shortest path on a graph) that lies between the two and combines their best features: geodesic semantics provide satisfactory completeness results like preferential orderings do and deal with iteration, as distance maps do. Further, and perhaps more important, geodesic semantics offer a novel, more natural representation of similarity using distinguishability.

# Definitive solutions of strategic games

In his dissertation of 1950, Nash based his concept of solution to a game on the principles that “a rational prediction should be unique, that the players should be able to deduce and make use of it.” In this paper, we address the issue of when Nash expectations of a definitive solution hold and whether the Nash Equilibrium (NE) solution concept is a match for such definitive solutions. We show that indeed, an existence of NE is a necessary condition for a definitive solution, and each NE σ is a definitive solution for some notion of rationality individually tuned for this σ. However, for specific notions of rationality, e.g., Aumann’s rationality, NE is not an exact match to definitive solutions, many games with NE do not have definitive solutions at all. In particular, strategic ordinal payoff games with two or more Nash equilibria, and even some games with a unique NE do not have definitive solutions. We also show that the iterated dominance approach is a better candidate for Nash’s definitive solution concept than the Nash Equilibrium.