An Introduction to Logic for Computer Science

Home , Finite model theory, Proof theory

An introduction to Logic for Computer Science

Sujata Ghosh ISI Chennai [email protected] What is logic? What is logic?

Reasoning is typically a human activity. What is logic?

Reasoning is typically a human activity.

There are inference rules which determine whether an argument is correct or not. What is logic?

Reasoning is typically a human activity.

There are inference rules which determine whether an argument is correct or not.

Logic is the study of reasoning, in particular, the study of these inference rules. Example argumentAn example in physics in Physics An example in Linguistics Intelligence without representation*

Rodney A. Brooks

MIT Artificial Intelligence Laboratory, 545 Technology Square, Rm. 836, Cambridge, MA 02139, USA

Received September 1987 Brooks, R.A., Intelligence without representation, Artificial Intelligence 47 (1991), 139–159.

* This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the research is provided in part by an IBM Faculty 9 Development Award, in part by a grant from the Systems Development Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 and in part by the Advanced Research Projects Agency under Office of Naval Research contract N00014-85-K-0124.

Abstract

Artificial intelligence research has foundered on the issue of representation. When intelligence is approached in an incremental manner, with strict reliance on interfacing to the real world through perception and action, reliance on representation disappears. In this paper we outline our approach to incrementally building complete intelligent Creatures. The fundamental decomposition of the intelligent system is not into independent information processing units which must interface with each other via representations. Instead, the intelligent system is decomposed into independent and parallel activity producers which all interface directly to the world through perception and action, rather than interface to each other particularly much. The notions of central and peripheral systems evaporateeverything is both central and peripheral. Based on these principles we have built a very successful series of mobile robots which operate without supervision as Creatures in standard office environments.

1. Introduction • We must incrementally build up the capabilities of Artificial intelligence started as a field whose goal intelligent systems, having complete systems at was to replicate human level intelligence in a each step of the way and thus automatically ensure Intelligencemachine. without representation*that the pieces and their interfaces are valid.

Early hopes diminished as the magnitude and • At each step we should build complete intelligent Rodneydifficulty A. Brooks of that goal was appreciated. Slow progress systems that we let loose in the real world with real was made over the next 25 years in demonstrating sensing and real action. Anything less provides a MIT Artificial Intelligence Laboratory, 545 Technology Square, Rm. 836, Cambridge, MA 02139, USA isolated aspects of intelligence. Recent work has candidate with which we can delude ourselves. Received tendedSeptember to concentrate1987 on commercializable aspects of Brooks, R.A.,"intelligent IntelligenceAn assistants" without example representation, for human workers.Artificial Intelligence 47in (1991), 139–159.WeArtificial have been following this approach and have built a series of autonomous mobile robots. We have * This reportNo describes one talks research about done replicating at the Artificial the Intelligence full gamut Laboratory of of thereached Massachusetts an unexpected Institute of Technology.conclusion Support (C) and for thehave a research ishuman provided intelligence in part by an any IBM more. FacultyIntelligence Instead 9 Development we see Award,a retreat in part by a grantrather from radical the Systems hypothesis Development (H). Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 and in part by the Advanced Research Projects Agencyinto specialized under Office ofsubproblems, Naval Research suchcontract as N00014-85-K-0124. ways to Intelligencerepresent knowledge, without natural languagerepresentation* understanding, (C) When we examine very simple level intelligence Abstract vision or even more specialized areas such as truth we find that explicit representations and models maintenance systems or plan verification. All the of the world simply get in the way. It turns out ArtificialRodney intelligence A. Brooks research has foundered on the issue of representation. When intelligence is approached in an incremental manner, with strict reliancework on in interfacing these subareasto the real isworld benchmarked through perception against and the action, reliance on representationto be better to disappears. use the world In this aspaper its own we outline model. our approachMITsorts Artificial to incrementallyofIntelligence tasks Laboratory, humans building 545 Technologycomplete do within Square, intelligent Rm. those 836, Creatures. Cambridge, areas. MA The 02139, fundamental USA decomposition of the intelligent system is not into independent information processing units which must interface with each other via representations. Instead, the intelligent system is decomposedReceivedAmongst into September independent the 1987 dreamers and parallel still activity in the producers field of whichAI (those all interface directly(H) Representation to the world through is perceptionthe wrong and unit action, of abstractionrather than interfaceBrooks,not R.A., dreaming to Intelligenceeach other about without particularly dollars,representation, much.that Artificial is), The Intelligencethere notions is 47 a of (1991),feeling. central 139–159. and peripheral insystems building evaporateeverything the bulkiest is partsboth central of intelligent and peripheral. Based on these principles we have built a very successful series of mobile robotssystems. which operate without supervision as Creatures in standard officethat oneenvironments. day all these pieces will all fall into place * This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the researchand is we provided will in seepart by"truly" an IBM Faculty intelligent 9 Development systems Award, emerge.in part by a grant from the Systems Development Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 andRepresentation in part by the Advanced has beenResearch the central issue in artificial 1. IntroductionProjects Agency under Office of Naval Research contract N00014-85-K-0124. However, I, and others, believe that human level • We mustintelligence incrementally work over build the uplast the 15 capabilitiesyears only becauseof ArtificialAbstractintelligence intelligence is too started complex as and a field little whose understood goal to be intelligentit has provided systems, an havinginterface complete between otherwise systems isolated at Artificialcorrectly intelligence decomposed research has foundered into theon the right issue of subpiecesrepresentation. When at theintelligence is approachedmodules in an and incrementa conferencel manner, with papers. was tostrict reliance replicate on interfacing human to the real level world through intelligence perception and in action, a reliance on representationeach step disappears. of the In thisway paper and we outlinethus automatically ensure ourmoment approach to incrementallyand that evenbuilding ifcomplete we knew intelligent the Creatures. subpieces The fundamental we decomposition of the intelligent system is not into machine.independent information processing units which must interface with each other via representations.that the Instead,pieces the and intelligent their systeminterfaces is are valid. decomposedstill wouldn't into independent know and parallel the activity right producers interfaces which all interface between directly to the world2. through The perception evolution and action, rather of intelligence than interface to each other particularly much. The notions of central and peripheral systems evaporateeverything is both central and Earlyperipheral. them. hopes Based Furthermore, diminishedon these principles we we as willhave the built never a magnitude very understandsuccessful series and ofhow mobile to robots• whichAt operate each withoutstep supervisionwe should as Creatures build in complete intelligent standard office environments. difficultydecompose of that goal human was levelappreciated. intelligence Slow until progress we've had a systemsWe that already we let loosehave in an the existence real world proof with realof, the was made1. lotIntroduction ofover practice the next with 25simpler years level in demonstratingintelligences. sensingpossibility and real ofaction. intelligent Anything entities: less provides human beings. a • We must incrementallyAdditionally, build up the many capabilities animals of are intelligent to some isolatedArtificial aspects intelligence of intelligence. started as a field Recent whose goal work has intelligent systems,candidate having with complete which systemswe can atdelude ourselves. tendedwas to toconcentrateIn replicate this paper human on commercializable I level therefore intelligence argue in aspects for a a ofdifferenteach step of the waydegree. and thus (This automatically is a subject ensure of intense debate, much of "intelligentmachine.approach assistants" to creating for human artificial workers. intelligence: that the piecesWe and have whichtheir been interfaces following really are valid.centers this approach around and a havedefinition built of Early hopes diminished as the magnitude and • At each stepa we series should of build autonomous complete intelligent mobile robots. We have No difficultyone talks of that about goal was replicating appreciated. theSlow fullprogress gamut of systems thatreached we let loose an inunexpected the real world conclusionwith real (C) and have a humanwas intelligence made over the any next more. 25 years Instead in demonstrating we see a retreat sensing andrather real action. radical Anything hypothesis less provides (H). a isolated aspects of intelligence. Recent work has candidate with which we can delude ourselves. into tendedspecialized to concentrate subproblems, on commercializable such aspects as waysof to represent"intelligent knowledge, assistants" natural for human language workers. understanding,We have been(C) following When this we approach examine and havevery built simple level intelligence vision or even more specialized areas such as trutha series of autonomouswe find mobile that robots. explicit We representations have and models No one talks about replicating the full gamut of reached an unexpected conclusion (C) and have a maintenancehuman intelligence systems any ormore. plan Instead verification. we see a retreat All therather radical hypothesisof the (H). world simply get in the way. It turns out work intoin thesespecialized subareas subproblems, is benchmarked such as ways against to the to be better to use the world as its own model. sorts represent of tasks knowledge, humans natural do language within understanding, those areas.(C) When we examine very simple level intelligence vision or even more specialized areas such as truth we find that explicit representations and models Amongstmaintenance the dreamers systems orstill plan in verification. the field Allof AI the (those of the world(H) simply Representation get in the way. is Itthe turns wrong out unit of abstraction not dreamingwork in these about subareas dollars, is benchmarked that is), there against is thea feeling. to be better toin use the building world as its the own bulkiest model. parts of intelligent that onesorts day of all tasks these humans pieces do will within all those fall areas. into place systems. and weAmongst will see the "truly"dreamers intelligentstill in the field systems of AI (those emerge. (H) Representation is the wrong unit of abstraction not dreaming about dollars, that is), there is a feeling. in building the bulkiest parts of intelligent that one day all these pieces will all fall into place systems.Representation has been the central issue in artificial However,and we will I, seeand "truly" others, intelligent believe systems that emerge. human level intelligence work over the last 15 years only because intelligence is too complex and little understood to beRepresentationit has has beenprovided the central an interfaceissue in artificial between otherwise isolated However, I, and others, believe that human level intelligence work over the last 15 years only because correctlyintelligence decomposed is too complex into and the little right understood subpieces to be at theit has providedmodules an interface and between conference otherwise papers. isolated momentcorrectly and decomposedthat even into if we the rightknew subpieces the subpieces at the wemodules and conference papers. still momentwouldn't and that know even theif we rightknew theinterfaces subpieces we between 2. The evolution of intelligence still wouldn't know the right interfaces between 2. The evolution of intelligence them.them. Furthermore, Furthermore, we we will never never understand understand how to how to decomposedecompose human human level level intelligenceintelligence until until we've we've had a had a We already Wehave already an existence have proof an of, existence the proof of, the lot of lotpractice of practice with with simpler simpler level level intelligences. intelligences. possibility possibility of intelligent entities: of intelligent human beings. entities: human beings. Additionally,Additionally, many animals aremany intelligent animals to some are intelligent to some In this paper I therefore argue for a different degree. (This is a subject of intense debate, much of In approach this paper to creating I thereforeartificial intelligence: argue for a differentwhich reallydegree. centers (This around is a asubject definition of intense of debate, much of approach to creating artificial intelligence: which really centers around a definition of ExampleAn argument example in law in Law

Ruling April 18th 2004, Court of The Hague, Lucia de B. 5.15 The child neurologist [Witness expert 4] has declared that a child does not die within a few moments. This takes a lot more time. Especially with children it is highly unusual to pass away quickly and without a certain cause of death. Usually the death of a child is anticipated beforehand or the cause of death is known.

5.16 Thereby the Court deduces that whenever a child dies suddenly and without a certain cause or through a life threatening event, one should more readily take the possibility of an unnatural cause of death into account than if this had happened to an adult. An example in daily life: 3 perfect logicians Logic in daily life: three perfect logicians First steps !!

Let’s first introduce Mathematical logic! Let’s first introduce Mathematical logic!

why ?! what?! how?! Let’s first introduce Mathematical logic!

why ?! what?! how?! Some viewpoints Some viewpoints

Dirichlet, Riemann, Dedekind: mathematical theories as general concepts, rather than formulas and calculations Some viewpoints

Dirichlet, Riemann, Dedekind: mathematical theories as general concepts, rather than formulas and calculations

Kronecker, Weierstrass: explicit representation and algorithmic treatment of mathematical entities Some viewpoints

Dirichlet, Riemann, Dedekind: mathematical theories as general concepts, rather than formulas and calculations

Kronecker, Weierstrass: explicit representation and algorithmic treatment of mathematical entities

Frege: pure logic system for mathematics to base on (introduction of first order logic) Some viewpoints (contd.)

Cantor, Hilbert: existential proofs, infinite sets, higher infinite, axiomatic characterisation of mathematical structures Some viewpoints (contd.)

Cantor, Hilbert: existential proofs, infinite sets, higher infinite, axiomatic characterisation of mathematical structures

Peano, Hadamard, Minkowski, Klein: supporters of the modern methodology Paris International Congress of Mathematics, 1900 Two problems from Hilbert’s list of problems

There does not exist any set whose cardinality lies between the set of integers and the set of real numbers

The set of axioms of arithmetic is consistent Russell’s paradox (1903) Russell’s paradox (1903)

Comprehension principle: Given a well- define logical or mathematical property p, there exists a set {x : p(x)} of all objects satisfying the property. Russell’s paradox (1903)

Comprehension principle: Given a well- define logical or mathematical property p, there exists a set {x : p(x)} of all objects satisfying the property.

A contradiction: Consider the property p(x) to be x ⍧ x. The comprehension principle yields the existence of the set R = {x: x ⍧ x}. What can we say about R? A popular version Hilbert’s Programme Hilbert’s Programme

Claiming the existence of a set of mathematical objects is same as proving that the corresponding axiom system is consistent, that is, does not lead to any contradiction. Hilbert’s Programme

Claiming the existence of a set of mathematical objects is same as proving that the corresponding axiom system is consistent, that is, does not lead to any contradiction.

Axiomatize the classical theories of mathematics and provide a finitistic proof of their consistency. Hilbert’s Programme (contd.)

Find suitable axioms and primitive concepts of a mathematical theory T Hilbert’s Programme (contd.)

Find suitable axioms and primitive concepts of a mathematical theory T

Finding axioms and inference rules for classical logic which make the passage from existing propositions to new propositions a syntactic formal procedure. Hilbert’s Programme (contd.)

Formalising T by means of the logical language, so that propositions of T are just strings of symbols and proofs are sequences of such strings obeying certain formal rules of inference Hilbert’s Programme (contd.)

Formalising T be means of the logical language, so that propositions of T are just strings of symbols and proofs are sequences of such strings obeying certain formal rules of inference

A finitary study of the formalised proofs in T that shows that it is impossible for a string of symbols that expresses a contradiction to be the last line of a proof Chronologically …..

1900: Hilbert’s 2nd problem - To prove the consistency of arithmetic.

1903: Russell’s paradox - more emphasis on axiomatisation and consistency, but no satisfactory logical system.

1903 - 1910: Russell and Whitehead - Theory of types (provided the logical formalisms to proceed).

1917 - 1921: Hilbert and his students provides a more sophisticated development of first order logic. Chronologically …..

1922: ε-calculus was developed by Hilbert and Bernays providing general techniques of consistency proof.

1924: A proof of consistency of analysis was presented by Ackermann. It had some errors.

1925: von Neumann corrected some errors and proved the consistency of ε-calculus without the induction scheme, building on which Ackermann devised a new ε-calculus.

1928: At ICM Bologna Hilbert claimed that the work of Ackermann and von Neumann constituted a proof of consistency of Arithmetic. Brouwer and Intuitionism Brouwer and Intuitionism

Role of intuition in mathematics was advocated by Klein and Poincaré Brouwer and Intuitionism

Role of intuition in mathematics was advocated by Klein and Poincaré

Brouwer rejected ‘Principle of Excluded Middle’: “p or not p” Brouwer and Intuitionism

Role of intuition in mathematics was advocated by Klein and Poincaré

Brouwer rejected ‘Principle of Excluded Middle’: “p or not p”

One can only state “p or q” if one can construct a proof of “p” or that of “q” Brouwer and Intuitionism

Role of intuition in mathematics was advocated by Klein and Poincaré

Brouwer rejected ‘Principle of Excluded Middle’: “p or not p”

One can only state “p or q” if one can construct a proof of “p” or that of “q” development of constructive mathematics Sunday, September 7, 1930

The venue: Königsberg. The event: a small conference on foundation of mathematics

Heyting has spoken on intuitionism

Carnap has spoken on logicism von Neumann has expounded on Hilbert’s proof theory

Hahn has spoken on his empiricist view Sunday, September 7, 1930

Heyting: I am satisfied with this meeting. It has clarified the relationship between formalism and intuitionism. Once the formalist has succeeded in Hilbert’s programme and shown finitely that the idealised mathematics proves no new meaningful sentences, even the intuitionists will fondly embrace the infinite.

A shy young man provided a firm critique to the Hilbert’s programme, though in a somewhat inarticulate manner, and no one present seemed to have understood it. Tuesday, September 15, 1931

Gödel presented his result in a meeting at Bad Elster

His paper was a paradigm of clarity unlike his earlier cautious remarks and nearly inexplicit announcement

Contained a detailed proof of First Incompleteness Theorem, a few related result and an announcement of the Second Incompleteness Theorem Incompleteness Theorems (popular versions)

First Incompleteness Theorem: Let T be a mathematical theory containing a certain amount of elementary arithmetic. Then T cannot be both consistent and complete, that is, if T is consistent then there is a statement in arithmetic which can neither be proved nor disproved in T. Incompleteness Theorems (popular versions)

Second Incompleteness Theorem: Let T be a consistent mathematical theory containing a certain amount of elementary arithmetic. Then T cannot prove its own consistency. Impact

Recall Hilbert’s programme: Formalising mathematical theories using a set of axioms and rules in a logical language and deduce theorems. Of course, such theories should be complete and encompass natural numbers at least. Impact

Gödel’s first theorem tells us that any such consistent theory cannot be complete. Thus there can be many undecidable statements. Impact

Gödel’s first theorem tells us that any such consistent theory cannot be complete. Thus there can be many undecidable statements.

Can it be the case that such statements are not really interesting for mathematicians? Impact

Recall Hilbert’s programme once again: There should be a finitary way to show that the formal system is consistent. Impact

Recall Hilbert’s programme once again: There should be a finitary way to show that the formal system is consistent.

Gödel’s second theorem tells us that even elementary arithmetic is not strong enough to prove its own consistency. What happened next? 1930’s era (Computability theory) 1930’s era (Computability theory)

Entscheidungsproblem (Hilbert): Given an axiom system and a formula φ, is there a general effective procedure to determine whether it is provable ? 1930’s era (Computability theory)

Entscheidungsproblem (Hilbert): Given an axiom system and a formula φ, is there a general effective procedure to determine whether it is provable ?

Gödel introduced the notion of general recursive functions 1930’s era (Computability theory)

Entscheidungsproblem (Hilbert): Given an axiom system and a formula φ, is there a general effective procedure to determine whether it is provable ?

Gödel introduced the notion of general recursive functions

Church introduced the notion of λ-definable functions 1930’s era (Computability theory)

Entscheidungsproblem (Hilbert): Given an axiom system and a formula φ, is there a general effective procedure to determine whether it is provable ?

Gödel introduced the notion of general recursive functions

Church introduced the notion of λ-definable functions

Turing introduced the Turing machines - effective procedure is same as calculable by Turing machines 1930’s era (Computability theory)

Entscheidungsproblem (Hilbert): Given an axiom system and a formula φ, is there a general effective procedure to determine whether it is provable ?

Gödel introduced the notion of general recursive functions

Church introduced the notion of λ-definable functions

Turing introduced the Turing machines - effective procedure is same as calculable by Turing machines

‘undecidable’ More on Turing’s paper

A notion of effective procedure is defined mathematically More on Turing’s paper

A notion of effective procedure is defined mathematically

The procedure can simulate any reasonable notion of computation and its formulation is also very instinctive - works as any intuitive computing machine More on Turing’s paper

A notion of effective procedure is defined mathematically

The procedure can simulate any reasonable notion of computation and its formulation is also very instinctive - works as any intuitive computing machine

It has its limitations as well - unsolvability of a problem More on Turing’s paper

A notion of effective procedure is defined mathematically

The procedure can simulate any reasonable notion of computation and its formulation is also very instinctive - works as any intuitive computing machine

It has its limitations as well - unsolvability of a problem

Universal machine can also be constructed that can take any other machine as input and behave like it 1930’s era (Proof theory)

Gentzen went on with the task of proving the consistency of arithmetic 1930’s era (Proof theory)

Gentzen went on with the task of proving the consistency of arithmetic

Analysed mathematical proofs - proposed Natural Deduction and Sequent Calculus 1930’s era (Proof theory)

Gentzen went on with the task of proving the consistency of arithmetic

Analysed mathematical proofs - proposed Natural Deduction and Sequent Calculus

Succeeded in giving a consistency proof of an axiom system of number theory in number theory + ‘something (transfinite)’ Compartmentalisation in Mathematical Logic

Proof theory

Set theory

Model Theory

Recursion Theory (Computability theory)

Logic and Algebra Compartmentalisation in Mathematical Logic

Proof theory

Set theory

Model Theory

Recursion Theory (Computability theory)

Logic and Algebra

Over the years, somewhat limited interaction! Rich interactions between logic and computer science developed, continuing till now! Logic in Computer Science

architecture

software engineering

theory of computation

algorithms

complexity

databases

programming languages semantics

artificial intelligence Let’s explore a few! Logic and algorithms

a huge range of algorithmic techniques Logic and algorithms

a huge range of algorithmic techniques search for uniform algorithms Logic and algorithms

a huge range of algorithmic techniques search for uniform algorithms

Courcelle’s theorem: A property of graphs of bounded tree-width is checkable in polynomial time iff it is definable in a monadic second order logic Logic and algorithms

a huge range of algorithmic techniques search for uniform algorithms

Courcelle’s theorem: A property of graphs of bounded tree-width is checkable in polynomial time iff it is definable in a monadic second order logic connections between fixed-parameter tractability and logical definability Logic and complexity

resources needed for solving a problem Logic and complexity

resources needed for solving a problem satisfiability of boolean formulas and NP- completeness Logic and complexity

resources needed for solving a problem satisfiability of boolean formulas and NP- completeness satisfiability of quantified boolean formulas and PSpace-completeness Logic and complexity

resources needed for solving a problem satisfiability of boolean formulas and NP- completeness satisfiability of quantified boolean formulas and PSpace-completeness interactive proofs and quantifier hierarchy Logic and complexity

Fagin’s theorem: NP and definability in existential second order logic Logic and automata

logic provides a description language for finite-state automata Logic and automata

logic provides a description language for finite-state automata

Büchi’s theorem: A (word/tree) language is regular iff it is definable in monadic second order logic on (words/trees) Logic and automata

logic provides a description language for finite-state automata

Büchi’s theorem: A (word/tree) language iff it is definable in monadic second order logic on (words/trees) formal verification of hardware and software systems Logic and automata

logic provides a description language for finite- state automata

Büchi’s theorem: A (word/tree) language iff it is definable in monadic second order logic on (words/trees) formal verification of hardware and software systems

Vardi - from philosophical logics to industrial logics Logic and database theory

logic as database query language to express questions asked against databases Logic and database theory

logic as database query language to express questions asked against databases

relational algebra and relational calculus: SQL (first order logic) Logic and database theory

logic as database query language to express questions asked against databases

relational algebra and relational calculus: SQL (first order logic)

query evaluation, query containment, query equivalence Logic and database theory

logic as database query language to express questions asked against databases

relational algebra and relational calculus: SQL (first order logic)

query evaluation, query containment, query equivalence

logic as specification language to express integrity constraints in databases Logic and database theory

logic as database query language to express questions asked against databases

relational algebra and relational calculus: SQL (first order logic)

query evaluation, query containment, query equivalence

logic as specification language to express integrity constraints in databases

inconsistent databases: semantics and dichotomy results Programming language semantics

denotational; axiomatic; operational Programming language semantics

denotational; axiomatic; operational

Dana Scott’s theorem: The computable functions are exactly the continuous functions over a certain topology. Programming language semantics denotational; axiomatic; operational

Dana Scott’s theorem: The computable functions are exactly the continuous functions over a certain topology. the Curry Howard isomorphism between proofs in intuitionistic logic and types in any programming language Programming language semantics denotational; axiomatic; operational

Hoare logic and automated theorem proving Programming language semantics denotational; axiomatic; operational

Hoare logic and automated theorem proving implementation; verification; language design Logic and AI

logics for action and agency Logic and AI

logics for action and agency logics for knowledge representation Logic and AI

logics for action and agency logics for knowledge representation logics for expressing plans, protocols and strategies Logic and AI

logics for action and agency logics for knowledge representation logics for expressing plans, protocols and strategies logics for ontologies Logic and AI

logics for action and agency logics for knowledge representation logics for expressing plans, protocols and strategies logics for ontologies automated reasoning systems; multi-agent interactive systems The present logical arena

Set theory

Model theory

Computability theory

Proof theory

Philosophical logic

Logic and natural language

Logic in computer science What this course is all about? basics of mathematical logic with theoretical computer science in mind both proof-theoretic and model-theoretic techniques a flavour of Gödel’s completeness and (depending on time and interest) incompleteness results an introduction to logics of computation Marks division

Mid-Semester/Home Assignments: 20

Project: 30

End-Semester: 50 Course plan

Week 1: Logic as foundations of mathematics and mathematical logic as foundations computer science

Week 2: Mathematical structures, language of logic, syntax and semantics of first order logic; Algorithm for finite satisfiability, zeroth-order logic, propositional logic

Week 3: Logical consequence, compactness theorem and its applications

Week 4: Hilbert-style deduction system for classical propositional logic and completeness theorem.

Week 5: Natural deduction system, sequent calculus and cut elimination

Week 6: An overview of proof theory in first-order logic; undecidability of satisfiability

Week 7: Isomorphism and elementary equivalence; EF Games, partial isomorphism; first-order (un)definability Course plan

Week 8: Introduction to modal logic; relational models, neighbourhood models, topological models

Week 9: Correspondence theory; canonical models, completeness theorem

Week 10: finite model theory, decidability and complexity

Week 11: Linear temporal logic, computational tree logic, alternating- time temporal logic

Week 12: completeness results, decidability

Week 13: Propositional dynamic logic, completeness, decidability

Week 14: Monadic second-order logic on words; Büchi-Elgot- Trakhtenbrot theorem