Sequent Calculus and the Specification of Computation Lecture Notes Draft: 5 September 2001 These notes are being used with lectures for CSE 597 at Penn State in Fall 2001. These notes had been prepared to accompany lectures given at the Marktoberdorf Summer School, 29 July – 10 August 1997 for a course titled “Proof Theoretic Specification of Computation”. An earlier draft was used in courses given between 27 January and 11 February 1997 at the University of Siena and between 6 and 21 March 1997 at the University of Genoa. A version of these notes was also used at Penn State for CSE 522 in Fall 2000. For the most recent version of these notes, contact the author. °c Dale Miller 321 Pond Lab Computer Science and Engineering Department The Pennsylvania State University University Park, PA 16802-6106 USA Office: +(814)865-9505
[email protected] http://www.cse.psu.edu/»dale Contents 1 Overview and Motivation 5 1.1 Roles for logic in the specification of computations . 5 1.2 Desiderata for declarative programming . 6 1.3 Relating algorithm and logic . 6 2 First-Order Logic 8 2.1 Types and signatures . 8 2.2 First-order terms and formulas . 8 2.3 Sequent calculus . 10 2.4 Classical, intuitionistic, and minimal logics . 10 2.5 Permutations of inference rules . 12 2.6 Cut-elimination . 12 2.7 Consequences of cut-elimination . 13 2.8 Additional readings . 13 2.9 Exercises . 13 3 Logic Programming Considered Abstractly 15 3.1 Problems with proof search . 15 3.2 Interpretation as goal-directed search .