Chip-scale programmable optical signal processors are often used to flexibly manipulate the optical signals for satisfying the demands in various applications, such as lidar, radar, and artificial intelligence. Silicon photonics has unique advantages of ultra-high integration density as well as CMOS compatibility, and thus makes it possible to develop large-scale programmable optical signal processors. The challenge is the high silicon waveguides propagation losses and the high calibration complexity for all tuning elements due to the random phase errors. In this paper, we propose and demonstrate a programmable silicon photonic processor for the first time by introducing low-loss multimode photonic waveguide spirals and low-random-phase-error Mach-Zehnder switches. The present chip-scale programmable silicon photonic processor comprises a 1×4 variable power splitter based on cascaded Mach-Zehnder couplers (MZCs), four Ge/Si photodetectors, four channels of thermally-tunable optical delaylines. Each channel consists of a continuously-tuning phase shifter based on a waveguide spiral with a micro-heater and a digitally-tuning delayline realized with cascaded waveguide-spiral delaylines and MZSs for 5.68 ps time-delay step. Particularly, these waveguide spirals used here are designed to be as wide as 2 µm, enabling an ultralow propagation loss of 0.28 dB/cm. Meanwhile, these MZCs and MZSs are designed with 2-µm-wide arm waveguides, and thus the random phase errors in the MZC/MZS arms are negligible, in which case the calibration for these MZSs/MZCs becomes easy and furthermore the power consumption for compensating the phase errors can be reduced greatly. Finally, this programmable silicon photonic processor is demonstrated successfully to verify a number of distinctively different functionalities, including tunable time-delay, microwave photonic beamforming, arbitrary optical signal filtering, and arbitrary waveform generation.